MECHANICAL CHARACTERIZATION OF HYDROXYAPATITE, THERMOELECTRIC
MATERIALS AND DOPED CERIA
By
Xiaofeng Fan

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Materials Science and Engineering - Doctor of Philosophy
2013

ABSTRACT
MECHANICAL CHARACTERIZATION OF HYDROXYAPATITE, THERMOELECTRIC
MATERIALS AND DOPED CERIA
By
Xiaofeng Fan
For a variety of applications of brittle ceramic materials, porosity plays a critical role
structurally and/or functionally, such as in engineered bone scaffolds, thermoelectric
materials and in solid oxide fuel cells. The presence of porosity will affect the mechanical
properties, which are essential to the design and application of porous brittle materials.
In this study, the mechanical property versus microstructure relations for bioceramics,
thermoelectric (TE) materials and solid oxide fuel cells were investigated. For the bioceramic
material hydroxyapatite (HA), the Young’s modulus was measured using resonant ultrasound
spectroscopy (RUS) as a function of (i) porosity and (ii) microcracking damage state. The
fracture strength was measured as a function of porosity using biaxial flexure testing, and the
distribution of the fracture strength was studied by Weibull analysis.
For the natural mineral tetrahedrite based solid solution thermoelectric material
(Cu10Zn2As4S13 - Cu12Sb4S13), the elastic moduli, hardness and fracture toughness were
studied as a function of (i) composition and (ii) ball milling time. For ZiNiSn, a
thermoelectric half-Heusler compound, the elastic modulus – porosity and hardness –
porosity relations were examined.
For the solid oxide fuel cell material, gadolina doped ceria (GDC), the elastic moduli
including Young’s modulus, shear modulus, bulk modulus and Poisson’s ratio were measured
by RUS as a function of porosity. The hardness was evaluated by Vickers indentation
technique as a function of porosity.

The results of the mechanical property versus microstructure relations obtained in this
study are of great importance for the design and fabrication of reliable components with
service life and a safety factor. The Weibull modulus, which is a measure of the scatter in
fracture strength, is the gauge of the mechanical reliability. The elastic moduli and Poisson’s
ratio are needed in analytical or numerical models of the thermal and mechanical stresses
arising from in-service thermal gradients, thermal transients and/or mechanical loading.
Hardness is related to a material’s wear resistance and machinability, which are two essential
considerations in fabrication and application.

Dedicated to my father, Jianmin Fan
You are, and will always be in my heart

iv

ACKNOWLEDGMENTS

I would like to give my sincere thanks to Professor Eldon Case, for his guidance and
support on my way of pursuing the doctoral degree. Your insightful advice and profound
knowledge enlightened me academically; your dedication to work, your ethicality and
integrity as a researcher influenced me and would be my lifetime treasure. I also thank
Professor Melissa Baumann, Professor Laura McCabe, Professor Tim Hogan and Professor
Jeffery Sakamoto for their help and suggestions during the completion of my degree.
Also, I have received countless help from my co-workers during my study at MSU.
I would like to thank Robert Schmidt, Jennifer Ni, Yutian Shu, Fei Ren, Andrew Morrison,
Travis Thompson, Qing Yang, Hui Sun, Xu Lu, Iulia Gheorghita and Luis Guerrero. I really
appreciated all your help and suggestions during my research.

v

TABLE OF CONTENTS

LIST OF TABLES………………………………………………………………..………….xii
LIST OF FIGURES……………………………………………………………….………...xvi
CHAPTER 1
INTRODUCTION…………………………………………………………………………1
REFERENCES……………………………………………………………………………3
CHAPTER 2
BACKGROUND
1. Hydroxyapatite and its applications………………………………………………..6
2. Thermoelectric materials……………………………………………………………7
3. Solid oxide fuel cells………………………………………………………………10
4. The importance of studying mechanical properties……………………………….11
5. Mechanical properties as functions of microstructure…………………………….12
5.1 Fracture strength as a function of porosity……………………………………15
5.2 Elastic moduli as functions of porosity, microcracking and composition…….17
5.2.1 Elastic moduli versus porosity………………………………………...18
5.2.2 Elastic moduli versus microcracking………………………………….19
5.2.3 Elastic moduli versus composition……………………………………20
5.3 Hardness as a function of porosity and indentation load……………………..21
5.3.1 Hardness versus porosity……………………………………………... 21
5.3.2 Load dependence on hardness…………………………………………22
5.4 Fracture toughness as a function of porosity………………………………….22
REFERENCES ……………………………………………………………………………..23
CHAPTER 3
EXPERIMENTAL PROCEDURES
1. Specimen preparation……………………………………………………………...34
2. Biaxial flexural testing…………………………………………………………….35
3. Weibull analysis…………………………………………………………………...39
4. Microstructural examination………………………………………………………42
5. Elastic modulus measurements……………………………………………………43
6. Hardness and fracture toughness measurements…………………………………..48
REFERENCES ……………………………………………………………………………. 51
CHAPTER 4
POROSITY DEPENDENCE OF THE WEIBULL MODULUS FOR HYDROXYAPATITE
AND OTHER BRITTLE MATERIALS
Abstract…………………………………………………………………………………..54
1. Introduction ……………………………………………..………………………...55
2. Experimental Procedure…………………………………………………...………57
2.1 Specimen preparation………………………………………………………….57
vi

2.2 X-ray diffraction………………………………………………………………58
2.3 Scanning electron microscopy………………………………………………...58
2.4 Biaxial flexural testing………………………………………………………...60
3. Results and Discussion…………………………………………………………….60
3.1 Phase analysis for the HA specimens in this study……………………………60
3.2 Microstructural evolution of HA specimens included in this study…………...63
3.3 Measurement of the Weibull modulus for the HA specimens in this study…..65
3.4 Porosity effect on Weibull modulus…………………………………………..67
3.4.1 Weibull modulus as a function of porosity for HA……………………67
3.4.2 Weibull modulus as a function of porosity for HA from this study, HA from
the literature and seven additional brittl e materials from the
literature……………………………………………………………….70
3.5 Physical interpretations of the three regions in the m versus P plots………….74
3.5.1 Region II……………………………………………………………….78
3.5.2 Region I………………………………………………………………..81
3.5.3 Region III……………………………………………………………...83
3.5.4 Transitions among the regions……………………………………….. 85
4. Summary and Conclusions……………………………………………………….. 86
5. Acknowledgements………………………………………………………………. 88
APPENDICES……………………………………………………………………………….90
APPENDIX A WEIBULL ANALYSIS…………………………………………………91
APPENDIX B FACTORS OTHER THAN POROSITY WHICH AFFECT THE
WEIBULL MODULUS…………………………………………………94
REFERENCES ……………………………………………………………………………. 98
CHAPTER 5
FRACTURE STRENGTH AND ELASTIC MODULUS AS A FUNCTION OF POROSITY
FOR HYDROXYAPATITE AND OTHER BRITTLE MATERIALS
Abstract…………………………………………………………………………………106
1. Introduction ……………………………………………..……………………….107
2. Background…………………………………………………….………...………108
2.1 Strength-porosity relationships for brittle solids ……………….……………108
2.2 Young’s modulus as a function of porosity …………………………………110
3. Experimental Procedure …………………………………………………………112
3.1 Specimen preparation and biaxial flexure testing ……………………...……112
3.2 Scanning electron microscopy ………………………………………………112
3.3 Elastic modulus measurements of HA ………………………………………113
4. Results and Discussion …………………………………………………………..113
4.1 Microstructural analysis of the initial HA powder…………………………...113
4.2 Strength versus porosity for HA and other brittle materials…………………115
4.2.1 Fracture strength versus P for the HA included in this study……..…115
4.2.2 Mechanisms for strength and Young’s modulus of green powder
compacts……………………………………………………………...121
4.2.3 Relative mean fracture strength, <f> /0, as a function of degree of
densification,  from literature………………………………………124
4.3 Relative Young’s modulus as a function of for HA and other brittle
vii

materials…………………………..………………………………………….126
4.4 Physical interpretation of the descriptions of <E> /E0 and as a function of
degree of densification, .................................................................................
4.5 Relationship between Young’s modulus and fracture strength for partially
sintered powder compacts……………………………………………………137
4.6 Relationship from literature for fracture toughness and porosity……………138
5. Summary and Conclusions ………………………………………………………140
6. Acknowledgements………………………………………………………..…….141
APPENDIX………………………………………………………………………………..143
REFERENCES ……………………………………………………………………………146
CHAPTER 6
THE EFFECT OF INDENTATION-INDUCED MICROCRACKS ON THE ELASTIC
MODULUS OF HYDROXYAPATITE
Abstract…………………………………………………………………………………151
1. Introduction ……………………………………………..……………………….152
2. Experimental Procedure…………………………………………………...……..158
3. Results and Discussion…………………………………………………………..162
3.1 Microstructure ……………………..………………………………………...163
3.2 Rationale for comparing this study’s HA indentation results with indentation
results for alumina from the literature ………….............................................163
3.3 Pre-indentation values of Young’s modulus and Poisson’s ratio ……………164
3.4 Comparing initial values of hardness and fracture toughness to literature values
for HA ……………………………………………………………………….165
3.5 Microcracking - elasticity relationship ………………………………………167
3.5.1 Models for microcracking-Young’s modulus relationship for aligned
surface limited cracks ………………………………………………..167
3.5.2 Young’s modulus – microcracking relationship for HA: experimental
results and model predictions ………………………………………..170
3.5.3 Poisson’s ratio– microcracking relationship for HA ………………..175
3.5.4 Relative change in Young’s modulus as a function of the relative change
in the Poisson’s ratio ………………………………………………..176
3.6 Comparison of Young’s modulus – microcracking results for HA (this study)
and alumina [1]………………………………………………………………177
3.7 The role of r in the modulus change as a function of microcrack damage…..179
4. Conclusions………………………………………………………………………182
5. Acknowledgements………………………………………………………………183
APPENDICES……………………………………………………………………………..186
APPENDIX A EQUATIONS TO CALCULATE SN (THE PRODUCT OF A CRACK
ORIENTATION AND A CRACK GEOMETRY PARAMETER IN
EQUATION (9)), FOR EACH MODULUS/MICROCRACK MODEL
WITH A PARTICULAR CRACK GEOMETRY EMPLOYED IN THIS
STUDY……………………………………………………………..…..187
APPENDIX B EQUATIONS USED TO PREDICT THE SLOPES, SP, OF THE ΔE/E0
VERSUS  PLOTS FOR HA SPECIMENS IN THIS STUDY AND
viii

ALUMINA [Kim 1993]………………………………………………..189
REFERENCES ……………………………………………………………………………191
CHAPTER 7
WEIBULL MODULUS AND FRACTURE STRENGTH OF HIGHLY POROUS
HYDROXYAPATITE
Abstract…………………………………………………………………………………197
1. Introduction ……………………………………………..……………………….198
2. Experimental Procedure…………………………………………………...……..203
2.1 Specimen preparation………………………………………………………..203
2.2 Mechanical testing and Weibull analysis ……………………………………205
2.3 Microstructural examination ………………………………………………...206
2.4 Statistical analysis……………………………………………………………206
3. Results and Discussion…………………………………………………………..208
3.1 Microstructural and phase analyses ..………………………………………...208
3.1.1 SEM analysis of initial powders and the sintered specimens….……..208
3.1.2 Evolution of relative density versus sintering temperature for the HA
specimens in the initial and intermediate sintering stages…………..208
3.1.3 X-ray diffraction results……………………………………………...212
3.2 Weibull modulus and fracture strength for HA specimens…………………..212
3.3 Fracture strength for HA specimens…………………………………………220
4. Summary and Conclusions ………………………………………………………226
5. Acknowledgements………………………………………………………………228
APPENDICES……………………………………………………………………………..229
APPENDIX A WEIBULL ANALYSIS……………….……………………………..230
APPENDIX B BIAXIAL FLEXURE TESTING…………………………………..233
REFERENCES ……………………………………………………………………………236
CHAPTER 8
ROOM TEMPERATURE ELASTIC PROPERTIES OF GADOLINIA-DOPED CERIA AS
A FUNCTION OF POROSITY
Abstract…………………………………………………………………………………242
1. Introduction ……………………………………………..……………………….242
2. Experimental Procedure…………………………………………………...……..244
2.1 Specimen preparation………………………………………………………..244
2.2 Elastic modulus measurements ………...……………………………………245
2.3 Microstructural examination procedures …………………………………....246
3. Results and Discussion…………………………………………………………..249
3.1 Microstructural analysis…………...………………………………………...249
3.2 Elastic modulus as a function of porosity …………………………………..251
3.3 Poisson’s ratio as a function of P ……………………………………………266
3.4 Elastic modulus-porosity for other materials in the literature………………..267
4. Summary and Conclusions ………………………………………………………268
5. Acknowledgements………………………………………………………………269
REFERENCES ……………………………………………………………………………270

ix

CHAPTER 9
ROOM TEMPERATURE HARDNESS OF GADOLINIA-DOPED CERIA AS A
FUNCTION OF POROSITY
Abstract…………………………………………………………………………………277
1. Introduction ……………………………………………..……………………….278
2. Experimental Procedure…………………………………………………...……..279
2.1 Specimen preparation………………………………………………………..279
2.2 Microstructural examination procedures..……………………………………280
2.3 Hardness measurements …………………………………..............................281
3. Results and Discussion…………………………………………………………..283
3.1 Microstructural and phase analysis …………...……………………………..283
3.2 Hardness as a function of porosity …………………………………………..283
3.3 Hardness as a function of load ………………………………………………289
3.4 Hardness versus grain size behavior ………………………………………...294
4. Summary and Conclusions ………………………………………………………296
5. Acknowledgements………………………………………………………………297
REFERENCES ……………………………………………………………………………298
CHAPTER 10
ROOM TEMPERATURE MECHANICAL PROPERTIES OF NATURAL MINERAL
BASED THERMOELECTRICS
Abstract…………………………………………………………………………………304
1. Introduction ……………………………………………..……………………….305
2. Experimental Procedure…………………………………………………...……..307
2.1 Specimen preparation………………………………………………………..307
2.2 Microstructural and phase characterization………………………………….308
2.3 Elasticity measurements…………………………………..............................308
2.4 Hardness and fracture toughness measurements…………………………….309
3. Results and Discussion…………………………………………………………..309
3.1 Microstructural and phase characterization …...……………………………..309
3.2 Elastic moduli, hardness and fracture toughness as a function of composition
…………………………………………………………………………......……..313
3.3 Elastic moduli, hardness and fracture toughness as a function of milling
time …………………………………………………………………………321
4. Summary and Conclusions ………………………………………………………321
5. Acknowledgements………………………………………………………………322
REFERENCES ……………………………………………………………………………323
CHAPTER 11
ROOM TEMPERATURE MECHANICAL PROPERTIES OF THERMOELECTRIC
INTERMETALLIC MATERIAL ZrNiSn
Abstract…………………………………………………………………………………326
1. Introduction ……………………………………………..……………………….327
2. Experimental Procedure…………………………………………………...……..329
2.1 Specimen preparation………………………………………………………..329
2.2 Pulsed electric current sintering (PECS)…………………………………….329
x

2.3 Microstructural analysis …………………………………..............................330
2.4 Elastic modulus measurement ……………………………………………….331
2.5 Hardness measurements……………………………………………………..331
3. Results and Discussion…………………………………………………………..333
3.1 Microstructural and phase analysis……….…...……………………………..333
3.2 Elastic modulus measurement ……………………………..………………..336
3.3 Hardness measurement ………………………………………………………339
3.4 Comparison to literature values for elastic moduli and hardness…………….339
4. Summary and Conclusions ………………………………………………………345
Acknowledgements…………………………………………………………..…………346
REFERENCES ……………………………………………………………………………347
CHAPTER 12
SUMMARY AND CONCLUSIONS…………………………………………………..351
REFERENCES ……………………………………………………………………………355
CHAPTER 13
FUTURE WORK…………………...…………………………………………………..357

xi

LIST OF TABLES

Table 4.1. For the disk-shaped HA specimens included in this study, the specimen label,
sintering temperature, T, sintering time, the specimen dimensions, the volume fraction
porosity P, the number of valid specimens N, the mean fracture strength <f>, the
characteristic strength c, and the Weibull modulus m. Each specimen was cold die pressed
at 33 MPa and sintered in air in an electrical resistance furnace…………………...……..…59
Table 4.2. For the HA studies from the literature: shape forming techniques, specimen
dimensions, grain size G.S., strength testing techniques, loading rate, surface finish, SF,
number of specimens N, porosity P, the mean fracture strength <f>, the characteristic
strength c and Weibull modulus m…………..……………………………………….…….68
Table 4.3. For the oxides other than HA from the literature: materials, shape forming
techniques, specimen dimensions, grain size G.S., strength testing techniques, loading rate,
surface finish, SF, number of specimens in each group N, porosity P, the mean fracture
strength <f>, the characteristic strength c and Weibull modulus m………...…………..…72
Table 4.4. For the non-oxides other than HA from the literature: powder specification, shape
forming techniques, specimen dimensions, grain size G.S., strength testing techniques,
loading rate, surface finish, SF, number of specimens in each group N, porosity P, the mean
fracture strength <f>, the characteristic strength c and Weibull modulus
m……………………………………………………………………………….…………….75
Table 4.5. For the m versus P data included in Region II (Figure 4.8), the number of m data
points, mN as well as the mean and standard deviation of m, <m> and mSTD, respectively.
The data ranges defined by <m> + mSTD overlap for every entry in the table, signifying that
there are no statistically significant differences among <m> for the various data groupings
listed in this table…………………………………………………...………………………..82
Table 5.1. Strength at zero porosity, 0, number of data points, N, fitting parameters, Aσ and
n, for the least-squares fit of <f>/0 versus degree of densification, , data to equation (4)
2

with coefficient of determination, R , for HA (this study), alumina (Nanjangud et al. 1995),
YSZ (Deng et al. 2002) and Si3N4 (Yang et al. 2003). The combined data set includes all of
the data from each of the four studies (this study, Deng et al. 2002, Nanjangud et al. 1995,
Yang et al. 2003)…………………………………………………………...……………….116
Table 5.2. Materials, particle size, shape forming technique, specimen dimensions, green
porosity, fracture testing technique and loading rate for the HA specimens in this study and
the materials from the literature (Deng et al. 2002, Nanjangud et al. 1995, Yang et al. 2003)
included in this study’s analysis on <f>/0 versus degree of densification, …………….125

xii

Table 5.3. Experimentally determined values of green porosity, PG, Young’s modulus
corresponding to theoretically dense materials, E0, number of data points, N, and leastsquares fitting parameters, AE and q, for the fit of E/E0 versus degree of densification, , to
2

equation (10) with coefficient of determination, R for HA (this study, Ren et al. 2009),
alumina (Nanjangud et al. 1995, Hardy et al. 1995, Ren et al. 2009), titanium (Oh et al.2003)
and YSZ (Deng et al. 2002)………………………………………………………………...127
Table 5.4a. Processing details for HA (Ren et al. 2009), alumina (Hardy et al. 1995, Ren et al.
2009) and titanium (Oh et al.2003) included in this study’s analysis E/E0 versus degree of
densification,. Processing information for additional data in the E/E0 analysis for alumina
(Nanjangud et al. 1995), YSZ (Deng et al. 2002) and HA (this study) is included in Table
5.2…………………………………………………………………………...………………130
Table 5.4b. Details of the Young’s modulus, E, measurement techniques for HA (this study)
and for the materials from the literature (HA (Ren et al. 2009), alumina (Nanjangud et al.
1995, Hardy et al. 1995, Ren et al. 2009), titanium (Oh et al.2003) and YSZ (Deng et al.
2002)) included this study’s analysis of E/E0 versus degree of densification,. The
measurement technique is significant since the isothermal moduli (determined by static or
quasistatic testing) is typically about 10 percent lower that the adiabatic moduli (determined
by dynamic testing) on the same material…………………………………………..………131
Table 5.5. Porosity range, strength at zero porosity, 0, Young’s modulus at zero porosity, E0,
2

number of data points, N, fitting parameters, A and r, and coefficient of determination, R for
the least-squares fit of the normalized strength - Young’s modulus data to equation (12), for
the three data sets (HA (this study), alumina (Nanjangud et al. 1995) and YSZ (Deng et al.
2002)) which included both and E data on the same material……..........................……..132
Table 6.1. The experimental values of SN for HA319 and HA513 indented at 9.8 N in this
study and Al2O3 (single and double sides indented at 196 N) [1] compared to the SN values
3

predicted by each model [1]. All SN values are in unit mm ………………………….……171
Table 6.2. For DBV model with modified half ellipse crack geometry, ratio of the depths of
the microcracked layer over the unmicrocracked layer, r, the number of points fitted, NP, the
2
least-squares fit values of slope, SE and the R value for the fit, the least-squares fit values of
2

slope through the zero point, SE0 and the R value for the fit, and the theoretically predicted
value of slope, SP…………………………………………………………………..……….180
Table 6.3. For DBV model with modified slit crack geometry, ratio of the depths of the
microcracked layer over the unmicrocracked layer, r, the number of points fitted, NP, the
2
least-squares fit values of slope, SE and the R value for the fit, the least-squares fit values of
2

slope through the zero point, SE0 and the R value for the fit, and the theoretically predicted
xiii

value of slope, SP…………………………………………………………………….……..181
Table 7.1a. Sintering temperature, T, and time, specimen thickness, t, and diameter, D,
volume fraction porosity, P, the number of valid specimens, N, the mean fracture strength
<f>, the characteristic strength, c, and the Weibull modulus m. Each specimen was hard
die pressed at 33 MPa and then sintered in air in an electrical resistance
furnace………………………………………………………………………………….…..204
Table 7.1b. Summary of the types of analysis done in this study along with the number of
specimen groups, number of individual specimens and the associated figures in this
study…………………………………………………………………………………….….209
Table 7.2. For the HA specimens included in this study, the fitting parameters for leastsquares fit of the mean fracture strength, <f>, versus the sintering temperature, Tsinter, to
equations (6), (7) and (8)……………………………………………………………….…..219
Table 8.1. Comparison of specimen fabrication techniques, elastic modulus measurement
techniques and porosity ranges for the present study, Selcuk and Atkinson [31], Amezawa et
al. [32], Kushi et al. [33] and Wang et al. [34]…………………………..…………………252
2

Table 8.2. Fitting parameters, E0 and R , for the least-squares fit of Young’s modulus data
for the present study, Selcuk and Atkinson [31], and the combined data set [the present study,
31-33] to equations (2) – (5)………………………………………………………….…….254
Table 8.3. The number of data points, ND, parameters from least-squares fit to equation (2),
2
2
2
E0, bE, G0, bG, B0, bB, and coefficient of determination, R (E), R (G) and R (B). E0, G0 and
B0 are intercepts of the least-squares fits with the y axis, respectively………………….…258
Table 8.4. The values of bE obtained from a least-squares fit of the Young’s modulus, E,
versus porosity, P, data to the equation (2) for the GDC10 specimens included in this study
and several oxides from the literature. The number of data points, ND, included in the leastsquares fit and the relative porosity range of the data is also specified………………….…261
Table 8.5. From the literature for the Young’s modulus for partially sintered powder compacts
including the GDC10 specimens in this study, three oxides and titanium metal, the values of
EPNG/E0, where EPNG is the Young’s modulus at P ~ PG and E0 is the Young’s modulus at P
= 0. EPNG/E0 is similar for each material, where EPNG/E0 is independent of the choice of E
versus P model. The degree of densification, is also given.
Table 9.1. For the GDC10 specimens included in this study and porosity-hardness data for
several oxides from the literature, the values of bH obtained from a least-squares fit of the
hardness, H, versus porosity, P, data to equation 2 (H = H0exp(-bHP)). The number of data
points, ND, included in the least-squares fit and the relative porosity range of the data is also
specified…………………………………………………………………….………………290
xiv

Table 10.1. For the specimens, the number of RUS resonant peaks measured, N, the RMS
error, the Young’s modulus, E, shear modulus, G, Poisson’s ratio, , bulk modulus, B, mass
density, , longitudinal and shear velocity, vL and vS……………………………...…..…..314
Table 10.2. For the composition-property study, the results of the least-squares fit the elastic
moduli, E, G, and B, and hardness, H, data to equation 3…………………………….……316
Table 11.1. For each specimen, the PECS temperature, T, and the relative porosity, P were
listed along with the mass and dimensions of the cut and polished bar specimens. The
pressure used in PECS for each specimen is 50 MPa, and the holding time is 20 minutes for
each specimen except for specimen HH-900, whose holding time is 10
minutes……………………………………………………………………………………...332
Table 11.2. For RUS analysis for each specimen, the number of resonant peaks, N, the rms
error, the Young’s modulus, E, the shear modulus, G, the bulk modulus, B, and Poisson’s ratio,
………………………………………………………………………………………...…..338
Table 11.3. Elastic modulus values for intermetallic compounds from the literature for the
ZrNiSn and related compositions………………………………………………………..….343
Table 11.4. Hardness data compared to the literature…………………………………344

xv

LIST OF FIGURES

Figure 2.1. Illustration of (a) a thermoelectric couple of an n -type and a p-type
semiconductor material connected together; (b) thermoelectric module of an array of
thermoelectric couples. For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation………..………….9
Figure 3.1. Cross-section view of the biaxial flexural testing set up……………………..….36
Figure 3.2. Digital image of the ring-on-ring fixture used in the biaxial flexure testing in this
study; (a) assembled fixture with support ring underneath and loading ring on top of the
specimen; (b) support ring (left), loading ring (right), rubber tape (midde, top), specimen
with adhesive Teflon tape (midde, bottom)………………………………….........................37
Figure 3.3. Digital image of valid and invalid fracture test……………………......………...38
Figure 3.4. An example of the Weibull plot. The slope of the linear least squares fit, m, is the
Weibull modulus, The characteristic strength, c, can be calculated from the least-squares fit
at lnln(1/(1- Pf)) = 0…………………………………………….......................……………..41
Figure 3.5. Digital images of (a) three-transducer configuration and (b) two-transducer
configuration of the RUS apparatus……………………………………………………..…...46
Figure 3.6. A typical RUS spectrum………………………………………………………....47
Figure 3.7. Diagram of the Vickers indentation impression and radial cracks system, (a)
viewed on top; (b) viewed by side……………………………………………………......….49
Figure 4.1. XRD patterns of the initial commercial HA powder and HA specimens sintered at
1100oC, 1200oC and 1360oC, where the arrow above the XRD pattern marks the position of
the faint peak associated with - TCP in the XRD pattern for initial powders. (A.U. stands
for arbitrary units)………………………………………………………………..…………..61
Figure 4.2. SEM micrographs of (a) fracture surfaces for HA specimens with porosity of 0.42;
(b) fracture surfaces for HA specimens with porosity of 0.30; (c) polished and thermally
etched surfaces for HA specimens with porosity of 0.24 and (d) polished and thermally
etched surfaces for HA specimens with porosity of 0.08……………………........................62
Figure 4.3. Grain size versus relative density,rel, trajectory for HA specimens included in
this study, representing eleven values of rel ranging from 0.38 to 0.91…………………....64
Figure 4.4. Digital photographs of specimens showing valid and invalid fracture patterns for
HA specimens fractured via biaxial flexure (the ring-on-ring technique) in this study. “Valid”
and “invalid” are defined in ASTM C1499-05 (ASTM 2003), where invalid ring-on-ring
fractures involve failure from edge flaws…………………………………………….……..64
xvi

Figure 4.5. For the HA specimens fractured by biaxial flexure in this study, the Weibull plots
(equation (1)) for each of the HA groups, where each group had a different volume fraction
porosity, P. (Table 4.1 lists the P values of each group of specimens in this
study)………………………………………………………………………………...……….66
Figure 4.6. Weibull modulus as a function of volume fraction porosity, P, for the HA
specimens in this study. (The numbers in parentheses indicate the total number of specimens
fractured in a given study)………………………………………………………………...…66
Figure 4.7. Weibull modulus as a function of P for HA (this study, Cordell et al.2009,
Meganck et al. 2005, Murray et al.1995, Lopes et al. 1999, Ruys et al.1995, Villora et
al.2004). (The numbers in parentheses indicate the total number of specimens fractured in a
given study)……………………………………………………………….………………….69
Figure 4.8. Combined data set for Weibull modulus as a function of P for HA (this study,
Cordell et al.2009, Meganck et al. 2005, Murray et al.1995, Lopes et al. 1999, Ruys et
al.1995, Villora et al.2004), oxides (alumina (Chao et al. 1991, Nanjangud et al. 1995),
zirconia (Pissenberger and Gritzner 1995), titania (Kishimoto et al. 1991), mullite (Vales et
al.1999)) and non-oxides (silicon (Borrero-Lopez et al.2009, Brodie and Bahr 2003, Paul et
al.2006), silicon nitride (Hirosaki and Akimune 1993, Nanjangud et al. 1995, Vales et
al.1999), LAST-T (Ren et al. 2006)) from literature. (The numbers in parentheses indicate
the total number of specimens fractured in a given study)…………………………………..79
Figure 5.1. The RUS spectrum obtained for as-sintered HA specimen sintered at 1200oC for
60 min. A.U. stands for arbitrary units……………………………………………………...114
Figure 5.2. SEM micrographs of initial HA powder showing rod-shaped particles and
agglomerates……………………………………………………………………….……….114
Figure 5.3. Average fracture strength, <f>, of HA (this study) as a function of porosity, P.
The solid line represents a least-squares fit of the <f> versus P data to equation (1) in the
entire P range of 0.08 < P < 0.62. The dashed line represents a least-squares fit of data to
equation (1) in the restricted P range of 0.08 < P < 0.55. For specimens with P values near PG,
the sintering temperature is indicated for each plotted data point. The formation and growth
of interparticle necks increase in the fracture strength while not appreciably changing P. Table
4.1 of Chapter 4 gives a complete list of the sintering temperatures and times for each group
of specimens shown in this figure…………………………………………………………..118
Figure 5.4. Relative fracture strength <f>/0 as a function of degree of densification, , for
HA (this study), and literature values of alumina (Nanjangud et al. 1995), YSZ (Deng et
al.2002) and silicon nitride (Yang et al.2003). Each dashed curve represents a least-squares
fit to equation (4) for each data set. The solid curve represents a least-squares fit to equation
(4) for the combined data sets. The parameters for the least-squares fit of the relative strength
versus degree of densification data to equation (4) are listed in Table
5.1…………………………………………………………...………………………………120

xvii

Figure 5.5. Relative Young’s modulus, <E>/E0, versus  for HA (this study), HA from
literature (Ren et al.2009) alumina (Hardy and Green 1995, Nanjangud et al. 1995, Ren et
al.2009), titanium (Oh et al. 2003) and YSZ (Deng et al.2002). Each dashed curve represents
a least-squares fit of <E>/ E0 versus  data to equation (10). The parameters for the leastsquares fit of the data sets to equation (10) are listed in Table 5.3…………………………129
Figure 5.6. The relative average fracture strength, <f>/0, and relative Young’s modulus,
<E>/E0, of HA (this study) as a function of degree of densification, , showing a very similar
trend in the relative strength and Young’s modulus versus . The dashed curve represents a
least-squares fit of <f>/0 to equation (4), and the solid curve represents a least-squares fit
of <E>/E0 to equation (10)………………………………………………………………….133
Figure 5.7. Normalized fracture strength <f>/0 versus normalized Young’s modulus,
<E>/E0, for HA (this study), alumina (Nanjangud et al. 1995) and YSZ (Deng et al.2002).
Each dashed line represents a least-squares fit to equation (12)…………..………………..136
Figure 6.1. Schematic (a) and SEM micrograph (b) of a Vickers indentation system,
including the indentation impression and radial cracks originating from the surface corners of
the indentation impression…………………………………………………………...……..154
Figure 6.2. RUS spectra obtained for a polished, unindented HA specimen bar sintered at
1360oC for 4 h. A.U. stands for arbitrary units………………………………..……………161
Figure 6.3. SEM micrographs of polished and thermally etched surface for specimen HA-GS
in this study…………………………………………………………………………………161
Figure 6.4. Side view of the indentation-crack systems for (a) a modified half ellipse crack
geometry [54] and (b) a modified slit crack geometry [54]………………………………...169
Figure 6.5. Relative Young’s modulus change as a function of volumetric crack number
density for (a) HA-391 and (b) HA-513. Each solid line represents the prediction given by
each model with particular crack geometry [54]. The dashed line represents a least-squares fit
to equation (9) for the HA data………………………………………........................……..172
Figure 6.6. Relative Poisson’s ratio change as a function of volumetric crack number density
for HA-391 and HA-513. The dashed line represents a least-squares fit to equation (4) for the
HA data………………………………………………………………………………..……174
Figure 6.7. Relative Young’s modulus change as a function of relative Poisson’s ratio change
for HA-391 and HA-513. The dashed lines represent least-squares fits to equation (10) for the
HA data, respectively…………………………………………………………….…………174
Figure 6.8. Relative Young’s modulus change as a function of the microcrack damage
parameter for HA-391, HA-513 as well as alumina specimens from Kim [1] using (a) the
DBV model with a modified half ellipse crack geometry and (b) the DBV model with a
xviii

modified slit crack geometry………………………………………….……………………184
Figure 6.9. Relative Young’s modulus change as a function of the microcrack damage
parameter for HA-391, HA-513 and alumina specimen indented under a 49 N load from Kim
[1] using (a) the DBV model with a modified half ellipse crack geometry and (b) the DBV
model with a modified slit crack geometry……………………………………………..….185
Figure 7.1. SEM micrographs of partially densified HA specimens sintered at 350oC (a) at
low magnification, (b) at higher magnification and HA specimens sintered at 1025oC at (c)
low magnification and (d) higher magnification. Note that 350oC is the minimum sintering
temperature in this study and 1025oC is the maximum sintering temperature in this
study…………………………………………………………………………............……..207
Figure 7.2. Relative density, RD, as a function of sintering temperature for HA specimens in
this study and a previous study (Fan et al. 2012 Parts I and II, Table 7.1b)……………..…211
Figure 7.3. Weibull plots for each of the 18 groups of specimens analyzed in this study (Table
7.1b)…………………………………………………………………………………….…..211
Figure 7.4a. The Weibull modulus versus porosity for HA specimens fabricated in this study
along with HA specimens from our previous study (Fan et al. 2012 Part I, Table 7.1b).…..214
Figure 7.4b. The Weibull modulus versus porosity for the combined data set of the HA data
shown in Figure 7.4a and Weibull modulus data from literature data for other brittle materials
(Fan et al. 2012 Part I)……………………………………………………………………...215
Figure 7.5. Weibull modulus, m, as a function of porosity for the HA specimens analysed in
this study (Table 7.1b). Each data point in this plot represents the valid fracture of roughly
15 to 20 individual specimens………………………………………………....................…218
Figure 7.6. Weibull modulus, m, as a function of sintering temperature, Tsinter, for the HA
specimens analysed in this study (Table 7.1b)……………………………….............……..219
Figure 7.7. The mean fracture strength, <f>, as a function of sintering temperature for the
HA specimens analysed in this study (Table 7.1b), where the solid curve represents a leastsquares fit of the data to equation (6), the dashed line represents a least-squares fit of the data
to equation (7) and the dotted curve represents a least-squares fit of the data to equation
(8)…………………………………………………………………………….......................224
Figure 7.A1. The Weibull plot of a group of HA specimens sintered in this study at 725oC
(Table 7.1a). The slope of the plot, m, is the Weibull modulus, which is a measure of the
scatter in fracture strength values for this group of specimens……………………......……231
Figure 7.B1. Digital image of (a) the ring-on-ring fixture showing the specimen support ring
and loading ring used in the biaxial flexure testing in this study; (b) the assembled ring-onring fixture used in the biaxial flexure testing in this study………………..................……234

xix

Figure 7.B2. Digital image of examples of valid and invalid fractures of HA specimens
included in this study. The crack patterns for the valid test specimens are discussed in the
ASTM standard for ring-on-ring testing (ASTM C1499 2003)……………………….........235
Figure 8.1. A portion of the RUS spectrum obtained for GDC10 specimen with P = 0.52 in
the present study (A.U. stands for arbitrary units)……………………………………..…...247
Figure 8.2. SEM micrographs of (a) fracture surface for GDC10 specimen #39 (P = 0.521);
(b) fracture surface for GDC10 specimen #11 (P = 0.262); (c) fracture surface for GDC10
specimen #3 (P = 0.102) and (d) thermally etched polished surface for GDC10 specimen #5
(P = 0.077)……………………………………………………………………………...…...248
Figure 8.3. Grain size-density trajectory for the GDC10 specimens in the present study. The
solid curves represents the least-squares fit of the grain size versus density data to equation
(1)………………………………………………………………………………….………..250
Figure 8.4. Young’s modulus, shear modulus and bulk modulus versus porosity for GDC10
specimens from the combined data set (the present study, Selcuk and Atkinson [45],
Amezawa et al. [46], and Kushi et al. [47]). The solid, dashed and dotted curves represent the
least-squares fit to equation (2) for the E-P, B-P and G-P data, respectively…………........256
Figure 8.5. Semi-log plot of the Young’s modulus versus porosity for GDC10 specimens
from the combined data set (the present study, Selcuk and Atkinson [45], Amezawa et al. [46],
and Kushi et al. [47]).The solid line represents the least-squares fit of E-P data with P up to
PC ~0.4 to equation (2), and the dashed line is the extrapolation of the solid line to higher
porosity values………………………………………………………………………….…..259
Figure 8.6. Poisson’s ratio for GDC10 specimens for the combined data set (the present study,
Selcuk and Atkinson [45], Amezawa et al. [46], and Kushi et al. [47]). The solid curves
represents the least-squares fit of the -P data to equation (3) and the dashed curves
represents the least-squares fit of the -P data to equation (2)…………….……………….263
Figure 9.1. SEM micrographs of (a) polished surface for the GDC10 specimen with P = 0.08,
(b) fractured surface for the GDC10 specimen with P = 0.10………………………..…….282
Figure 9.2. XRD patterns of the as-received GDC10 powder and GDC10 specimen sintered
at 1400oC (P = 0.10). ……………………………………………………………………….284
Figure 9.3. Hardness versus porosity for a combined GDC10 data set (this study and
Mangalaraja et al. [71]), where the curve represents a least-squares fit to equation 2. Note
that in this study the indentation load was 4.9 N for all the specimens except two with the
highest porosities and in Mangalaraja’s study [71] the indentation load was 5 N. For P > 0.2,
the error bars on the H data for this study are smaller than the symbol size……………….285
Figure 9.4. Semi-logarithmic plot of the hardness versus porosity for GDC10 specimens in
this study. The solid line represents a least-squares fit of data to equation (2) for the P range
of 0.08 < P < 0.4………………………………………………………………...………….288
xx

Figure 9.5. The hardness versus applied indentation load for three selected GDC10
specimens in this study. The horizontal lines represent the average of the H values as a
function of load, namely the solid line for P = 0.38, the dashed line for P= 0.23 and the
dotted line for P = 0.08…………………….……………………………………………….292
Figure 10.1. SEM micrographs of fracture surfaces of specimens included in both the
composition and milling time studies. For the composition study, the microstructures are
shown in figures (a) – (c) for the weight fraction of synthetic phase, x, where (a) x = 0, (b) x
= 0.5, (c) x = 1. For the milling time study, the microstructures correspond to (d) 1 h milling
time, (e) 9 h milling time………………………………………………………………..….310
Figure 10.2. Grain size as a function of (a) the weight fraction of the synthetic phase; (b) the
milling time…………………………………………………………………………………312
Figure 10.3. Young’s, shear, bulk modulus and Poisson’s ratio as a function of the weight
fraction of the synthetic phase for the mineral TE materials in this study. The solid, dashed,
dotted and dash-dotted curves represent respectively the least squares fit of equation (3) to
the Young’s modulus, shear modulus, bulk modulus and Poisson’s ratio
data……………………………………………………………………………….………...316
Figure 10.4. Normalized Young’s, shear modulus and hardness as a function of the weight
fraction of the synthetic phase (in each case, the data were normalized by dividing the each
coefficient in equation (3) by A0). The solid curve represents a least squares fit of all the
normalized E, G and H data to equation (3)……………………………………….……….317
Figure 10.5. Hardness and fracture toughness as a function of the weight fraction of the
synthetic phase for the mineral TE materials in this study. The solid curve represents a least
squares fit of the hardness data to equation (3); the dashed line represents the mean value of
Kc over the entire weight fraction of the synthetic phase…………………………………..318
Figure 10.6. Young’s, shear, bulk modulus and Poisson’s ratio as a function of milling
time………………………………………………………………………..………………..320
Figure 10.7. Hardness and fracture toughness as a function of milling time……..………...320
Figure 11.1. SEM micrographs of powder processed ZrNiSn, for (a) CGSR powders, (b) ball
milled powders……………………………………….............……………………………..334
Figure 11.2. SEM micrographs of fractured surface for (a) HH-1000, (b) HH-800, (c) HHBM-1000……………………………………………………………………….............…...335
Figure 11.3. XRD spectra for specimens HH-900 and HH-950. The faint peak indicated by
the blue arrow is not yet identified……………………………………………………..…..337
Figure 11.4. Young’s modulus, shear modulus and Poisson’s ratio versus porosity for
specimens fabricated from CGSR powders (open symbol) and specimen fabricated from ball
milled powders (X). The solid curves for each data set represent a least-squares fit to
xxi

equation 2…………………………………………………………………..……………….340
Figure 11.5. Hardness as a function of load for specimens fabricated from CGSR powders
(solid symbol) and specimen fabricated from ball milled powders (open symbol)……..….341
Figure 11.6. Hardness as a function of porosity for specimens fabricated from CGSR
powders (open symbol) and specimen fabricated from ball milled powders (solid symbol)
indented at load = 4.9N…………………………………………………………………..…341

xxii

CHAPTER 1
INTRODUCTION

For a wide variety of brittle ceramic material applications, porosity plays a critical role
structurally and/or functionally. Porous ceramics are widely used in filtration [Zievers 1991,
Guizard 2002], absorption [Drisko 2009], separation [Araki 2005], catalyst supporting [Opre
2005], thermal insulation [Clyne 2006] and as lightweight structural components [Fu 2011].
For example, porous bioceramic hydroxyapatite is used as bone scaffold materials [Hench
1991], and highly porous ceramics such as doped ceria are critical components used in solid
oxide fuel cells [Samson 2012].
The presence of porosity in ceramics will affect the mechanical properties, including
fracture strength [Park 2009], Young’s modulus [Fan 2012, Ni 2009], shear modulus [Ren
2009, Ni 2009], hardness [Hoepfner 2003] and fracture toughness [Rice 1996]. These
mechanical properties determine the capability of a material to resist external or internal
stresses. For instance, the thermal stress produced due to temperature difference is
proportional to the Young’s modulus [Kaliakin 2002]. Hardness determines how fast the
material will be worn out in service [Leyland 2000]. The brittleness of the material, which is
a function of the Young’s modulus, hardness and toughness, is directly related to the
machinability of the material [Quinn 1997]. Thus, a thorough understanding of the
mechanical properties as a function of porosity is essential to the design and application of
porous brittle ceramics.
This dissertation includes mechanical property – microstructure relations for bioceramic,
solid oxide fuel cell material and thermoelectric (TE) materials. Chapters 4 – 7 focused on
the bioceramic material, hydroxyapatite (HA). For the first time, the Weibull modulus for HA
1

with porosity ranging from green porosity to nearly full density was investigated. Relations
for the fracture strength – porosity, Young’s modulus – porosity, and Young’s modulus –
microcracking damage state were studied for HA specimens. Chapters 8 and 9 explored the
elastic modulus – porosity and hardness – porosity relations for gadolina doped ceria used in
solid oxide fuel cells. Chapters 10 and 11 focused on the thermoelectric materials, namely
the natural mineral tetrahedrite based material and the half-Heusler compound, respectively.
The elastic moduli, hardness and fracture toughness of the tetrahedrite based solid solution
material were studied as a function of composition and ball milling time. The elastic modulus
– porosity and hardness – porosity relations for half-Heusler compound was examined in
Chapter 11.

2

REFERENCES

3

REFERENCES

[Araki 2005] K. Araki, J. W. Halloran, Porous ceramic bodies with interconnected pore
channels by a novel freeze casting technique, J. Am. Ceram. Soc. 88, 1108–1114, 2005.
[Clyne 2006] T.W. Clyne, I.O. Golosnoy, J.C. Tan, A.E. Markaki, Porous materials for
thermal management under extreme conditions, Philos. T. Roy. Soc. A 364, 125-146, 2006.
[Drisko 2009] G. L. Drisko, V. Luca, E. Sizgek, N. Scales, R. A. Caruso, Template synthesis
and adsorption properties of hierarchically porous zirconium titanium oxides, Langmuir, 25,
5286–5293, 2009.
[Fan 2012] X. Fan, E.D. Case, F. Ren, Y. Shu, M.J. Baumann, Part II: Fracture strength and
elastic modulus as a function of porosity for hydroxyapatite and other brittle materials. J.
Mech. Behav. Biomed. 8, 99-110, 2012.
[Fu 2011] Q. Fu, E. Saiz, A. P. Tomsia, Bioinspired strong and highly porous glass scaffolds,
Adv. Funct. Mater. 21, 1058–1063, 2011.
[Guizard 2002] C. Guizard, A.Ayral, A. Julbe, Potentiality of organic solvents filtration with
ceramic membranes. A comparison with polymer membranes, Desalination, 147, 275–280,
2002.
[Hench 1991] L. L. Hench, J. K. West, Principles of Electronic Ceramics, Wiley, New York,
190.
[Hoepfner 2003] T.P. Hoepfner, E.D. Case, The influence of the microstructure on the
hardness of sintered hydroxyapatite, Ceramics International. Ceram. Int. 29, 699-706, 2003.
[Kaliakin 2002] V.N. Kaliakin, In Introduction to Approximate Solution Techniques,
Numerical Modeling, and Finite Element Methods, Marcel Dekker, New York, 2002.
[Leyland 2000] A. Leyland, A. Matthews, On the significance of the H/E ratio in wear
control: a nanocomposite coating approach to optimised tribological behaviour, Wear, 246,
1–11, 2000.
[Ni 2009] J.E. Ni, F. Ren, E.D. Case, E.J. Timm, Porosity dependence of elastic moduli in
LAST (Lead-antimony-silver-tellurium) thermoelectric materials. Mater. Chem. Phy. 118,
459 – 466, 2009.
[Opre 2005] Z. Opre, J.D. Grunwaldt, M. Maciejewski, D. Ferri, T. Mallat, A. Baiker,
Promoted Ru-hydroxyapatite: designed structure for the fast and highly selective oxidation of
alcohols with oxygen. J. Catal. 230, 406-419, 2005.
[Park 2009] Y.H. Park, T. Hinoki, A. Kohyama, Development of multi-functional NITEporous SiC for ceramic insulators, J. Nuclear Mater. 386–388, 1014-1017, 2009.
[Quinn 1997] J.B. Quinn, G.D. Quinn, Indentation brittleness of ceramics: A fresh approachJ
4

Mater. Sci. 32, 4331-4346, 1997.
[Ren 2009] F. Ren, E.D. Case, A. Morrison, M. Tafesse, M.J. Baumann, Resonant ultrasound
spectroscopy measurement of Young’s modulus, shear modulus and Poisson’s ratio as a
function of porosity for alumina and hydroxyapatite, Philos. Mag. 89, 1163-1182, 2009.
[Rice 1996] R. W. Rice, Evaluation and extension of physical property-porosity models
based on minimum solid area, J. Mater. Sci. 31, 102-118, 1996.
[Samson 2012] A.J. Samson, P. Hjalmarsson, M. Sogaard, J. Hjelm, N. Bonanos, Highly
durable anode supported solid oxide fuel cell with an infiltrated cathode, J. Power Sources
216, 124-130, 2012.
[Zievers 1991] J.F. Zievers, P. Eggerstedt, E.C. Zievers, Porous ceramics for gas filtration,
Am. Ceram. Soc. Bull. 70, 108-111, 1991.

5

CHAPTER 2
BACKGROUND

1. Hydroxyapatite and its applications
Hydroxyapatite (HA, Ca10(PO4)6(OH)2) is widely studied as a candidate for hard tissue
(bone and teeth) engineering scaffold due to its similarity to natural bone minerals [Hench
1991]. HA can form a direct bond to the bone, showing excellent biocompatibility,
osteoconductivity and bioactivity [Hench 1991].
Both dense and porous HA have extensive scopes of applications due to HA’s
particular physical and chemical properties. In HA crystals, the positive calcium ions and
negative phosphate groups provide multiple sites for specific adsorption of gases, proteins,
viruses and bacteria [Mahabole 2005, Ohtsuki 2010, Yang 2007 a, 2007 b]. Silver
nanoparticles deposited onto porous bulk HA have been used to determine the concentration
of adsorbed protein via changes in the localized surface plasmon resonance of the silver
nanoparticles [Ohtsuki 2010]. Porous HA with a bimodal pore size distribution (large pores:
40 m; small pores: 3 m) can filter microorganisms from water and gas [Yang 2007 a, Yang
2007 b]. When water or gas flows through the highly permeable porous HA filters, microbes
are absorbed on the walls of large pores or blocked by the small pores [Yang 2007 a, Yang
2007 b]. In terms of the gas sensing properties of HA, Mahabole [Mahabole 2005] found that
-

OH groups in HA are adsorption sites for CO, so that when CO gas flows through a porous
HA pellet, the electrical conductivity changes with adsorption level of CO on HA [Mahabole
2005].
The calcium ions in HA can actively exchange with other cations such as Cd, Pb, Co
6

and Cr [Yin 2010], thus making HA useful in environmental, sensing and catalytic
applications. Three dimensional HA structures consisting of spherical macropores with
uniform 0.8 m diameters were fabricated by burning out polystyrene spheres [Srinivasan
2006]. The structures were then immersed in waste water containing cadmium and lead
[Srinivasan 2006]. Cadmium was captured via ion exchange with calcium, while lead formed
an insoluble phase with HA by a dissolution/precipitation reaction [Srinivasan 2006].
In another study, the concentration of CO2 gas was monitored by electrical conductivity
changes in porous cast HA films [Nagai 1988]. The change of electrical conductivity as a
function of CO2 concentration was assumed to be due to the formation of surface layers with
high electrical resistance on the HA pores [Nagai 1988]. In an additional application, yttrium
-

ions exchanged with calcium ions in porous HA generate OH vacancies. The concentration
-

OH vacancies changes as a function of the ambient water vapor concentration, thus changes
in electrical conductivity are directly related to changes in humidity [Owada 1989].
Ion exchanged HA also is widely studied as support platforms for transition metal ions
catalysts [Opre 2005, Mori 2003, Mori 2004, Venugopal 2003]. The advantages of using HA
as support are: (i) active transition metal ions can be immobilized in HA due to ion exchange
and (ii) the HA support is stable and avoids possible side reactions induced by the support
itself [Opre 2005, Mori 2003, Mori 2004, Venugopal 2003]. In a study by Wang [Wang 2010],
HA nanowire arrays were prepared by template-assisted electrodeposition, then horseradish
peroxidise (HRP) cyanide sensors were cast onto the surfaces of multiple glassy carbon
substrates.
2. Thermoelectric materials
7

Thermoelectric (TE) materials have attracted great attention in recent years due to their
potential to convert waste heat into electricity. The safety, sustainability, and reliability make
TE materials stand out compared to other energy sources such as fossil fuels, nuclear energy,
wind and solar energy.
The power generation function of TE materials is based on the Seebeck effect [Chen
2003]. When two different materials are joined together with a temperature gradient between
them, then an electrical current will be developed such that the magnitude of the current is
proportional to the temperature difference. As demonstrated in Figure 2.1a, a thermoelectric
couple of a n -type and a p-type semiconductor material connected together, an electric
current is formed due to the moving of the electrons and the holes in the n -type and p-type
semiconductors respectively under the temperature gradient.
Conversely, a temperature gradient can be developed when a potential is applied to the
same n-type and p-type configuration (Figure 2.1a). This phenomenon is called the Peltier
effect [Snyder 2002], which is the basis for TE refrigeration devices.
When TE materials are incorporated into devices, they are built into modules that
contain dozens of the thermoelectric couples (Figure 2.1b) arranged into an array that is
electrically in series and thermally in parallel.
The efficiency of TE materials is characterized by a dimensionless figure of merit (ZT,
ZT = S2σT/κ), where S is the Seebeck coefficient, σ is the electrical conductivity, T is the
absolute temperature, and κ is the thermal conductivity [Zebarjadi 2012]. More efficiency
can be achieved with higher ZT. For conventional bulk TE materials studied, the ZT value is
less than or around unity [Zebarjadi 2012].
The concept of an ideal TE material is an “electron crystal, phonon glass”, which

8

(a)

(b)

Figure 2.1. Illustration of (a) a thermoelectric couple of an n -type and a p-type
semiconductor material connected together; (b) thermoelectric module of an array of
thermoelectric couples. For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation.

9

maximizes the electrical transport while hinders the thermal transport [Slack 1995].
Strategies towards improving ZT include tuning the electronic properties by doping [Hsu
2004] and band engineering [Heremans 2008]; lowering the thermal conductivity by
nanostrucuring [Poudel 2008]. Nanostructuring has been studied intensively recently due
to its ability to decouple the Seebeck coefficient and the electrical conductivity, and offer
additional phonon scattering to reduce the thermal conductivity [Kanatzidis 2010].
Nanostructuring has been performed over length scales from atomic-scale lattice defects like
dislocations and vacancies [He 2010], nanoscale precipitates [Pei 2011a], nano-inclusions
[Zhao 2008] to mesoscale interfaces like grain boundaries [Joshi 2008].
Current studied TE materials include, but are not limited to, chalcogenides [Pei 2011b],
clathrates [Shi 2010], oxides [Koumoto 2006], skutterudites [Nolas 1999], half-Heuslers
[Uher 1999], and silicides [Yani 2007]. Each TE material system has an optimal performance
over a certain temperature range with individual merits compared to other systems.
3. Solid oxide fuel cells
Solid oxide fuel cells (SOFCs) are capable of producing electricity through
electrochemical reactions [Minh 1993]. Similar to other types of fuel cells, SOFCs consist of
two electrodes and the electrolyte materials. The difference between SOFCs and other fuels
cell is that SOFCs have solid oxide electrolyte materials (ceramics). Compared to other fuel
cells, the advantages of SOFCs include long-term stability, high efficiency and less
environmental effect [Zhu 2003]. However, the operating temperatures for SOFCs are
relatively high, which poses challenges to maintain mechanical and chemical stability.
Doped cerium dioxides are studied for potential application in SOFCs due to the high
ionic conductivity and low cost. Gadolinia doped ceria (GDC), is commonly used in SOFCs.

10

Its dense form is applied as membrane or diffusion blocking material; the porous GDC, finds
applications in anode catalyst [Sauvet 2001], anode mixed ionic electronic conductor (MIEC)
[Goodenough 2007], oxygen storage material [Trovarelli 1996], composite cathode oxygen
ion conductor [Murray 2001] in SOFCs. Nano-composite cathodes (NCC’s) of porous GDC
surface-decorated with nano-sized MIEC catalyst particles have received much interest due
to their high electrochemical performance [Nicholas 2010, 2012] and
o

electrochemical/microstructural durability [Samson 2012] at intermediate (500-700 C)
temperatures.
4. The importance of studying mechanical properties
In general, before any material or any component is put into application, the mechanical
properties should be studied thoroughly to ensure the mechanical integrity during use. In
applications, stresses will arise both internally and externally, due to thermal expansion
mismatch [Jeon 2000], thermal gradient [Kamaya 2011], thermal transient [Zhou 2011], and
mechanical loading and manufacturing [Zhou 2012]. For example, when HA scaffolds are
implanted in human bodies, the stress from everyday activity like running and walking may
induce microcracks in scaffolds [Zioupos 1994]. In waste heat harvesting applications, TE
materials are operated at cycles of low to high temperatures. The stresses arising from cyclic
thermal gradients, thermal transients, thermal expansion mismatch among TE materials,
interconnects and heat sources, may generate and growth of cracks, degradation of properties,
and consequently failure of the entire TE system [Case 2012]. In addition, when used in
automobile applications, the mechanical vibrations will also cause mechanical degradation or
even failure. In SOFC applications, the mismatch of coefficient of thermal expansion (CTE)
mismatch between the cell layers (electrolyte, cathode, current collector) may lead to stress

11

increase or even fracture [Jiang 2012]. Also, the materials in this study (HA, TE materials
and GDC) are all brittle materials, which means that relatively little energy will be absorbed
before the fracture of the materials when subjected to stress.
For commercial use, good mechanical reliability and predictable performance is
required, especially when failure may cause fatal damage. Fracture strength is a direct
indicator of how the material will survive under certain load [Wachtman 2009]. In designing
a component, knowing the scatter in fracture strength is of equal importance as knowing the
mean fracture strength, since the whole system will fail in the weakest part [Wachtman 2009].
The Weibull modulus is a gauge for the distribution of fracture strength [Trustrum 1979]
(Chapter 3, Section 3).
Young’s modulus and Poisson’s ratio are crucial parameters in the stiffness matrix in
modelling the materials subject to thermal and mechanical loading, and to model load
transfer between different stacked layers [Larson 2013]. Hardness is directly related to the
material’s machinability [Boccaccini 1997] and wear resistance [Leyland 2000], which are
two important considerations in design and manufacturing. Fracture toughness is the measure
of the material’s resistance to crack propagation, and is extremely important in designing any
brittle materials [Wachtman 2009].
5. Mechanical properties as functions of microstructure
The mechanical properties, including elastic moduli, fracture strength, Weibull modulus,
hardness and fracture toughness are not only functions of the material itself, but are also
affected by the microstructure of the material, such as porosity, composition, microcracking,
and other factors such as damage state and temperature. Porosity is commonly observed in a
wide range of ceramic materials. Some pores are left behind after sintering, while some pores

12

are introduced in on purpose by methods like partial sintering [Fan 2012], addition of
sacrificial fugitives [Ren 2005] and fabrication using replica templates [Carbajo 2002].
Porous ceramics are of significant interest both for functional or structural applications in
areas such as filtration [Zievers 1991, Guizard 2002], absorption [Drisko 2009], separation
[Araki 2005], catalyst supports [Opre 2005, Mori 2003], thermal insulation [Clyne 2006] and
lightweight structural components [Fu 2011].
For synthetic HA bone scaffolds, it is of great importance to incorporate the bimodal
porosity similar to natural bones to enhance bone ingrowth [Hsu 2007]. Bimodal porosity is
observed in natural bones, with macropores at the diameters of several hundred microns and
micropores at the diameters of tens of microns [Hench 1991]. These bimodal pores are
biologically essential for blood and nutrient circulations. Studies have shown that early
attachment of bone cells to the bioceramic scaffold was enhanced as a result of the
microporosity in the scaffold [El-Ghannam 2004]. Moreover, the newly formed natural bone
cells tend to grow into the macropores of the scaffold and finally replace the resorbable
scaffold [El-Ghannam 2004].
A number of mechanical properties are sensitive to porosity, including the flexural
fracture strength [Park 2009], compressive fracture strength [Sammis 1986, Nielsen 1990],
the Young’s modulus [Fan 2012b, Ni 2009], shear modulus [Ren 2009, Ni 2009] and
hardness [Hoepfner 2003].
In addition to the mechanical properties, porosity also affects a variety of physical
properties, including thermal and electrical properties. For example, thermal conductivity
[Park 2009, Sulistyo 2010] and thermal diffusivity [Park 2009] are both functions of porosity.
For electrical properties, the dielectric constant [Hoepfner 2002, Geis 2002], dielectric

13

breakdown voltage [Hench 1990, Geis 2002], and electrical conductivity [Shimizi 2009,
Sulistyo 2010] are all functions of porosity.
In sintered ceramic materials, generally microcracking can be formed due to three
mechanisms: (i) thermal expansion anisotropy (TEA) [Case 2005], (ii) thermal expansion
mismatch (two phases) [Todd 2004] and (iii) externally applied mechanical loads [Zhou
2012]. TEA applies to non-cubic polycrystalline materials with different thermal expansion
coefficients along crystallographic axes. Grains expanding and shrinking at different rate
during heating and cooling generate stress. The interplay of stored elastic strain energy and
surface energy determines microcracking formation. Microcracks are generated when grain
size reaches the critical grain size. The critical grain size, Gcr, can be estimated [Cleveland
1978]
Gcr =

14.4 f

(1)

E(T) 2 ( 2 )
max

where γf is the fracture surface energy, E is the Young’s modulus of the material, ΔT is the
temperature difference during sintering, and Δαmax is the maximum difference in the thermal
expansion coefficients along different crystallographic axes.
For example, HA belongs to the hexagonal system with a space group P63/m, with
thermal expansion coefficients that are different along the “a” axis and “c” axis of HA. Using
2

-6

γf of 0.48 J/m , E of 113 GPa, ΔT of 1000oC and Δαmax of 7.4x10 /oC, Hoepfner and Case
estimated the critical grain size of HA as 1.1 m [Hoepfner 2004]. The presence of
microcracks affects a wide range of properties, including mechanical properties such as
Young’s modulus [Fan 2012c], hardness [Perera 2010], fracture strength [Diaz 2008] and
14

fracture toughness [Kratschmer 2011]. Other physical properties such as electrical
conductivity [Hilpert 2004], thermal diffusivity [Deroo 2010] and dielectric constant [Morito
2005] are also a function of microcracking damage state.
5.1 Fracture strength as a function of porosity
Fracture strength is the stress at which a test specimen fractures [Wachtman 2009]. The
experimental fracture strength is usually determined from the stress-strain curve during a
fracture test (tensile test, compression test, flexure test). For brittle materials, the stress-strain
curve usually remains linear before the fracture takes place, meaning that no apparent plastic
deformation before fracture [Wachtman 2009].
The theoretical strength of a single crystal material can be approximated as [Wachtman
2009]

 th 

E
a0

(2)

where E is the Young's modulus of the material,  is the surface energy, and a0 is the
equilibrium distance between atomic centers.
For brittle polycrystalline materials, fracture strength is affected by flaw size which is
related to porosity [Park 2009, Fan 2012], microcracking [Diaz 2008], grain size [Hirosaki
1993], surface finish [Nakamura 2010] of the test specimens. In addition, fracture strength is
affected by fracture testing technique [Cordell 2009].
The discussion on the fracture strength – porosity relationship dates back to the fifties.
In 1953, Ryshkewitch [Ryshkewitch 1953] suggested the following equation to describe the
porosity dependence of the strength, ,
 = 0 exp(-nP)

(3)
15

where 0 is the strength for nonporous body, P is the volume fraction porosity and n is a
unitless, material-dependent constant.
In 1958, Schriller [Schiller 1958] gave a porosity-relative strength relationship that
included the concept of a critical porosity, Pc, such that [Schiller 1958],
q
n
P 
   P  
 1     1  k  
P  
 0   Pc   
 c 
   

(4)

where k, q and n are empirically determined parameters. Pc is the critical porosity that for P >
Pc, the above equation is no longer valid.
In 1959, Millard [Millard 1959] proposed a relationship such that


P  Pcr
 = S0  1 
 P  P

cr 


(5)

where S0 is a constant with unit of the strength, Pcr is the critical porosity defined as the
porosity of the loose powder.
In 1972, Carnaglia [Carnaglia 1972] suggested that the strength-porosity relationship
could be written in terms of a function of the relative Young’s modulus E/E0, such that
1/ 2

 
E 
 (1  P) 
0 
E0 

(6)

Based on mathematical model, Hrma [Hrma 1974] gave the relative strength as a
function of the volume fraction porosity normalized by a critical porosity, Pc, such that

P

 1  
P 
0
 c

n

(7)

Equations 3-7 can all describe the strength-porosity relationship very well to a critical
16

porosity value, Pc, where for P > Pc, the strength decreases more rapidly than is predicted by
equations 3-7.
A slightly modified equation provides better fitting to the strength-porosity data [Fan
2012b]
n



P 
 A  1    A  
 Pg 
0



(8)

where  is fracture strength of the porous ceramic, 0 is fracture strength of the theoretically
dense ceramic, A and n are dimensionless constants. The degree of densification, is defined
as (1-P/Pg), where Pg is the porosity at green porosity.
5.2 Elastic moduli as functions of porosity, microcracking and composition
Elastic moduli are fundamental properties of a material, measuring the stiffness of the
material. Theoretically, Young’s modulus represents the second derivative of the binding
energy with respect to strain [Wachtman 2009]. Experimentally, Young’s modulus can be
measured from the linear portion of a stress-strain curve during a mechanical testing.
Young’s modulus is a fourth rank tensor, which consists of 81 stiffness constants [Nye
1985]. The 81 stiffness constants can be reduced to 21 independent stiffness constants after
applying the symmetry considerations. The number of independent stiffness constants can be
further reduced when considering the symmetry of a particular crystal structure. For example,
in cubic crystal structure, the 4 threefold rotation axis reduces the number of independent
stiffness constants to three [Nye 1985].
As mentioned above, since the Young’s modulus represents the second derivative of the
binding energy, the theoretical Young’s modulus is only a function of the atomic binding

17

energy [Wachtman 2009]. In real materials, the elastic modulus is also a function of many
other factors, including porosity [Fan 2012], microcracking [Fan 2012c] and composition
[Radovic 2006].
For polycrystalline materials, the stress generated in a specimen subjected to a strain
arising from mechanical loading can be expressed as [Wachtman 2009]
 = E (9)
For TE materials used in harvesting waste energy, the temperature-dependent Young’s
modulus, E(T), Poisson’s ratio, (T), and thermal expansion, (T), are required for both the
analytic and finite element analysis of the stress, induced by thermal gradients or thermal
transients [Kaliakin 2002, Martin 1973] such that

( T ) 

E(T)(T)T
1  (T) 

(10)

where T is the temperature gradient or the temperature change during the thermal cycling.
5.2.1 Elastic moduli versus porosity
In terms of characterizing the behaviour of Young’s modulus as a function of porosity,
the equation proposed by Spriggs in 1961 has widely been used to predict the Young’s
modulus of porous brittle materials [Spriggs 1961]
E = E0 exp (-bEP)

(11)

where E and E0 are the Young’s modulus at porosity P and zero, and bE is a material constant
that is dependent on the pore character in the materials [Rice 1998].
In 1987, a new semi-empirical equation was derived by Phani [Phani 1987] to describe
the porosity dependence of Young's modulus
n

E = E0 (1 - aP)

(12)
18

where a and n are both dimensionless constants, where a = 1/Pc and Pc as the critical porosity
at which Young’s modulus is zero. According to Phani [Phani 1987], equation 12 is valid for
any range of porosity.
Lam [Lam 1994] in 1994 and Hardy [Hardy 1995] in 1995 suggested that the relative
Young’s modulus, E/E0, of porous ceramics can be described as

E 
P 
 1  
E 0  Pg 



(13)

where E is Young’s modulus of the porous ceramic, E0 is the Young’s modulus of the
theoretically dense ceramic, and Pg is the porosity at green density. The right hand side of
equation (equation 13) is defined as the degree of densification.
A slightly modified equation provides better fitting to the modulus-porosity data, in a
similar manner to the relative fracture strength (equation 8)


E
P 
 A  1  
 Pg 
E0



n

(14)

where A and n are dimensionless constants. The degree of densification, is defined as (1P/Pg), where Pg is the porosity at green porosity.
For porosity dependence of Poisson’ ratio, , here is not a single trend found to work in
general. The compilation of Poisson’s ratio versus porosity data for fifteen different
materials including oxides and non-oxides showed that as the porosity increased, Poisson’s
ratio may increase, decrease or stay almost constant [Boccaccini 1994].
5.2.2 Elastic moduli versus microcracking

19

Near surface microcracking can be generated by external mechanical loads and post
processing such as grinding, cutting and polishing [Zhou 2012]. During sintering of
polycrystalline materials, microcracks can be formed through the whole body due to thermal
expansion anisotropy [Case 2005], thermal expansion mismatch [Todd 2004] and
Martensitic-type crystallographic phase change [Deville 2004].
The presence of microcracking affects the Young’s modulus of the material in a linear
manner. Theoretical research by Budiansky and O'Connell [Budiansky 1976], Salganik
[Salganik 1974], Hoenig [Hoenig 1979] and Laws and Brockenbrough [Laws 1987] have
suggested the following relationship to describe the effect of microcracking on Young's
modulus, such that
E = EN (1 – f )

(15)

where EN is the Young's modulus for non-microcracked specimen, f is the crack orientation
function, and  is the crack damage parameter, which is a function of the crack number
density and the crack radius. For circular microcracks of uniform radius c, is defined as =
3

N c , where N is the volumetric crack number density.
5.2.3 Elastic moduli versus composition
In a solid solution system, the replacement of one atom by another element will change
the binding energy, which in turn changes the elastic moduli. Studies on the elastic moduli
dependence on composition in solid solution system have shown linear [Ren 2007] (also
known as the Vegard’s law [Vegard 1921]) or parabolic relationships [Mokhtari 2004,
Kanoun 2005, Ren 2007].
An empirical equation has been successfully used to describe the property change as a

20

function of fractional composition, x (0 ≤ x ≤ 1), in solid solution system, such that
A = A0 + (A1 – A0)x + k x (1-x)

(16)

where A0 = the value of A at x = 0, A1 = the value of A at x = 1, and k is a parameter
(sometimes called the “bowing parameter”) which has the same unit as A, where A can be
material properties including Young’s modulus, shear modulus and hardness. k is a measure
of the deviation of A from the linear rule of mixture model. The k values for elastic modulus
depend on the lattice mismatch strain in the solid solution system [Bernard 1986, Mokhtari
2004].
5.3 Hardness as a function of porosity and indentation load
Hardness shows how a material will react to plastic deformation when a force is applied
[Wachtman 2009]. Hardness is directly related to the two essential factors for fabrication and
mechanical stability, namely, the wear resistance [Zeng 2005, Krakhmalev 2006] and
machinability [Kamboj 2003, Wang 2002]. For example, when HA is used as dental
restoration materials, the wear resistance requires careful consideration [Angker 2006]. For
thermoelectric devices, the TE material will be diced into hundreds of small pillars and
assembled in array to form a TE module [Wesolowski 2012]. The machinability of the TE
materials is an important concern in fabrication.
As is the case with many other material properties, hardness is affected by porosity
[Hoepfner 2003], grain size [Xu 1995] and composition [Ren 2008]. Inclusions [Canakci
2011] and precipitates [Liao 2011] are also commonly discussed microstructural features that
affect hardness.
5.3.1 Hardness versus porosity
An empirical exponential relationship similar to equation 11 is widely used to
21

characterize the hardness dependence on porosity [Hoepfner 2003]
H = H0exp(-bHP)

(17)

where H0 is the hardness for zero porosity materials, bH is a material constant measuring the
degree of hardness change with respect to porosity.
5.3.2 Load dependence on hardness
The indentation load can be varied for each measurement. In some cases, the hardness
value is independent on the indentation load [Ni 2010, White 2011]. For some ceramics [Bull
1989, Sangwal 2009] and metals [Pharr 2010], the measured hardness values decreased with
increasing applied load. This phenomenon is called the indentation size effect (ISE) [Nix
1998], which is caused by elastic recovery and work hardening during indentation, surface
dislocation pinning in metals [Gong 1999]. Reversed indentation size effect (RISE) has also
been reported [Sangwal 2000] where hardness decreases with decreasing load. In RISE, the
specimens undergo relaxation instead of elastic recovery in unloading [Sangwal 2000].
5.4 Fracture toughness as a function of porosity
Fracture toughness is a measure of material's resistance to crack initiation and
propagation [Wachtman 2009]. When the stress intensity factor is great than fracture
toughness, rapid crack growth will take place. It is an important property in design and
application for brittle materials. A study by Rice [Rice 1996] showed that fracture toughness
decreased exponentially with increasing porosity, similar to other properties such as elastic
moduli and hardness. For fracture toughness determined by Vickers indentation, as the
porosity exceeds 0.1, no radial cracks will be generated during the indentation due to the
presence of porosity. In these cases, tensile test, bend test and double-cantilever beam test
can be performed to determine the fracture toughness.
22

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CHAPTER 3
EXPERIMENTAL PROCEDURES

This chapter is a general outline of the techniques and equipments used in this study.
1. Specimen preparation
All of the hydroxyapatite (HA, Ca10(PO4)6(OH)2) specimens discussed in Chapters 4, 5,
6 and 7 in this dissertation were made from biomedical grade commercial HA powders
(Taihei Chemical Industrials Co., Osaka, Japan), with a vendor-specified average powder
particle size of 1 – 3 m. Each specimen was fabricated by cold die uniaxially pressing
(Carver Inc., Wabash, IN) of ~2.5 grams of HA powder into disk-shaped compact at 33 MPa
for 1 minute using a steel pellet die with a diameter of 32 mm.
The as-pressed HA specimens were followed by pressure-less sintering in air in an
electrical resistance furnace (CM Inc., Model #0100277, Bloomfield, NJ), at a
heating/cooling rate of 10oC/min. The sintering temperatures and sintering times were varied
deliberately in order to obtain specimens with a wide range of porosities, from near zero
porosity to near green porosity. During the sintering process, specimens were placed on an
alumina crucible covered with 2 – 3 mm thick layer of HA loose powder, and the HA
specimens were covered with a layer of 2 – 3 mm HA loose powder on top. The powder bed
assured an even thermal distribution and helped to avoid warping of specimens during
sintering.
The volume fraction porosity, P, of the as-sintered specimens was calculated from the
specimen mass along with the average specimen diameter and thickness, assuming a
3

theoretical density of 3.156 g/cm for HA [CRC 2009]. The specimen mass was measured
34

using an electronic balance (Denver Instrument Company, Arvada, CO), with a vendorspecified precision of ±0.001 gram. The diameter and thickness were measured three times
using digital caliper (Davis Calibration, Carol Stream, IL, with a vendor-specified precision
of ±0.01 mm) and an average value was calculated and used in the porosity calculation.
For grain size evaluation and indentation studies, the as-sintered HA specimens were
then polished using 600 grit SiC abrasive paper followed by polishing using diamond paste
with grit size from 90 m, 30 m, 15 m, 9 m, 6 m, 1 m down to 0.5 m (Buehler
Ecomet 3 variable speed grinder/polisher, Lake Bluff, IL), until a mirror-like finish was
obtained. During the polishing process, specimens were ultrasonically cleaned (VWR
B3500A-MT, Radnor, PA) for 10 minutes before shifting down to a smaller grit.
2. Biaxial flexural testing
As-sintered HA specimens were fractured using biaxial flexural testing (Figure 3.1)
performed on a universal testing machine (Model 4206, Instron, Norwood, MA) via a ringon-ring configuration (Figures 3.2a and 3.2b, ASTM 2003). The diameter of the support ring
is 20 mm, and the diameter of the loading ring is 10 mm (Figures 3.2a and 3.2b). A piece of
rubber tape was placed between the support ring and the test specimen in order to eliminate
any contact stress or friction (Figures 3.2a and 3.2b). An adhesive Teflon tape was attached
onto the upside face of the specimen, for the purpose to hold the remnants of the specimen
after fracture (Figures 3.2a and 3.2b). The crosshead loading speed was 1 mm/min.
After fracture, the remnants of the specimens were used for fractographic examination to
determine the validity of the fracture testing. The crack pattern corresponding to a valid test
is recognized as cracks initiating from the inner area of the loading ring; while an invalid
fracture testing is recognized as a diametral crack across the specimen that initiated from the

35

Figure 3.1. Cross-section view of the biaxial flexural testing set up.

36

(a)

(b)

Figure 3.2. Digital image of the ring-on-ring fixture used in the biaxial flexure testing in this
study; (a) assembled fixture with support ring underneath and loading ring on top of the
specimen; (b) support ring (left), loading ring (right), rubber tape (middle, top), specimen
with adhesive Teflon tape (middle, bottom).
37

Figure 3.3. Digital image of specimens showing crack patterns that correspond to valid and
invalid fracture test.

38

specimen edge (Figure 3.3, ASTM 2003). Only data of validly fractured specimens were
included in the study.
The fracture strength, , was calculated using the following equation [ASTM 2003]
2

DS  D 2
3F 
L  1   ln DS 



1   
D 
2h 2 
2D 2
 L 



(1)

where F is the load at failure, h and D are the thickness and the diameter of the test specimen,
DS and DL are the diameters of the support ring and the loading ring,  is the Poisson’s ratio
of the material.
Compared to other flexure test methods, including three point and four point flexure
tests, the advantages of the biaxial flexure test include the following: no extensive polishing
is needed before testing; the fracture is not affected by edge flaws or the slight warping of
specimens during sintering [Wachtman 2009].
3. Weibull analysis
In fracturing a group of brittle specimens fabricated in the same manner, the fracture
strength distribution does not follow a normal distribution. Instead, a Weibull distribution
typically fits the data well, Weibull statistics are based on the weakest link theory
[Wachtman 2009]. The distribution of the largest flaws (the weakest link) determines the
distribution of the fracture strengths in a brittle material. While knowing the mean strength is
important, knowing the distribution of strength is even critical in the design of products
fabricated from brittle materials.
The Weibull modulus is related to an applied stress at which a given fraction of
specimens will fail. For example, if one is designing products made from brittle materials,
and requires that less than 1 specimen out of 1000 specimens will fail in use, then the

39

Weibull modulus can be used to estimate the stress that would cause this fraction of failure.
Weibull modulus is directly linked to material reliability, in a way that the less scatter in the
strength leads to a higher Weibull modulus, meaning the more reliable a component is.
A high Weibull modulus value, for example, m = 20, indicates a narrow distribution of
the largest flaws and the fracture strengths [Wachtman 2009]. Carefully processed structural
ceramics such as SiC [Zhan 1999] and Si3N4 [Hirosaki 1993] can exhibit high m values. On
the other hand, a low Weibull modulus value, say, 5 and below, implies that strengths can
vary over a wide range, which is associated with a broad distribution of flaw size. In the
extreme, a Weibull modulus value will be infinite if the fracture strength of each specimen is
identical. On the contrary, a Weibull modulus value approaching zero would correspond to an
infinite scatter among the strength values of the group of specimens. The Weibull modulus is
affected by many factors such as porosity [Chao 1992], surface finish [Nakamura 2010a],
and test techniques [Cordell 2009], which is discussed in detail in Chapter 4, Appendix B.
The general form of Weibull distribution for fracture strength is expressed in the
following equation [Wachtman 2009]:
   m 
Pf  1  exp     dV 
  c 

 V 


(2)

where Pf is the cumulative probability of failure, V is the specimen volume subject to stress,
the scale parameter c is called the characteristic strength, the shape parameter m is named
the Weibull modulus [Wachtman 2009]. For specimens with constant volume, Equation (2)
can be simplified to

40

2
1

lnln(1/(1-Pf))

0
-1
-2

m

-3

ln c

-4
0.7

0.8

0.9

ln

1.0

1.1

Figure 3.4. An example of the Weibull plot. The slope of the linear least-squares fit, m, is the
Weibull modulus. The characteristic strength, c, can be calculated from the least-squares fit
at lnln(1/(1- Pf)) = 0.

41

   m 
Pf  1  exp    
  c  
   

(3)

By taking the double logarithm of equation (3), the two-parameter Weibull equation can
be rewritten as

 1
ln ln
1 P
f



  m ln   m ln  c



(4)

To determine the Weibull modulus of a group of specimens (all specimens have the
same volume and are tested under identical conditions), the fracture strength of each
specimen are ranked in order, starting from the lowest to the highest. If we assign ‘i’ as the
rank order for each specimen and ‘N’ is the total number of the specimens in the group, and
then the cumulative probability of failure, Pf , can be expressed as (i – 0.5)/N. Plotting
lnln(1/(1 - Pf)) versus lnwill result in a straight line with the slope m, given that the data
follow Weibull distribution (Figure 3.4). The characteristic strength, c, can be calculated
from the least-squares fit at lnln(1/(1- Pf)) = 0. At characteristic strength, c, the value of Pf
is 0.63, which means 63% of specimens will fail at c.
4. Microstructural examination
Microstructural examination of specimens was conducted via scanning electron
microscopy (SEM) (JEOL 6400V, JEOL Corp. Japan or Carl Zeiss Microscopy, LLC,
Thornwood, NY) on selected surfaces of selected specimens, including fractured surfaces,
polished surfaces, thermally etched surfaces and indented surfaces. Prior to examination, a
gold coating was sputtered on the examined surfaces of non-conductive specimens using a

42

gold sputter (EMSSCOPE SC500, Ashford, Kent, UK) to enhance electrical conductivity. An
accelerating voltage of 15 – 20 kV and a working distance of 5 - 15 mm were used.
Thermal etching was done on polished specimens by heating at temperatures 100oC
below the sintering temperatures for 1 hour in an electrical resistance furnace (CM Inc.,
Model #0100277, Bloomfield, NJ). The small grooves appearing at the grain boundaries are
due to surface diffusion.
The grain size was determined using the linear-intercept method. Multiple straight lines
were randomly superimposed over the SEM micrograph. Approximately 200 intercepts of the
lines and the grain boundaries were counted. Grain size was obtained via dividing the true
line length by the number of intercepts. A stereographic projection factor of 1.5 was used in
the mean grain size calculation [Case 1981, Underwood 1968].
5. Elastic modulus measurements
There are many techniques in the literature to measure the elastic moduli [Wachtman
2009, Radovic 2004]. Basically, the techniques can be categorized into two groups: the static
methods and the dynamic methods.
Static methods are based on the direct measurements of the stress – strain curve during
mechanical testing (tensile, compressive, flexural) [Wachtman 2009]. Then the elastic
moduli are determined from the slope of the linear part on the stress – strain curve. The
disadvantages of the stress – strain curve method include (i) a required specimen shape and
dimension, (ii) required careful surface processing, (iii) relative poor precision compared to
dynamic methods [Radovic 2004].
Another widely used static method is the nanoindentation technique [Oliver 2004]. The
elastic moduli can be determined from the slope of the linear unloading part of the load –

43

displacement curve. High precision can be achieved on well-prepared, relatively hard
specimens with homogenous microstructure [Radovic 2004]. However, the pre-obtained
Poisson’s ratio value is required in the Young’s modulus calculation using nanoindentation,
which poses a difficulty that either the Poisson’s ratio value is not accessible or the value is
not accurate.
Dynamic methods can also be categorized into two groups: the pulse methods [Rogers
2000] and the resonance methods [Migliori 1996, 2005]. Dynamic methods possess
remarkable merits compared to the static methods, with the advantages of minimum
specimen preparation, ease of specimen shapes and sizes, and high precision.
Pulse methods are based on the relationship between the sound velocities traveling in a
material and the elastic constants of the material [Rogers 2000]. By measuring the transit
time for longitudinal and transversal ultrasonic waves along with the dimensions and mass of
the specimens, the elastic moduli can then be calculated. The elastic moduli determined this
way are in a given direction or set of directions.
To date, the resonant ultrasound spectroscopy (RUS) method has been considered a very
good method for measuring the elastic moduli. In principle, the specimen is excited by a
range of driving frequencies, and the natural resonant vibrations of the specimen are recorded.
The entire matrix of elastic constants can then be determined with high precision from the
single resonant frequency spectrum.
RUS has been successfully used to measure the elastic moduli for a number of materials
including single-crystals [Yoneda 2012], polycrystalline bulk metals [Hurley 2010, Kaplan
2009], ceramics [Ren 2009, Seiner 2012], composites [Gorsse 2003, Vdovychenko 2006] and
thin film materials [Nakamura 2010b, Tan 2010].

44

The application of RUS method requires some experience. Some aspects need to be
considered during RUS analysis. First, an initial guess of the elastic moduli is required to
ensure converge in the final solution. A close initial guess can be made by reference of
similar materials in the literature, and the initial guess can be modified based on the shift
between measured frequency peaks and predicted frequency peaks. Another concern lies in
the accuracy of the specimen dimensions. Since the calculation of the mechanical vibration
modes is based on the geometry of the specimens (in this study, either disk-shaped or barshaped), it is of great importance to cut and polish the specimen as close to a perfect disk or
bar shape.
In this study, the Young’s, shear, bulk modulus and Poisson’s ratio of the specimens
were measured by RUS (RUSpec, Quasar International, Albuquerque, NM, USA). Two types
of configuration were used, namely, the three-transducer configuration (Figure 3.5a) mainly
used for disk-shaped specimens and the two-transducer configuration (Figure 3.5b) for barshaped specimens. For the three-transducer configuration, a disk-shaped specimen was places
on the tripod, where one piezoelectric transducer emits an ultrasonic signal through a range
of frequencies, while the other two piezoelectric transducers pick up and record the
mechanical resonance peaks of the specimen as a function of frequency (Figure 3.5a). For the
two-transducer configuration, the body diagonal or the face diagonal of the bar-shaped
specimen was lined up and held between the two piezoelectric transducers, where one
transducer excites and the other being the pick up transducer. The two transducers are spring
loaded, which allow free vibration of the specimen (Figure 3.5b).
In each measurement, about 30 to 50 sharp mechanical resonant peaks were recorded for
analysis (Figure 3.6). After each measurement, the specimen was rotated a small angle on the

45

(a)

(b)

Figure 3.5. Digital images of (a) three-transducer configuration and (b) two-transducer
configuration of the RUS apparatus.
46

7

Amplitude (Arbitrary Unit)

6
5
4
3
2
1
0
5

50

100

Frequency (kHz)

Figure 3.6. A typical RUS spectrum.

47

150

200

tripod or held along another face/body diagonal on the two-transducer configuration before
another measurement was performed. The purpose of doing so is to detect possible missing
vibrational modes due to the placement of the transducers at the vibration antinode. For diskshaped specimens, some vibration frequency peaks are degenerate due to the geometric
symmetry of the specimens.
The elastic stiffness tensor was then calculated using the resonant frequencies along
with the mass, geometry and dimensions of the specimen. For all the specimens discussed in
this dissertation, they are polycrystalline solids with randomly oriented grains, which implies
isotropic elastic behavior (two independent elastic constants). The independent elastic
constants were calculated via a commercial software (RPModel, Quasar International,
Albuquerque, NM, USA) that performs an algorithm using a weighted error minimization
between the initial input and calculated resonance frequencies until it is converged. The
program also generated the root mean square (RMS) error of the fit, and only measurements
with RMS error less than 0.5 % were used to ensure a relative accurate calculation.
6. Hardness and fracture toughness measurements
Both the hardness, H, and the fracture toughness, Kc, were measured using the Vickers
indentation technique. Vickers indentation is a non-destructive technique that requires
minimum sample preparation, and has been widely used in hardness and fracture toughness
evaluation [Barsoum 1997, Quinn 1997]. A diamond indenter, with square-based pyramid
shape was applied.
Hardness shows a material’s resistance to plastic deformation. The hardness measured
by Vickers indentation can be calculated from the following relationship [Wachtman 2009]

48

(a)

(b)

Figure 3.7. Diagram of the Vickers indentation impression and radial cracks system, (a)
viewed on top; (b) viewed by side.

49

H=

1.8544PL

(5)

(2a ) 2

where PL is the indentation load, is the calibration factor and 2a is the diagonal length of
the indentation impression (Figures 3.7a and 3.7b).
In this study, a Vickers indenter (Shimadzu HMV 2000, Kyoto, Japan) with load range
from 5 g to 2000 g was used. Each hardness value was calculated from the average of 10
indentations. The Vickers indenter was calibrated using a HV 790 steel standard calibration
block (Yamamoto Scientific Tools Lab Co. LTD, Chiba, Japan). The calibration factors, 
were determined from the average hardness value obtained from ten indentations on the
calibration block for each load.
In brittle materials, the residual stresses during the indentation will cause the formation
of the semi-elliptical cracks [Lube 2001], known as radial cracks (Figures 3.7a and 3.7b).
Using the crack length, the fracture toughness can be estimated via the Vickers indentation
technique using equation (6) [Anstis 1981]
1/ 2

E
Kc =   
H

PL

(6)

c3 / 2

where E is the Young’s modulus determined from RUS, H is the hardness calculated from
equation (5), PL is the indentation load and 2c is the radial crack length (Figures 3.7a and
3.7b) and the calibration constant, 0.016 [Anstis 1981].

50

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51

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[ASTM 2003] ASTM C1499-05, Standard test method for monotonic equibiaxial flexural
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Inc., New York, 1997.
[Case 1981] E.D. Case, J.R. Smyth, V. Monthei, Grain-size determinations, J. Am. Ceram.
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a sintered silicon-nitride failing from pores, J. Am.Ceram. Soc. 75, 2116-2124, 1992.
[Cordell 2009] J.M. Cordell, M.L. Vogl, A.J.W. Johnson, The influence of micropore size on
the mechanical properties of bulk hydroxyapatite and hydroxyapatite scaffolds, J. Mech.
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[Hirosaki 1993] N. Hirosaki, Y. Akimune, M. Mitomo, Effect of grain growth of β-silicon
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1894, 1993.
[Hurley 2010] D.H. Hurley, S.J. Reese, S.K. Park, Z. Utegulov, J.R. Kennedy, K.L. Telschow,
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property evolution, J. Appl. Phys. 107, 063510, 2010.
[Kaplan 2009] G. Kaplan, T.W. Darling, K.R. McCall, Resonant ultrasound spectroscopy and
homogeneity in polycrystals, Ultrasonics 49, 139-142, 2009.
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[Migliori 2005] A. Migliori, J. D. Maynard, Implementation of a modern resonant ultrasound
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experimental techniques for determination of elastic properties of solids, Mater. Sci. Eng.
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spectroscopy measurement of Young's modulus, shear modulus and Poisson's ratio as a
function of porosity for alumina and hydroxyapatite, Phil. Mag. 89, 1163-1182, 2009.
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53

CHAPTER 4
POROSITY DEPENDENCE OF THE WEIBULL MODULUS FOR
HYDROXYAPATITE AND OTHER BRITTLE MATERIALS
X. Fan, E. D. Case, F. Ren, Y. Shu and M. J. Baumann
Department of Chemical Engineering and Materials Science, Michigan State University,
East Lansing, MI, 48824
Published in: J Mechanical Behavior of Biomedical Materials, 8:21-36, 2012.

Abstract
Porous brittle materials are used as filters, catalyst supports, solid oxide fuel cells and
biomedical materials. However the literature on the Weibull modulus, m, versus volume
fraction porosity, P, is extremely limited despite the importance of m as a gauge of
mechanical reliability. In Part I of this study, m is determined for 441 sintered hydroxyapatite
(HA) specimens fractured in biaxial flexure for 0.08 < P < 0.62. In this study, we analyze a
combined data set collected from literature that represents work from a total 17 different
research groups (including the present authors), eight different materials and more than 1560
oxide and non-oxide specimens, the m versus P plot is “U - shaped” with a wide band of m
values for P < 0.1 (Region I) and P > 0.55 (Region III), and a narrower band of m values in
the intermediate porosity region of 0.1 < P < 0.55 (Region II). The limited range of m ( 4 <
m < 11) in Region II has important implications since Region II includes the P range for the
majority of the applications of porous brittle materials. Part II of this study focuses on the P
dependence of the mean fracture strength, <f>, and the Young’s modulus E for the HA

54

specimens tested in Part I along with literature data for other brittle materials. Both <f> and
E are power law functions of the degree of densification, , where  = 1 - P/PG and PG is the
green (unfired) porosity.

1. Introduction
Most thermal, electrical and mechanical properties of solids are affected by porosity. For
example, porosity affects thermal conductivity [Hu 2010a, Park 2009, Sulistyo 2010] and
thermal diffusivity [Park 2009]. For electrical properties, the dielectric constant [Case 2006,
Geis 2002, Hoepfner and Case 2002, Yang 2010], dielectric breakdown voltage [Geis 2002,
Hench and West 1990], and electrical conductivity [Geis 2002, Shimizi 2009, Sulistyo 2010]
are all functions of porosity. In terms of mechanical properties, the flexural fracture strength
[Park 2009], compressive fracture strength [Hu and Wang 2010b, Nielsen 1990, Sammis and
Ashby 1986], fracture surface energy [Vandeperre, 2004, Case and Smyth, 1981], Young’s
modulus [Ni 2009, Ren 2009], shear modulus [Ni 2009, Ren 2009] and hardness [Hoepfner
and Case 2003] also are porosity dependent.
However, the literature data on the P dependence of the Weibull modulus, m, is quite
limited, despite the fact that m is an important gauge of the mechanical reliability of brittle
materials. While many studies in the literature evaluate m for brittle specimens with volume
fraction porosities, P, of roughly 0.1 or less, few studies give Weibull modulus versus P
results for P > 0.1 and the data is especially sparse for P > 0.3.
For a very wide range of brittle material applications, porosity is required for the
55

materials to function in the desired manner. In fact, for many applications of porous materials,
P ranges from about 0.2 to 0.5 [Liu2010]. For engineered bone materials, porosity is required
for the HA to mimic the function of natural bone [Baumann2008, Ren2005]. Other example
of applications of porous HA include use as microbe filters [Yang 2007a, Yang 2007b],
sensors [Mahabole2005, Nagai1988, Owada1989] and catalytic supports [Ohtsuki2010],
where a number of these environmental, sensing and catalytic applications of porous HA [da
Silva2006, Mori2003, Mori2004, Nishikawa 2003, Opre2005, Srinivasan2006, Venugopal
and Scurrell 2003, Yin and Ellis 2010] are based on HA’s ion exchange capabilities. In
general, coatings and dielectric layers (both dense and porous) of HA also have been
explored for biomedical and microwave applications [Hontsu1997, Silva2003, Silva2005],
such as biocompatible coatings on implanted medical devices [Hontsu1997].
In addition to HA, porous brittle materials such as alumina, silica, mullite, zeolites and
Ni-YSZ cermets also have been used for gas separation [Sommer and Melin 2005,
Zeng2009], water filtration [Corneal2010, Karnik2005], heavy metal immobilization [Barba
and Callejas 2006, Twigg and Richardson 2007], catalytic supports [Bein 1996, de
Lange1995, Groschel 2005, Lambert and Gonzalez 1991] and anodes for solid oxide fuel
cells (SOFC) [Jung2006].
This study is presented in two parts. Part I deals with the porosity dependence of the
Weibull modulus, m, of brittle materials. Part II examines the average fracture strength, <f>,
and the Young’s modulus, E, of brittle materials as functions of porosity. For both Part I and
Part II, HA powders were sintered to achieve a range of volume fraction porosity, P, from P =
56

0.08 to P = 0.62. All HA specimens in the experimental phase of this study were fractured in
biaxial flexure using a ring-on-ring (ROR) fixture.
The Weibull modulus, m, (Part I), the average fracture strength, <f> and Young’s
modulus, E, (Part II) were analyzed in conjunction with appropriate data from the literature.
The number of HA specimens (n = 540) fractured in this study is more than twice the
combined number of HA specimens in porosity-Weibull modulus studies previously reported
in the literature.
Although individual m versus P studies in the literature were not as extensive (in terms
of numbers of specimens fractured), this study analyzes literature data from 17 different
research groups for eight different materials, including HA in this study, six literature studies
of HA [Cordell2009, Meganck 2005, Murray1995, Lopes 1999, Ruys1995, Villora2004] as
well as data from other oxides: zirconia (ZrO2, [Pissenberger and Gritzner 1995]), titania
(TiO2, [Kishimoto 1991]), alumina (Al2O3, [Chao 1991], [Nanjangud 1995]) and mullite
(2SiO2•3Al2O3, [Vales1999]) and non-oxides (silicon, [Borrero-Lopez2009], [Brodie and
Bahr 2003], [Paul2006]), silicon nitride (Si3N4, [Hirosaki and Akimune 1993], [Vales1999])
and LASTT (Ag0.9Pb9Sn9Sb0.6Te20 semiconducting thermoelectric materials, [Ren 2006]).
2. Experimental Procedure
2.1 Specimen preparation
All the hydroxyapatite (HA) specimens in this study were fabricated from biomedical
grade commercial powder (Taihei Chemical Industrials Co., Osaka, Japan). The powder was
uniaxially pressed into disk-shaped compacts at 33 MPa and then sintered in air from 550oC
57

to 1360oC from 12 minutes to 4 hours (Table 4.1). The specimen mass was measured using
an electronic balance (Denver Instrument Company, Arvada, CO, with a vendor-specified
precision of ±0.001 gram). For each specimen, three measurements of the specimen’s
diameter and thickness were made using digital caliper (Davis Calibration, Carol Stream, IL,
with a vendor-specified precision of ±0.01 mm). The porosity of the sintered specimens
(Table 4.1) was calculated from the specimen mass along with mean specimen diameter and
3

thickness, assuming a theoretical density of 3.156 g/cm for HA [CRC 2009]. The current
study includes 23 groups of HA specimens with volume fraction porosity, P, values ranging
between approximately 0.08 and 0.62 (Table 4.1). From 20 to 24 specimens were prepared
for each porosity group, yielding a total number of 540 specimens.
2.2 X-ray diffraction
To examine the phase purity of the HA specimens, X-ray diffraction (XRD) was
performed on the as-received commercial powder as well as specimens sintered at 1100oC for
2 h, 1200oC for 1 h and 1360oC for 4 h. Specimens sintered at 1360oC were examined as both
bulk and pulverized specimen. All XRD patterns were collected on a rotating anode
o

diffractometer (Rigaku 20B, Rigaku Corp., Japan) using Cu K radiation between 10 and
o

o

o

60 2with a step size of 0.02 . Bulk specimens were scanned at a rate of 0.5 2θ per
o

minute, whereas powder specimens were scanned at a rate of 2 2θ per minute.
2.3 Scanning electron microscopy
Microstructural examination was conducted on polished and thermally etched surfaces
of HA specimens via scanning electron microscopy (SEM) (JEOL 6400V, JEOL Corp.
58

Table 4.1. For the disk-shaped HA specimens included in this study, the specimen label,
sintering temperature, T, sintering time, the specimen dimensions, the volume fraction
porosity P, the number of valid specimens N, the mean fracture strength <f>, the
characteristic strength c, and the Weibull modulus m. Each specimen was cold die pressed
at 33 MPa and sintered in air in an electrical resistance furnace.

Label T (oC)
P-08 1360
P-09 1350
P-13 1360
P-17 1360
P-21 1250
P-24 1220
P-30 1200
P-33 1200
P-36 1200
P-35 1200
P-39 1175
P-42 1150
P-46 1125
P-46 1100
P-51 1075
P-55 1050
P-60 1000
P-62 900
P-62 800
P-62 700
P-62 600
P-62 550
P-61 green

Time Thickness Diameter
(mins)
(mm)
(mm)
240
120
27
12
85
60
110
60
60
30
55
40
60
120
120
120
120
120
120
120
120
120
0

1.84±0.05
1.85±0.02
1.89±0.02
1.92±0.01
2.00±0.03
1.92±0.02
2.00±0.03
2.09±0.03
2.13±0.05
2.03±0.04
2.14±0.03
2.08±0.02
2.15±0.03
2.19±0.02
2.27±0.02
2.32±0.03
2.42±0.02
2.47±0.02
2.49±0.02
2.50±0.02
2.48±0.03
2.47±0.03
2.50±0.02

P

23.4±0.1
23.8±0.1
24.0±0.1
24.3±0.1
24.6±0.1
25.2±0.2
25.9±0.1
26.2±0.1
26.4±0.1
26.8±0.2
27.2±0.1
28.1±0.2
28.6±0.3
28.4±0.1
29.2±0.1
30.1±0.1
32.2±0.03
32.0±0.02
32.2±0.01
32.2±0.01
32.2±0.01
32.1±0.01
32.2±0.01

0.08±0.01
0.33±0.01
0.13±0.01
0.17±0.01
0.21±0.01
0.24±0.01
0.30±0.01
0.33±0.01
0.36±0.02
0.35±0.01
0.39±0.01
0.42±0.01
0.46±0.02
0.46±0.01
0.51±0.01
0.55±0.01
0.60±0.01
0.62±0.01
0.62±0.01
0.62±0.01
0.62±0.01
0.62±0.01
0.61±0.01

59

N

<f>

20
18
18
24
14
15
17
18
18
12
19
19
20
20
23
21
21
21
19
19
21
22
22

(MPa)
91±16
62±5
58±7
69±10
42±5
39±8
33±4
33±4
34±8
29±4
23±3
22±3
23±4
17±2
12±2
8±1
4.5±0.5
3.5±0.3
3.3±0.3
2.0±0.2
1.4±0.1
1.3±0.1
0.6±0.03

c (MPa)

m

97±15
6.8
63.6±4.9 15.0
61±9
9.2
74±8
8.4
44.4±4.3 9.8
42.2±8.2 6.1
34.5±3.5 9.8
35.2±4.0 8.8
37.7±6.5 4.4
31.3±3.9 7.5
24.3±3.7 10.5
22.7±3.0 9.4
24.5±4.3 6.6
17.3±1.9 11.5
12.6±1.6 8.2
8.4±1.0
9.5
4.8±0.3 10.1
3.6±0.2 15.5
3.4±0.2 12.2
2.1±0.2 10.4
1.4±0.1 12.7
1.3±0.2 10.6
0.6±0.05 21.2

Japan). Thermal etching was done by heating the specimens in air in a conventional box
furnace at temperatures 100oC below the sintering temperatures for 1 hour. After thermal
etching, a gold coating of approximately 20 nm thick was sputtered on the HA specimens
using a gold sputter (EMSSCOPE SC500, Ashford, Kent, UK) before SEM examination.
Each SEM image was taken using a working distance of 15 mm and an accelerating voltage
of 15 kV. The grain size analysis was performed using the linear-intercept method. Multiple
straight lines were randomly superimposed over the SEM micrographs and approximately
200 intercepts between the lines and the grain boundaries were counted to determine the
mean grain size. A stereographic projection factor of 1.5 was used in the mean grain size
calculation [Case1981, Underwood 1968].
Microstructural examination was also conducted on fracture surfaces of HA specimens
via SEM using the same procedure described above.
2.4 Biaxial flexural testing
Biaxial flexural testing (BFT) was performed on a universal testing machine
(Model 4206, Instron, Norwood, MA) using a ring-on-ring (ROR) test fixture [ASTM 2003].
The diameters of the loading ring and the supporting ring were 10 mm and 20 mm,
respectively. In this study, each as-sintered specimen was fractured at a crosshead speed of 1
mm/min.
3. Results and Discussion
3.1 Phase analysis for the HA specimens in this study
The XRD patterns for the initial HA powder as well as bulk specimens sintered at
60

Intensity (A.U.)

2000
1500
1360oC powder

1000

Initial powder
1100oC bulk

500

1200oC bulk
1360oC bulk

0
10

20

30
40
2deg)

50

60

Figure 4.1. XRD patterns of the initial commercial HA powder and HA specimens sintered at
1100oC, 1200oC and 1360oC, where the arrow above the XRD pattern marks the position of
the faint peak associated with - TCP in the XRD pattern for initial powders. (A.U. stands
for arbitrary units).

61

(a)

(b)

(c)

(d)

Figure 4.2. SEM micrographs of (a) fracture surfaces for HA specimens with porosity of 0.42;
(b) fracture surfaces for HA specimens with porosity of 0.30; (c) polished and thermally
etched surfaces for HA specimens with porosity of 0.24 and (d) polished and thermally
etched surfaces for HA specimens with porosity of 0.08.

62

1100oC, 1200oC and 1360oC showed no detectible secondary phases (Figure 4.1). The XRD
pattern of specimens sintered at 1360oC and ground into powder showed a faint peak at 2 ~
o

31 (indicated by the arrow in Figure 4.1). This additional peak is likely associated with
-tricalcium phosphate (-TCP), where -TCP is one of the minor products of the
decomposition of HA at high temperature [Sueter 1972].
3.2 Microstructural evolution of HA specimens included in this study
In general, during solid state sintering of powder compacts, the pore size and
morphology evolves through three stages: (1) an initial stage when necks start to form and
grow between powder particles, (2) an intermediate stage where tubular pores appear along
the grain boundaries and (3) a final stage at which the tubular pores break up into isolated
quasi-spherical pores typically located at the triple points of grains [Barsoum 1997, German
1996].
This typical microstructural evolution was observed for the HA powders sintered in this
study (Figures 4.2a –4.2d). At P = 0.42, during the initial sintering stage (Figure 4.2a), the
powder particles begin to neck and an open network of pores is present. At P = 0.30, when
the sintering was between the initial stage and the intermediate stage (Figure 4.2b), a
relatively open pore network still exists between particles. During the intermediate sintering
stage, not only the most densification occurred, but also the pore size, number density of
pores, and pore morphology underwent significant changes. At P = 0.24, tubular pores were
observed along the grain boundaries (Figure 4.2c), which corresponds to the intermediate
sintering stage [Barsoum 1997, German 1996]. In the final sintering stage, isolated spherical
63

Grain size (m)

8
6
4
2
0
0.4

0.6

rel

0.8

1.0

Figure 4.3. Grain size versus relative density,rel, trajectory for HA specimens included in
this study, representing eleven values of rel ranging from 0.38 to 0.91.

Invalid fracture

Valid fracture

Figure 4.4. Digital photographs of specimens showing valid and invalid fracture patterns for
HA specimens fractured via biaxial flexure (the ring-on-ring technique) in this study. “Valid”
and “invalid” are defined in ASTM C1499-05 [ASTM 2003], where invalid ring-on-ring
fractures involve failure from edge flaws.
64

or quasi-spherical pores were observed at grain boundaries or within grains (Figure 4.2d),
while the porosity decreased to 0.08.
The grain size versus density trajectory of the HA specimens showed a typical behavior
[Barsoum 1997, Berry and Harmer 1986], with limited grain growth for specimens with
relative densities less than about 0.8 to 0.9 (Figure 4.3). In this study, the grain size
measurements (Figure 4.3) covered the entire P range of to 0.62, which corresponds to
the relative density, rel, range of 0.91 – 0.38. For rel values between approximately 0.38 and
0.8, the grain size slowly increases from about 0.5 m to 1.5 m (Figure 4.3), and this
density range includes 19 out of the 23 sets of specimens included in this study (Table 4.1).
Thus, for about 85 percent of the HA data in this study, the grain size of the specimens is
within the relatively restricted range from 0.5 m to 1.5 m.
3.3 Measurement of the Weibull modulus for the HA specimens in this study
For the 23 groups of HA specimens in this study (Table 4.1), a total number of 540 disks,
tested using the ROR configuration [ASTM 2003], yielded 441 valid tests (Figure 4.4).
According to ASTM C1499-05 [ASTM 2003], fractures that initiated from the specimen
edges and extended diametrally across the specimen (Figure 4.4) were invalid and were not
included in the strength calculations in this study. Figure 4.4 also gives an example of a valid
ROR fracture pattern [ASTM 2003].
A two-parameter Weibull analysis was applied to the fracture strength data in this study
using the following equation

65

2
1

P = 0.61

lnln(1/(1-Pf))

0

P = 0.60
P = 0.51
P = 0.42
P = 0.35
P = 0.30
P = 0.21
P = 0.13

-1
-2
-3

P = 0.62
P = 0.55
P = 0.46
P = 0.39
P = 0.33
P = 0.24
P = 0.17
P = 0.09

-4
-1

0

1

2
3
lnf

4

5

Figure 4.5. For the HA specimens fractured by biaxial flexure in this study, the Weibull plots
(equation (1)) for each of the HA groups, where each group had a different volume fraction
porosity, P. (Table 4.1 lists the P values of each group of specimens in this study).

Weibull modulus

20
16
12
8
4
0.0

0.1

0.2

0.3

0.4

0.5

0.6

Porosity

Figure 4.6. Weibull modulus as a function of volume fraction porosity, P, for the HA
specimens in this study. (The numbers in parentheses indicate the total number of specimens
fractured in a given study).
66

 1
ln ln 
 1  Pf



  m ln  f  m ln  c



(1)

where Pf is the probability of failure, m is the Weibull modulus, σf is the fracture strength and
σC is the characteristic strength [Wachtman 2009]. Weibull plots of the 23 groups of HA
specimens are shown in Figure 4.5; while details of the Weibull analysis is given in Appendix
A.
The average fracture strength, <f>, the characteristic strength, c, and the Weibull
modulus, m, of the 23 groups of HA specimens were summarized in Table 4.1. Both <f>
and c decreased monotonically with increasing porosity (Table 4.1), while m varied from
4.4 to 21.2 (Figure 4.6). In Part II of this study, we discuss the porosity dependence of
fracture strength and Young’s modulus for the HA specimens included in this study along
with other HA and non-HA specimens from the literature.
3.4 Porosity effect on Weibull modulus
3.4.1 Weibull modulus as a function of porosity for HA
In addition to the HA specimens sintered and fractured in this study, we used HA data
from six other research groups [Cordell2009, Meganck 2005, Murray1995, of volume
fraction porosity, P. Details of the materials, processing and testing of HA from literature are
summarized in Table 4.2. It is worthwhile to note that the number of HA specimens fractured
in this study is greater than the total number of specimens fractured in all six studies
currently available in the literature [Cordell2009, Meganck 2005, Murray1995, Lopes 1999,
Ruys1995, Villora2004].
67

Table 4.2. For the HA studies from the literature: shape forming techniques, specimen
dimensions, grain size G.S., strength testing techniques, loading rate, surface finish, SF,
number of specimens N, porosity P, the mean fracture strength <f>, the characteristic
strength c and Weibull modulus m.
Shape
forming
tech.

Ref.

Meganck

Cold
Pressed

Cold
Pressed

Lopes

Testing
Specimen G.S. tech. &
dimension (m) loading
rate

32 mm

b

25 x 2.5
mm

13 x

Ruys

Villora

Cold
Pressed
8x~4x
Extrusion
~8 mm
Slip
Cast

Cordell

N/S
0.9
1.2
2.4

c

Cold
Pressed

Cold
pressed

7.9

2 mm

c

N/S

N/S

~6.48 x
13 mm

a

N/S
3 x 4 x 45
Extruded
mm bar

Murray

Uniaxial
cold
pressed

16.7 x
1.2 - 1.5
mm c

N/S

SF

N

Piston-on
-3 ball
1
10
0.5
m
mm/min
12
Ring-onring
N/S 12
5
12
mm/min
12
20
20
Ball-on- 5
ring
m 20
20
Diametr
28
-al
d
comp.
28
10
28
mm/min
Comp.
30
Strain
rate:
30
0.002 s -1 15
4-p bend m
30
Strain
rate:
30
0.002 s-1
15
Standard d
shell test
15
68

P

<f >
(MPa)

0.02

130±
12

c
(MPa)

134.9
11.9
±0.4

0.12 ~28
0.07
0.05
0.06
0.45
0.24
0.07
0.09

~50
~55
~46
~28
~73
~80
~55

m

5.8
N/S
~ 50
N/S

5.2
3
5.6
6.9
7.9
6.5
3.4

0.42 35

39

3.7

0.55 24

25

6

0.65 27

32

2

0.4
0.41
0.46
0.46
0.28

110±
18
70.9±
8.8
21.8±
2.3
18.6±
2.5
69.7

0.06 170.6

118.2 5.8
73.9

8.7

22.7

10.3

19.1

7.2
7.6

N/S

13.5

Table 4.2 (cont'd).
a

In [Cordell 2009], PMMA spheres with diameters of 5.96 m and 16.2 m added to the

powders burned out during sintering leaving large pores with diameters similar to the PMMA
sphere diameters. This results in a bimodal pore size distribution. Specimens in the other
references in this table have unimodal pore size distribution.
b
c
d

Diameter of disc-shaped specimens
Diameter and thickness of disc-shaped specimens
As-fabricated

Weibull modulus

20

HA (441) (this study)
HA (10) (Meganck et al. 2005)
HA (48) (Lopes et al. 1999)
HA (80) (Ruys et al. 2005)
HA (84) (Villora et al. 2004)
HA (120) (Cordell et al. 2009)
HA (30) (Murray et al. 1995)

I

16

III

II

12
8
4
0.0

0.1

0.2

0.3
0.4
Porosity

0.5

0.6

Figure 4.7. Weibull modulus as a function of P for HA (this study, Cordell2009, Meganck
2005, Murray1995, Lopes 1999, Ruys1995, Villora2004]. (The numbers in parentheses
indicate the total number of specimens fractured in a given study).

69

For the HA data we can divide the m versus P behavior into three regions (Figure 4.7). In
Region I (P < 0.1), there is considerable scatter in the m versus P data (Figure 4.7). In this
study, for P = 0.08, the Weibull modulus was 15 while for P = 0.08, the Weibull modulus was
6. In region II (0.1 < P < 0.55), the m values fall into a relatively narrow band from ~ 4 to ~
11. In region III (P > 0.55), m tends to increase with increasing P but there is scatter in the
data especially near the green density P value of 0.62 (Table 4.1). For green (unfired) HA
powder compacts, the Weibull modulus is as high as 21 (this study).
3.4.2 Weibull modulus as a function of porosity for HA from this study, HA from the
literature and seven additional brittle materials from the literature
The combined data set includes the m versus P data for HA [this study, Cordell2009,
Meganck 2005, Murray1995, Lopes 1999, Ruys1995, Villora2004] plus the m versus P data
for other oxides (zirconia [Pissenberger and Gritzner 1995], titania [Kishimoto 1991],
alumina [Chao 1991, Nanjangud 1995] and mullite [Vales1999]) and non-oxides (silicon
[Borrero-Lopez2009, Brodie and Bahr 2003, Paul2006], silicon nitride [Hirosaki and
Akimune 1993, Nanjangud 1995, Vales1999] and LAST-T semiconducting thermoelectric
materials [Ren 2006]) from the literature (Figure 4.8). Details of the materials, processing
and testing of the other oxides [Kishimoto 1991, Nanjangud 1995, Pissenberger and Gritzner
1995, Vales1999] and the non-oxides [Borrero-Lopez2009, Brodie and Bahr 2003, Hirosaki
and Akimune 1993, Nanjangud 1995, Paul2006, Ren 2006, Vales1999] are summarized in
Tables 4.3 and 4.4, respectively.
It is extremely important to note that for the combined data set, the mean fracture

70

strengths, <f>, of the dense forms of these materials can differ from one another by up to
more than a factor of 5. For example, the fracture strength for HA with P = 0.023 is 130 MPa
[Meganck 2005] (Table 4.2), while the fracture strength of silicon nitride with P = 0.01 is 689
MPa [Hirosaki and Akimune 1993] (Table 4.4). Nevertheless, despite the large range in the
<f> values, the m versus P values of the combined data set (Figure 4.8) are consistent with
the trends observed for the HA (Figure 4.7). This underscores the fact that m versus porosity
trends depicted in Figure 4.8 apply to an extremely diverse set of materials.
The Weibull modulus versus P data for the combined data set (Figure 4.8) shows a large
scatter in region I (P < 0.1) and less scatter in region II (0.1 < P < 0.55). In Region II the band
of m values between 4 and 11 includes data from this study (14 m values) as well as data
from eight additional studies including four different materials: HA [Lopes one m value
[Lopes 1999], Ruys two m values [Ruys1995], Villora one m value [Villora2004], Cordell
four m values [Cordell2009], Murray one m value [Murray1995]), Nb and Ta doped zirconia
(Pissenberger one m value [Pissenberger and Gritzner 1995]), titania (Kishimoto three m
values [Kishimoto 1991]) and alumina (Nanjangud two m values [Nanjangud1995]). All of
the m data for the non-oxides lies in Region I (P < 0.1) with m values ranging from ~ 2 to 53
(Figure 4.8).
The m versus P behavior observed in Region II and Region III (Figures 4.6 –4.8) has
not been previously reported. For HA and other brittle materials, the m versus P data
available in the literature for P > 0.47 is especially limited. The authors are only aware of two
groups of Weibull modulus data for P > 0.47, namely specimens from the Villora HA study
71

Table 4.3. For the oxides other than HA from the literature: materials, shape forming
techniques, specimen dimensions, grain size G.S., strength testing techniques, loading rate,
surface finish, SF, number of specimens in each group N, porosity P, the mean fracture
strength <f>, the characteristic strength c and Weibull modulus m.

Ref.

Testing
Materials and
Specimen
tech. &
shape
G.S.
dimen.
loading
forming tech.
(m)
rate

SF

N

P
0.38

<f >
or
m
c
(MPa)
25.9
8.4
c

0.27
4-p bend
20
25.4
-30
N/S
0.08
mm/min

111.4

0.03

Nanjangud

Commercial
62 mm x
available
10 mm x
alumina
N/S
4 mm
extruded bars

293.4

c

262.7
c

8.1
9.3

c

Commercial
titanium
dioxide
Kishimoto
powders.
Uniaxially
Pressed
Niobia and
tantala doped
zirconia.
Isostatically
pressed at 8
kPa
10.0%, 0.0%,
Pissenberger
1.99 m a
14.2%, 0.0%,
15.2 m a
18.0%, 0.0%,
1.91 m a
0.0%, 10.0%,
1.61 m a

5 mm x
20 mm x
0.19 mm

1.1
1
1

0.4
0.3
0.2
0.1

1

4 mm x 3
mm x 30 N/S
mm

3-p bend

d

N/S

63 f
71 f
129 f

8.4
5.3
5.7

288 f

34
32
21
3-p bend
N/S 31

13

4.3

0.24

4.9
48

f

0.06

5.1
172

f

0.1

5.2
f

0.05

72

126
127 f

6.2

Table 4.3 (cont'd).
0.0%, 14.2%,
1.89 m a
0.0%, 18.0%,
1.71 m a
10.0%, 0.0%,
0.38 m a
14.2%, 0.0%,
0.43 m a
18.0%, 0.0%,
0.44 m a
0.0%, 10.0%,
0.34 m a
0.0%, 14.2%,
0.47 m a
0.0%, 18.0%,
0.55 m a

0.02
0.03
0.05
0.02
0.01
0.01
0.01
0.02

Vales

Chao

a
b
c
d
f

Commercial
mullite

45x4x3
mm

45x4x3
mm
Commercial45x4x3
grade
mm
sintered
6
50.8x
alumina
b
3.175 in

~5

4-p bend
d
5
mm/min

20

0.075

Biaxial
flexure

Weight percentage of doped niobia and tantala, particle size
Diameter and thickness of disc-shaped specimens
The characteristic strength c
Polished
The mean fracture strength <f>

73

199 f
184 f
216 f
194 f
210 f
224 f
217 f

N/S

10
4.9
5.9
8.5
9.2
8.8
14
14

13

385.9

3-p bend
4-p bend d

173 f

c

~3
3

~
0.07

353.4
c

338.8
c

5
5.4
3.8
2.3

[Villora2004] (representing 56 fractured specimens) and data for nine groups of specimens
from this study (representing 189 fractured specimens). If one extends the P range out to
Region III (P > 0.55), then only a single m value from Villora’s study [Villora2004] remains.
However, Villora’s specimens are multiphase specimens of HA and SiO2 that also include
animal bones, waste diatom filters and paper industrial sludge [Villora2004]. Thus, while the
other porous brittle materials analyzed in this study consisted of partially sintered single
phase materials (this study, Borrero-Lopez2009, Brodie and Bahr 2003, Chao 1991,
Cordell2009, Hirosaki and Akimune 1993, Kishimoto 1991, Lopes 1999, Meganck 2005,
Murray1995, Nanjangud1995, Paul2006, Pissenberger and Gritzner 1995, Ruys1995,
Vales1999, Villora2004], Villora’s specimens are in contrast quite heterogeneous. The
multiphase nature of Villora’s specimens could lead to microcracking which could also affect
the flaw populations and hence the value of the Weibull modulus.
3.5 Physical interpretations of the three regions in the m versus P plots
A critical question here concerning the m versus P behavior shown in Figures 4.6 –4.8 is
“What physical mechanisms may be responsible for the observed behaviors?” For the entire
range of P in Figures 4.6 –4.8, the Weibull modulus behavior is almost certainly
fundamentally linked to the evolution of the porosity during sintering as discussed briefly in
Section 3.2. Given that, then what may be the particular links between the evolution of
porosity and the m versus P behavior? It is important to note that if the Weibull modulus
versus porosity data (Figure 4.8) were to be plotted for each material in isolation, there are
too few data points to determine a trend.
74

Table 4.4. For the non-oxides other than HA from the literature: powder specification, shape
forming techniques, specimen dimensions, grain size G.S., strength testing techniques,
loading rate, surface finish, SF, number of specimens in each group N, porosity P, the mean
fracture strength <f>, the characteristic strength c and Weibull modulus m.

Ref.

Borrero
-Lopez

Ren

Chao

Materials and
Specimen
shape
G.S.
dimen.
forming tech.

Poly-silicon
wafer. Cut
from
commercial
wafer

12.5 cm
x 12.5
cm x
199±4
m

mm

<f>
SF

Tens
of mm

4-p
bend
5 mm /
min

N

P

30

~0

As-c
ut
30

75

1.6

f

~0
161.2

~
0.01

~15

15.3

~
~
30 0.09

3.2

c

768.5
600
grit

f 11.5

f

4-p
bend

Biaxial
flexure

m

14.4

f

131

76

Ball-on
-ring
1 m 15
0.5 mm
/ min

3-p
bend

c

or

(MPa)

Twist
test
5 mm /
min

Ag0.9Pb9Sn9 22 mm
x
700
-Sb0.6Te20 3 mma m
Cast
45 x 4 x
3.175
mm
34 x 4 x
Commercial3.175
grade silicon
N/S
mm
nitride
50.8 mm
x3.17
a

Testing
tech. &
loading
rate

908.1

717.1

c

10.1

c

15.7

c

12.0

Table 4.4 (cont'd).
-Si3N4
powder
Die pressed
Hirosaki at 20 MPa,
isostatically
pressed at
200 MPa

6x6x
50 mm

Matrix
0.2-2 x
1-5 ma
Elongat
ed
grains
2-20
x10-300
m

Paul

Vales

4 % residual
Si

3-p
bend
800
0.5 mm grit
/s

15

15

~
0.01
~
0.01

~
0.01

689
599

515

f

53

f

25

f

19

c

3.5

c

2.5

c

4.5

c

4.5

a

5 x 10
mm
x
525 m
5 x 10
mm x
(100) p-type 250 m Single
silicon wafer 5 x 10
crystal
mm x
100m
5 x 10
mm x
50m
Si3N4 with ~

15

247

3-p
bend

435
N/S

25

0
648

726

4-p
Acicular,
45 x 4 x
bend
Polis1 to 5
20
3 mm
5 mm / hed
m
min

76

0.037

N/S

7

Table 4.4 (cont'd).
b

7.1

b

6.1

b

4.8

1 x 10
3 x 15

Brodie

Commercial
bulk polysilicon,
Chemical
vapor
deposition

2 x 10

24.6
b
2 x 10
mm x
~5
b
3 x 15
a
mm

Tension
test
N/S
0.01
mm/s

b

b

f

N/S
6.9

8.0

Diameter and thickness of disc-shaped specimens
The grain aspect ratios (relative lengths minor axis X major axis) listed for the elongated

grains in specimens
c

0

6.7

5 – 20

b

20

4.8

5 x 30
~3

a

5.0

The characteristic strength c
The mean fracture strength <f>

77

In this discussion, we begin with Region II since it plays a key role in the m versus P
behavior and since it’s important to many of the applications of brittle, porous materials
3.5.1 Region II
Region II (0.1 < P < 0.55, Figures 4.6 - 4.8) is important in several ways.

First, the P

range from roughly 0.2 to 0.5 includes the P values that are employed in most applications of
porous brittle materials, including filters, SOFC, catalyst supports and biomedical materials
[Liu 2010].
Also, the details of the m versus P behavior of m in Region II reveal phenomena that have
not been discussed earlier in the limited m versus P literature. In particular, the combined
data set depicted in Figure 4.8 includes a total of 32 m values: 14 m values for HA [this
study], 10 m values for other HA studies, and 8 values for three other oxides, namely zirconia,
titania and alumina (Table 4.5). In contrast to Regions I and III, in Region II the m values
occur in a relatively narrow band from about 4 to 11. The “banded” nature of the m versus P
data in Region II is underscored by the fact that there are no statistically significant
differences among <m>, the mean m values for each of the data groupings in Table 4.5. The
statistical analysis was carried out using one-way ANOVA with p < 0.03 considered to be
statistically significant.
Although there are no statistically significant differences in the <m> for Region II (Table
4.5 and Figure 4.8), we note that the processing, testing, and the materials themselves differ
considerably (Tables 4.1 –4.4). For the HA specimens in this study (Table 4.1), the material,
the material vendor, the initial powder particle size, the powder pressing technique, the
78

55

Nb and Ta doped zirconia (NA) (Pissenberger)
Titania (118) (Kishimoto)
HA (441) (this study)
Alumina (20-30) (Nanjangud)
HA (10) (
Meganck)
Si (60) (Borrero-Lopez)
HA (48) (
Lopes)
Mullite (20) (Vales)
HA (80) (
Ruys)
Silicon Nitride (20) (Vales)
HA (84) (
Villora)
Alumina (~ 100) (Chao)
Silicon Nitride (45) (Hirosaki)
HA (120) (
Cordell)
Ag0.9Pb9Sn9Sb0.6Te20 (15) (Ren)
HA (30) (
Murray)
Silicon nitride (~ 90) (Chao)
(1 0 0) p-type silicon wafer (99) (Paul)
III
Polycrystalline silicon (>160) (Brodie)

I

Weibull modulus

50
25
20
15

II

10
5
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

Porosity

Figure 4.8. Combined data set for Weibull modulus as a function of P for HA [this study,
Cordell2009, Meganck 2005, Murray1995, Lopes 1999, Ruys1995, Villora2004], oxides
(alumina [Chao 1991, Nanjangud 1995], zirconia [Pissenberger and Gritzner 1995], titania
[Kishimoto 1991], mullite [Vales1999]) and non-oxides (silicon [Borrero-Lopez2009, Brodie
and Bahr 2003, Paul2006], silicon nitride [Hirosaki and Akimune 1993, Nanjangud 1995,
Vales1999], LAST-T [Ren 2006]) from literature. (The numbers in parentheses indicate the
total number of specimens fractured in a given study).

79

sintering furnace, the specimen geometry, testing machine and loading rate were all kept
constant during the entire study, whereas in contrast for the other data in Region II (Table
4.5), many of these variables differed among the research groups (Tables 4.2 –4.4).
How then can we reconcile the banded nature of the Region II data despite differences in
the materials, testing and processing, especially since in Appendix B we cite literature that
demonstrates that m is in fact a function of the materials, testing and processing?

First, as is

true for the vast majority of the research involving m, the particular studies that focus on
processing and testing effects on m (Appendix B) were performed almost entirely for
specimens with P < 0.1 (Region I). Thus information for the effects of processing and testing
on m is not currently available in the literature for specimens with P values within Regions II
and III.
A second key point is that the universality in the m versus P behavior (manifest in the
banded nature of Region II, Figure 4.8) may reflect the universality of microstructural
evolution that occurs during the powder sintering process. This universality is consistent with
the striking similarities among the microstructure evolution observed during sintering of
many ceramic [Barsoum 1997] and metallic [German 1994, 1996] powders. In particular,
during the intermediate stage of sintering (which roughly corresponds to Region II), tubular
pores along grain boundaries evolve from the connected pore phase in Region III and toward
isolated spherical pores in Region I [Barsoum 1997, German 1994, 1996]. Thus, the
3-dimensional arrangement of tubular pores associated with the intermediate sintering stage
(and hence Region II) may overwhelm effects of processing and testing, including such
80

parameters as processing-induced surface flaws, along with the temporal (loading rate, for
example) and spatial distributions of stress within the specimen. Therefore, the results seem
to agree well with established theories of microstructural development in ceramics.
Also, in general a high m value (say, m > 20) indicates relatively little scatter in fracture
strength while a low m value (for instance, m less than 3 or 4) indicates significant scatter in
f, which degrades mechanical reliability [Wachtman 2009]. Thus in particular, the banded
nature of Region II (4 < m < 11) requires designers of such components to include high
safety factors. Furthermore, the broad-ranging nature of the Region II database (Table 4.5),
which includes HA data from this study, HA data from five additional literature studies and
data for alumina, titania, and zirconia may indicate a difficulty in obtaining any brittle,
porous material sintered from powdered compacts with Weibull moduli outside the range of 4
< m < 11. However, m values between 2 and 4 have been reported for dense commercial
silicon wafers [Borrero-Lopez2009, Paul2006], thus designers in the microelectronics
industry have been able to produce serviceable products from brittle components that have
even lower Weibull moduli than the porous, brittle materials in Region II.
3.5.2 Region I
In Region I, the m versus P behavior is dramatically different than the banded nature of
Region II, since m is as large as 53 and as small as 2 for the data included in Region I (Figure
4.8). The spread in the m data in Region I is likely influenced by (i) the various processing
and testing parameters discussed in Appendix B and (ii) defects such as surface scratches or
cracks caused by grinding and cutting.
81

Table 4.5. For the m versus P data included in Region II (Figure 4.8), the number of m data
points, mN as well as the mean and standard deviation of m, <m> and mSTD, respectively.
The data ranges defined by <m> + mSTD overlap for every entry in the table, signifying that
there are no statistically significant differences among <m> for the various data groupings
listed in this table.

Material

mN

<m>

mSTD

HA

14

8.55

1.88

HA from other
studies

10

6.99

1.82

Zirconia, titania,
alumina

8

6.29

1.72

18

6.68

1.76

a

32

7.50

2.02

b

All data, except
HA from study
Entire data set in
Region II
a

References
This study
Cordell2009, Lopes 1999,
Murray1995,
Ruys1995,
Villora2004
Zirconia [Pissenberger and
Gritzner
1995],
titania
[Kishimoto 1991],
alumina
[Nanjangud1995]

The references include each of those listed in the table for the HA from other studies plus

zirconia, titania and alumina.
b

The references include each of those indicated in footnote a plus this study.

82

Thus for the relatively dense specimens in Region I (P < 0.1), surface cracks arising from
processing (machining and cutting) rather than pores may be the dominant flaws although in
Region II pores likely dominate the m versus P behavior (Section 3.5.1).
Processing-induced flaws are not porosity dependent and are unpredictable from specimen to
specimen, which coupled with the effects of differences in test and processing techniques
(Tables 4.2 - 4.4) make the significant scatter of the Weibull modulus data in Region I
reasonable.
3.5.3 Region III
Region III, where P > 0.55, corresponds to an initial sintering stage in which only limited
interparticle necking occurs rather than densification. [Barsoum 1997, German 1996]. A
relative uniformity in the dominate flaw distribution in Region III (the connected pore phase
that permeates the space between the “necked” particles) could thus lead to less scatter in the
fracture strength data itself and hence higher Weibull modulus values than the more
anisotropic grain-edge, tubular pores that dominate the pore phase in much of Region II
(intermediate sintering stage).
However, in contrast to Region II, where the m values fall between approximately 4 and
11 (Figure 4.8 and Table 4.5), in Region III the “band” of m values is from about 11 to 23
(Figure 4.8 and Table 4.1). The wider band of values of m values in Region III is likely
related to powder processing issues such as the presence or absence of agglomerates.
Furthermore, the mean fracture strength decreases monotonically as P increases (Table 4.1),
again emphasizing the de-coupled nature of the mean strength (a measure of the center of the
83

strength distribution) and the Weibull modulus, which is a function of the dispersion in the
individual strength values.
For Region III, it is important to note that except for one m value from a study of
HA/SiO2 composites by Villora [Villora2004], the current authors were unable to find in the
literature Weibull modulus data for P > 0.55 for HA or any brittle material. Thus all but one
of the m versus P data in Region III (Figure 4.8) was obtained from the HA specimens
fractured in this study. The increase in m in Region III from the values observed in Region II
is at first surprising, but one must recall that the average strength and m are in general
decoupled.
The evolution of dominant flaws associated with the evolution of porosity between
regions determines the observed m versus P behavior. In Region I, surface cracks developed
during processing may be the dominant flaws. These porosity independent flaws coupled
with the effects of differences in test and processing techniques lead to the significant scatter
of the Weibull modulus data in Region I. In region II, the tubular pores along the grain
boundaries likely overwhelm effects of processing and testing and become the dominant
flaws. In Region III, the connected open pores result in a relative uniformly distributed flaws
and thus higher Weibull modulus. In addition, pore clustering and pore anisotropy may have
significant effects which could be the subject of future studies.
In contrast to the behavior of m versus P for the entire spectrum of P (Regions I, II and
III), in Table 4.1 (and as will be discussed in far more detail in Part II of this paper), we
observe that the average fracture strength and Young’s modulus (Part II) decrease
84

monotonically with increasing P. For the HA specimens sintered and fractured in this study,
the uniformity in the material, processing and testing of the HA specimens in Region III
makes it unlikely that the parameters discussed in Appendix B are responsible for the
relatively wide range of m values observed in this study. Also, as was the case for the
specimens in Region II, no studies currently exist in the literature to indicate whether or not
the studies on specimens from Region I (Appendix B) have any applicability to the m versus
P behavior for specimens in Region III.
3.5.4 Transitions among the regions
The transition from Region I to Region II at P ≈ 0.1 may be related to a change in
fracture mode. For example, Flinn [Flinn 2000] found that for 0.15 < P < 0.40 (where P =
0.40 was the highest P value in their study), the fracture mode for partially sintered alumina
specimens was predominantly intergranular.

However, for P < 0.15, the fraction of

transgranular fracture increased as P decreased [Flinn 2000]. Thus the P value corresponding
to the transition in fracture mode observed by Flinn [Flinn 2000] for their porous alumina
specimens corresponds roughly to the P value for the transition from Region I to Region II
behavior (Figures 4.7 and 4.8 in this study) Thus a change in fracture mode may be linked to
the Region I – Region II boundary, although further study is needed to explore this concept.
At P  0.55 there is a second transition in m versus P behavior, where the limited m
range of Region II changes to the rapidly increasing m values of Region III (Figures 4.7 and
4.8). The Region II Region III transition behavior may reflect the transition from the initial
sintering stage (Region III) to the intermediate sintering stage (Region II). The change in
85

pore morphology may also be related to the fracture mode. Perhaps in the very high porosity
ranges (P > 0.55), the fracture surfaces tend to be somewhat “ rougher” than in Region II,
which may lead to a return (in Region III) toward the higher m values that were characteristic
of Region I. In the future, the authors plan to use confocal laser scanning microscopy (CLSM)
to quantify the surface roughness of fracture surfaces over a broad range of P for the HA
specimens fractured in this study.
Furthermore, in Part II we shall show that a single mathematical form involving a power
law function of (1- P) describes both the fracture strength and elastic modulus as a function
of P for P values extending from unfired powder compacts to near theoretical density. Thus,
the current study involving the “U-shaped” behavior for m versus P in porous materials is in
contrast to the behavior of both fracture strength and Young’s modulus versus P (Part II). The
overall scarcity of m versus P data in the literature is likely the reason that the Region II and
Region III behavior has not been identified until now.
For the full data set, the demarcations between Regions I and II as well as between
Regions II and III were determined by eye based on the trends shown in Figure 4.8. Future
work could include determining a statistical or analytical basis for selecting values of P that
demarcates the Regions I, II and III.
4. Summary and Conclusions
In this study, a total of 540 HA specimens were sintered and fractured. In addition, more
than 1560 specimens data of HA, other oxides and non-oxides from the literature were
compared and analyzed, making this study the most extensive Weibull modulus study in the
86

open literature (Figure 4.8). It should be emphasized that in Figure 4.8, the m versus P
literature data is included from 17 different research groups for a total of 8 different
materials.
A three-region, “U” shaped plot of m versus P data was observed for HA and a combined
data set including HA, other oxides and non-oxides from the literature which shows a wide
band of m values for Region I (P < 0.1) and Region III (P > 0.55), and a narrower band of m
values in Region II (0.1 < P < 0.55) (Figure 4.8). Since Figure 4.8 includes HA data from this
study as well as data for brittle oxides and non-oxides for 16 other research groups, this trend
may hold for a wide range of brittle materials.
The experimental data from this study and eight other research groups with m versus P
data in Region II (Figure 4.8) show that Weibull modulus values fall in the range between
about 4 and 11 regardless of the composition, grain size, testing techniques or surface finish
of the specimens. Thus, in Region II the Weibull moduli are likely not dominated by the
processing and testing parameters discussed in Appendix B, since the specimens included in
Region II (Figure 4.8, Table 4.5) do vary widely in terms of both processing and testing
(Tables 4.2 and 4.3). In particular, unlike the more dense specimens in Region I, failure of the
Region II specimens is likely decoupled from the statistics of surface flaws that might be
induced by handling since again the limited range in m in Region II would seem to preclude a
large influence from such effects. Thus, instead of processing or testing techniques
(Appendix B), the banded nature of the m versus P behavior for the several materials given in
Figure 4.8 and Table 4.5 is may stem from the near universality in the pore evolution during
87

sintering, namely the evolutions of anisotropic (tubular) pores during the intermediate
sintering stage.
Nested within Region II is the P range (from about 0.2 to 0.4) which includes the P
values required for many of the biomedical and other applications of brittle, porous materials
(Section 1). Thus the m versus P behavior in Region II has very significant technical
implications. For volume fraction porosities between 0.1 and 0.55, a medium to high scatter
in the fracture strength is to be expected, thus requiring the designers of such components to
include high safety factors in their designs.
For the precursor to Region II, namely Region III (P > 0.55), the authors were able to
find only one data point in the literature that corresponds to Region III, namely a single m
value from a study by Villora [Villora2004], for a HA/silica composite material.

The

general trend that for Region III, the m values are greater than in Region II may be related to
evolution of pore morphology in the initial and intermediate sintering stages, respectively, or
to differences in fracture mode in term associated with the differing pore morphologies. The
trend of higher m values in Region III than in Region II is somewhat surprising and further
study is needed to explore this behavior.
5. Acknowledgements
The authors acknowledge the financial support of the National Science Foundation,
Division of Materials Research Grant DMR-0706449. In addition, the authors acknowledge
the use of the Electron Microscopy facilities at the Center for Advanced Microscopy,
Michigan State University. Ring-on-ring loading fixtures were fabricated in the Physics
88

Department Machine Shop in Michigan State University.

89

APPENDICES

90

APPENDIX A

WEIBULL ANALYSIS
Weibull statistics have been widely used to analyze fracture strength data of brittle
materials, as well as the spread in the distribution of data for other physical quantities. The
Weibull distribution has been applied to distributions of other types of mechanical property
data, including Young’s modulus [Chen2010, Sarkar 2011] and hardness [Sarkar 2011] in
addition to applications as diverse as wind speed distributions in meteorology [Gabriel2011].
In this study we apply Weibull analysis only to fracture strength data.
The general form of the Weibull distribution for fracture strength can be expressed as
[Wachtman 2009]:

 
Pf  1  exp  
 V C
 

m 

 dV 





(A1)

where Pf is the probability of failure, V is the specimen volume subject to (tensile) stress, the
scale parameter c is called the characteristic strength, and the shape parameter m is the
Weibull modulus [Wachtman 2009]. For tensile tests performed on specimens with constant
volume, equation (A1) can be simplified to

   m 
Pf  1  exp   
 c  
   

(A2)

or

91

 1
ln ln 
 1  Pf



  m ln   m ln  c



(A3)

Under multiaxial stress conditions (as in the case of an ROR test, where the stress is
equibiaxial), equation (A1) becomes much more complicated, which requires numerical
evaluation of the integral term [Chao 1991, Chao and Shetty 1992). However, if the
specimens fail due to surface flaws in a given test configuration, equation (A3) can then be
rewritten as

m 
 
 
 dS 
Pf  1  exp  
 c 


 S


(A4)

where S is the surface area subject to stress. For ROR testing, the equibiaxial stress is
constant within the loading ring perimeter on the tension side of the specimen.

If the

fracture origins fall within the loading ring perimeter on the tension surface, then equation
(A4) can be simplified as
m 
 
 
 S0 
P f  1  exp 
 c 


 


(A5)

where S0 is the surface area circumscribed by the loading ring. For a fixed loading ring
diameter, S0 in equation (A5) can be “absorbed” into c, which in turn results in equation
(A2).
Thus, the two-parameter Weibull analysis can be applied to the ROR test with a fixed
loading ring diameter if the fractures are caused by surface distributed flaws. Also, all of the
HA specimens fractured in this study were fractured in the as-sintered condition, that is, none
92

of the specimens were polished. The lack of polishing might increase the probability of
specimen failure via surface flaws.

93

APPENDIX B

FACTORS OTHER THAN POROSITY WHICH AFFECT THE WEIBULL
MODULUS
The Weibull modulus varies with porosity as described by the “U” shaped m versus P
plot (Figure 4.8) in Section 3.4.2. Tables 4.2 through 4.4 in this study show the wide range of
processing and testing conditions associated with the literature data presented in Figure 4.8.
This range of processing and testing conditions needs to be considered, since in general the
scatter in m values at a given porosity (Figure 4.8) may result from specimen agglomerates
[Murray1995], processing techniques and microstructure [Villora2004], strength testing
techniques [Chao 1991, Cordell2009], pore size and pore shape [Chao 1992], spatial
distribution of pores [Chao 1991], loading rate [Andrews2010], grain size [Kim1995] as well
as surface finish [Nakamura2010]. In this appendix, we briefly discuss the magnitude of the
changes in m that have been observed for each of the factors listed above.
Eliminating agglomerates via emulsion-refining before sintering improved the
mechanical reliability of sintered HA specimens [Murray1995], where the as-sintered
agglomerate-free specimens had an m of 13.5, compared to the dramatically lower m value of
7.6 of sintered specimens with agglomerates [Murray1995]. In another microstructure and
processing technique study, Villora [Villora2004] formed specimens from powdered animal
bones, waste diatom filters and paper industrial sludge using cold pressing, extrusion and slip
casting.

Then with a single time-temperature sintering profile Villora [Villora2004]

94

obtained partially densified specimens with porosities of 0.42, 0.55 and 0.65, respectively
[Villora2004]. The m values for the cold pressed, extruded and slip cast specimens were 3.7,
6 and 2, respectively. Differences in composition and processing likely lead to
microstructural variations that in turn resulted in the observed range of m values
[Villora2004].
Cordell demonstrated the importance of the strength testing technique on m via (i)
compression and (ii) four-point bend tests on HA specimens that included fugitive (pore
forming) PMMA microspheres [Cordell2009]. Compression tested HA specimens sintered
with 16.2 m PMMA microsphere inclusions had an m value of 8.7 compared to m = 5.8 for
HA specimens sintered with 5.96 m PMMA microsphere inclusions [Cordell2009],
indicating a higher m value for larger pores. Conversely, for the four-point bend test,
specimens with larger pores had m = 7.24 compared to m = 10.3 for specimens with smaller
pores. The different m values for four-point bend and compression may result from the
differing stress distributions induced within the specimens during testing.
A silicon nitride specimens studied by Chao indicated a dependence of m on pore size
and shape [Chao 1992], where larger, irregularly shaped pores were more likely to act as
fracture origins compared to smaller, equiaxed pores, although no numerical values of m
were reported [Chao 1992]. A second study by Chao explored the relationship of m and the
spatial distribution of pores in alumina and silicon nitride [Chao 1991] for specimens
fractured using three-point bend, four-point bend and biaxial flexure testing. For alumina, the
three strength testing techniques agreed well with each other, such that m = 23.8 for
95

four-point bend, m = 25.4 for three-point bend and m = 22.3 for biaxial flexure testing [Chao
1991]. However, for silicon nitride specimens, m = 10.1 for four-point bend, m = 15.7 for
three-point bend and m = 12.0 for biaxial flexure testing [Chao 1991]. Chao [Chao 1991]
noted that fracture originated from surface flaws in alumina but from subsurface pores in
silicon nitride, thus the spatial distribution of pores likely influences the link between m and
a given test technique.
Andrews found that when fracture is induced by surface flaws in dense silicon nitride,
loading rates of 30 MPa/s, 0.3 MPa/s and 0.003 MPa/s gave m values of 26.7 to 15.4 and 6.9,
respectively [Andrews2010]. Also, as the loading rate decreased, the specimens with surface
flaws became more susceptible to slow crack growth [Andrews2010].
Kim [Kim1995] showed a coupling between grain size and m. Using identical test
techniques, silicon nitride specimens with grains one m in diameter and 5.3 m long had m
values ranging from 23 to 30, while m was 17 for specimens with grains 2 m in diameter
and 12 m long. Kim [Kim1995] attributed the measured differences in m to a narrower
critical flaw size distribution for the smaller grain size.
Nakamura’s study of porcelain specimens with glazed surfaces and surfaces roughened
using 1000, 600 and 100 abrasive grit [Nakamura2010] shows that m depends on surface
finish. For the glazed specimens, m ~16 and for the abraded specimens, the m values were
from 11 to 13 [Nakamura2010]. However, among the abraded specimens no obvious
correlation between m and grit size was reported [Nakamura2010].
Each mechanism discussed in this appendix may affect m differently for this study and
96

for each of the eighteen studies from the literature (Tables 4.2, 4.3, and 4.4) that are plotted
as the combined data set in Figure 4.8. Thus, despite this considerable diversity in the
processing, testing and material properties among the studies in the combined data set (that
represents a total of more than 1560 specimens including HA, other oxides and non-oxides),
a universal three-region U-shaped behavior was observed in m versus P plot (Figure 4.8),
within which we observed a strip of m values between about 4 and 11 for a wide intermediate
porosity region (Region II, Figure 4.8).
Furthermore, for this study, the composition, initial powder particle size, the furnace and
furnace atmosphere used to densify the specimens, the specimen geometry, the test technique,
specimen surface finish and the loading rate were all held fixed. Nevertheless, the scatter in
m versus P in this study is comparable to the scatter in m versus P observed in the data from
the literature, implying that at least for Region II, the porosity rather than the processing or
test conditions are the dominant factor determining the m value.

97

REFERENCES

98

REFERENCES

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Annual Conference on Composites, Advanced Ceramics, Materials, and Structures: B:
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& Sons, Inc., Hoboken, NJ, USA.
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105

CHAPTER 5
FRACTURE STRENGTH AND ELASTIC MODULUS AS A FUNCTION OF
POROSITY FOR HYDROXYAPATITE AND OTHER BRITTLE MATERIALS

X. Fan, E. D. Case, F. Ren, Y. Shu and M. J. Baumann
Department of Chemical Engineering and Materials Science, Michigan State University,
East Lansing, MI, 48824
Published in: J Mechanical Behavior of Biomedical Materials, 8:99-110, 2012.

Abstract
Part I of this paper discussed the Weibull modulus, m, versus porosity, P, behavior of
brittle materials, including HA. While the Weibull modulus m deals with the scatter in
fracture strength data, this paper (Part II) focuses on two additional key mechanical
properties of porous materials, namely the average fracture strength, <f>, and the Young’s
modulus, E, for P in the interval from P ≈ zero to P ≈ PG (the porosity of the unfired
compacts). The <f> versus P data for HA from this study and the literature data for alumina,
yttria stabilized zirconia (YSZ) and silicon nitride are described well by functions of , where
 = 1 – P/PG = the degree of densification. A similar function of  applies to the <E> versus
P behavior of HA from this study and data from the literature for alumina, titanium and YSZ.
All of the data analyzed in this study (Part II) are based on partially and fully sintered powder
compacts (excluding green powder compacts), thus the <f>/0 versus  and <E>/E0 versus
 relationships may apply only to such specimens.

106

1. Introduction
In previous chapter, the Weibull modulus, m, versus volume fraction porosity, P, behavior
was examined for 441 cold-die pressed and sintered hydroxyapatite (HA) disk-shaped
specimens. The HA specimens were fractured in biaxial flexure using a ring-on-ring (ROR)
fixture for 0.08 < P < 0.62. A “U” shaped m versus P plot was obtained for this study’s
fractured HA specimens as well as for combined data sets that included HA data from the
literature along with oxide and non-oxide data from the literature.
In this study we examine the porosity dependence of the average fracture strength, <f>,
and the average Young’s modulus, <E>. For <f> versus P, we analyze data for HA (this
study], alumina [Nanjangud 1995], yttria stabilized zirconia (YSZ) [Deng 2002] and silicon
nitride [Yang 2003]. In addition, Young’s modulus versus P data are examined for HA [this
study, Ren 2009], alumina [Hardy and Green 1995, Nanjangud 1995, Ren 2009], titanium
[Oh 2003], and YSZ [Deng 2002]. Both the fracture strength and Young’s modulus
dependence on P can be expressed in terms of a power law dependence on the degree of
densification, where(1 –P/PG) and PG is the porosity of a green (unfired) powder
compact.
Of key importance is the complementary nature of Parts I and II of this study. Part I
considered the Weibull modulus, m, of a wide range of materials where m is a measure of the
spread of the distribution of strength values. In addition to m, the average strength (the mean
of the strength distribution) and the Young’s modulus (which characterizes the stress-strain
response of a brittle material prior to fracture) also are fundamental mechanical properties.
107

Construction of the stiffness matrix for finite element method calculation of stress and strain,
for example, requires Young’s modulus data [Martin 1973]. Unlike the Weibull modulus, the
average strength and Young’s modulus decrease smoothly and monotonically with increasing
P.
2. Background
2.1 Strength-porosity relationships for brittle solids
Nearly sixty years ago, Ryshkewitch [Ryshkewitch 1953] and Duckworth [Duckworth
1953] suggested that the porosity dependence of the fracture strength, f, of brittle materials
could be represented as
f = 0 exp(-bP)

(1)

where 0 is the fracture strength for the theoretically dense (P = 0) material, P is the volume
fraction porosity and b is a unitless, material-dependent constant.
Rice [Rice 1998] stated that the constant b in equation (1) typically ranges from about 4 to
6 in part of a comprehensive review text on porosity in ceramics. In addition, Rice [Rice 1998]
emphasized that equation (1) describes very well the strength-porosity relationships governing
the strength of ceramics, but only up to a critical porosity, PC. For P > PC, Rice [Rice 1998]
noted that the strength decreases more rapidly than is described by equation (1). However, Rice
did not present a method for predicting PC.
Prior to Rice’s work [Rice 1998], a number of researchers considered a similar concept
of a critical porosity. In 1958, Schriller [Schriller 1958] gave a porosity-relative strength
relationship based on a mathematical model that included the concept of a critical porosity,
108

PC, such that
P
 f  q log C
P

(2)

where f is the fracture strength, q is a quality factor depending on the preparation of the mix
and impurities. In this case, the critical porosity corresponds to zero fracture strength of a
material.
In 1959, Millard [Millard 1959] also employed the concept of a critical porosity in a f
versus P relationship such that


P  PC
 f  S 0 1 
 P  P

C


(3)

where Millard [Millard 1959] defined S0 as a constant “having units of strength” and PC as a
critical porosity equal to porosity of a loose powder.
In this study, we fit the relative average fracture strength, <f>/0, to the following
function form

 f 
P
 A  1 
 P
o
G


n


  A  n



(4)

where <f> is average fracture strength of a group of porous ceramics with a fixed value of P,
0 is fracture strength of the theoretically dense (P = 0) ceramic, PG is the porosity of the
powder compact and (1 –P/PG) is defined as the degree of densification.
In the work by Rice [Rice 1998], Schriller [Schriller 1958] and Millard [Millard 1959],
there is an implicit assumption that the green or unfired state corresponds to a critical or
initial point of strength in the partially sintered specimens. However, the bonding

109

mechanisms among particles that bind together the green powder compact are different than
the bonding between grains in partially sintered/fully sintered bodies [Barsoum 1997, Flatt
2004, Hornbaker 1997, Libowitzky 1999]. Thus, the “initial point” of strength (G at P = PG)
may be difficult to define unambiguously.
The P = 0 value of the fracture strength, 0, is a function of parameters such as the defect
state, the nature of loading, and other factors [Rice 1998]. However, for the partially sintered
bodies examined in this study, even when P  0, equation (4) and the corresponding
equation for Young’s modulus (equation (10) that is discussed in the next section) describe
remarkably well the porosity dependence of the relative strength and Young’s modulus data,
respectively.
2.2 Young’s modulus as a function of porosity
In an analogy to Ryshkewitch’s [Ryshkewitch 1953] and Duckworth’s [Duckworth 1953]
expression for the P dependence of the relative fracture strength (equation (1)), in 1961
Spriggs proposed that the porosity dependence of Young’s modulus could be written as
[Spriggs 1961]
E/E0 = exp (-bP)

(5)

where E is the Young’s modulus at a given volume fraction porosity P, E0 is the value of E
for theoretically dense materials and b is a material-dependent constant. For bP << 1, E/E0 =
exp (-bP) can be approximated as
E/E0 = (1 - bP)

(6)

In 1987 Phani [Phani and Niyogi 1987] proposed a semi-empirical equation
110

n

E/E0 = (1 - aP)

(7)

where a is defined as the unitless "packing geometry factor" and the constant n depends on
grain morphology and pore geometry. Phani’s [Phani and Niyogi 1987] Young’s modulus
versus P equation (equation (7)) is thus roughly analogous in form to equation (3), the
strength-porosity equation proposed in 1959 by Millard [Millard 1959]. Phani [Phani and
Niyogi 1987] defined the constant in equation (7) as a = 1/PC and PC as the critical porosity
at which Young’s modulus is zero. According to Phani [Phani and Niyogi 1987], E = E0 (1 n

aP) is valid for any range of porosity. Note that equations (5), (6) and (7) are all essentially
identical for small values of P and for n near unity. As we shall discuss in Section 4, n is in
fact near unity for the materials analyzed in this study, and the predictions of equations (5)
and (7) diverge with increasing P.
In 1994 Lam [Lam 1994] developed a linear relationship of the relative Young’s modulus,
E/E0, of porous ceramics based on Phani’s work [Phani and Niyogi 1987], such that

E 
P
 1 
 P
E0 
G


 =  

























(8)

where E is Young’s modulus of the porous ceramic, E0 is the Young’s modulus of the
theoretically dense ceramic, P is the volume fraction porosity. PG and  are as defined
following equation (4).
Nanjangud [Nanjangud 1995] and Hardy [Hardy and Green 1995] in 1995 referred to
Lam’s 1994 paper [Lam 1994] to obtain the expression

111

E  E' 
P
 1 
 P
E 0  E' 
G


 =  



























(9)

where E’ is the Young’s modulus value for the green, cold-pressed porosity state.
2

In this study, we use a slightly modified equation that provides a higher R (coefficient
of determination) for the least-squares fit to a given <E> versus P data set, namely

E
P
 A E 1 
 P
E0
G


q


  A E q



(10)

where <E> is the average Young’s modulus of a group of porous ceramics with a fixed value
of P. Also AE and q are dimensionless constants. Equation (10) is equivalent to Phani’s
equation [Phani and Niyogi 1987] if one sets AE = 1 and a = 1/PG.
3. Experimental Procedure
3.1 Specimen preparation and biaxial flexure testing
Details of specimen preparation and biaxial flexure testing technique were given in Part
I of this paper. The fracture strength, f , was calculated using the following equation as
suggested by the ASTM standard C1499-05 [ASTM 2003]

f 

 D 
D2  D2
3F 
L 
 1    ln  S 
1    S
D 
2h 2 
2D 2
 L 



(11)

where F is the load at failure, h and D are the thickness and the diameter of the disk-shaped
test specimen, DS and DL are the diameters of the supporting ring and the loading ring,
respectively. A Poisson’s ratio, , of 0.27 [Ren 2009] was used in the above calculation.
3.2 Scanning electron microscopy
Scanning electron microscopy (SEM) was used to examine the size and morphology of

112

the initial HA powders. A small amount of the HA powder was dispersed for 10 minutes in
ethanol (190 Proof, Becon Laboratory, Inc.) using an ultrasonic bath (Bransonic ultrasonic
cleaner, 1510R-MT, Branson Ultrasonics Corporation, CT). Immediately after removing the
HA/ethanol mixture from the ultrasonic bath, a droplet of the mixture was placed onto a
metal stub covered with carbon tape. The ethanol was allowed to evaporate over 24 hours
before the specimen surfaces were sprayed with pressurized air to remove loose powder
particles. The powder sample was then gold-coated and examined by SEM. Details of the
gold-coating and the SEM experimental procedures were given in Part I of this paper.
3.3 Elastic modulus measurements of HA
The Young’s modulus of the as-sintered HA specimens were measured by Resonant
Ultrasound Spectroscopy (RUS) (RUSpec, Quasar International, Albuquerque, NM, USA).
The HA specimens were placed on a tripod arrangement of three piezoelectric transducers. A
driving transducer (operating over a frequency range of 10 to 250 kHz in this case) generated
a sinusoidal mechanical vibration that was sensed by the two pick-up transducers [Ren 2008].
The peaks recorded by the pick-up transducers (Figure 5.1) correspond to the mechanical
resonance modes of the specimen. Using the recorded resonance frequencies and the mass,
shape and dimensions of the specimens, the Young’s modulus, E, as well as the shear
modulus, G, and Poisson’s ratio, were calculated using a commercial software package
®

(RPModel , Quasar International, Albuquerque, NM, USA).
4. Results and Discussion
4.1 Microstructural analysis of the initial HA powder
113

7
6

Amplitude (A.U.)

5
4
3
2
1
0
10

50

100

150

200

250

Frequency (kHz)

Figure 5.1. The RUS spectrum obtained for as-sintered HA specimen sintered at 1200oC for
60 min. A.U. stands for arbitrary units.

Figure 5.2. SEM micrographs of initial HA powder showing rod-shaped particles and
agglomerates.
114

SEM analysis showed the initial HA powder particles to be approximately 0.8 m in
length and rod-shaped, with an aspect ratio of ~ 3 (Figure 5.2). These primary powder
particles formed irregularly-shaped agglomerates ranging from several microns to roughly 10
microns across (Figure 5.2).
Based on measurements of the specimen mass and dimensions, the green density of the
HA specimens was approximately 38% of theoretical density, which corresponds to a volume
fraction porosity of PG = 0.62. While PG for nearly equiaxed powders is typically between
roughly 0.4 and 0.5 [German 1994], the relatively high value of PG = 0.62 for the HA powder
compacts in this study is likely a consequence of the rod-like shape of the starting HA
powders (Figure 5.2).
Rod-like particles do not pack as closely as spherical or quasi-spherical particles, thus
powder compacts formed from rod-shaped particles tend to have fewer initial particle
contacts, larger initial pores and a higher green porosity [German 1994]. In addition to the
rod-shaped particles, the irregularly-shaped agglomerates observed in the as-received
powders (Figure 5.2) also likely increase PG. Thus, given the rod-like powder particles and
the agglomerates included in the as-received HA powder, the measured PG value of 0.62 is
reasonable.
4.2 Strength versus porosity for HA and other brittle materials
4.2.1 Fracture strength versus P for the HA included in this study
For the HA specimens included in this study, the average fracture strength was plotted
on a logarithmic scale as a function of porosity in order to show clearly the changes in
115

Table 5.1. Strength at zero porosity, 0, number of data points, N, fitting parameters, Aσ and
n, for the least-squares fit of <f>/0 versus degree of densification, , data to equation (4)
2

with coefficient of determination, R , for HA [this study], alumina [Nanjangud 1995], YSZ
[Deng 2002] and Si3N4 [Yang 2003]. The combined data set includes all of the data from
each of the four studies [this study, Deng 2002, Nanjangud 1995, Yang 2003].
References

Material

0

2

N

Aσ

n

R

a

17

1.00 ± 0.004

1.08 ± 0.007

0.989

b

11

1.14 ± 0.02

1.04 ± 0.03

0.997

c

8

1.01 ± 0.03

1.53 ± 0.11

0.992

d

(MPa)
HA in this study

80

Nanjangud 1995

Alumina

380

Deng 2002

YSZ

800

Yang 2003

Si3N4

850

8

0.85 ± 0.21

0.77 ± 0.18

0.860

Combined data set
a

HA

All listed
above

e

44

0.95 ± 0.03

1.02 ± 0.06

0.948

The0 value was estimated from the extrapolation of the strength versus porosity plot

(Figure 5.3 in this study)
b

The 0 value was estimated from the extrapolation of the strength versus porosity plot

(Figure 5.1 in [Nanjangud 1995])
c

The0 value was estimated from the extrapolation of the strength versus porosity plot

(Figure 5.2a in [Deng 2002])
d
e

The0 value was given in [Yang 2003]
The normalized strength of each data set of the four materials listed above versus degree of

densification was combined into a single data set and the combined data set were
least-squares fit to equation (4)

116

strength at high porosities (Figure 5.3). In this study, the HA specimens were sintered at
temperatures from 550oC to 1360oC with sintering times from 12 min to 4 hours (Table 5.1,
Part 1). In particular, for the HA specimens sintered between 550oC and 1000oC, the volume
fraction porosity, P, was very close to the green porosity, PG. Although the P value changed
little over the sintering temperature range from 550oC and 1000oC, the strength increased by
a factor of 3 due to the development of imterparticle necking. The concept of the initial
formation and growth of interparticle necks with little or no increase in density is consistent
with the observations and models discussed by Barsoum [Barsoum 1997], German [German
1996] and Kinergy [Kinergy 1976]. Thus, the mean fracture strength decreases monotonically
with increasing P for HA in this study, while in contrast for Part I of this study, a “U –
shaped” Weibull modulus, m, versus P trend was found for HA as well as seven other oxides
and non-oxides in the literature.
The differences between the <f> versus P trend and the m versus P trend reflect the
fundamental differences in the statistical meaning of the mean strength <f>, which is a
measure of the central tendency of the strength distribution, and the Weibull modulus, m,
which measures the spread in a strength distribution. Thus, <f> and m can vary
independently. In an entirely analogous way, if one determines the average of a distribution,
then one does not necessarily have information about the standard deviation, since the mean
and standard deviation, in general, are independent quantities, with one related to the
“center” of the distribution and the other related to the spread of the distribution. Appendix A

117

Average fracture strength (MPa)

55
20
o

1000 C

7

o

900 C
o

800 C

3

o

700 C

1
0.0

o

Specimens with P values near PG are

600 C
o

sintered at as-labeled temperatures for 2 h

0.1

0.2

0.3
0.4
Porosity

0.5

550 C
Unfired

0.6

0.7

PG

Figure 5.3. Average fracture strength, <f>, of HA [this study] as a function of porosity, P.
The solid line represents a least-squares fit of the <f> versus P data to equation (1) in the
entire P range of 0.08 < P < 0.62. The dashed line represents a least-squares fit of data to
equation (1) in the restricted P range of 0.08 < P < 0.55. For specimens with P values near PG,
the sintering temperature is indicated for each plotted data point. The formation and growth
of interparticle necks increase in the fracture strength while not appreciably changing P. Table
4.1 of Chapter 4 gives a complete list of the sintering temperatures and times for each group
of specimens shown in this figure.

118

discusses particular studies that highlight the different behaviors for the <f> versus porosity
and m versus porosity.
Nine groups of HA specimens with high porosities (0.47 < P < 0.62) were fabricated,
fractured and analyzed in this study. To the authors’ knowledge, no strength data for HA with
P > 0.47 are available in the literature except for two data points from Villora’s study [Villora
2004], where Villora’s specimens were composites containing both crystalline HA and SiO2
phases.
In this study a least-squares fit of all the fracture strength HA data to equation (1) yielded
2

a poor coefficient of determination, R , value of 0.786 (Figure 5.3).

However, a
2

least-squares fit restricted to a subset of data (with P < 0.6) yielded a higher R value of
0.915 (Figure 5.3).
However, in contrast to using equation (1) for the least-squares fit of <f> versus P for
this study’s HA data, when the least-squares fit to equation (4) was instead performed on
2

<f>/ 0 versus  for the entire porosity range, an R value of 0.989 was obtained (Figure
5.4). Thus as demonstrated by comparing Figures 5.3 and 5.4, equation (4) describes the
<f>/ 0 versus P behavior of HA [this study] over the entire range of porosity included in
this study; while the exponential strength-porosity relationship represented by equation (1)
only fits <f>/ 0 versus P data well for P values less than about 0.6.
For the HA specimens in this study that were sintered in air at temperatures between
550oC and 1000oC (Part I, Table 5.1), the specimens’ volume fraction porosity was
essentially constant at 0.62. However, the mean specimen strength more than tripled,
119

1

1

0.1

 
f

0.1

HA (This study)
Alumina (Nanjangud et al.1995)
YSZ (Deng et al. 2002)
Si3N4 added with Yb2O3 (Yang et al. 2003)

0.01

0.0

0.2

0.4

0.6

0.8

0.01

1.0

Degree of densification

Figure 5.4. Relative fracture strength <f>/0 as a function of degree of densification, , for
HA [this study], and literature values of alumina [Nanjangud 1995], YSZ [Deng2002] and
silicon nitride [Yang2003]. Each dashed curve represents a least-squares fit to equation (4)
for each data set. The solid curve represents a least-squares fit to equation (4) for the
combined data sets. The parameters for the least-squares fit of the relative strength versus
degree of densification data to equation (4) are listed in Table 5.1.

120

from 1.3 MPa for specimens fired at 550oC to 4.5 MPa for specimens fired at 1000oC (Table
5.1 in Part I of this paper). The increase in specimen strength without an accompanying
increase in specimen density likely results from powder particle necking, where necking is a
common phenomenon associated with the initial stage of sintering [Barsoum 1997, German
1996]. With increasing sintering temperature, the dimensions of the interparticle necks
progressively increase [Barsoum 1997, Kingery 1976]. As suggested by Nanjangud
[Nanjangud 1995] in their study of partially sintered alumina, the increasing dimension of
interparticle necks is consistent with increasing fracture strength.
4.2.2 Mechanisms for strength and Young’s modulus of green powder compacts
The HA specimens in this study that were tested in the green (unfired) state (P~0.61) had
a very low average fracture strength of 0.6 MPa. In contrast to the partially sintered HA
specimens, interparticle necking does not occur in the unfired powder compact since at room
temperature the mass diffusion needed to develop interparticle necks [Barsoum 1997] is
negligible. Thus, instead of interparticle necking, other mechanisms must account for the
observed fracture strength of the green powder compacts. These mechanisms likely include
van der Waals attractions [Flatt 2004], hydrogen bonding [Libowitzky 1999], electrostatic
interactions as well as the cohesive forces between particles generated through capillarity
[Hornbaker 1997]. In addition to these molecular level mechanisms, there may also be
contributions to mechanical strength based on mesoscopic level mechanisms, such as a
mechanical interlocking among powder particles [German 1984]. Such mechanical effects
may be enhanced by rod-shaped HA powder particles included in this study.
121

Powder compacts and porous specimens exposed to atmospheric humidity may absorb
moisture from the air. Particle-particle interactions between dry (especially microscopic-scale
and smaller) particles are dominated by van der Waals bonding [Flatt 2004, Fung2009,
Quintanilla 2006, Yang 2007, Yu 2003] while interactions between wet particles are
dominated by capillarity [Fung2009, Yang 2007, Yu 2003]. For unfired particle compacts in
materials with crystal structures (such as HA) that contain hydroxyl groups and oxygen ions,
hydrogen bonds may also be present [Libowitzky 1999, Stelte 2011]. Electrostatic
interactions also may contribute to bonding between particles of dielectric (not electrically
conducting) materials such as HA, where the electrostatic interaction between the Ca ions
and the phosphate groups likely play a role in binding green HA powder compacts.
In this study, an indication that water may play a role in binding together the particles in
the green compacts arises from the fact that the green (unfired) HA specimens had a volume
fraction porosity of about 0.61, while the specimens sintered at 550oC to 1000oC had a
volume fraction porosity of about 0.62. The slightly lower apparent volume fraction porosity
for the green HA compacts aged in air compared to the partially sintered specimens is likely
due to water forming liquid bridges at particle contact points and/or accumulating in the
voids between particles [Mitarai and Nori 2006].
Thus, there are many possible binding mechanisms for brittle, non-conducting powder
compacts such as HA compacts included in this study. However, these binding modes are
fundamentally different from the mechanism of interparticle necking produced by diffusive
transport during sintering. Therefore, while it makes physical sense that the powder compacts
122

possess a non-zero fracture strength, the fracture strength of green compacts is likely
dependent on the ambient environment and should not be considered as a “zero point” in
the evolution of strength versus sintering temperature and time since the process of
strengthening via diffusion (sintering) involves the growth of necks between adjacent
particles and thus is physically very different than processes such as Van der Waals bonding
and capillarity that contribute to the strength of green powder compacts.
Recall that in Sections 2.1 and 2.2 of this study, each of equations 2 - 4 for the
dependence of strength on P as well as equations 7 - 10 for the E versus P behavior involved
either the green density, PG, or a critical porosity, PC. Furthermore, PC (equations 2 and 3)
was defined somewhat similarly to PG. Also, although not specifically included in either
equation (1) or equation (5), Rice [Rice 1998] discussed the concept of a critical porosity, PC,
such that for P > PC the exponential equations (1) and (5) no longer described the strength or
Young's modulus versus P behavior, respectively, although Rice did not specify a means of
determining PC. This section points out although PG or PC is fundamentally connected to the
models described in Section 2, the physical mechanisms that give rise to the strength and
elastic modulus for green (unfired) specimens are almost certainly different than the
mechanisms that are important after interparticle necking begins. Nevertheless, the P value at
which the strength and elastic modulus values are likely dominated by the formation and
growth of interparticle necks is likely very close to PG, so that using the porosity of the green
compact as PG likely does not cause significant errors specifically in equations (4) and (10),

123

which were used in this paper to describe the P () dependence of fracture strength and
elastic modulus.
4.2.3 Relative mean fracture strength, <f> /0, as a function of degree of densification,
 from literature
Among the limited number of f versus P studies in the literature that include P values
ranging from roughly green (unfired) density to near full density are: Nanjangud’s alumina
study [Nanjangud 1995], the current authors’ HA research [this study], Deng’s YSZ work
[Deng 2002] and Yang’s silicon nitride study [Yang 2003] (Figure 5.4). Since none of these
four data sets [Deng 2002, Nanjangud 1995, Yang2003, this study] included experimental
data for theoretically dense specimens, the 0 values (where for each material, 0 is the f
value at P = 0) were estimated from the extrapolation of the strength versus porosity plots
(Table 5.1) in each reference [Deng 2002, Nanjangud 1995, Yang2003, this study].
When the combined data are plotted in terms of log (<f>/0) versus degree of densification,
, the four data sets [Deng 2002, Nanjangud 1995, Yang 2003, this study] follow the same
trend (Figure 5.4). A least-squares fit of the combined average strength versus  data to
2

equation (4) yielded A = 0.95 + 0.03, n = 1.02 + 0.06 and R = 0.948 (Table 5.1 and Figure
5.4).
For the <f>/0 versus data plotted in Figure 5.4 [Deng2002, Nanjangud 1995,
Yang2003, this study], information concerning the material, porosity, shape forming
technique, sintering conditions, fracture testing technique and loading rate are summarized in
Table 5.2. Since each of the factors given in Table 5.2 can affect fracture strength, it is
124

Table 5.2. Materials, particle size, shape forming technique, specimen dimensions, green
porosity, fracture testing technique and loading rate for the HA specimens in this study and
the materials from the literature [Deng 2002, Nanjangud 1995, Yang 2003] included in this
study’s analysis on <f>/0 versus degree of densification, .
References

Materials

Nanjangud Commercial
1995
green extruded
alumina

Particle Shape
size
forming
technique
(m)

Specimen
dimensions

N/S

Rods: ~ 7.5 ~ 42%
x 75 mm

Deng 2002 Yttria - stabilized 0.028
tetragonal ZrO2
granules

Yang 2003 95.5 %  -

Extruded

0.55

Si3N4, 1.3 wt. %

Single ended
pressing at
30 MPa
and 75
MPa
Uniaxially
pressed at
20 MPa

Green
porosity
( PG)

a

Bars: 62 x
10 x 4 mm
3 x 4 x 40
mm

60 % at
30 MPa
and
56 % at
75 MPa

3 x 4 x 40 - 55 50 mm
58 %

oxygen, 1 to 7.5

Fracture
test tech.
& loading
rate
4 point
bend
25.4
mm/min
3 point
bend
0.5
mm/min

3 point
bend
0.5
mm/min

wt. % Yb2O3
added
This study Commercial HA 10.20
powder

a

Uniaxially
cold
pressed at
27 MPa

Diameter and length of rod-shaped specimens

b

Diameter and thickness of disc-shaped specimens

125

23.8 - 32.2
x 1.85 2.55 mm

b

~ 62 %

Biaxial
flexure
1 mm/min

important to consider the differences among the studies [Deng 2002, Nanjangud 1995, Yang
2003, this study] in terms of these processing and testing factors.
4.3 Relative Young’s modulus as a function of for HA and other brittle materials
As is the case for <f> as a function of P, there are a limited number of studies in the
literature that provide Young’s modulus versus P data for values of P extending from near
green density to approximately full density.

The few materials for which there are studies

that do include a wide range of <E> versus P data are alumina [Hardy and Green 1995,
Nanjangud 1995, Ren 2009], HA [Ren 2009], titanium [Oh 2003] and YSZ [Deng 2002].
In this study, we least-squares fit the relative Young’s modulus, <E>/E0, versus the
degree of densification, data to equation (10) from four different materials: alumina [Hardy
and Green 1995, Nanjangud 1995, Ren 2009], HA [this study, Ren 2009], titanium metal [Oh
2003] and 3 mol% YSZ [Deng 2002] (Figure 5.5). The materials’ green porosity PG, the
Young’s modulus at theoretical density, E0, and the fitting No single study in the current
literature provides a comparison among <E>/E0 versus  data for several different materials.
However, in this study we demonstrate that equation (10) describes well the <E>/E0 versus 
data behavior of multiple data sets (Figure 5.5). In particular, Ren’s HA specimens [Ren 2009]
were fabricated and tested in the same way as the HA specimens in this study and as expected,
the <E>/E0 versus  results (Figure 5.5) are quite similar between this study and Ren’s study
[Ren 2009]. However, the Ren study included <E> versus P data only for P < 0.5 while this
study included data for P up to 0.62.

126

Table 5.3. Experimentally determined values of green porosity, PG, Young’s modulus
corresponding to theoretically dense materials, E0, number of data points, N, and
least-squares fitting parameters, AE and q, for the fit of E/E0 versus degree of densification, ,
2

to equation (10) with coefficient of determination, R for HA [this study, Ren 2009], alumina
[Nanjangud 1995, Hardy 1995, Ren 2009], titanium [Oh 2003] and YSZ [Deng 2002].
Reference

120

HA

0.62

120

Alumina

0.47

402.8

Alumina

0.46

402.8

Nanjangud
1995

Alumina

0.42

402.8

Oh2003

Titanium

0.36

Oh2003

Titanium

0.36

105

Deng 2002

e

0.62

Hardy 1995

d

HA

Ren 2009

c

E0 (GPa)

Ren 2009

b

PG

This study

a

Material

YSZ

0.56

N

234.8

a

10

a

15

2

AE

q

R

0.84 ± 0.07

1.20 ± 0.12

0.979

0.95 ± 0.01

1.38 ± 0.02

0.998

b

9

0.93 ± 0.03

0.92 ± 0.07

0.979

b

13

1.01 ± 0.08

0.83 ± 0.11

0.905

b

35

0.97 ± 0.02

0.90 ± 0.03

0.988

114.6

c

25

0.78 ± 0.03

0.64 ± 0.05

0.948

d

25

0.85 ± 0.03

0.64 ± 0.05

0.948

7

0.87 ± 0.02

1.19 ± 0.06

0.995

e

Aggregate average Young's modulus from single crystal data [Landolt and Bornstein 1979].
Aggregate average Young's modulus from single crystal data [Wachtman 1960].
Aggregate average Young's modulus from single crystal data [Fisher and Renken 1964].
0 was estimated from the extrapolation of the E vs. P plot (Figure 5.3 in [Oh 2003])
Calculated from aggregate average E from single crystal zirconia stabilized with 2.8 mol%

and 3.4 mol % yttria [Ingel and Lewis 1988].

127

In the three studies of partially sintered alumina by Nanjangud [Nanjangud 1995],
Hardy [Hardy and Green 1995] and Ren [Ren 2009], the Young’s modulus data was
normalized by E0, value of E at P = 0. Hardy [Hardy and Green 1995] and Nanjangud
[Nanjangud 1995] fit their <E> versus P data to equation (9), while Ren [Ren 2009] fit the
<E> versus P data to equations (5) and (6). Nevertheless, the studies by [Nanjangud 1995],
Hardy [Hardy and Green 1995] focused only on the partially sintered alumina included in
their own studies. Ren [Ren 2009] not only studied partially sintered alumina and HA but
also compared (i) the <E>/E0 versus P behavior for their alumina specimens to literature data
for alumina and (ii) the <E>/E0 versus P behavior for their HA specimens to literature data
for HA. However, Ren [Ren 2009] did not systematically compare their alumina and HA
data.
4.4 Physical interpretation of the descriptions of <E> /E0 and as a function of degree of
densification, 
In terms of the success that equations (10) and (4) have, respectively, in describing the
Young’s modulus, <E> /E0 versus behavior (Figures 5.5) and the fracture strength, <f>/0
versus behavior (Figure 5.4) for each of the materials analyzed in this study, the physical
interpretation of that success must begin with a consideration of the data that has been
examined in this study. Each of the materials included in the fracture strength (Figure 5.4)
and the Young’s modulus (Figures 5.5) analysis were single phase, partially sintered powder
compacts without pore forming agents (Tables 5.2 and 5.4a), indicating that the specimens in
each study likely had a unimodal pore size distribution [Rice 1998].
128

E/E0

1

1

0.1

HA (This study)
HA (Ren et al. 2009)
Alumina (Ren et al. 2009)
Alumina (Hardy et al. 1995)
Alumina (Nanjangud et al.1995)
Titanium (Oh et al.2003)
YSZ (Deng et al. 2002)

0.01
0.0

0.2

0.4
0.6
Degree of densification

0.8

0.1

0.01
1.0

Figure 5.5. Relative Young’s modulus, <E>/E0, versus  for HA [this study], HA from
literature [Ren 2009] alumina [Hardy and Green 1995, Nanjangud 1995, Ren 2009], titanium
[Oh 2003] and YSZ [Deng 2002]. Each dashed curve represents a least-squares fit of <E>/E0
versus  data to equation (10). The parameters for the least-squares fit of the data sets to
equation (10) are listed in Table 5.3.

129

Table 5.4a. Processing details for HA [Ren 2009], alumina [Hardy 1995, Ren 2009] and
titanium [Oh 2003] included in this study’s analysis E/E0 versus degree of densification, .
Processing information for additional data in the E/E0 analysis for alumina [Nanjangud 1995],
YSZ [Deng 2002] and HA [this study] is included in Table 5.2.

References

Materials, purity
and vendor

Commercial HA
HA
powder (Taihei
Ren 2009 Chemical Industrials
Co., Osaka, Japan)
Commercial
alumina powders
Alumina
(Baikowski
Ren 2009
RC-HP, Malakoff,
TX)
Commercially
available alumina
Alumina (Malakoff
Hardy 1995 Industries, TX)
>99.99% purity

Ti
Oh 2003

Commercially
available Ti
powders fabricated
by plasma rotating
electrode process
and the gas
atomization
process. Gas
atomized Ti
powders supplied
by Sumitomo
Titanium Corp.

Powders

Green
Shape
porosity
forming
technique
( PG )

Sintering
temperatures

Average
particle size: 1 ~ 62 %
- 3 m

Uniaxially
1125oC to
cold
1360oC for 1 h
pressed at
to 6 h
33 MPa

Vendor-specif
ied average
powder
~ 47 %
particle size
of 0.3m

Uniaxially
1200oC to
cold
1475oC for 1 h
pressed at
to 4 h
23 MPa

Median
particle size
of 0.38 m

Uniaxially
800oC to
pressed at
1600oC for 2 h
45 MPa

~ 46 %

300–500 m
(mean
diameter 374
m); 150–
250 lm (mean
diameter 189 ~ 36 %
m) and 45–
150 lm (mean
diameter 65
m)

130

Pressed at 70 MPa for 0.6
ks then pressurelessly
sintered in vacuum at
1173 K, 1373 K and
1573 K, for 7.2 ks. Also
sintered under uniaxial
punch pressure of 5 or 10
MPa at 1173 K for 7.2 ks
and 1 MPa at 1223 K for
7.2 ks.

Table 5.4b. Details of the Young’s modulus, E, measurement techniques for HA [this study]
and for the materials from the literature (HA [Ren 2009], alumina [Nanjangud 1995, Hardy
1995, Ren 2009], titanium [Oh 2003] and YSZ [Deng 2002]) included this study’s analysis of
E/E0 versus degree of densification, . The measurement technique is significant since the
isothermal moduli (determined by static or quasistatic testing) is typically about 10 percent
lower that the adiabatic moduli (determined by dynamic testing) on the same material.
Reference

Materials

This study
HA
Ren 2009
HA
Ren 2009
Alumina
Hardy 1995
Alumina
Nanjangud 1995 Alumina
Oh2003

Ti

Deng 2002

YSZ

E measurement techniques
Resonant ultrasound spectroscopy
Resonant ultrasound spectroscopy
Resonant ultrasound spectroscopy
Ultrasonic velocity technique
Ultrasonic velocity technique
Evaluated via the compression test in an
elastic range of deformation up to a
maximum 1000 N with an applied load
of 0.5 N per step
Pulse-echo method

131

Type of E
measurement
Dynamic
Dynamic
Dynamic
Dynamic
Dynamic
Quasistatic
Dynamic

Table 5.5. Porosity range, strength at zero porosity, 0, Young’s modulus at zero porosity, E0,
2

number of data points, N, fitting parameters, A and r, and coefficient of determination, R for
the least-squares fit of the normalized strength - Young’s modulus data to equation (12), for
the three data sets (HA [this study], alumina [Nanjangud 1995] and YSZ [Deng 2002]) which
included both and E data on the same material.
Reference

Material

P range

0

E0 (GPa) N

A

r

R

2

(MPa)
This study

HA

0.21 – 0.62

80

120

10

Nanjangud
1995

Alumina

0.07 – 0.42

380

402.75

10

Deng 2002

YSZ

0.02 – 0.49

800

234.8

7

132

0.954 ± 0.856±
0.990
0.050
0.046
1.282 ± 1.201±
0.995
0.039
0.053
0.962 ± 1.253±
0.996
0.019
0.059

 and E/E0
f

1

1

0.1

0.1

0.01

0.01
Relative strength
Relative Young's modulus

0.0

0.2

0.4

0.6

0.8

1.0

Degree of densification

Figure 5.6. The relative average fracture strength, <f>/0, and relative Young’s modulus,
<E>/E0, of HA [this study] as a function of degree of densification, , showing a very similar
trend in the relative strength and Young’s modulus versus . The dashed curve represents a
least-squares fit of <f>/0 to equation (4), and the solid curve represents a least-squares fit
of <E>/E0 to equation (10).

133

As was discussed in Part I and by German [German 1994] and Barsoum [Barsoum 1997],
during sintering of ceramic [Barsoum 1997] or metal [German 1994] powder compacts, there
is a geometrically similar evolution of pore morphology through initial, intermediate and
final stages. This near universality of powder-particle-size independent, geometrically-similar
pore shape evolution for sintered powder compacts [Barsoum 1997, German 1994, Part I)
likely contributes to the ability of any single mathematical form being able to describe the
porosity dependence of <E> /E0 versus  and <f> /0 versus for the range of materials
considered here.
In addition, the fracture testing (Table 5.2) in each study was performed on as-sintered,
non-polished specimens, though the testing techniques were different in the four studies
(ring-on-ring for HA [this study], four-point bend for alumina [Nanjangud 1995] and
three-point bend for YSZ [Deng 2002] and silicon nitride [Yang 2003]). Also, for the Young’s
modulus data, the E measurements were by dynamic techniques (Table 5.4b) with the
exception of the Ti study [Oh 2003] where E was measured by quasistatic compression
testing. For a given material, E values determined by quasistatic techniques are often roughly
10 percent lower than measurements made by dynamic techniques.
In terms of the physical interpretation of the results of this study, it is helpful to consider
the “boundary values” for equations (4) and (10). Consider equation (4), which is

 f 
P
 A  1 
 P
0
G


n


  A  n



When P  zero ( one) <f/0 approaches A regardless of the value of the exponent n.

134

When P  zerothe specimen strength approaches the strength at zero porosity (f 0),
which means A 1. When P approaches PG ( approaches zero), then <f/0  zero
independently of the numerical values of Aor n. For Figure 5.4, the <f/0 data does (i)
approach 0 as approaches 0 and (ii) approach 1 as approaches 1. Similarly, for equation
(10)

E
P
 A E 1 
 P
E0
G


q


  A E q



note that for P  0 (that is,  1), then <E>/E0  A for all values of exponent q. Since E
 E0 at P  0, then Aalso 1. In addition, if P  PG, then <E>/E0  0 regardless of the
numerical values of Aand q. As was the case for the <f/0 data and equation (4), the
<E>/E0 data does approach the asymptotic values expected from equation (10) for  0 and
 1 (Figure 5.5).
Note that the numerical values of Agiven inTable 5.1 for each of the materials (HA
[this study], alumina [Nanjangud 1995], YSZ [Deng 2002] and Si3N4 [Yang 2003]) are
relatively close to unity, except for alumina where A= 1.14 ± 0.02. Physically, for
theoretically dense materials  1 and <f/0 approaches Ahus an Avalue other than
unitymay indicate that the value of 0 (determined here by extrapolation of the <f> versus 
plot, see footnotes for Table 5.1) may be greater or smaller than the “true” value of 0 for the
material. In practice, 0 can be problematic to determine since in addition to the included
porosity, the actual values of 0 depends on the flaw population present in a given specimen
[Case 2002, Case in press].

135

1.0

0.8

f

0.6

0.4

0.2

0.0
0.0

HA (This study)
Alumina (Nanjangud et al. 1995)
YSZ (Deng et al. 2002)

0.2

0.4

0.6

0.8

1.0

E / E0

Figure 5.7. Normalized fracture strength <f>/0 versus normalized Young’s modulus,
<E>/E0, for HA [this study], alumina [Nanjangud 1995] and YSZ [Deng 2002]. Each dashed
line represents a least-squares fit to equation (12).

136

Also, for equation (10), the A values (Table 5.3) are close to, but in most cases, smaller
than unity. Values of E0 are less problematic than 0 values to determine, since for
polycrystalline materials E0 can be estimated from the aggregate average of single crystal
Young’s modulus data from the literature (footnotes for Table 5.3). It is not clear why the A
values in Table 5.3 are not closer to unity.
For equations (4) and (10), the predictions that <f/0 and <E>/E0 go to zero for P 
PG, is somewhat ambiguous [Barsoum 1997, Flatt 2004, Hornbaker 1997, Libowitzky 1999].
For green compacts (for which P = PG by definition) both the strength and the elastic moduli
are dominated by mechanisms that are independent of the diffusional processes [Barsoum
1997, German 1994] that lead to the formation of solid contacts between particles.
4.5 Relationship between Young’s modulus and fracture strength for partially sintered
powder compacts
Since the relative fracture strength and relative Young’s modulus for HA [this study]
show very similar trends when plotted on the same normalized property versus degree of
densification plot (Figure 5.6), it is natural to inquire about the direct relationship between
fracture strength and relative Young’s modulus. Combining equations (4) and (10) yields
 E
 f 
 A 
E A
0
 0 E

n

q
 E 
  A

 E 


 0 

r

(12)

r

where A = A / (AE) and r = n/q. A least-squares fit of the <f0 versus the <E>/E0 data
to equation (12) gives r = 0.86 ± 0.05 for the HA [this study], r = 1.20 ± 0.05 for alumina
[Nanjangud 1995] and r = 1.25 ± 0.06 for YSZ [Deng 2002], with coefficients of
137

2

determination, R , of 0.990, 0.995 and 0.996, respectively (Figure 5.7). Thus, equation (12)
describes the porosity dependence of <f0 versus the <E>/E0 very well over the range of
porosities included in the data (Table 5.5) for HA [this study], alumina [Nanjangud 1995] and
YSZ [Deng 2002].
From an empirical point of view, the exponent r in equation (12) should be (and in fact
is) close to unity since n and q are both close to unity. In addition, the factor A in equation
(12) is a proportionality constant between fracture strength and Young’s modulus. When
considering the physical meaning of this proportionality, it must be emphasized that the data
(Figure 5.7) represents partially sintered powder compacts over a range of volume fraction
porosity P. In addition to being functions of porosity, both E and f are functions of the
damage state, including microcracking, where microcracks can be introduced into a specimen
by various mechanisms such as thermal expansion anisotropy [Case 1981, Case 2002],
thermal expansion mismatch [Cleveland 1978], phase transitions [Case 1984]. Also, the
functional forms for E and f have differing dependencies on crack size [Case 2002, Case in
press, Rice 1998]. For brittle materials E is a function of the third moment of crack radius
and f depends on the largest flaw [Case 2002, Case in press], thus if microcracks or other
damage were introduced into the partially sintered bodies, it is extremely likely that equation
(12) would no longer hold.

Nevertheless, equation (12) does describe extremely well the

<f0 versus the <E>/E0 data for the three brittle ceramics (HA, alumina and YSZ)
depicted in Figure 5.7.
4.6. Relationship from literature for fracture toughness and porosity
138

For the partially sintered alumina specimens, Flinn [Flinn 2000] found the fracture
toughness, KC can be expressed as


P
 1  
 P 
K c  tip0 
1
K c  tip

j

(13)

where Kc-tip = crack tip fracture toughness, Kc-tip0 = crack tip fracture toughness for a
theoretically dense material, and exponent j is an empirically determined constant. Equation
(13) and equations (4) and (10) have a similar functional form.
Using a least-squares fitting technique, Flinn [Flinn 2000] found j = 1.15 for their
alumina powders, where the exponent j is similar in magnitude to the exponent n in equation
(4) and q in equation (10) (Tables 5.1 and 5.3). For Flinn, the volume fraction porosity P1, in
equation (13) is not PG. Instead Flinn [Flinn 2000] made P1 in equation (13) an adjustable
parameter, obtaining a P1 value of 0.45 via a least-squares fit of their alumina data.
Flinn [Flinn 2000] also found a linear relationship between E and Kc-tip that is
analogous to the nearly linear E versus fracture strength behavior described in equation (12)
in this study. However, an important difference between this study and the study by Flinn is
that Flinn used pore formers to create bimodal pore size distributions. For specimens that are
densified from powder compacts containing particles that are approximately equiaxed, the
pore size distribution tends to be unimodal [Rice 1998]. To generate bimodal pore size
distribution, researchers often use pore formers, for example: (1) fugitive phases in the form
of solid particles that burn out during firing [Yuan 2008] or (2) liquid additives such as
hydrogen peroxide volatilize during firing [Ren 2005]. In this study the green HA compacts

139

were sintered without pore formers (Tables 5.2 and 5.4a), however the HA powders in this
study are rod shaped with an aspect ratio of ~ 3, thus the details of the pore size distribution
likely depends on the degree of alignment of the powder particles. Nevertheless the Flinn
data follows trends similar to those observed in this study.

Thus, a comparison of this study

with Flinn’s study [Flinn 2000] implies that the relationships presented here for the
dependence of <f0 (equation 4) and <E>/E0 (equation 10) for materials with unimodal
pore size distributions may also apply to materials with bimodal pore size distributions, but
further study is needed to explore that topic.
5. Summary and Conclusions
In this study, the fracture strength and Young’s modulus were measured for 540
polycrystalline HA specimens. In Part I of this study, the Weibull modulus, m, versus P
behavior was examined for the same HA specimens as well as a combined data set with more
than 1560 oxides and non-oxides, showing a 3-region, “U” – shaped trend. The Weibull
modulus, m, is an important measure of the scatter in fracture strength (mechanical
reliability). Also the fracture strength values themselves and the Young’s modulus are
important where, for example, the analysis of stress and strain by analytical or numerical
means requires knowledge of the Young’s modulus. Thus, the mechanical behaviors
discussed in both Parts I and II are crucial to the design of brittle components.
Equation (4) was applied to describe the  dependence of the normalized fracture
strength data for polycrystalline HA [this study], alumina [Nanjangud 1995], 3 mol% yttria
stabilized zirconia (YSZ) [Deng 2002] and silicon nitride [Yang 2003] (Figure 5.4). Also, this
140

study uses equation (10) to fit the Young’s modulus versus  data for alumina [Nanjangud
1995, Hardy and Green 1995, Ren 2009], HA [Ren 2009, this study], titanium [Oh 2003] and
YSZ [Deng 2002] (Figures 5.5).
Equations (4) and (10) are similar power law functions of with the exponents ≈ 1 in
both cases (Tables 5.1 and 5.3). Also both equations (4) and (10) are excellent descriptors of
the dependence of Young’s modulus and fracture strength (Figures 5.4, 5.5 and 5.6)
respectively over the entire range of for specimens of approximately green density
(to specimens that are nearly theoretically dense (.While earlier researchers
[Hardy and Green 1995, Nanjangud 1995] demonstrated a similar equation could be applied
to alumina, this study shows that the results apply to many partially sintered brittle materials.
In addition, the similarity of equations that describe the  dependence of <f> (equation
(4)) and <E> (equation (10)), leads to a nearly linear relationship between <f>/0 and
<E>/E0 (equation (12) and Figure 5.7) for HA from this study, alumina [Nanjangud 1995]
and yttria (3 mol%) - stabilized zirconia (YSZ) [Deng 2002].
6.0 Acknowledgements
The authors acknowledge the financial support of the National Science Foundation,
Division of Materials Research Grant DMR-0706449. In addition, the authors acknowledge
the use of the Electron Microscopy facilities at the Center for Advanced Microscopy,
Michigan State University.

The purchase of the Resonant Ultrasound Spectroscopy

apparatus and the laser scattering apparatus equipment used in this study was funded by the
Defense University Research Instrumentation Program (DURIP) Grant No.
141

N00014-07-1-0735.

142

APPENDIX

143

APPENDIX

DIFFERING TRENDS FOR THE MEAN FRACTURE STRENGTH AND WEIBULL
MODULUS
In previous chapter, the Weibull modulus versus P for a large collection of brittle
materials followed a three-region, U-shaped function of the volume fraction porosity, P,
which begs the question “Since two of the parameters that characterize the fracture of brittle
materials are fracture strength, f, and Weibull modulus, m, then do f and m, follow the
same trends as a function of P?” If we examine the fracture strength, f versus P and m
versus P behaviors for six studies taken from the literature that include five different brittle
materials, namely studies of TiO2 by Kishimoto [Kishimoto1991], Al2O3 by Nanjangud
[Nanjangud 1995], niobia-doped ZrO2 and tantala-doped ZrO2 by Pissenberger [Pissenberger
and Gritzner 1995] and three studies of HA (one study each by Lopes [Lopes 1999], Ruys
[Ruys 1995] and Villora [Villora 2004]), the m versus P trend is notably different than the
fracture strength versus P behavior for each study. The f versus P data tends to decrease
monotonically with increasing P, however, for the m versus P data, there are no apparent
trends in part because there are relatively few data points for m versus P in each study
[Kishimoto1991, Lopes 1999, Nanjangud 1995, Pissenberger and Gritzner 1995, Ruys1995,
Villora2004].
The difference between the physical nature of m and the fracture strength in terms of the
characterization of brittle materials is further highlighted in a study by Sharpe [Sharpe 2008]

144

in which single crystal SiC micro-specimens were fractured in tension. For SiC specimens
with circular or elliptical holes, the average tensile fracture strength was quite high, ranging
from 0.71 to 1.36 GPa depending on the geometry of the holes. On the other hand, the values
of Weibull modulus for the SiC specimens were modest (between 3.7 and 5.9). Thus, high
fracture strength does not necessarily correspond to a high Weibull modulus.
However, in contrast to the SiC specimens in Sharpe’s study [Sharpe 2008], the HA
specimens in this study tested in the green (unfired) state had a very low average fracture
strength at 0.6 MPa but a surprisingly high Weibull modulus of 21.2. Again, average fracture
strength and Weibull modulus are often not correlated but characterize different physical
properties, such that m measures the spread in the strength values and the mean strength
indicates the “center” of the strength distribution.

145

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146

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150

CHAPTER 6
THE EFFECT OF INDENTATION-INDUCED MICROCRACKS ON THE ELASTIC
MODULUS OF HYDROXYAPATITE

X. Fan, E. D. Case and M. J. Baumann
Department of Chemical Engineering and Materials Science, Michigan State University,
East Lansing, MI, 48824
Published in: J Materials Science, 47:6333-6345, 2012.

Abstract
The presence of microcracks in materials affects a wide range of mechanical properties
including elastic modulus, Poisson’s ratio, fracture strength and fracture toughness. The
microcrack-induced reductions of the Young’s modulus, E, and Poisson’s ratio, , are
functions of the size, geometry and number density of microcracks. In this study, an array of
Vickers indentation microcracks was placed on the surfaces of two hydroxyapatite (HA)
specimens with totals of 391 and 513 indentations per specimen. This study tests the validity
of theoretical studies of microcrack-damage-induced changes in E and , where the changes
are expressed either by (i) the volumetric crack number density, N and (ii) the crack damage
parameter, . All elasticity measurements were done via resonant ultrasound spectroscopy.
For both the HA specimens included in the study and alumina specimens indented in an
earlier study [Kim 1993], E and decreased approximately linearly with increasing
microcrack damage. The slopes of the E and  versus N and  are also computed and
compared to the available theoretical models.

151

1. Introduction
Applications of hydroxyapatite (HA, Ca10(PO4)6(OH)2), both in dense and porous
forms, include filters [Misra 2005, da Silva 2006], catalytic supports [Opre 2005, Jamwal
2008, Rakap 2011] and dielectric coatings [Silva 2003, Silva 2005]. In addition, HA is used
in many types of sensors, including those for toxic metal ions [El Mhammedi 2009], CO gas
[Mahabole 2005], proteins [Ohtsuki 2010] and humidity [Owada 1989]. HA is also widely
studied as a candidate for engineered hard tissue (bone and teeth) due to its chemical
similarity to natural bone mineral and its excellent biocompatibility [Hench 1991, Burg 2000,
Karageorgiou 2005].
The Young's modulus, E, and Poisson’s ratio, , are two fundamental mechanical
properties of materials. For example, for bodies with complex geometries, analytical methods
to calculate stress and strain often become mathematically unwieldy, thus numerical
techniques such as finite element techniques are needed [Martin 1973]. The construction of
the stiffness matrix for finite element calculations and simulations requires Young’s and
Poisson’s ratio data [Martin 1973].
Brittle materials including HA often incur microcrack damage and in general
microcracks affect a wide range of mechanical properties including Young’s modulus [Smith
1995, Perera 2010, Chotard 2008, Patapy 2009, Yousef 2005], Poisson’s ratio [Case 1984,
Wang 2009], fracture strength [Rokhlin 1993, Diaz 2008], hardness [Perera 2010] and
fracture toughness [Evans 1986, Kratschmer 2011, Hubner 1977, Meschke 1997]. Other
physical properties such as electrical conductivity [Ota 1986, Hilpert 2004], thermal
152

diffusivity [Hasselman 1993, Deroo 2010], thermal expansion [Chotard 2008] and dielectric
constant [Kimura 1990, Morito 2005] are also functions of the microcrack damage state.
In particular, the median and radial microcracks induced by Vickers indentation (Figures
6.1a and 6.1b) are widely used to model the flaws produced in manufacturing processes such
as grinding, cutting and polishing [Rice 2002]. Cracks induced by Vickers indentation are
also used as model cracks for thermal shock studies [Lee 2002, Collin 2002], crack healing
studies [Wilson 1997a, Wilson 1999, Wilson 1997b, Kese 2006], R-curve evaluation [Braun
1992], residual stress studies [Salomonson 1996] and in the design of dental crowns [Kim
2006].
In addition to the experimental studies linking microcracks with physical properties, a
number of theoretical studies have explored the changes in properties due to microcracking.
In particular, theoretical work on the dependence of Young’s modulus on microcrack damage
includes research by Budiansky and O'Connell [Budiansky 1976], Salganik [Salganik 1974],
Hoenig [Hoenig 1979] and Laws and Brockenbrough [Laws 1987]. In each of these studies
[Budiansky 1976, Salganik 1974, Hoenig 1979, Laws 1987], the effect of microcrack damage
is described by a linear function of Young's modulus on the crack parameter, and crack
orientation function f, such that
E = E0 (1 - f)

(1)

where E0 is the Young's modulus of non-microcracked specimen. For circular microcracks of
3

uniform radius c, the crack damage parameter is defined as = N c , where N is the
volumetric crack number density.
153

(a)

2a

2c
(b)

Figure 6.1. Schematic (a) and SEM micrograph (b) of a Vickers indentation system,
including the indentation impression and radial cracks originating from the surface corners of
the indentation impression.
154

For most experimental studies of microcrack-damaged materials, the N and c are not
measured directly. Instead, if microcracking mechanisms such as thermal expansion
anisotropy, thermal expansion mismatch or Martensitic-type crystallographic phase changes
are present, then the microcracking damage state is inferred from changes in physical
properties.
In addition to changes in elastic moduli, changes in electrical conductivity due to
microcracking were explored by Hoenig [Hoenig 1979] such that for crack damage parameter,
ε


 1  g
0

(2)

where  is the electrical conductivity of a cracked body and is the electrical conductivity
of a non-cracked body. For randomly oriented microcracks, g = grandom = 8/9. For an array
of aligned cracks, g = galigned = 8/3. Similarly, for thermal conductivity, k, Hoenig obtained
k
 1  g
k0

(3)

where k is the thermal conductivity of a cracked body and kis the thermal conductivity of a
non-cracked body and ε is the crack damage parameter. The orientation parameter g takes on
exactly the same numerical values as for electrical conductivity in the randomly oriented and
aligned cases. Thus for a fixed value of the crack damage parameter, the microcracking
induced changes are three times greater for the aligned case compared to the randomly
oriented case for both electrical conductivity and thermal conductivity.
An expression by Hasselman [Hasselman 1979] for the relative change in thermal
155

-1

conductivity due to randomly oriented microcracks, namely k/k0 = (1+8/9) , is nearly
identical to Hoenig’s expression [Hoenig 1979] for small values of crack density since by
-1

power series expansion, (1 + x) ≈ 1- x for small values of x.
The Poisson’s ratio change with respect to the volume crack number density was studied
by Walsh [Walsh 1965], and is rewritten here as


 S  N
0

(4)

where Δ is the Poisson’s ratio change, 0 is the Poisson’s ratio of unindented specimen, N is
the volumetric number density of microcracks, and S is the product of a crack orientation
and a crack geometry parameter. For 3-dimensional, randomly oriented cracks, Walsh [Walsh
3

1965] found a S value of 4πc /3 for randomly oriented microcracks.
As discussed above, in order to test the various microcracking-property change theories
[Budiansky 1976, Salganik 1974, Hoenig 1979, Laws 1987, Hasselman 1979, Walsh 1965],
knowledge of both the crack size, c, and crack number density, N, are needed. However, for a
specimen with a 3-dimensional microcrack distribution, the N and c values obtained from
surface measurements on the specimen are not representative of the 3-dimensional
microcracks throughout the specimen bulk. This is in part because the specimen surface is
subjected to biaxial stress while the bulk is typically under a triaxial stress state. Therefore a
representative number and size of microcracks will not necessarily intercept the specimen
surface [Case 1984]. A second factor that makes it difficult to determine N and/or c from
surface measurement alone is that post-processing procedures, such as polishing and cutting,

156

may induce surface microcracks that are not related to internal microcracking mechanisms
such as thermal expansion anisotropy or phase changes [Case 1984]. Thus, the surface crack
population likely includes surface microcracks created by specimen processing.
In order to calculate the crack damage parameter equation (1)) and determine the
validity of equation (1), the values of N and c need to be determined. There are at least two
possible solutions: (1) induce surface cracks by indentation (for example) so that N and c can
be measured directly [Kim 1993, Kim 1993b, Case 1993], or (2) use a scattering technique
such as small angle neutron scattering (SANS) in which penetrating radiation (neutrons)
scatter from volumetrically distributed microcracks [Case 1984].
In this study, an array of microcracks was placed on the polished surfaces of two HA
specimens using Vickers indentation in order to control the number density, size and location
of the surface microcracks. The process of adding Vickers indentation cracks to the
specimens was periodically interrupted in order to measure the Young’s modulus, E, and
Poisson’s ratio, , via Resonant Ultrasound Spectroscopy (RUS). A total of 391 indentations
were made on the first specimen and a total of 513 indentations were made on the second
specimen.
In the literature, there are very few studies in which changes in elasticity are directly
measured for an array of microcracks where both the size and the number of microcracks are
known. In this study, the observed changes in Young’s modulus and Poisson’s ratio were
compared to the theoretical predictions from the literature, as discussed in the next section of
this paper. In addition, we also compare our HA Young’s modulus results to a similar study
157

performed using Vickers indentation generated microcracking in commercial alumina
specimens [Kim 1993, Kim 1993b, Case 1993]. Thus, in this study, we consider the effect of
microcrack damage on two very widely used brittle materials: (1) HA which has many
biomedical, filter and sensor applications (as discussed above), and (2) alumina which is used
in a great variety of applications requiring high temperature, chemical stability and resistance
to erosion [Dorre 1984]. In addition, as will be discussed in Section 3.2, alumina and HA
have very different mechanical properties including the elastic moduli, hardness and fracture
toughness.
2. Experimental Procedure
For each of the three HA specimens included in this study, approximately 2.5 grams of
biomedical grade commercial HA powder (Taihei Chemical Industrials Co., Osaka, Japan)
were uniaxially pressed to form disk-shaped compacts at 33 MPa and then sintered in air at
1360oC for 4 hours with a heating/cooling rate of ~ 10oC/minute. For each of the three HA
specimens included in this study, the as-sintered disc diameter was 23.4 mm with a thickness
of approximately 1.9 mm.
The three sintered HA specimens were then polished using 600 grit SiC abrasive paper
followed by polishing with diamond paste down to a grit size of 0.5 m. Two specimens
(labeled as HA-391 and HA-513) were used to study the Vickers indentation damage-induced
changes in Young’s modulus and Poisson’s ratio. A third specimen (labeled as HA-GS),
(processed from the same powder batch with nominally identical processing parameters as
HA-391 and HA-513) was thermally etched to determine grain size.
158

After polishing, each of the two disk-shaped HA specimens used for indentation was cut
into a bar using a commercial milling machine (Michigan State University Physics Shop),
yielding one specimen with dimensions of 13.4 mm x 10.3 mm x 1.6 mm (HA-391) and one
specimen with dimensions of 15.8 mm x 12.2 mm x 1.8 mm (HA-513). Specimen HA-GS
was not sectioned following polishing, but instead remained in the disk form.
Microcracks were induced by Vickers indentation (Buehler, Lake Bluff, Illinois, US)
using a load of 9.8 N, a load time of 5 seconds and a loading rate of 70 m/s. Indentations
were placed across the surface in a grid pattern with 0.5 mm spacing between indentation
centers. A 23 x 17 grid pattern was placed on specimen HA-391, and a 27 x 19 grid pattern
was placed on specimen HA-513. The diagonal length of the indentation impression, 2a, and
the radial crack length, 2c, for each specimen were measured for the first row of indentations
(23 indents for HA-391 and 27 indents for HA-513) using the optical microscope mounted on
the Vickers indenter.
The microstructure of each of the three HA specimens was examined using scanning
electron microscopy (SEM, JEOL 6400V, JEOL Corp. Japan). A roughly 20 nm thick gold
coating (EMSSCOPE SC500, Ashford, Kent, Great Britain) was sputtered on each specimen
before SEM examination. SEM was conducted using a 15 mm working distance and a 15 kV
accelerating voltage.
The specimen used for grain size evaluation (HA-GS) was thermally etched by
annealing the as-densified specimen in air at 1300oC for 2 hours. The mean grain size was
evaluated via a linear intercept method using at least 200 intercepts [Case 1981]. For the
159

indented specimens, micrographs of the indentation impression and radial crack system were
obtained.
The Young’s modulus and Poisson’s ratio of specimens HA-391 and HA-513 were
measured via RUS [Migliori 1997] using commercial equipment (RUSpec, Quasar
International, Albuquerque, NM, USA). RUS is a non-destructive technique widely used in
the last decade to measure the elastic moduli of solid materials [Ren 2009, Schmidt 2010, Ni
2010, Zhang 2010, Gladden 2010, Ni 2009, Zhang 2007]. During the RUS measurements,
the specimen was placed on a three-transducer configuration. One of the three transducers
generates a mechanical vibration (a frequency range of 5 to 250 kHz in this case) and the
other two transducers detect the specimen’s mechanical resonance modes (Figure 6.2). The
Young’s modulus, and Poisson’s ratio were calculated from the resonance frequencies and
the mass, shape and the dimensions of the HA specimens using a commercial software
package (RPModel, Quasar International, Albuquerque, NM, US). For both HA-391 and
HA-513, the Young’s modulus, E0, and Poisson’s ratio, 0 were measured prior to indentation
(where the subscript “0” denotes the values of Young’s modulus and Poisson’s ratio without
Vickers-indentation induced microcrack damage). Then seventeen rows of indentations with
23 indentations in each row were made in HA-391, E and  were measured by RUS after
completing the first row of 23 indentations. Subsequent E and  measurements for specimen
rd

th

th

th

th

th

th

th

HA-391 were made after the 3 , 5 , 7 , 9 , 11 , 13 , 15 and 17 rows of indentations
were completed. For specimen HA-513, nineteen rows of indentations with 27 indentations
in each row were placed on the specimen, then E and  were measured by RUS after each
160

50

Amplitude (A.U.)

40

30

20

10

0
50

100

150

200

250

Frequency (kHz)

Figure 6.2. RUS spectra obtained for a polished, unindented HA specimen bar sintered at
1360oC for 4 h. A.U. stands for arbitrary units.

Figure 6.3. SEM micrographs of polished and thermally etched surface for specimen HA-GS
in this study.
161

row of indentations was completed (that is, following every 27 indentations).
In addition to the elasticity measurements, the hardness and fracture toughness of the HA
specimens were measured in this study in order to compare the detailed mechanical
properties of our HA specimens (HA-391 and HA-513) to HA data from the literature.
Both the hardness, H, and fracture toughness, KC, evaluations of the HA specimens
were calculated for the first 20 indentations on each of the two indented HA specimens in
order to compare the H and KC values from this study to H and KC data in the literature for
HA specimens without extensive microcrack damage.
The hardness, H, of the HA specimens, was calculated from the relationship [Wachtman
2009]
H=

1.8544P

(5)

(2a ) 2

where P is the applied load and 2a is the diagonal length of the indentation impression.
Also, the fracture toughness of the HA specimens was estimated via the Vickers indentation
method using equation (6) [Wachtman 2009]
1/ 2

E
KC =   
H

F
c

(6)

3/ 2

where E is the Young’s modulus of the HA specimens, which was determined by RUS in this
study, H is the hardness of the specimens calculated using equation (5). F is the applied load
and c is half of the radial crack length. The calibration constant  was set to 0.016, according
to Anstis [Ansis 1981].
3. Results and Discussion
162

3.1 Microstructure
The volume fraction porosity, P, for the sintered HA specimens was approximately 0.06
as determined from the measured mass and volume of the specimen. SEM examination
revealed quasi-spherical pores, approximately 1 to 3 microns in diameter located at triple
points, grain boundaries and within the grains (Figure 6.3). The grains were approximately
equiaxed with a mean grain size of 8.6 m, as determined from the linear intercept method
(Figure 6.3).
The indentation impression diagonal length 2a and the radial crack length 2c were
measured for the first 20 indentations on each specimen (Figure 6.1b). For specimen HA-391
the mean indentation diagonal length, <2a> was 66.2 ± 4.6 m the mean radial crack length,
<2c> was 253.8 ± 17.0 m. For HA-513, <2a> was 64.2 ± 1.9 m and <2c> was 252.2 ±
31.0 m.
3.2 Rationale for comparing this study’s HA indentation results with indentation results
for alumina from the literature
In this study, we chose to compare our HA data to the alumina data by Case and Kim
[Kim 1993, Case 1993, Kim 1993b]. The reason for this important comparison is that
alumina and HA are both widely used brittle materials although their mechanical properties
are quite different. For the commercial alumina specimens used by Case and Kim [Kim
1993], the hardness, H, Young’s modulus, E, and toughness, KC, for undamaged specimens
were 12.9 GPa, 320 to 340 GPa and 3.1 MPa•m

1/2

, respectively. For the HA samples in this

study, we measured H ~4.2 GPa, E ~100 GPa and KC ~0.5 MPa•m

163

1/2

. In addition to the

differences in material properties, the indentation load and the number density of microcracks
in alumina (49 N to 196 N) [Kim 1993] and HA (9.8 N) were different. Thus, the test of
theories for the microcrack-induced changes in Young’s modulus [Budiansky 1976, Salganik
1974, Hoenig 1979, Laws 1987] and Poisson’s ratio [Walsh 1965] becomes more rigorous if
we compare the theories’ predictions with experimental results from two materials with
dissimilar mechanical properties, indented using dissimilar loads.
3.3 Pre-indentation values of Young’s modulus and Poisson’s ratio
In order to assure that the HA specimens in this study have mechanical properties
consistent with typical HA specimens in the literature, we compared our HA specimens to
HA specimens in the literature in terms of the Young’s modulus and Poisson’s ratio (this
section) and hardness, fracture toughness (Section 3.4).
In this study, the Young's modulus determined by RUS for unindented HA-391 and HA513 were 103.5 ± 0.3 GPa and 104.1 ± 0.3 GPa. In order to make comparisons among
specimens with differing volume fraction porosities P, we can use the empirical porosity
dependence of Young’s modulus as given by [Ren 2009, Spriggs 1961]
E = ED exp (-bEP)

(7)

where E is the Young’s modulus at a volume fraction porosity P, ED is the value of E for
theoretically dense materials (P = 0) and bE is a material-dependent constant that describes
the rate of decrease of E with increasing P. Using equation (7) and the values ED =128 GPa
and bE = 3.5 obtained by Ren [Ren 2009] for a combined data set including HA specimens
from studies by Ren [Ren 2009], Liu [Liu 1998], Arita [Arita 1995] and He [He 2008], as
164

well as the aggregate Young’s modulus [Landolt 1979], the estimated Young's modulus for
the HA specimens included in this study (P = 0.06) is 103.7 GPa. Thus, the measured
Young's modulus for unindented HA specimens in this study is consistent with the modulus
value of 103.7 GPa reported in the literature [Ren 2009].
The Poisson’s ratio for unindented HA-391 and HA- 513 were 0.268 ± 0.003 and 0.265
± 0.003, respectively. The Poisson’s ratio of HA (P = 0.06) measured by Ren [Ren 2009] was
approximately 0.27, which is also in agreement with the data from this study.
3.4 Comparing initial values of hardness and fracture toughness to literature values for
HA
We also wish to compare the Vickers hardness, H, and the fracture toughness, KC, from
this study to those found in the literature. The H values of specimens HA-391 and HA-513 (P
= 0.06 for both specimens) were 4.15 ± 0.45 GPa and 4.41 ± 0.21 GPa, respectively. In the
literature, we used studies by Hoepfner [Hoepfner 2004] and Dey [Dey 2011] who measured
the porosity-dependent hardness of polycrystalline HA for comparison. Hoepfner [Hoepfner
2004] measured H for cold die pressed and sintered HA with porosities of 0.02 to 0.31. Dey
[Dey 2011] measured the hardness using nanoindentation on porous microplasma-sprayed
HA coatings with porosities from 0.05 to 0.16. Thus the range of porosity in Hoepfner’s
study [Hoepfner 2004] and Dey’s study [Dey 2011] brackets the P value measured for
specimens in this study.
Both Hoepfner [Hoepfner 2004] and Dey [Dey 2011] did a least-squares fit of their H
versus P results to the following relationship

165

H = HD exp (-bHP)

(8)

where HD is the value for theoretically dense materials (P = 0) and bH is a
material-dependent constant. A least-squares fit by Dey et al. of their H versus P data [Dey
2011] to equation (8) yielded values of HD = 5.92 and bH = 5.9. Hoepfner and Case’s
least-squares fit of their H versus P data to equation (8) [Hoepfner 2004] gave least-squares
parameters of HD = 6.00 and bH = 6.03.
From equation (8), for P = 0.06 (which corresponds to the P values of our indented
specimens), using Hoepfner’s [Hoepfner 2004] least-squares fit parameters gives H = 4.18
GPa. Using Dey’s [Dey 2011] parameters gives H = 4.16 GPa. In addition, Hoepfner
[Hoepfner 2004] obtained an H value of 4.15 ± 0.25 GPa from 7 to 10 indentations made on
the HA specimen at P = 0.06 with a mean grain size of 5.9 m. In this study, the porosity of
the HA specimens was 0.06 with a mean grain size of 8.6 m. Thus the microstructure of the
HA specimens and mean H values measured by Vickers indentation (4.15 ± 0.45 GPa and
4.41 ± 0.21 GPa for this study’s specimens) agree very well with the results by Hoepfner
[Hoepfner 2004]. Also, Dey’s microplasma-sprayed HA coatings had P ranging from 0.05 to
0.16, however, the grain size was not specified but instead only an average splat size of 63 ±
21m [Dey 2011]. Nevertheless, the hardness values of Dey’s specimens and HA specimens
in this study agreed quite well.
The measured fracture toughness of specimens HA-391 and HA-513 was 0.55 ± 0.03
MPa•m

1/2

and 0.54 ± 0.05 MPa•m

1/2

, respectively, which agrees well with the fracture

toughness values reported by both Wang [Wang 2009] and Dey [Dey 2011]. Wang measured
166

toughness of bulk HA specimens with P = 0.01 and grain size of 0.73 m using the Vickers
indentation method, yielding a toughness value of 0.61 ± 0.04 MPa•m

1/2

. With

nanoindentation loads of 100 mN, 300 mN, 500 mN, 700 mN and 1000 mN, Dey [Dey 2011]
measured the fracture toughness of the same porous microplasma-sprayed HA coatings that
were used for Dey’s hardness measurements. As the applied indentation load increased, the
fracture toughness decreased slightly from ~0.59 ± 0.02 MPa•m

1/2

to 0.50 ± 0.06 MPa•m

1/2

[Dey 2011].
Thus, the experimentally determined mechanical properties of hardness, fracture
toughness, Young’s modulus and Poisson’s ratio of the HA specimens included in this study
agree well with the corresponding experimental values for HA from the literature [Ren 2009,
Hoepfner 2004, Dey 2011, Wang 2009].
3.5 Microcracking - elasticity relationship
3.5.1 Models for microcracking-Young’s modulus relationship for aligned surface
limited cracks
In a series of studies [Kim 1993, Case 1993, Kim 1993b], Case et al. used arrays of
Vickers indentation-induced microcracks to show that microcracks decrease the effective
Young’s modulus of materials, and experimental data were acquired to test the
microcrack-modulus relationships based on theory developed by Budiansky [Budiansky
1976], Salganik [Salganik 1974], Hoenig [Hoenig 1979] and Laws [Laws 1987]. However, in
Kim and Case’s studies [Kim 1993, Case 1993, Kim 1993b], experiments were conducted on
aligned and surface-limited cracks in contrast to the 3-dimensional, randomly distributed

167

cracks assumed in theoretical work [Budiansky 1976, Salganik 1974, Hoenig 1979, Laws
1987].
Using Vickers indentation to induce surface microcracks on polycrystalline alumina
allowed the direct measurement of the crack length, c, and the crack number density, N, [Kim
1993, Case 1993, Kim 1993b] and provided the opportunity for the first direct test of the
microcracking-modulus theories [Budiansky 1976, Salganik 1974, Hoenig 1979, Laws 1987].
However, by studying the microcracks induced by Vickers indentation on the specimen
surface, Kim and Case [Kim 1993, Case 1993, Kim 1993b] modified the
microcracking-modulus theories so to apply them to surface-limited microcracks.
The effect of surface-limited microcracks on the overall Young’s modulus change was
evaluated using a rule of mixtures (ROM) model [Case 1993] and a dynamic beam vibration
(DBV) model [Case 1993]. The details of these models are discussed in Appendix A.
The magnitude of the modulus decrement in both models [Case 1993] is a function of
the size, geometry and number density of microcracks, such that

E
 SN  N
E0

(9)

where ΔE is the modulus change, E0 is the modulus of unindented specimen (no microcrack
damage), N is the volumetric number density of microcracks, and SN is the product of a crack
orientation and a crack geometry parameter [Kim 1993b].
In this study, the crack geometries (Appendix A) used to evaluate the effect of Vickers
indentation induced microcracks on the Young’s modulus include the unmodified/modified

168

(a)

(b)

Figure 6.4. Side view of the indentation-crack systems for (a) a modified half ellipse crack
geometry [Kim 1993b] and (b) a modified slit crack geometry [Kim 1993b].

169

half ellipse [Kim 1993b] (Figure 6.4a) and the unmodified/modified slit [Kim 1993b] (Figure
6.4b). In contrast to the unmodified geometries [Kim 1993b], for the modified geometries
[Kim 1993b], the surface area of the deformation zone was subtracted and the perimeter of
the deformation zone was added, as was discussed by Kim and Case [Kim 1993b].
Calculation of SN for both ROM and DBV models that include a number of specific crack
geometries [Kim 1993b] are given in Appendix A.
Case and Kim [Kim 1993, Case 1993, Kim 1993b] did not report changes in Poisson’s
ratio as a function of microcrack damage. However, in this study we compared the
measurements of Poisson’s ratio as a function of N to Walsh’s equation (equation (4)) [Walsh
1965]. Although the expression by Walsh was derived for randomly oriented cracks, equation
(4) still is consistent with this study’s observed linear decrease in Poisson’s ratio with
microcracking damage (Figure 6.6, Section 3.5.3). Thus, we only analyzed the changes in
Poisson’s ratio as a function of N in this study since no expression for changes in  for
aligned cracks in the available literature.
3.5.2 Young’s modulus – microcracking relationship for HA: experimental results and
model predictions
In this study, a total number of 391 and 513 Vickers indentations were placed onto two
HA bars, respectively, so that the number density, size and spatial location of the induced
surface-limited microcracks was well controlled. For HA-391, as the number of indentations
increased from zero to 391, the Young's modulus decreased, from 103.5 GPa for unindented
specimen to 101.4 GPa. For HA-513, E decreased from 104.1 GPa to 102.4 GPa. The

170

Table 6.1. The experimental values of SN for HA319 and HA513 indented at 9.8 N in this
study and Al2O3 (single and double sides indented at 196 N) [Kim 1993] compared to the SN
values predicted by each model [Kim 1993]. All SN values are in unit mm3.
Experiment/models

HA319

Al2O3

HA513

(double)
Experiment

1.05 x 10

DBV modified half ellipse

0.99 x 10

DBV modified slit

1.27 x 10

DBV unmodified half ellipse

1.40 x 10

DBV unmodified slit

1.67 x 10

ROM modified half ellipse

0.39 x 10

ROM modified slit

0.49 x 10

ROM unmodified half ellipse

0.54 x 10

ROM unmodified slit

0.65 x 10

171

-2
-2
-2
-2
-2
-2
-2
-2
-2

-2

1.16 x 10

-2

1.13 x 10

-2

1.17 x 10

-2

1.31 x 10

-2

1.56 x 10

-2

0.43 x 10

-2

0.45 x 10

-2

0.50 x 10

-2

0.60 x 10

1.773 x 10
1.465 x 10
1.906 x 10
2.304 x 10
2.855 x 10
0.815 x 10

-2
-2
-2
-2
-2
-2

-2

1.06 x 10

1.283 x 10
1.590 x 10

-2
-2

Relative change in Young's modulus
E/E0

0.030

Unmodified slit DBV

(a) HA391

Unmodified half ellipse DBV

0.025

Modified slit DBV

0.020

Experiment
Modified half ellipse DBV

0.015

Unmodified slit ROM

0.010
Modified slit ROM

0.005
0.000
0.0

Modified half ellipse ROM
Unmodified half ellipse ROM
0.4

0.8

1.2

1.6

2.0

Volumetric Crack Number Density N(mm

Relative change in Young's modulus
E/E0

0.025

(b) HA513

)

-3

Unmodified slit DBV
Unmodified half ellipse DBV

0.020

Modified slit DBV
Experiment

0.015
Modified half ellipse DBV

0.010

Unmodified slit ROM

0.005

Modified slit ROM
Modified half ellipse ROM
Unmodified half ellipse ROM

0.000
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Volumetric Crack Number Density N(mm

-3

)

Figure 6.5. Relative Young’s modulus change as a function of volumetric crack number
density for (a) HA-391 and (b) HA-513. Each solid line represents the prediction given by
each model with particular crack geometry [Kim 1993b]. The dashed line represents a
least-squares fit to equation (9) for the HA data.
172

effective Young's modulus decrement, ΔE/E0, was plotted as a function of number density of
microcracks, N (Figures 6.5a and 6.5b). The load and the crack orientation of Vickers
indentation were fixed, so SN in equation (9) was also fixed since SN is the product of a crack
orientation and a crack geometry parameter. As expected from equation (9), ΔE/E0 was a
3

linear function of N (Figures 6.5a and 6.5b), with SN = 0.0105 mm (Table 6.1) and the
2

3

coefficient of determination R = 0.968 for HA-391. For HA-513, SN = 0.0116 mm (Table
2

6.1) and R = 0.951. Each solid line in Figures 6.5a and 6.5b represents the prediction given
by each model with particular crack geometry. Since each parameter in each model is known,
each solid line is plotted using a fixed SN value (not from least-squares fitting).
Among all the combinations between models and crack geometries [Kim 1993b], the
best correlations with experimental data for specimen HA-391were the DBV model with
3

modified slit crack geometry (SN = 0.0127 mm , Table 6.1) and the DBV model with
3

modified half ellipse crack geometry (SN = 0.0099 mm , Table 6.1, Figure 6.5a).
Also, as was the case for specimen HA-391, the experimental data for specimen
3

HA-513 correlated best with the DBV models with the modified slit (SN = 0.0117 mm ,
3

Table 6.1) and the modified half ellipse (SN = 0.0113 mm , Table 6.1) crack geometries
(Figure 6.5b).
The Young’s modulus versus microcrack number density data in this study was
bracketed by the predictions for the modified slit and half ellipse crack geometries for the
DBV models (Figures 6.5a and 6.5b), implying that the microcrack geometry in this study
was possibly a slit shape with truncated round corners.
173

HA-513
HA-391

0.08

/0

0.06
0.04
0.02
0.00
0.0
0.4
0.8
1.2
1.6
2.0
-3)
Volumetric Crack Number Density N(mm

Figure 6.6. Relative Poisson’s ratio change vs. volumetric crack number density for HA-391
and HA-513. The dashed line represents a least-squares fit to equation (4) for the HA data.

HA-513
HA-391

0.020

E/E0

0.015
0.010
0.005
0.000
0.00

0.02

0.04

0.06

0.08

/0

Figure 6.7. Relative Young’s modulus change vs. relative Poisson’s ratio change for HA-391
and HA-513. The dashed lines represent least-squares fits to equation (10) respectively.

174

In this study, the Young’s modulus was measured in a dynamic ultrasound resonance
method, so it is reasonable that the DBV models [Kim 1993b] agreed better with the
experimental data compared to the ROM models [Kim 1993b], since DBV models [Case
1993] are based on the dynamic beam vibration equation and the ROM represent a static
mixing approximation.
Besides being a function of microcrack damage, E is also a function of porosity [Ren
2009, Ni 2009, Rice 1998, Jang 2005] and composition [Diaz 2008, Ren 2007, Krstic 2008].
However in this study, both the composition and the porosity of the specimens are fixed.
Also, E is generally considered to be independent of grain size [Rice 1998, Ren 2009b],
except for grain sizes smaller than about 100 nm [Zhou 2008], which is not the case for the
materials included in this study and from the literature.
3.5.3 Poisson’s ratio– microcracking relationship for HA
The Poisson’s ratio values for the unindented HA-391 and HA- 513 were 0.268 ± 0.003
and 0.265 ± 0.003, respectively. After the introduction of 391 and 513 indentations per
specimen, the Poisson’s ratio for HA-391 and HA- 513 dropped to 0.245 ± 0.006 and 0.246 ±
0.008, respectively.
The effective Poisson’s ratio decrement, Δ/0, for both HA specimens were plotted as a
function of N (Figure 6.6), similar to the ΔE/E0 versus N plots discussed in earlier (Figures
6.5a and 6.5b). As was predicted by Walsh [Walsh 1965] (equation (4)), a linear relationship
between Poisson’s ratio and crack number density was observed for the HA specimens in this

175

2

2

study, with coefficients of determination given by R = 0.972 for HA-391 and R = 0.908 for
HA-513. The least-squares fit of Δ/0 for aligned cracks to equation (4) gave the slope S =
3

3

3

0.044 mm for HA-391 and S = 0.044 mm for HA-513. Using S = 4πc /3 (as proposed by
Walsh [Walsh 1965] for randomly oriented cracks) the calculated value of S in equation (4)
3

is 0.009 mm .
3.5.4 Relative change in Young’s modulus as a function of the relative change in the
Poisson’s ratio
In Case’s paper [Case 1984], the four microcracking – elasticity models considered in
this study [Budiansky 1976, Salganik 1974, Hasselman 1979, Walsh 1965] were expressed in
terms of E and . Walsh [Walsh 1965] proposed a linear relationship between ΔE/E0 and
Δ/0. The other three studies [Budiansky 1976, Salganik 1974, Hasselman 1979] predicted
a nearly linear ΔE/E0 versus Δ/0 behavior.
Also, the relative Young’s modulus change (equation (9)) and the relative Poisson’s ratio
change (equation (4)) for the HA specimens demonstrated similar linear trends when plotted
as a function of volume crack number density, N (Figures 6.7). Thus, combining equations (4)
and (9) gives

E S N 

E 0 S  0

(10)

which supports Walsh’s [Walsh 1965] statement that ΔE/E0 changes linearly with Δ/0.
The linear relationship between ΔE/E0 and Δ/0 proposed by Walsh [Walsh 1965] had
a value of slope SN/S = 1 (equation (10)). The nearly linear ΔE/E0 versus Δ/0 behavior
proposed by the other three studies [Budiansky 1976, Salganik 1974, Hasselman 1979] had
176

SN/S (equation (10)) close to 1, especially for low Δ/0.
The nearly linear relationship between ΔE/E0 and Δ/0 is shown for specimens
2

HA-391 and HA-513 with R of 0.979 and 0.924, respectively, as predicted by the literature
[Budiansky 1976, Salganik 1974, Hasselman 1979, Walsh 1965]. The least-squares fit of
ΔE/E0versus Δ/0 to equation (10) yielded slopes SN/S of 0.23 and 0.22 for HA-391 and
HA-513, respectively (Figure 6.7).
In order to compare to slopes of ΔE/E0 versus Δ/0for randomly oriented microcracks
induced by thermal expansion anisotropy or phase transformation, Case [Case 1984]
examined five sets of experimental data from the literature. The slopes of ΔE/E0 versus Δ/0
for the data presented by Case [Case 1984] were extracted using the software Datathief (B.
Tummers, DataThief III. 2006, http://datathief.org). The slopes are 0.46 ± 0.38 for HfO2
[Dole 1978], 0.15 ± 0.42 for Gd2O3 [Case 1981], 1.53 ± 0.23 for Nb2O5 [Manning 1973],
0.19 ± 0.13 for Eu2O3 [Suchomel 1976] and 1.26 ± 0.21 for YMgxCr1-xO3 [Case 1984]. The
large uncertainty is likely due to the large scatter in the Poisson’s ratio measurements. Thus,
while the data presented in the Case study [Case 1984] from several researchers [Case 1984,
Dole 1978, Case 1981, Manning 1973, Suchomel 1976] represents specimens with randomly
oriented cracks, the scattered in the slope values from 0.15 to 1.53 precludes a clear
comparison with the predicted slope for ΔE/E0 versus Δ/0.
3.6 Comparison of Young’s modulus – microcracking results for HA (this study) and
alumina [Kim 1993]
In the study by Case and Kim [Kim 1993], a 46 x 7 grid pattern of Vickers indentations

177

were placed on single surface or both surfaces of polycrystalline alumina bars ( ~ 69 x 12 x
1mm) using loads of 49, 98 and 196 N. The Young’s modulus was measured using the
dynamic sonic resonance technique [Kim 1993]. The mean crack lengths ranged from 227.6
± 20.3 micron (49N), 370.1 ± 35.1 micron (98N) to 664.8 ± 31.9 micron (196N). The
effective Young’s modulus versus the microcrack number density of the indented alumina
specimen showed a linear relationship as stated in equation (9).
HA and alumina are quite different in mechanical properties including hardness, Young’s
modulus and fracture toughness (Section 3.2). Also, the load used for indentation and number
density of microcracks in alumina [Kim 1993] and HA are different (The range of loads used
for Vickers indentation were 49 N to 196 N for alumina [Kim 1993, Kim 1993b, Case 1993]
and 9.8 N for the HA in this study). However, the Young’s modulus - microcrack relationship
for both alumina and HA showed the same linear trend of decreasing Young’s modulus with
-2

3

increasing crack number density with different SN (equation (9), SN = 1.05 x 10 mm for
HA-391, SN = 1.16 x 10

-2

3

-2

mm for HA-513, SN = 1.61 x 10
-2

3

mm for alumina indented on

3

single side and SN = 1.77 x 10 mm for alumina indented on both sides [Kim 1993] (Table
6.1)). The difference in slopes for the relative modulus change versus N is a function of load
and also due to the large difference in toughness for the two materials (Section 3.2). The
alumina results [Kim 1993, Kim 1993b, Case 1993] and the HA results (this study) lead to
the question “how do we compare the alumina and HA results since the physical properties
are much different for the two materials?” We will address this question by plotting all of the

178

alumina data and the HA data together on a single plot of ΔE/E0 versus (Figures 6.8a and
6.8b), such that

E
 SE  
E0

(11)

Information on calculation of  and theoretical prediction of the slopes, Sp for different crack
geometries is given in Appendix B.
For each HA specimen, the predicted slopes, Sp, of the Young’s modulus change, ΔE/E0,
versus crack damage parameter, , by both the DBV models with modified half ellipse and
modified slit crack geometries, are similar to the SE values obtained by a least-squares fit of
ΔE/E0versus (Tables 6.2 and 6.3)The slopes SE (Figures 6.8a and 6.8b) obtained from the
least-squares fit are within 10% of the predicted values of slopes, Sp (Tables 6.2 and 6.3). For
alumina specimens with microcracks induced by a 49 N Vickers indentation load [Kim 1993],
the values of SE in equation (11) are quite close to the predicted values, Sp using DBV
models with modified half ellipse crack geometry (Table 6.2); while for DBV models with
modified slit crack geometry, SE are within 20% of Sp (Table 6.3). Thus, both the DBV
models with modified half ellipse and modified slit crack geometries predict reasonably well
the Young’s modulus change as a function of microcrack damage state, with DBV models
with modified half ellipse crack geometry being a better model in this case.
3.7 The role of r in the modulus change as a function of microcrack damage
For alumina specimens indented using loads of 98 N and 196 N [Kim 1993], the slopes,
Sp calculated from both the DBV models with modified half ellipse and modified slit crack

179

Table 6.2. For DBV model with modified half ellipse crack geometry, ratio of the depths of
the microcracked layer over the unmicrocracked layer, r, the number of points fitted, NP, the
2

least-squares fit values of slope, SE and the R value for the fit, the least-squares fit values of
2

slope through the zero point, SE0 and the R value for the fit, and the theoretically predicted
value of slope, SP.
Specimens

r

NP

SE

R

HA - 391
HA - 513

0.086
0.075

10
20

1.15 ± 0.07
1.03 ± 0.04

Al2O3 49N

0.13

7

0.13

2

2

SE0

R

SP

0.97
0.98

1.18 ± 0.04
1.17 ± 0.03

0.97
0.95

1.005
0.905

1.27 ± 0.12

0.96

1.14 ± 0.07

0.95

1.372

8

1.49 ± 0.10

0.97

1.62 ± 0.06

0.96

1.372

0.23

8

0.87 ± 0.04

0.99

0.85 ± 0.02

0.99

1.905

0.23

5

0.89 ± 0.04

0.99

0.88 ± 0.02

0.99

1.905

0.47

10

1.41 ± 0.05

0.99

1.43 ± 0.03

0.99

2.426

0.47

6

1.38 ± 0.02

0.99

1.39 ± 0.01

0.99

2.426

both sides
Al2O3 49N
single side
Al2O3 98N
both sides
Al2O3 98N
single side
Al2O3 196N
both sides
Al2O3 196N
single side

180

Table 6.3. For DBV model with modified slit crack geometry, ratio of the depths of the
microcracked layer over the unmicrocracked layer, r, the number of points fitted, NP, the
2

least-squares fit values of slope, SE and the R value for the fit, the least-squares fit values of
2

slope through the zero point, SE0 and the R value for the fit, and the theoretically predicted
value of slope, SP.
Specimens

r

NP

SE

R

HA - 391
HA - 513

0.086
0.075

10
20

0.84 ± 0.05
0.75 ± 0.03

Al2O3 49N

0.13

7

0.13

2

2

SE0

R

SP

0.97
0.98

0.86 ± 0.03
0.86 ± 0.02

0.97
0.95

0.924
0.832

0.90 ± 0.08

0.96

0.81 ± 0.05

0.95

1.266

8

1.05 ± 0.07

0.97

1.14 ± 0.04

0.96

1.266

0.23

8

0.62 ± 0.03

0.99

0.61 ± 0.02

0.99

1.758

0.23

5

0.64 ± 0.03

0.99

0.63 ± 0.01

0.99

1.758

0.47

10

1.03 ± 0.04

0.99

1.04 ± 0.02

0.99

2.239

0.47

6

1.01 ± 0.01

0.99

1.02 ± 0.01

0.99

2.239

both sides
Al2O3 49N
single side
Al2O3 98N
both sides
Al2O3 98N
single side
Al2O3 196N
both sides
Al2O3 196N
single side

181

geometries are almost twice the slope values, SE obtained from the least-squares fit of the
experimental data, respectively (Tables 6.2 and 6.3).
The mismatch between the models and the experimental data may be due to the limited
thickness of the alumina specimens. For example, in developing a Vickers indentation
technique for direct measurement of fracture toughness for bar-shaped specimens of brittle
materials, Anstis et al. [Anstis 1981] noted that “to avoid interactions with the lower free
surface, the crack dimensions were not allowed to exceed one-tenth of the thickness”. In this
study and in the work by Kim and Case [Kim 1993, Case 1993, Kim 1993b], the ratio of
crack depth to specimen thickness was expressed in terms of r, where r = depth of the
microcracked layer/depth of the unmicrocracked layer L/(t-L).
If we express the Anstis criteria in terms of r, then r is less than 0.11. In Tables 6.2 and
6.3, for indented HA specimens HA-391 and HA-513, also for the Case and Kim’s data, the
experimental value of r very nearly satisfies the Anstis criteria [Anstis 1981] for indentation
cracks made with Vickers indentations loads of 49 N (Tables 6.2 and 6.3, Figures 6.9a and
6.9b). For alumina specimens indented under loads of 98 N and 196 N [Kim 1993], the r
values are 0.23 and 0.47, respectively. Thus, the values of r for the of 98 N and 196 N
indentations in alumina [Kim 1993] represent relative crack depth/specimen thickness ratios
that are far higher than the upper bound of 0.11 suggested by Anstis [Anstis 1981] and
therefore the stress fields indentation crack lengths are likely significantly perturbed by
proximity of the free specimen surface to the penetrating indentation crack.
4. Conclusions
182

For Young's modulus and Poisson’s ratio changes due to surface-limited microcrack
damage in HA (as a function of the volumetric crack number density, N), this study confirms
the theoretical work by Case and Kim and Walsh. Based on the ΔE/E0 versus N results
presented in this study, we recommend that the Case and Kim’s DBV model with modified
half ellipse crack geometry be used for analyzing the relative change in Young’s modulus as a
function of N since this model provides the best fit to the experimental data. In addition, this
study allows one to predict the changes in normalized Young's modulus versus normalized
Poisson's ratio.
In terms of the Young's modulus change as a function of crack damage parameter, the
experimental work on HA included this study and the Case and Kim’s alumina work from the
literature demonstrates the linear ΔE/E0 versus  trend predicted from earlier theoretical work
by Laws and Brockenbough and Budiansky and O'Connell, as modified for surface-limited
damage by the Case and Kim model. Since alumina and HA have quite different mechanical
properties and were indented using considerably different loads, this result supports the
conclusion that the ΔE/E0 versus  trend may apply to a wide variety of brittle materials and
loading conditions.
5. Acknowledgements
The authors acknowledge the financial support of the National Science Foundation,
Division of Materials Research Grant DMR-0706449.

183

0.020

(a) Modified half ellipse

E/E0

0.015
HA-391
HA-513
Alumina 49N both sides
Alumina 49N single side
Alumina 98N both sides
Alumina 98N single side
Alumina 196N both sides
Alumina 196N single side

0.010

0.005

0.000
0.000

0.005

0.010



0.015

0.020

(b) Modified slit

0.020

E/E0

0.015
HA-391
HA-513
Alumina 49N both sides
Alumina 49N single side
Alumina 98N both sides
Alumina 98N single side
Alumina 196N both sides
Alumina 196N single side

0.010

0.005

0.000
0.000

0.005

0.010

0.015

0.020

0.025

0.030



Figure 6.8. Relative Young’s modulus change vs. the microcrack damage parameter for
HA-391, HA-513 and alumina specimens from Kim [1993] using (a) DBV model with
modified half ellipse crack geometry and (b) DBV model with modified slit crack geometry.

184

0.020

(a) Modified half ellipse

E/E0

0.015

0.010
HA-391
HA-513
Alumina 49N both sides
Alumina 49N single side

0.005

0.000
0.000

0.020

0.005

0.010



0.015

0.020

(b) Modified slit

E/E0

0.015

0.010
HA-391
HA-513
Alumina 49N both sides
Alumina 49N single side

0.005

0.000
0.000

0.005

0.010

0.015



0.020

0.025

0.030

Figure 6.9. Relative Young’s modulus change vs. microcrack damage parameter for HA-391,
HA-513 and alumina specimen indented under a 49 N load from Kim [1993] using (a) DBV
model with modified half ellipse geometry and (b) DBV model with modified slit geometry.
185

APPENDICES

186

APPENDIX A

EQUATIONS TO CALCULATE SN (THE PRODUCT OF A CRACK ORIENTATION
AND A CRACK GEOMETRY PARAMETER IN EQUATION (9)), FOR EACH
MODULUS/MICROCRACK MODEL WITH A PARTICULAR CRACK GEOMETRY
EMPLOYED IN THIS STUDY
In the ROM model [Case 1983], the overall modulus was taken as the sum modulus of
the undamaged layer and the damaged layer. The strain in the undamaged layer was set equal
to the strain in the damaged layer. In the DBV model [Case 1983], the Bernoulli – Euler
beam equation was applied with the boundary conditions that both the bending moments and
shear force were zero at the beam ends.
In the ROM model [Case 1983], SN (equation (9)) was calculated using equation (A1)

2  A2 
SN  f 
P

(A1)

In the DBV model [Case 1983], SN was expressed as equation (A2)

SN 

r2  3
(r  1) 2

f 

2  A2 
P

(A2)
2

In both equations (A1) and (A2), f is the microcrack orientation parameter, <A > is the
mean of the square of the crack surface area and <P> is the mean crack perimeter. In
equation (A2), r is the ratio between the half the crack length and the specimen thickness
[Case 1983].
Equations to calculate SN for different crack geometries in the ROM model are
summarized in equation (A3) – (A6).
187

ROM model with unmodified slit crack geometry

 2 (1  u 0 2 ) 2(2c 2 ) 2
SN 
2
4c

(A3)

ROM model with modified slit crack geometry

SN 

 2 (1  u 0 2 ) 2(2c 2  a 2 / 2) 2
2
(4c  a )

(A4)

ROM model with unmodified half ellipse crack geometry

SN 

16(1  u 0 2 ) 2(c 2 / 2) 2
3
2c

(A5)

ROM model with modified half ellipse crack geometry

SN 

16(1  u 0 2 ) 2(c 2 / 2  a 2 / 2) 2
3
(c  a )

(A6)

The value of SN for DBV model is calculated via multiplying each corresponding SN value
for ROM model by

r2  3
(r  1) 2

, where r is the ratio between the half the crack length and the

specimen thickness.

188

APPENDIX B

EQUATIONS USED TO PREDICT THE SLOPES, SP, OF THE ΔE/E0 VERSUS 
PLOTS FOR HA SPECIMENS IN THIS STUDY AND ALUMINA [Kim 1993]
The relative change in Young’s modulus, ΔE/E0 in DBV model (from equations (11)
and (28) in Case’s paper [Case 1983]) is given by
ΔE/E0 =

r2  3

r
f G Nl
(r  1) 2 r  1

(B1)

where f is the microcrack orientation parameter, for aligned microcracks in penny, half penny,
ellipse and half ellipse crack shapes (from Table II in Kim and Case’s paper [Kim 1993b]), f
is defined as
2

f = 16(1 - 0 )/3

(B2)

for aligned slit crack shapes (from Table II in Kim and Case’s paper [Kim 1993b])
2

2

f = π (1 - 0 )/2

(B3)

G is the geometry parameter (from equation (4) in Kim and Case’s paper [Kim 1993b])
G=

2  A2 
P

(B4)

For DBV model with modified half ellipse crack geometry (from Table II in Kim and
Case’s paper [Kim 1993b])
G=

(a  c)( c  a ) 2
2

(B5)

For the DBV model with a modified slit crack geometry (from Table II in Kim and Case

189

[Kim 1993b])
G=

2(4c 4  2a 2 c 2   2 a 4 / 4)

(B6)

4c   2 a

Nl is the microcrack number density in the microcracked surface layer
Nl = the number of microcracks/(specimen length x specimen width x depth of the
microcrack damaged layer)
r = depth of the microcracked layer/depth of the unmicrocracked layer
In the plot of ΔE/E0 versus for HA specimens in this study and alumina specimens
[Kim 1993], is defined as GNl. The theoretically predicted slope, Sp of ΔE/E0 versus is
expressed as
Sp =

r2  3

r
f
(r  1) 2 r  1

(B7)

190

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191

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196

CHAPTER 7

WEIBULL MODULUS AND FRACTURE STRENGTH OF HIGHLY POROUS
HYDROXYAPATITE
1

1

2

X. Fan , E. D. Case , I. Gheorghita and M. J. Baumann

1

1

Department of Chemical Engineering and Materials Science, Michigan State

University, East Lansing, MI, 48824
2

Dr. Phillips High School, Orlando, FL 32819

Published in: J Mechanical Behavior of Biomedical Materials, 20:283-295, 2013.

Abstract
Porous hydroxyapatite (HA) is used in a variety of applications including biomedical
materials such as engineered bone materials and microbe filters. Despite the utility of the
Weibull modulus, m, as a gauge of the mechanical reliability of brittle solids, there have been
very few studies of m for porous HA. A recent study of porous HA that included the current
authors (Fan, X., Case, E. D., Ren, F., Shu, Y., Baumann, M. J., 2012. J. Mech. Behav.
Biomed. 8, 21-36) showed increases in m for porosity, P, approaching PG, the porosity of the
green (unfired) specimen. In this paper, eighteen groups of highly porous HA specimens (12
groups fabricated in this study and 6 groups from Fan 2012 Part I) were analyzed with P
values from 0.59 to 0.62, where PG = 0.62. The partially sintered HA specimens were
fractured in biaxial flexure using a ring-on-ring test fixture. The fracture strength decreased
monotonically with decreasing sintering temperature, Tsinter, from 4.8 MPa for specimens
197

sintered at 1025oC to 0.66 MPa for specimens sintered at 350oC. However, the Weibull
modulus remained surprisingly high, ranging from 6.6 to 15.5. In comparison, for HA
specimens with intermediate values of P, from about 0.1 to 0.55, the Weibull modulus tended
to be lower (ranging from about 4 to 11) than the highly porous specimens included in this
study.
1. Introduction
Porous hydroxyapatite (HA, Ca10(PO4)6(OH)2) is widely studied and used as a
candidate material for hard tissue replacement due to its similarity to the natural minerals
found in bones and teeth [Hench 1991]. HA can directly bond to bone, showing excellent
biocompatibility and osteoconductivity [Hench 1991]. To allow for bone in-growth and
circulation of body fluid and nutrients [Baumann 2007], the porosity of the artificial bone
scaffold is made to be equivalent to the natural cancellous bone, which has a relative porosity
of 0.3 to 0.9 [Wagoner Johnson and Herschler 2011].
In addition to its use as tissue engineering materials, porous HA is used in other
biomedical applications. For example, porous HA with a bimodal porosity having 0.22 < P <
0.5 has been studied as a filter to separate microorganisms from water and gas [Yang 2007a,
2007b]. Also, studies of porous bulk HA scaffolds used for sensor applications show that
changes in the localized surface plasmon resonance of embedded silver nanoparticles are
proportional to the concentration of absorbed protein [Ohtsuki 2010].
Additional health-related applications of HA include three dimensional HA structures
consisting of spherical macropores that capture cadmium and lead from waste water
198

[Srinivasan 2006]. In use as a CO sensor, the electrical conductivity of porous HA pellets
change with the adsorption level of flowing CO gas, enabling the monitoring of CO in the
environment [Mahabole 2005].
Porous brittle materials other than HA are also applied as both biomedical and
non-biomedical materials. For example, Zhou [2011] used nanoporous silica to deliver an
anticancer drug to cells in vitro. Layered porous alumina (with 0.35 < P < 0.6) is used in gas
separation and dehydration applications [Sommer and Melin 2005]. NiO/YSZ ceramics with
0.15 < P < 0.5 are being studied as anodes in solid oxide fuel cells [Liu 2011].
In addition to current and potential applications involving partially densified ceramic
powder compacts, there are a number of uses for extremely porous cellular materials with
porosities up to 0.9 or greater [Gibson 2010]. Porous HA scaffolds with P ~ 0.89 that were
functionalized with poly(lactic-co-glycolic acid) (PLGA) microspheres have been studied as
(i) in vivo and in vitro drug delivery platforms, and (ii) enhanced bone regeneration surfaces
[Son 2011]. Alumina and mullite substrates with P from 0.75 to 0.9 have been used to
support transition metal catalysts [Twigg and Richardson 2007].
This broad range of medical and non-medical applications for porous brittle materials
and for hydroxyapatite in particular means that it is critical to determine how the physical
properties of these materials, including the mechanical properties, are affected by porosity.
For example, many researchers have studied the porosity dependence of the elastic moduli of
brittle materials including alumina [Hardy and Green 1995, Nanjangud 1995, Ren 2009],
partially stabilized zirconia [Deng 2002], and hydroxyapatite [Fan 2012 Part II, Ren 2009].
199

Also, the fracture strength of porous brittle materials has been studied extensively, including
porous HA specimens under compressive, flexural and tensile loading for relative porosities
from roughly 0.03 to 0.85 [Akao 1981, He 2008, Le Huec 1995, Pramanik 2007, Yao 2005].
In terms of the mechanical properties of brittle materials, the Weibull modulus, m, is
very important, where m may be defined in terms of the two-parameter Weibull distribution

 1
ln ln
1 
f



  m ln  f  m ln  c



(1)

In equation (1), Λf is the cumulative probability of failure and m is the Weibull shape
parameter, or the Weibull modulus. Also, f is the fracture strength and C is the Weibull
scale parameter, known as the characteristic strength [Wachtman 2009]. The Weibull
modulus, m, is a measure of mechanical reliability, for example, a high Weibull modulus (say,
m > 20) indicates a very narrow flaw size population and hence a narrow range of fracture
strengths. Conversely, a low Weibull modulus (for example, m < 2) means that the spread in
flaw size and strength is very wide. The relationship between the Weibull modulus and
distribution of fracture strengths for a given group of specimens is discussed in more detail in
Appendix A, where it is important to note that for a nominally identical group of specimens,
the Weibull modulus (a single value per group) characterizes the spread in fracture strength.
In this study (Sections 2.1 and 2.2) and our previous study [Fan 2012 Part I], we varied
the sintering temperature while keeping the processing and testing parameters fixed in order
to study the effect of microstructural evolution of the powder compact (porosity and the
initial necking between adjacent grains). The vast majority of the literature on the Weibull

200

modulus of brittle materials is limited to relatively dense specimens (P > 0.1), however the
recent study by Fan [Fan 2012 Part I] did measure the Weibull modulus for HA specimens
with 0.08 ≤ P ≤ 0.62 (where P = 0.62 corresponds to the porosity of the HA powder
compacts in this study). One of the findings of the Fan study [Fan 2012 Part I] and a key
motivation for this study was that m increased for P > 0.55, a result which has not been
reported for any brittle material, perhaps in part because there also have been so few Weibull
modulus studies for P > 0.55.
In addition to microstructure (porosity and interparticle necking), the Weibull modulus
may be affected by powder processing techniques [Villora 2004], strength testing techniques
[Cordell 2009], loading rate [Andrews 2010], grain size and shape [Kim 1995] and surface
finish [Nakamura 2010]. However, in this study and our previous study [Fan 2012 Part I], we
varied the sintering temperature to achieve microstructural differences while keeping
constant the specimen preparation and testing parameters (Sections 2.1 and 2.2) in order to
study the Weibull modulus behaviour in the porosity regime for P > 0.55. In particular, we
concentrated on the P range between P = 0.59 and 0.62, which is also called the initial
sintering stage, as will be discussed next.

Also, in Section 3.2 we will review, discuss and

compare Weibull modulus values from the literature, this study and from our previous study
[Fan 2012 Part I].
In both our recent Weibull study [Fan 2012 Part I] and the current study, the evolution of
microstructure during sintering is of central importance. As powder compacts are heated,
their microstructure evolves via (i) an initial, (ii) an intermediate and then (iii) a final
201

sintering stage. The initial sintering stage extends from the unfired powder compacts to
partially sintered specimens with shrinkage of roughly 3% to 10 % [Coble 1961, German
1996, Barsoum 1997]. Thus the initial sintering stage corresponds to the porosity range of
interest (approximately 0.59 ≤ P ≤ 0.62) in this study. The initial sintering stage includes
two subdomains: first necks grow between adjacent powder particles without specimen
densification and then interparticle neck growth continues with densification [Coble 1961,
German 1996, Barsoum 1997]. During the first part of the initial sintering stage, although
densification does not occur, the elastic moduli and fracture strength increase as the neck size
between adjacent particles increases [Fan 2012 Part II]. By the intermediate stage, the
porosity has evolved into tubular pores along the grain edges and in the final sintering stage
the pores are quasi-spherical and located at grain boundaries or triple points [German 1996,
Barsoum 1997].
Given the results of our earlier study [Fan 2012 Part I], in order to further explore the
Weibull modulus behaviour of HA specimens in the initial sintering stage, we first examine
the Weibull modulus for 18 different groups of specimens (12 groups fabricated in this study
and 6 groups from Fan 2012 Part I) of highly porous HA specimens with P from 0.59 to 0.62
(Table 7.1a). Note that each group of specimens was processed at a different sintering
temperature (Table 7.1a), where each group yields a single value of Weibull modulus, since
the Weibull modulus is a measure of the scatter in the fracture strength for that group
(Appendix A). We will also compare the Weibull modulus behaviour for HA specimens with
0.59 < P < 0.62 with (i) trends in Weibull modulus for HA for 0.08 ≤ P ≤ 0.62 and a
202

number of additional materials (0 < P ≤ 0.62, Table 7.1b).

In addition, we consider the

fracture strength as a function of sintering temperature for HA specimens with 0.59 < P <
0.62 (Table 7.1b).
2. Experimental Procedure
2.1 Specimen preparation
For this study and the previous Weibull modulus study [Fan 2012 Part I], each HA
specimen was made from ~2.5 grams of the same lot of biomedical grade commercial HA
powders (HAP-200, Lot # is 10040501) with a vendor-specified purity of Cl ≤ 20ppm; S
≤ 50ppm; Heavy metals ≤ 5ppm; As ≤ 1ppm; Fe ≤ 10ppm; Mg≤ 50ppm; Mn ≤
20ppm (Taihei Chemical Industrials Co., Osaka, Japan). Also, for this study and the earlier
study [Fan 2012 Part I], each specimen was fabricated from HA powder discs compacted
uniaxially in a cylindrical steel die using a hydraulic press (Carver Inc, Wabash, IN) at 33
MPa for 1 minute. All specimens fabricated in this study and the previous study [Fan 2012
Part I] were sintered in air for in an electrical resistance furnace (CM Inc. Bloomfield, NJ)
with a heating/cooling rate of 10oC/min.
Each group of specimens in this study (Table 7.1a and Table 7.1b) and the previous
study [Fan 2012 Part I] was sintered at a different temperature. The differences in processing
between this study and the previous study [Fan 2012 Part I] were that in this study the
sintering was performed at temperatures between 350oC and 1025oC, producing specimens
with 0.59 ≤ P ≤ 0.62 (Table 7.1a) while in the previous study, the sintering temperatures
ranged from 550oC to 1360oC yielding specimens with 0.08 ≤ P ≤ 0.62.
203

Table 7.1a. Sintering temperature, T, and time, specimen thickness, t, and diameter, D,
volume fraction porosity, P, the number of valid specimens, N, the mean fracture strength
<f>, the characteristic strength, c, and the Weibull modulus m. Each specimen was hard
die pressed at 33 MPa and then sintered in air in an electrical resistance furnace.

T (oC)

Time
(mins)

t
(mm)

D
(mm)

P

N

<f>

c

(MPa)
(MPa)
350
120
2.41 32.15 0.62 18 0.66 ± 0.11 0.71
400
120
2.42 32.15 0.62 16 0.78 ± 0.07 0.82
450
120
2.44 32.15 0.62 15 0.95 ± 0.10
1.0
500
120
2.44 32.15 0.62 16
1.0 ± 0.1
1.1
550*
120
2.47 32.14 0.62 22
1.3 ± 0.1
1.3
600*
120
2.48 32.15 0.62 21
1.4 ± 0.1
1.4
650
120
2.49 32.20 0.62 17
1.6 ± 0.2
1.6
700*
120
2.50 32.17 0.62 19
2.0 ± 0.2
2.1
725
120
2.46 32.19 0.62 20
2.6 ± 0.2
2.7
750
120
2.44 32.21 0.62 19
2.9 ± 0.4
3.1
775
120
2.51 32.18 0.62 17
2.9 ± 0.3
3.1
800*
120
2.49 32.16 0.62 19
3.3 ± 0.3
3.4
850
120
2.44 32.13 0.62 22
3.5 ± 0.3
3.7
900*
120
2.47 32.00 0.62 21
3.5 ± 0.3
3.6
950
120
2.40 31.74 0.60 19
3.6 ± 0.4
3.7
975
120
2.41 31.58 0.60 16
3.9 ± 0.4
4.1
1000*
120
2.42 32.16 0.60 21
4.5 ± 0.5
4.8
1025
120
2.42 30.91 0.59 16
4.9 ± 0.6
5.1
* Specimens were from our previous Weibull study [Fan 2012 Part I]

204

m
6.6 ± 0.4
11.7 ± 0.9
10.8 ± 0.5
14.8 ± 1.9
10.6 ± 0.8
12.7 ± 1.1
11.5 ± 0.7
10.4 ± 0.8
14.7 ± 0.3
9.4 ± 0.5
11.9 ± 0.7
12.2 ± 0.8
13.7 ± 0.4
15.5 ± 0.4
10.9 ± 0.9
10.9 ± 0.5
10.1 ± 0.7
11.9 ± 0.8

Thus, using the same processing techniques (the same cold pressing pressure and time,
the same sintering furnace and heating/cooling rate) allow us to directly compare the results
between this study and the our previous study [Fan 2012 Part I]. Also, within each group of
HA specimens, the porosity of sintered specimens was quite uniform with a standard
deviation in volume fraction porosity, P, of less than 0.005 for the specimens within the
group.
The porosity of the sintered specimens was calculated from the specimen mass along
with average specimen diameter and thickness, assuming a theoretical density (a density
corresponding to P = 0) of 3.156 g/cm3 for HA [CRC 2009].
2.2 Mechanical testing and Weibull analysis
In a Weibull modulus study, the specimens should be fractured using a standard test
technique such as the four-point bend test or a biaxial flexure text [Wachtman 2009, ASTM
2003]. In this study the biaxial flexure test (BFT) was used to fracture the partially sintered
HA specimens. The BFT technique has been employed for fracture studies of a wide variety
of biomedical materials including bone cement [Boyd 2008], dental ceramics [Addison 2012,
Yoshimura 2012, Gonzaga 2011] and hydroxyapatite [Meganck 2005, Fan 2012 Part II Fan
2012 Part II]. As noted by Addison [2012], Wachtman [2009] and others, an advantage of
BFT technique is a reduced sensitivity to specimen edge flaws which may be introduced
during the cutting or processing of a specimen.

Additional details of the BFT technique are

given in Appendix B.
In this study and the previous study [Fan 2012 Part I], the as-sintered HA specimens
205

were tested using identical procedures. Each HA specimen was fractured in biaxial flexural
testing, with the loading axis perpendicular to the circular face of the disk shaped specimens,
on a universal testing machine (Model 4206, Instron, Norwood, MA) via a ring-on-ring
configuration ([ASTM 2003], Appendix B). The crosshead speed was 1 mm/min.
The fracture strength, f , was calculated using the following equation [ASTM 2003]

f 


D2  D2
3F 
L  1    ln DS 


1    S
D 
2h 2 
2D 2
 L 



(2)

where F is the load at failure, h and D are the thickness and the diameter of the test specimen;
DS and DL are the diameters of the supporting ring and the loading ring (Appendix B). In this
study, DS = 20 mm and DL = 10 mm. A Poisson’s ratio, , of 0.27 [Ren 2009] was used.
The remnants of the fractured specimens were examined to determine whether or not
each of the fractured specimens represented a valid ring-on-ring (ROR) test. The definitions
of valid and invalid ROR tests are discussed in Appendix B. Only the data for validly
fractured specimens were used in the analysis of Weibull modulus and fracture strength.
2.3 Microstructural examination
SEM on fractured surfaces of two HA specimens sintered at 350oC and 1025oC. For the
two HA examined by SEM, a 21 nm thick gold coating was sputtered on the examined
surfaces using a gold sputter (EMSSCOPE SC500, Ashford, Kent, UK) prior to SEM
examination. An accelerating voltage of 15 kV and a working distance of 15 mm were used.
2.4 Statistical analysis
A one-way ANOVA analysis was used to compare Weibull modulus values as a

206

(a)

(b)

(c)

(d)

Figure 7.1. SEM micrographs of partially densified HA specimens sintered at 350oC (a) at
low magnification, (b) at higher magnification and HA specimens sintered at 1025oC at (c)
low magnification and (d) higher magnification. Note that 350oC is the minimum sintering
temperature in this study and 1025oC is the maximum sintering temperature in this study.

207

function of porosity for HA specimens fabricated in this study along with data from our
earlier study [Fan 2012 Part I].
3. Results and Discussion
3.1 Microstructural and phase analyses
3.1.1 SEM analysis of initial powders and the sintered specimens
SEM showed the initial HA powders consisted of irregularly-shaped agglomerates
composed of rod-shaped particles approximately 0.8 m in length with an aspect ratio of ~ 3
(Figure 7.2 in Fan 2012 Part I]. The rod-like geometry of the agglomerates in the initial
powder particles is consistent with the high green porosity, PG of 0.62 [German 1994]. In
contrast, the green density is about 0.4 to 0.5 for powder compacts composed of nearly
equiaxed powder particles [German 1994]. For this study, micrographs of fractured surfaces
showed a morphology that is typical of the initial sintering stage [German 1994, Barsoum
1997], with (i) the start of necking between adjacent powders and (ii) the presence of an open
network of pores (Figure 7.1a –7.1d). The arrows in Figure 7.1b indicate the connected pore
channels with diameter of ~2 m is observed wandering through the entire microstructure.
Micrographs of the specimen sintered at 1025oC (Figure 7.1c and 7.1d) showed a more
uniform distribution of pores, where in Figure 7.1d the arrows indicate isolated pores with
diameters of ~ 1 m. The uniformity of pore distribution after sintering at 1025oC may be
due to the rearrangement of powder particles during initial sintering stage [Henrich 2007].
3.1.2 Evolution of relative density versus sintering temperature for the HA specimens in
the initial and intermediate sintering stages

208

Table 7.1b. Summary of the types of analysis done in this study along with the number of
specimen groups, number of individual specimens and the associated figures in this study.
Analysis
done
Relative
density, RD

Weibull
modulus
(high P
analyzed in
this study)

Weibull
modulus
(HA
fabricated in
this study
and from
Fan 2012
Part I)
Weibull
modulus
(combined)

Fracture
strength

Material

Number of
groups

HA

23 (12 from
this study and
11 from Fan
2012 Part I)

HA

18 (12 from
this study and
6 from Fan
2012 Part I)

Number of specimens
Corresponding
in total and with valid
figures
fractures
552 in total (288 from
this study and 264
from Fan 2012 Part I)
432 in total (288 from
this study and 144
from Fan 2012 Part I)
334 with valid
fractures (211 from
this study and 123
from Fan 2012 Part I)

7.2
7.3 (Weibull
probability
plots),
7.5 (Weibull vs.
porosity),
7.6 (Weibull
modulus versus
Tsinter)

HA

35 (12 from
this study and
13 from Fan
2012 Part I)

828 in total (288 from
this study and 540
from Fan 2012 Part I);
652 with valid
fractures (211 from
this study and 441
from Fan 2012 Part I)

7.4a

HA, alumina
titania, mullite,
LAST-T,
silicon nitride
silicon

96

More than 1771 valid
fractures

7.4b

18 (12 from
this study and
6 from Fan
2012 Part I)

432 in total (288 from
this study and 144
from Fan 2012 Part I);
334 with valid
fractures (211 from
this study and 123
from Fan 2012 Part I)

7.7

HA

209

In general, for sintered ceramic powder compacts, the relative density versus sintering
temperature may be represented in terms of a sigmoidal relationship (Ewsuk and Ellerby
2006, Ouyang 2012], as given by equation (3) such that
RD = RDf -

RD i  RD f
 T  Tconst1 

1  exp
 T

 const2 

(3)

where RDf is the final relative density, RDi is the initial relative density. The temperatures
Tconst1 and Tconst2 are two constants defining the inflection points on the sigmoidal curve,
both with units of oC.
In examining the Weibull moduli of porous HA specimens in this study, we employed
the HA specimens fabricated in this study along with specimens from our previous study
[Fan 2012 Part I]. Although, as discussed in Sections 2.1 and 2.2, the specimens in the two
studies were fabricated and tested using the same procedures, it is worthwhile to determine
whether or not the densification behavior for the HA specimens fabricated in this study and
in the previous study [Fan 2012 Part I] follow the expected sigmoidal behavior (equation 3).
For the RD versus sintering temperature evaluation (Figure 7.2), we expanded the scope of
our examination of the HA specimens to include a total of 23 specimen groups, of which 12
groups consisted of HA specimens fabricated in this study (Table 7.1b, Figure 7.2) along and
11 groups were specimens from a previous study ([Fan 2012 Part I], Table 7.1b, Figure 7.2).
The 12 specimen groups fabricated in this study were each sintered for 2 hours. The 11
porosity groups selected from the Fan 2012 study were sintered for 2 hours except for one
group that was sintered for 110 minutes [Fan 2012 Part I]. The remaining groups from Fan
210

1.0

Specimens fabricated in this study
Specimens from Fan et al. 2012

Relative density, RD

0.9
0.8
0.7

Intermediate
sintering stage

Initial sintering stage

0.6
0.5
0.4
0.3
300

450

600

750 900 1050 1200 1350
Tsinter (oC)

Figure 7.2. Relative density, RD, as a function of sintering temperature for HA specimens in
this study and a previous study [Fan 2012 Parts I and II] (Table 7.1b).

2
T = 350oC

lnln(1/(1-Pf))

T = 400oC

450oC

T = 500oC

T = 550oC

T = 600oC

650oC

T = 700oC

T = 725oC

T = 750oC

775oC

T = 800oC

T = 850oC

1

T = 900oC

950oC

T = 975oC

T=

0

T=

-1

T=

-2

T=

T = 1000oC

-3

T = 1025oC

-4
-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

lnf

Figure 7.3. Weibull plots for the 18 groups of specimens analyzed in this study (Table 7.1b).
211

2012 study were not included in the RD analysis in this study because their sintering times
differed significantly from 2 hours [Fan 2012 Part I].
A least-squares fit of these 23 porosity groups (Table 7.1b and Figure 7.2) shows that
equation (3) describes the relative density versus sintering temperature extremely well with
the fitting parameters RDf = 0.95 ± 0.02, RDi = 0.38 ± 0.003, Tconst1 = 1176 ± 7oC,
2

Tconst2 = 67 ± 4 oC, and a coefficient of determination, R = 0.995 (Figure 7.2). Thus, as
expected, it is confirmed that the sintering behavior for the combined data set (Table 7.1b) is
consistent with equation 3 (Figure 7.2) which in turn supports the use of the combined data
from this study and the previous study (Table 7.1b) in considering the trends in Weibull
modulus as a function of microstructural evolution.
3.1.3 X-ray diffraction results
X-ray diffraction (XRD) results from our earlier study [Fan 2012 Part I] using the same
powders and sintering procedures as in this study did not show the presence of

secondary

phases either for the initial powders (prior to sintering) or the bulk specimens sintered at
1100oC, 1200oC and 1360oC. Thus, we assume that the HA specimens included in this study
sintered from 350oC to 1025oC should also be single phase.
3.2 Weibull modulus and fracture strength for HA specimens
A two-parameter Weibull analysis (equation (1)) was used to analyze the fracture data
for each of the validly fractured specimens in each group (Figure 7.3, Table 7.1b). A total of
432 specimens yielded 334 valid fractures (Table 7.1b). Except for the data set with Tsinter =
350oC, the Weibull probability plots for the specimens fabricated in this study (Figure 7.3)
212

were relatively linear, indicating that likely one type flaw dominated the fracture process
which was also the case for the Weibull data in our paper [Fan 2012 Part I].
Figures 7.4a and 7.4b show the Weibull modulus as a function of porosity for HA data
included in this study, HA data from our previous study (Figure 7.4a), and a combined data
set including the HA data along with Weibull modulus data from the literature for a number
of other brittle materials [Fan 2012 Part I]. Figure 7.4b shows a Weibull modulus trend
pointed out in our earlier paper [Fan 2012 Part I], namely that of a U-shaped distribution as a
function of porosity. We shall first discuss Figure 7.4a, then Figure 7.4b and then discuss
details concerning the three porosity regions (Regions I, II and III, Figure 7.4b).
Figure 7.4a includes the (i) Weibull moduli in the range from HA specimens 0.59 < P
< 0.62 for specimens fabricated in this study (filled circles in Figure 7.4a) along with the
entire set of HA data from a previous study [Fan 2012 Part I]. A one-way ANOVA analysis
with p < 0.0005 showed that the m values in Region III (P > 0.55) were
significantly higher than those in Region II (0.1 < P < 0.55). Thus, the additional data
provided by this study (Figure 7.4a) bolsters our earlier observation [Fan 2012 Part I] that
Weibull modulus values for the partially sintered HA (0.59 < P < 0.62) tends to be higher
than that for Region II (0.1 < P < 0.55).
In addition, Figure 7.4b compares Weibull moduli from this study and our previous
study of HA [Fan 2012 Part I] to Weibull modulus data from a variety of brittle materials in
addition to HA. Included in Figure 7.4b are valid fracture data (Appendix B) from 1560
specimens (441 HA specimens from Fan 2012 Part I, 372 HA specimens from the
213

25

HA (211) (Fabricated in this study)
HA (441)(Fan et al. 2012)

I
Weibull modulus

20

III
II

15
10
5
0
0.0

0.1

0.2

0.3
0.4
Porosity

0.5

0.6

Figure 7.4a. The Weibull modulus versus porosity for HA specimens fabricated in this study
along with HA specimens from our previous study ([Fan 2012 Part I], Table 7.1b);

214

55

HA (211) (this study)
HA (813) (Fan 2012)
Nb and Ta doped zirconia (N/S) (Pissenberger)
Titania (118) (Kishimoto)
Alumina (20-30) (Nanjangud)
Si (60) (Borrero-Lopez)
Mullite (20) (Vales)
HA specimens
Silicon Nitride (20) (Vales)
fabricated in this study
Alumina (~ 100) (Chao)
Silicon Nitride (45) (Hirosaki)
LAST-T (15) (Ren)
Silicon nitride (~ 90) (Chao)
III
(1 0 0) p-type silicon wafer (99) (Paul)
Polycrystalline silicon (>160) (Brodie)

I

Weibull modulus

50
25
20
15
10
5

II

0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

Porosity
Figure 7.4b. The Weibull modulus versus porosity for (a) HA specimens fabricated in this
study along with HA specimens from our previous study ([Fan 2012 Part I], Table 7.1b); (b)
the combined data set of the HA data shown in Figure 7.4a and Weibull modulus data from
literature data for other brittle materials [Fan 2012 Part I].

215

literature and data for more than 747 specimens for seven brittle materials other than HA.
Recall that each Weibull modulus value in the literature represents roughly 15 to 20 or
more fractured specimens (Appendix A), such that from the literature Figure 7.4b includes a
total number of m values for the specimens include 37 m values for HA from Fan [2012 Part
I], 12 m values for zirconia, 7 m values for titania, 4 m values for alumina, 15 m values for
silicon, one m value for mullite, 7 m values for silicon nitride, one m value for LAST-T
thermoelectrics.
When the additional HA data from the current study is combined with the previous
study [Fan 2012 Part I], then Figure 7.4b represents the fracture of a total of 1771 brittle
specimens, including the 652 HA specimens from this study and the previous Weibull study
[Fan 2012 Part I]. For the combined data set (Figure 7.4b, Table 7.1b), the Weibull modulus
data shows a three region, U-shaped trend over the entire porosity range including the data
from the previous Weibull study [Fan 2012 Part I] and the data from this study for 0.59 < P <
0.62.
In terms of the nature of the three regions in Figures 7.4a and 7.4b, it is important to
note that the particular values of P used to delimit each region, Pboundary, are empirical and
were determined solely from an examination of the m versus P behavior. For example, in
Region I (P ≤ 0.1), the Weibull moduli showed a wide range of values with 2 < m < 53
(Figure 7.4b). The spread in m values is due to the various processing and testing techniques
as well as the surface defects caused during processing. The value of Pboundary = 0.1 marks

216

an apparent change in the distribution of m values as a function of P, so Pboundary = 0.1 is
assigned as a boundary between Regions I and II (Figures 7.4a and 7.4b).
For other powder compacts, with values of PG that are different than the HA powders in
this study and in our previous Weibull study [Fan 2012 Part I], the Pboundary values may be
somewhat different than shown in Figure 7.4b. It may be preferable to plot m versus the
degree of densification, where (1 –P/PG) rather than m versus P (Figures 7.4a and 7.4b)
in order to account for differences in PG among powder compacts. However, PG is not stated
in the Weibull modulus literature for either (i) HA work performed by research groups other
than the current authors and [Fan 2012 Part I] or for (ii) m data for materials other than HA.
Thus, re-plotting Figure 7.4b in terms of m versus must await additional research on the
Weibull modulus of porous brittle materials.
Region II (0.1 < P ≤ 0.55) displays a behavior not reported in the literature prior to our
earlier paper [Fan 2012 Part I], namely a band of m values, with 4 < m < 11 (Figures 7.4a and
7.4b). The narrow range of m values in Region II results from the near universal evolution of
pore morphology during the intermediate of sintering stage of sintering powder compacts,
where the specimen porosity dominates the flaw population and diminishes the influence of
differing processing and testing techniques.
For Region III (P > 0.55) (Figures 7.4a and 7.4b), the relatively high m values for the
HA specimens may stem from the relatively open, interconnected pore structure. We expect
that materials other than HA sintered from powder compacts may exhibit a similar behaviour
in Region III, due to the very similar evolution of interparticle necking and pore modification
217

Green porosity

16

Data from this study
Data from Fan et al. 2012

Weibull modulus

14

o

900 C
o

o

725 C,500 C
o

850 C
o

o

12

1025 C

o

o

975 C
o
950 C

1000 C

10

600 oC
800oC
o
775o C,400 C
650 oC
450 C
o
o 550 C
700 C
o
750 C

8
o

350 C

6
0.56

0.58

0.60
Porosity

0.62

0.64

Figure 7.5. Weibull modulus, m, as a function of porosity for the HA specimens analysed in
this study (Table 7.1b). Each data point in this plot represents the valid fracture of roughly 15
to 20 individual specimens.

218

Table 7.2. For the HA specimens included in this study, the fitting parameters for
least-squares fit of the mean fracture strength, <f>, versus the sintering temperature, Tsinter,
to equations (6), (7) and (8).
Equation
number

ai (MPa)

bi

5

-0.282 ± 0.759

0.001 ± 0.002

6

-1.93 ± 0.26

0.0062 ± 0.0004

7

0.363 ± 0.059

0.0025 ± 0.0002

2

ci
a

3.69e-6 ± 1.62e-6

R
c

0.964

a

N.A.

0.951

b

N.A.

0.945

a

Units of b1 (equation (6)) and b2 (equation (7)) are MPa/oC

b

Units of b3 (equation (8)) are oC

-1 c

;

2

Units of c1 are MPa/(oC)

Data from this study
Data from Fan et al. 2012

16

Weibull modulus

14
12
10
8

Range of Weibull modulus
included in Region II

6
4
300

400

500

600

700 800
Tsinter (oC)

900

1000 1100

Figure 7.6. Weibull modulus, m, as a function of sintering temperature, Tsinter, for the HA
specimens analysed in this study (Table 7.1b).

219

observed during the initial sintering stage for wide range of metal and ceramic materials
[German 1996, Barsoum 1997]. Thus, this study confirms the apparent trend that the Weibull
modulus tends to be surprisingly high for P > 0.55 (Figures 7.4a, 7.5, Table 7.1b), as was also
observed in the study by Fan 2012 Part I. It should be noted that except for this study and the
earlier study by Fan [2012] Part I, only two Weibull modulus values were available in the
literature for P > 0.47 for any brittle material [Villora 2004]. In particular, those two
literature m data values at P = 0.55 and 0.65 were obtained from diametral compression
testing on composite HA specimens that included diatoms and industrial sludge [Villora
2004].
In addition to m as a function of P, we also consider m as a function of the sintering
temperature, Tsinter (Figure 7.6). A band of m values from roughly 9 to 15 (except one m
value at ~7) from this study displays considerable scatter as a function of Tsinter (Figure 7.6,
Table 7.1b), although the m values are consistently high. Since for sintering between 350oC
and 900 oC, there is no densification, Figure 7.6 shows a spread in the m values from 7 to 15
at fixed value of P = 0.62, which corresponds to the porosity of the green powder compacts.
Changes in the Weibull modulus for specimens sintered in the temperature interval between
350oC and 900oC is likely related to an increase in the relative interparticle neck size
increased with increasing Tsinter, as will be discussed further in the following section on
fracture strength.
3.3 Fracture strength for HA specimens

220

Since the Weibull moduli of the highly porous (P > 0.55) HA specimens in both this
study and the Fan 2012 Part I study show behavior that has not been reported previously, it is
of interest to also consider the behavior of the mean fracture strength for these specimens.
However, first we shall very briefly review the highlights of fracture strength versus P
relationships for brittle materials in order to put the behavior of our materials into context.
A widely used empirical strength-porosity relationship [Ryshkewitch 1953, Duckworth
1953] describes the decrease in modulus with increasing P such that
f = 0exp(-bP)

(4)

where 0 is the strength for the theoretically dense material, P is the volume fraction porosity
and b is a unitless, material-dependent constant measuring the rate of decrease in f with
increasing P. As reviewed by Rice [1998], equation (4) generally describes well the
strength-porosity behaviour for brittle materials up to a critical porosity, PC, but for P > PC,
the fracture strength may decrease more rapidly than equation (4) predicts [Rice 1998].
Equation (5), a modified Phani equation [Phani and Niyogi 1987], has been used by Fan
[2012 Part II] as an alternative to equation (4). Using equation (5), the relative fracture
strength, <f>/0, can be described as a function of the degree of densification
over the entire range of P from P near zero to P near PG, such that


 f 
P
 A  1 
 P
o
G


n


  A  n



(5)

where <f> is the average fracture strength of a group of porous ceramics with a fixed value
of P, PG is the porosity of the green (unfired) powder compact,  (1 –P/PG) is defined as
221

the degree of densification, A and n are dimensionless constants.
In our previous papers [Fan 2012 Part I and Part II], the porosity dependence of the
fracture strengths of partially sintered HA specimens was described well by equation (5).
This study deals with fracture of highly porous HA specimens, namely from 0.59 to 0.62, so
that the sintering process is limited to the initial sintering stage in which interparticle necks
develop. Moreover, in this study, for sintering temperatures between 350oC and 900oC, there
is no densification. Since the specimens do not densify, the porosity is fixed at P = PG (where
PG = 0.62 in this study). Thus, although equations (4) and (5) are extremely useful in
describing strength-porosity relationships when the entire data set includes only specimens
with P < PG, equations (4) and (5) are of very limited utility in describing the changes in
strength that are observed with changes in sintering temperature over the sintering range
from 350oC and 900oC where the porosity is fixed at P = PG.
However, if we consider the mean fracture strength <f> versus the sintering
temperature, Tsinter, there is a monotonic decrease in <f> with decreasing Tsinter (Figure 7.7,
Table 7.1b). The <f> versus Tsinter trends were examined via a least-squares fit of the data
to each of three empirical functions given by equations (6) through (8)
<f> = a1 + b1 Tsinter + c1 (Tsinter)

2

(6)

<f> = a2 + b2 Tsinter

(7)

<f> = a3 exp(b3Tsinter)

(8)

where the numerical values and physical units of each of the empirical fitting parameters a1,

222

b1, c1, a2, b2, a3, and b3 are given in Table 7.2 along with the coefficient of determination, R

2

2

for the least squares fit of the data to each equation (Figure 7.7). Based on the R values
given in Table 7.2, the quadratic empirical relationship (equation (6)) best describes the <f>
versus Tsinter behavior observed in this study, although the quadratic relationship (equation
(6)) is only marginally better than the linear relationship (equation (7)).
In addition to the porosity and sintering temperature, it is important to consider whether
other factors might affect the measured <f> values. For example, since the strength of brittle
materials can be affected by factors such as grain size [Barsoum 1997, Ren 2006, Wachtman
2009] and microcracking [Hoepfner and Case 2004, Rokhlin 1993, Sridhar 1994, Tomczyk
2012], we shall briefly consider the role that the grain size and microcracking might play in
the measured state of the <f> values. In general, during final sintering stage (P < 0.05) the
grain size, G, typically increases dramatically, P is relatively constant, and fracture strength ≈
-0.5

G

[Barsoum 1997]. However, during the initial sintering stage, from the measured grain

size-density trajectory for the HA specimens in the Fan 2012 Part I (which are identical to the
powders used in the current study), the grain size ranged from about 0.5 micron to 1.5
microns for HA with P from about 0.62 to 0.2. Furthermore, in the range of P included in this
study (0.59 to 0.62), the grain size was essentially fixed at approximately 0.5 microns [Fan
2012 Part I] and thus the functional dependence of strength upon G is likely not a significant
consideration in this study.
HA typically occurs in the hexagonal form [Case 2005], which along with all non-cubic
crystalline materials, making HA susceptible to microcracking due to thermal expansion
223

6

Data from this study
Data from Fan et al. 2012 Part I

<f > (MPa)

5
4
3
2
1
300

400

500

600 700 800
Tsinter (oC)

900 1000 1100

Figure 7.7. The mean fracture strength, <f>, as a function of sintering temperature for the
HA specimens analysed in this study (Table 7.1b), where the solid curve represents a
least-squares fit of the data to equation (6), the dashed line represents a least-squares fit of
the data to equation (7) and the dotted curve represents a least-squares fit of the data to
equation (8).

224

anisotropy (TEA). TEA-induced microcracking for grain sizes larger than a
material-dependent critical grain size, GC, can occur upon cooling from an elevated
temperature due to the unequal thermal expansion along the crystallographic axes in a
polycrystal [Hoepfner and Case 2004, Sridhar 1994, Tomczyk 2012].
SEM observations by Case [2005] indicate microcracking due to TEA does occur in
polycrystalline HA with P ≈ 0.02 and a mean grain size of 7.9 + 1.5 microns, although the
same study did not observe microcracking in the non-cubic materials tricalcium phosphate
(TCP) and biphasic calcium phosphate (BCP, a two- phase polycrystalline material
containing both HA and TCP). In the Case study [2005], the grain sizes for the TCP and BCP
were 2.9 ± 0.6 microns and 1.3 microns respectively, with P > 0.37 for both TCP and BCP
[Case 2005]. In interpreting the microcracking results from the Case study [2005], it is
important to consider that (i) P > 0.2 to 0.3 can strongly suppress microcracking in brittle
materials [Manning 1972] and (ii) for G < GC microcracking does not occur. Given the very
high P values and small grain sizes (about 0.5 micron) of the HA specimens in this study, it is
unlikely that microcracking plays a significant role.
Thus, grain size and TEA-induced microcracking should have a limited role in
determining the <f> versus Tsinter behavior of the HA specimens in this study. Likewise,
given that 14 of the 18 groups of specimens in this study have a fixed porosity of P = PG, the
results of <f> are not a strong function of porosity. Thus, the monotonic decrease in the
<f> versus Tsinter is likely due to the development of interparticle necking of adjacent
grains (Figure 7.7).
225

4. Summary and Conclusions
Since the Weibull modulus is a measure of the spread or scatter in fracture strength, the
Weibull modulus is extremely important as a gauge of mechanical reliability (Appendix A).
Nevertheless, Weibull modulus data for highly porous materials is extremely limited in the
open literature. This study explored the porosity dependence of the Weibull modulus and
fracture strength of a total of 432 HA specimens that were sintered from 350oC to 1025oC for
2 hours resulting in a range of volume fraction porosity from 0.59 to 0.62, indicating only
limited densification occurred (since the porosity of the unfired powder compacts, PG, was
0.62). The specimens were then fractured via biaxial flexural testing on a ring-on-ring test
fixture.
A key result of this study is that the Weibull modulus ranged from 6.6 to 15.5 for these
highly porous, partially sintered HA specimens (0.59 < P < 0.62), which represents
surprisingly small scatter in strength for such specimens. However, in contrast to the Weibull
modulus behavior, the mean fracture strength drops monotonically as the sintering
temperature decreases (Figure 7.7), indicating that the extent of interparticle necking
decreases as Tsinter decreases. Considering the Weibull modulus and fracture strength data
together, we conclude that as Tsinter decreases, the interparticle neck dimensions decrease as
we would expect from the evolution of necking during sintering while the open,
interconnected porosity leads to a relatively narrow flaw size distribution and thus a
correspondingly high Weibull modulus.
This study and our earlier Weibull modulus study [Fan 2012 Part I] constitute the first
226

systematic work on the Weibull modulus of high porous brittle materials. In particular, this
study confirms that for HA, the Weibull modulus remains high for partially sintered
specimens with P at or near the green density, PG (Region III in Figures 7.4a and 7.4b).
It is likely the Weibull modulus versus porosity behavior in Region III (P near or equal
to PG) for other brittle materials will be similar to that observed in this study for HA. This
prospect is supported by the fact that in Region II (0.1 < P ≤ 0.55, Figure 7.4b) the band of
32 m versus P values includes data from four different materials, namely alumina (three m
values), zirconia (two m values), titania (three m values) and HA (24 m values). Thus if the
behavior in Region II (Figure 7.4b) is similar for various brittle materials, that similarity may
carry over to Region III also. However, except for this study and the Fan 2012 Part I paper,
the current lack of Weibull modulus data in the literature for brittle materials with P >
roughly 0.2 makes wide-ranging comparisons difficult to make.
As discussed in the Introduction, many of the current uses of highly porous materials are
for porosity ranges between 0.2 and 0.5. However, from the results of study, we may
conclude that knowing the Weibull modulus for these materials in the initial sintering stage
may stimulate the development of both biomedical and non-biomedical applications of
sensors, catalyst supports and filters. Despite their low strength, the mechanical reliability
(low scatter in strength values) indicated by the surprisingly high Weibull modulus of the
materials may indicate to designers that new, practical applications are possible.
The extremely diverse range of biomedical and non-biomedical applications and
potential uses for HA and other highly porous brittle materials makes these findings on
227

Weibull modulus very interesting, due to the direct connection between Weibull modulus and
mechanical reliability. Future work should explore the porosity dependence of the Weibull
modulus for highly porous specimens of biomaterials other than HA as well as additional
brittle materials.
5. Acknowledgements
The authors acknowledge the financial support of the National Science Foundation,
Division of Materials Research Grant DMR-0706449. The authors would like to thank
Jennifer Ni (graduate student, Chemical Engineering and Materials Science Department,
Michigan State University) for her help in taking the SEM images. The authors also thank the
Michigan State University High School Honors Science/Mathematics/Engineering Program
(HSHSP) for support received by I. Gheorghita during this research.

228

APPENDICES

229

APPENDIX A

WEIBULL ANALYSIS
Weibull analysis has been widely used in to characterize the scatter in a measured
property, such as fracture strength [Meganck 2005, Ren 2006, Wachtman 2009, Morrison
2012]. Thus, the Weibull modulus is directly linked to material reliability, since the less
scatter in a property such as strength, the more reliable a component is.
Typically Weibull analysis is performed on a group of specimens that are nominally
identical. In this study and in our earlier study [Fan 2012 Part I], a group of specimens
consisted of 24 specimens that were fabricated and tested in the same way (Sections 2.1 and
2.2), except that the sintering time and temperature differed among the groups which resulted
in specimens with differing stages of microstructural development [German 1996, Barsoum
1997].
As an example of the application of Weibull analysis to a particular group of fractured
specimens, let us consider the group of HA specimens from this study which was sintered at
725oC with 20 valid fractures (Table 7.1a). Figure 7.A1 is a Weibull probability plot, where
the Weibull modulus, m, is the slope of the least-squares fit of lnln(1/(1- Λf)) versus lnf
(Figure 7.A1, equation 1). The characteristic strength, C, (Table 7.1a) can be calculated
from the least-squares fit at lnln(1/(1- Λf)) = 0. At characteristic strength, C, the value of Λf
is 0.63, which means 63% of specimens will fail at or below C.

230

2
1

o

HA sintered at 725 C
P = 0.62

lnln(1/(1-f))

0
-1
-2
m = 14.7 for this group
of HA specimens

m

-3

ln c

-4
0.7

0.8

0.9

lnf

1.0

1.1

Figure 7.A1. The Weibull plot of a group of HA specimens sintered in this study at 725oC
(Table 7.1a). The slope of the plot, m, is the Weibull modulus, which is a measure of the
scatter in fracture strength values for this group of specimens.

231

It is important to comment on the nature of both the Weibull modulus and the
characteristic strength. If the fracture strength was exactly the same for each specimen of a
given group then the plotted line in Figure 7.A1 would be vertical and the Weibull would be
infinite. Conversely, a Weibull modulus value approaching zero would correspond to an
infinite scatter among the strength values of the group of specimens.
In addition, the characteristic strength is akin to the median strength, where for the
median strength, 50% of the fractures would fall below the median. Often the median and the
mean of a distribution are similar but the difference between the median and the mean
depends on the details of the distribution. Likewise, the characteristic strength and the mean
strength are typically similar but not equal.

232

APPENDIX B

BIAXIAL FLEXURE TESTING
Biaxial flexure is used widely for fracture testing of biomedical and non-biomedical
materials. In biaxial flexure testing, typically a disc specimen is placed on a support ring
(Figure 7.B1a) which the load applied by either a sphere (referred to as “ball on ring testing”,
[Wachtman 2009]) or by a loading ring (termed “ring on ring testing”, ASTM C1499 2003).
In this study, ring on ring testing was used (Figures 7.B1a and Figure 7.B1b).
In a valid ring-on-ring test (ASTM C1499 2003), the cracks initiate on the specimen
surface between the support ring and loading ring (Figure 7.B2). A common type of invalid
fracture test is one in which the specimen fails via a diametral crack, indicating that failure
initiated from the specimen edge (Figure 7.B2). Only the data for validly fractured specimens
were included in the Weibull modulus and fracture strength studies.

233

Figure 7.B1. Digital image of (a) the ring-on-ring fixture showing the specimen support ring
and loading ring used in the biaxial flexure testing in this study; (b) the assembled
ring-on-ring fixture used in the biaxial flexure testing in this study.

234

Figure 7.B2. Digital image of examples of valid and invalid fractures of HA specimens
included in this study.

The crack patterns for the valid test specimens are discussed in the

ASTM standard for ring-on-ring testing (ASTM C1499 2003).

235

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241

CHAPTER 8
ROOM TEMPERATURE ELASTIC PROPERTIES OF GADOLINIA-DOPED CERIA
AS A FUNCTION OF POROSITY

X. Fan, E. D. Case, Qing Yang and Jason D. Nicholas
Department of Chemical Engineering and Materials Science, Michigan State University,
East Lansing, MI, 48824
Accepted by: Ceramics International

Abstract
Gadolinia doped ceria (GDC) is commonly used in chemical reactors, gas sensors, gas
separation membranes, and solid oxide fuel cells (SOFCs). In the present study, the room
temperature elastic properties of GDC10 (Ce0.9Gd0.1O1.95) were measured as a wide
function of porosity for the first time. GDC10 specimens with volume fraction porosities
ranging from 0.07 to 0.60 were produced by hard die pressing and sintering GDC10 powders
in air at temperatures ranging from 825 oC to 1475oC. The room temperature Young’s
modulus, shear modulus, bulk modulus, Poisson’s ratio were measured using resonant
ultrasound spectroscopy. The elastic moduli decreased exponentially with increasing
porosity.

1. Introduction
Dense gadolinia doped ceria (GDC) is commonly used as a membrane and/or diffusion
blocking material in oxygen sensors, oxygen separators, and Solid Oxide Fuel Cells (SOFCs)

242

[Steele 2001, Fergus 2006, Bouwmeester 1997]. In contrast, porous GDC is commonly used
as an anode catalyst [Sauvet 2001, Trovarelli 1996], an anode mixed ionic electronic
conductor (MIEC) [Goodenough 2007], an oxygen storage material [Trovarelli 1996], a
mechanical support [Liu 2002], and/or a composite cathode oxygen ion conductor [Murray
2001] within these same devices. In fact, nano-composite cathodes (NCC’s) of porous GDC
surface-decorated with nano-sized MIEC catalyst particles have received much attention in
the literature due to their high electrochemical performance [Nicholas 2010, Nicholas 2012,
Shah 2009, Shah 2008, Samson 2011] and electrochemical/microstructural durability
[Samson 2012] at intermediate (500-700oC) temperatures.

Like conventional SOFCs,

SOFCs utilizing NCC’s [Jiang 2006, Vohs 2009] must be designed so that coefficient of
thermal expansion (CTE) mismatch between the cell layers (electrolyte, cathode, current
collector, etc.) do not lead to fracture [Sun 2009, Lin 2007, Montross 2002]. Unfortunately,
the porous GDC elastic properties needed to model load transfer into the MIEC-GDC NCC
layer are not available in the literature (to a first order approximation, the mechanical
response of a NCC is determined solely by that of the porous, structurally-supporting,
ionic-conducting scaffold).
In fact, the mechanical property literature for SOFC materials has been almost entirely
limited to materials with relatively low porosity. For example, hardness and fracture
toughness have been measured for samaria-doped ceria [Shemilt 1997] and gadolina-doped
ceria [Fu 2010, Morales 2010] but only for specimens with P < 0.08. In terms of the fracture

243

strength, f, of GDC10 in particular, the values available in the literature also include only
low porosity specimens, with f values from 200 to 300 MPa (P < 0.08) [Ishida 2005] and
from 150 to 175 MPa (P < 0.04) [Reddy 2005]. While no creep data for GDC10 is available
in the literature, Ikuma et al. studied the creep of undoped ceria with 0.01 < P < 0.08 [Ikuma
1997]. Two recent reviews of the mechanical properties of SOFC materials by Nakajo et al.
[Nakajo 2012a, Nakajo 2012b] included creep data on SOFC materials other than GDC, but
no creep data on GDC. Routbort et al [Routbort 1997] studied GDC creep above 1200oC, but
only for samples with P > 0.08. In addition, Nakajo et al [Nakajo 2012a] noted that creep
data was extremely sparse for any porous SOFC material but suggested that finite element
models that deal with the deformation of struts can perhaps be “adequate for preliminary
studies, because of the actual uncertainties on the creep behaviour of SOFC materials”. Lastly,
as will be discussed in detail in Section, 3.2, the available literature data on elastic modulus
and Poisson’s ratio of GDC10 includes only low porosity specimens. Therefore, the present
paper focuses on the room temperature elastic moduli and Poisson’s ratio of GDC10 over a
wide range of porosity (0.07 < P < 0.60).
2. Experimental Methods
2.1 Specimen preparation
2

All GDC10 specimens were prepared from a single batch of 99.9% pure, 30.7 m /g (~27
nm primary particle size) Anan Kasei Ce0.9Gd0.1O1.95 (GDC10) powder obtained from
Rhodia, Inc. (Cranbury, NJ). Light scattering measurements performed by the vendor yielded

244

th

a 240 nm d50 (i.e. the diameter of particle agglomerates in the 50 volumetric percentile)
particle size. This value was consistent with previous transmission electron microscopy, x-ray
2

diffraction, and light scattering measurements on a separate lot of 30-35 m /g Rhodia
GDC10 powder [Nicholas 2007] which showed ~30 nm equiaxed primary particles grains
agglomerated into lenticular masses 75 nm wide and 600 nm long. Vendor analysis on this
previous lot of GDC10 powder showed it to have less than 10 ppm SiO2 [Nicholas 2007]. All
powder processing in the present study was done in the absence of Pyrex® glassware and
other potential sources of Si contamination.
Cylindrical specimens 2.7 ± 0.3g in weight were prepared for mechanical testing by cold
pressing the GDC10 nano-powders to 27.3 MPa in a three quarter inch stainless steel die.
These specimens were fired in air with a 4oC/min heating rate and held for 5 hours at various
temperatures in order to produce specimens of varying porosities.
2.2 Elastic modulus measurements
The resonant ultrasound spectroscopy (RUS) technique [Migliori 1993, Migliori 2005]
was used to measure the elastic moduli of as-sintered GDC10 specimens which were fired to
825, 850, 875, 900, 950, 975, 1000, 1050, 1100, 1200, 1450, or 1475oC and therefore had
sintered porosities of 60, 55, 48, 45, 38, 33, 27, 23, 15, 18, 8 and 6%, respectively. RUS is a
non-destructive technique that has been used to measure the elastic moduli for a number of
different oxides including hydroxyapatite [Fan 2012], alumina [Ren 2009] and lithium
lanthanum zirconia oxide [Ni 2012] as well as non-oxide ceramics [Seiner 2012],
thermoelectrics [Morrison 2012, Schmidt 2012], metals [Hurley 2010, Kaplan 2009] and
245

composites [Gorsse 2003, Vdovychenko 2006].
The RUS equipment (RUSpec, Quasar International, Albuquerque, NM) used in the
present study consists of a computer controlled transceiver and a tripod arrangement of PZT
(lead zirconate titanate) transducers that includes a single driver transducer and two pick-up
transducers. Each specimen was placed on the tripod transducer stage where the driver
transducer excited the mechanical vibrations in the specimen and the other two transducers
detected the acoustic response of the specimen. The frequency of the driver transducer was
swept from 10 kHz to 500 kHz in 29999 steps. The resonant vibrational frequencies of the
specimen were sensed and recorded (Figure 8.1). Using the frequencies of the resonant peaks
(Figure 8.1) along with the specimen mass and dimensions, the elastic moduli of the
disc-shaped specimens were calculated using a commercial software package (CylModel,
Quasar International, Albuquerque, NM).
2.3 Microstructural examination procedures
An Auriga Dual Column focused ion beam-scanning electron microscope (FIB-SEM)
(Carl Zeiss Microscopy, LLC, Thornwood, NY) with a nominal x-y resolution of 1 nm was
used to perform microstructural examinations on four fractured surfaces and one thermally
etched polished surface. The thermally etched specimen was prepared by polishing down
specimen#5 (P = 0.077) to a final grit of 1 m and holding the specimen in air for 2 hours at
1350oC (1350oC is 100oC lower than the 1450oC sintering temperature used to produce this
specimen). In order to suppress electrostatic charging during SEM analysis, a 15 nm thick
gold coating was sputtered onto all specimens surfaces prior to the SEM examination. The
246

22

Amplitute (A.U.)

20
18
16
14
12
10
15

50

100
150
Frequecy (kHz)

200

250

Figure 8.1. A portion of the RUS spectrum obtained for GDC10 specimen with P = 0.52 in
the present study (A.U. stands for arbitrary units).

247

(a)

(b)

(c)

(d)

Figure 8.2. SEM micrographs of (a) fracture surface for GDC10 specimen #39 (P = 0.521);
(b) fracture surface for GDC10 specimen #11 (P = 0.262); (c) fracture surface for GDC10
specimen #3 (P = 0.102) and (d) thermally etched polished surface for GDC10 specimen #5
(P = 0.077).

248

mean grain size was evaluated from the SEM micrographs via the linear intercept technique
using a total of 150 to 200 intercepts per micrograph with a stereographic projection factor of
1.5 [Underwood 1968].
3. Results and Discussion
3.1 Microstructural analysis
For the GDC10 specimens included in the present study, the evolution of the
microstructure as a function of P is depicted in Figure 8.2. For P = 0.521(specimen #39,
Figure 8.2a), clusters of agglomerates appear with aspect ratios of roughly 1.1 to 1.7 and
agglomerate dimensions of ~ 350 nm x ~200 nm to ~150 nm x ~ 100 nm. For P = 0.262
(specimen #11, Figure 8.2b), relatively dense islands of grains with tubular pores along the
grain boundaries are separated by relatively porous regions. For P = 0.102 (specimen #3,
Figure 8.2c) and P = 0.177 (specimen #14, micrograph not shown), the fracture mode is
predominately intergranular. These more dense microstructures are more uniform, with pores
ranging from several to hundreds of nanometers across located at triple points. For P = 0.077
(specimen #5, Figure 8.2d), this thermally etched specimen shows equiaxed grains and
isolated pores of up to hundreds of nanometers across located at triple points or grain
boundaries.
A grain-size density trajectory for the GDC10 specimens in the present study is typical
of that observed for a variety of oxide ceramics [Fan 2012b, Berry 1986], namely there is
limited grain growth from green density up to relative density, RD, of roughly 0.8 to 0.85
(corresponding to a volume fraction porosity of ~ 0.2 – 0.15). For RD > 0.8, the grain size of
249

1.2
Fractured surfaces
Thermally etched surface

Grain size (m)

1.0
0.8
0.6
0.4
0.2
0.0
0.5

0.6

0.7

0.8

0.9

1.0

Relative density, RD

Figure 8.3. Grain size-density trajectory for the GDC10 specimens in the present study. The
solid curves represents the least-squares fit of the grain size versus density data to equation
(1).

250

the GDC10 specimens increased rapidly from about 0.3 m to 1.1 m for RD = 0.93 (P =
0.07, Figures 8.2d and 8.3). This implies that approximately two thirds of the 21 GDC10
specimens included the present study had grain sizes from roughly 0.1 m and 0.3 m
(Figure 8.3).
The grain-size density trajectory (Figure 8.3) for the GDC10 specimens was
least-squares fit to the following empirical relationship
GS = A1exp(RD/B1) + GS0

(1)

where A1 and B1 define the shape of the exponential curve. A1 has units of grain dimension,
B1 is dimensionless and GS0 is the initial powder particle size. The least-squares fit of the
grain size versus relative density data to equation (1) gives A1 = 0.00002 m, B1 = 0.085 and
2

GS0 = 0.046 m, with a coefficient of determination, R = 0.963. Equation (1) describes the
grain size-density trajectory very well for the GDC10 specimens included in the present
study as well as a grain size-density trajectory from the literature for sintered hydroxyapatite
[Fan 2012b]. Taken together, these results represent the first time detailed microstructural
analyses of GDC10 grain size, pore shape, fracture mode, and porosity have been reported
along with elastic property measurements.
3.2 Elastic modulus as a function of porosity
Several studies in the literature report on the elastic moduli of relatively dense bulk
GDC10 specimens [Selcuk 1997, Amezawa 2011, Kushi 2009, Wang 2007]. Among these
studies [Selcuk 1997, Amezawa 2011, Kushi 2009, Wang 2007], only Selcuk and Atkinson
[Selcuk 1997] report the Young’s modulus as a function of volume fraction porosity, P. Using
251

Table 8.1. Comparison of specimen fabrication techniques, elastic modulus measurement
techniques and porosity ranges for the present study, Selcuk and Atkinson [Selcuk 1997],
Amezawa et al. [Amezawa 2011], Kushi et al. [Kushi 2009] and Wang et al. [Wang 2007].
Elastic modulus
measurement
technique

References

Specimen fabrication
technique

The
present
study
Selcuk and
Atkinson
[Selcuk
1997]

Hard die pressed at 27.3 MPa,
sintered from 825 oC to 1450
o
C for 5 h
Circular disk cut from
extruded green tape, binder
burn out at 400oC, densified at
1500oC for 1 h

Amezawa
et al.
[Amezawa
2011]

Powder prepared by
co-precipitation,
hydrostatically pressed at 150
MPa, sintered at 1550oC for
5h

Relative
porosity
range

Resonant ultrasound
spectroscopy

0.066 – 0.595

Impulse excitation
technique

0.02 – 0.05

Sonic resonance
measurement.
Room temperature,
one specimen under
0.01% O2, one under

0.035 and
0.033

99.15% H2 and 0.85%
H2O

Powder prepared by
Kushi et al.
co-precipitation,
[Kushi
hydrostatically pressed at 150
2009]
MPa, sintered at 1550oC
Powder mixed with 3 wt %
Wang et al. PVB, uniaxially pressed then
[Wang
CIPed at 250 MPa. Held at
2007]
400oC for 1 h and sintered at
1550oC for 20 h

Sonic resonance
measurement. Room
temperature under
pure Argon

0.051

Nanoindentation, with
assumed Poisson’s
< 0.02
ratio of 0.3

252

the impulse excitation technique (IET), Selcuk and Atkinson [Selcuk 1997] measured the
Young’s modulus, E, shear modulus, G, and Poisson’s ratio, , for 26 disc-shaped
polycrystalline GDC10 specimens over a very limited P range, namely 0.02 < P < 0.05 (Table
8.1). (The number of specimens was not directly stated by Selcuk and Atkinson [Selcuk 1997]
but here 26 data points have been extracted from Selcuk and Atkinson’s plots). The Amezawa
et al. [Amezawa 2011] and Kushi et al. [Kushi 2009] GDC10 elasticity measurements were
performed using the sonic resonance technique in a low pO2 atmosphere (Table 8.1). As will
be discussed later in this section, heating in a low pO2 atmosphere can induce atomic scale
defects which can in turn decrease the elastic modulus [Wang 2007, Wachtel 2011]. Wang et
al.’s E data [Wang 2007] was obtained by nano-indentation on a > 98% dense GDC10
specimen cooled from 800oC and various pO2 levels (Table 8.1).
In the present study, the RUS technique was employed to measure the E, G,  and the
bulk modulus, B, for 21 disk-shaped, as-sintered GDC10 specimens with P values ranging
from 0.07 to 0.60, where the porosity of the green (unfired) powder compacts, PG, was
0.62. The porosity dependence of E, G and B, was assumed to have the same functional
forms (equations (2) – (5)) employed by Selcuk and Atkinson [Selcuk 1997], namely
M = M0exp (-bMP)

(2)

M = M0 (1 - bMP)

(3)



bM P
M = M0 1 

 1  (bM  1) P 

(4)

2

M = M0 (1-P) (1 + bMP)

(5)

253

2

Table 8.2. Fitting parameters, E0 and R , for the least-squares fit of Young’s modulus data
for the present study, Selcuk and Atkinson [Selcuk 1997], and the combined data set [the
present study, Selcuk 1997, Amezawa 2011, Kushi 2009, Wang 2007] to equations (2) – (5).
2

2

2

2

E0 from E0 from E0 from E0 from R from R from R from R from
Ref.

equation equation equation equation equation equation equation equation
(2) (GPa) (3) (GPa) (4) (GPa) (5) (GPa) (2)
(3)
(4)
(5)

The
present 235.2
study
Selcuk
and
a
Atkinson 217.8
[Selcuk
1997]
Combined
data set
[the
present
study,
Selcuk
1997,
226.7
Amezawa
2011,
Kushi
2009,
Wang
2007]
a

161.3

292.0

218.1 a

217.0

200.3

239.8

a

175.7

218.5

a

206.4

2

0.989

0.886

a

0.992

0.931

0.886

a

0.942

E0 and R values are taken from Selcuk and Atkinson [Selcuk 1997].

254

0.969

0.885

a

0.987

0.963

0.885

a

0.986

where M represents the elastic moduli (E, G, or B in this case), M0 is the elastic moduli at P =
0 and bM is a unitless, material-dependent constant that measures the rate of decrease in M
with increasing P. For P  zero, equations (2), (4) and (5) are approximated well by the
linear relationship given by equation (3). Equation (2) is an empirical relationship widely
used to describe the elastic modulus change as a function of porosity [Rice 1998]. Equation
(4) is modified from equation (2) to treat materials with dilute distribution of spherical pores
[Hasselman 1962]. Equation (5) is based on the theoretical composite sphere model, where
spherical pores were treated as a second phase randomly distributed in the matrix material
[Ramakrishnan 1993].
Equations (2) – (5) were used to fit the Young’s modulus for GDC10 specimens in the
present study. Among equations (2) – (5), the exponential relationship (equation (2)) best
described the E versus P data. Since equations (3) - (5) are best suited to low porosity, P, and
the exponential relationship (equation (2)) is approximated by the linear form (equation (3)),
it is not surprising that equations (2) – (5) fit the Selcuk and Atkinson elasticity data (for
which 0.02 < P < 0.5) about equally well (Table 8.2). For equations (2) through (5), Table 8.2
2

compares the least-squares fitting parameters and R for (i) the data from the present study,
(ii) the Selcuk and Atkinson data [Selcuk 1997] and (iii) the combined data set (the present
study, and references [Selcuk 1997, Amezawa 2011, Kushi 2009, Wang 2007]). When E data
from the present study and the combined data set are least-squares fit to equations (2) to (5),
2

then based on the R values in Table 8.2, equation (2) provides the best description of the P

255

1.0
250

0.9

(E)
(E)
(E)
(E)

200

E, G and B (GPa)

Relative density, RD
0.8
0.7
0.6
(G)
(G)
(G)
(G)

(B)
(B)
(B)
(B)

0.5

0.4

This study
Selcuk 1997 [31]
Kushi 2009 [33]
Amezawa 2011 [32]

150

E

100
B
50
G
0
0.0

0.1

0.2
0.3
0.4
Relative porosity, P

0.5

0.6
PG

Figure 8.4. Young’s modulus, shear modulus and bulk modulus versus porosity for GDC10
specimens from the combined data set (the present study, Selcuk and Atkinson [Selcuk 1997],
Amezawa et al. [Amezawa 2011], and Kushi et al. [Kushi 2009]). The solid, dashed and
dotted curves represent the least-squares fit to equation (2) for the E-P, B-P and G-P data,
respectively.

256

dependence of the Young’s, shear and bulk moduli (Table 8.2). Also, the values of E0
predicted by equations (3) and (5) are lower than all of the 26 points in the Selcuk and
Atkinson study (Figure 8.4), thus, the E0 values from equations (3) and (5) are not physically
reasonable. Also, the E0 value of 292 GPa predicted by equation (4) seems unreasonably high
(Figure 8.4). In contrast, the E0 value predicted by equation (2) fits well the E versus P trend
(Figure 8.4).
2

The E0 and R values provided by Selcuk [Selcuk 1997] were similar for each equation
2

(equations (2) – (5)). However, the R values were lower than the present study or the
combined data set (the present study, and references [Selcuk 1997, Amezawa 2011, Kushi
2

2009, Wang 2007]) The lower R values for the Selcuk and Atkinson study [Selcuk 1997]
may be related to the small porosity range (0.02 to 0.05) in that study, thus a small error in
the measured porosity, P, can lead to a large relative error in the (E, P) data values.
Equation (2) was also used to fit the E, G and B values for the combined data set (with
data from the present study, and references [Selcuk 1997, Amezawa 2011, Kushi 2009, Wang
2007]) (Figure 8.4). Although Selcuk and Atkinson [Selcuk 1997], Amezawa et al.
[Amezawa 2011] and Kushi et al. [Kushi 2009] did not report B values in their studies, for
isotropic polycrystalline specimens, B can be calculated from the E and G data supplied by
Selcuk and Atkinson [Selcuk 1997], Amezawa et al. [Amezawa 2011] and Kushi et al.
[Kushi 2009] using equation (6), which allowed E, G, and B to be plotted for the combined
data set (Figure 8.4).

257

Table 8.3. The number of data points, ND, parameters from least-squares fit to equation (2),
2

2

2

E0, bE, G0, bG, B0, bB, and coefficient of determination, R (E), R (G) and R (B). E0, G0 and
B0 are intercepts of the least-squares fits with the y axis, respectively.
Reference

Relative
ND
bE R (E)
bG R (G)
bB R (B) porosity
(GPa)
(GPa)
(GPa)
range

The present
21
study
Selcuk and
Atkinson
26
[Selcuk 1997]
Combined data
set [the present
study, Selcuk
1997,
50
Amezawa
2011, Kushi
2009]

E0

2

G0

2

B0

2

235.2 4.7 0.989 88.4 4.5 0.987 190.7 5.4 0.986

0.07–0.60

217.8 2.9 0.886 82.2 3.0 0.871 211.1 2.5 0.788

0.02–0.05

226.7 4.5 0.992 85.0 4.3 0.993 233.7 6.5 0.950

0.02–0.60

258

Relative density, RD
1.0

0.9

0.8

0.7

0.6

0.5

0.4

E (GPa)

100

PC
10
PG
1

0.0

This study
Selcuk 1997 [31]
Kushi 2009 [33]
Amezawa 2011 [32]

0.1

0.2
0.3
0.4
Relative porosity, P

0.5

0.6

Figure 8.5. Semi-log plot of the Young’s modulus versus porosity for GDC10 specimens
from the combined data set (the present study, Selcuk and Atkinson [Selcuk 1997], Amezawa
et al. [Amezawa 2011], and Kushi et al. [Kushi 2009]).The solid line represents the
least-squares fit of E-P data with P up to PC ~0.4 to equation (2), and the dashed line is the
extrapolation of the solid line to higher porosity values.

259

B=

EG
3(3G  E )

(6)

The least-squares fit parameters obtained for the 50 pairs of modulus-porosity data
points for E, G and B from the combined dataset (the present study, and references [Selcuk
1997, Amezawa 2011, Kushi 2009])) to equation (2) are E0 = 226.7 ± 4.4 GPa, bE = 4.5 ± 0.2;
G0 = 85.0 ± 0.7 GPa, bG = 4.3 ±0.1; B0 = 233.7 ± 6.6 GPa, bB = 6.5 ± 0.5. The coefficients
2

of determination, R , were 0.992, 0.993 and 0.950 for E, G and B, respectively (Table 8.3).
E0, G0 and B0 (the extrapolations of the E, G and B data to P = 0) obtained from the
least-squares fit are useful in that they can be compared directly to the aggregated average
moduli from single crystal data [Simmons 1971] or the theoretical elastic moduli values
calculated from the density functional theory [Kanchana 2006] or the statistical moment
method [Van Hung 2011].
Except for the Young’s modulus measured by nano-indentation [Wang 2007], the moduli
obtained from the three different experimental techniques: RUS (the present study), sonic
resonance [Amezawa 2011, Kushi 2009] and IET [Selcuk 1997] agree extremely well, even
accounting for a possible shift in modulus due to low pO2 for GDC10 from the Amezawa et
al. [Amezawa 2011] and Kushi et al. [Kushi 2009] studies. In general, a low pO2 induces
atomic scale defects (mainly oxygen vacancies) which in turn lower the observed elastic
moduli in GDC10 specimens [Amezawa 2011, Kushi 2009, Hashida 2009] and 7wt-% yttria
stabilized zirconia [Huang 2008].
Wang et al. [Wang 2007] computed the E value from their nano-indentation data using

260

Table 8.4. The values of bE obtained from a least-squares fit of the Young’s modulus, E,
versus porosity, P, data to the equation (2) for the GDC10 specimens included in this study
and several oxides from the literature. The number of data points, ND, included in the
least-squares fit and the relative porosity range of the data is also specified.
Ref.

Material

bE value

E
E
Processing
Relative ND
range measurement technique
porosity
(GPa) method
range
The
4.7 ± 0.2 9.5 – Resonant
Cold pressed at 0.066 – 21
Ce0.9Gd0.1O1.95
present
158.3 ultrasound 27.3 MPa,
0.595
study
spectroscopy sintered 825oC
to 1450oC for 5h
Ren et
3.4 ± 0.2 12 - Resonant
Cold pressed at 0.05 – 15
Ca10(PO4)6(OH)2
al.
107 ultrasound 33 MPa, 1125oC 0.51
spectroscopy to 1360oC for 1
to 6 h
3.4 ± 0.1 105 - Resonant
Uniaxially cold 0.06 – 9
Al2O3
340 ultrasound pressed at 23
0.39
o
spectroscopy MPa, 1200 C to
1475oC for 1 to
4h
Luo
3Y-TZP
3.69
50 - Ultrasonic Single action die 0 –
12
and
220
velocity
pressed followed 0.38
Steve
method
by CIP. Sintered
ns
at 1150–1450oC
Oduleye (Bi-Pb)SrCaCuO 2.77
30 - Bending
Extruded.
0.2 –
17
et al.
75
test
Sintered at
0.53
o
o
810 C - 855 C
Petrak CoO
4.51
103 - Resonant
Pressed, sintered 0.09 – 8
et al. [
159
sphere
at 1400oC in Ar. 0.20
technique Also hot pressed
at 1300oC
4.36
137 - Resonant
Pressed, sintered 0.05 – 8
Co0.5Mg0.5O
206
sphere
at 1650oC 0.15
o
technique 1700 C in air
5.39
124 - Resonant
Pressed, sintered 0.03 – 10
CoAl2O4
234
sphere
at 1650oC 0.15
o
technique 1700 C in air
261

the Oliver –Pharr equation [Oliver 1992], such that
2

1 (1  indenter ) (1  sample )


Er
E indenter
E sample
2

(7)

where Er is the reduced modulus, which is functions of both the modulus of the indenter and
the specimen. indenter and sample are the Poisson’s ratios of the diamond indenter and the
specimen, respectively. Eindenter and Esample are the elastic modulus of the diamond indenter
and the specimen, respectively. Assuming indenter = 0.07, Eindenter = 1140 GPa for the
diamond indenter and sample = 0.3, the E value for 98% dense GDC10 [Wang 2007] is
254.6 GPa. Several research groups have noted differences between the elastic moduli
obtained via nano-indentation and the elastic moduli measured by dynamic methods such as
RUS, sonic resonance and IET. For example, in a study by Radovic et al. [Radovic 2004]
that compared E measurements by the nano-indentation and RUS techniques, the E values
measured by nano-indentation were 10% higher than E measured by RUS. Similarly, for
skutterudite thermoelectric materials, Schmidt et al. [Schmidt 2010] reported that E
measured via nano-indentation was 8% higher than E measured by RUS.
While equation (2) fits the elastic modulus versus P data relatively well, it has been
observed empirically that the fit at higher porosity (say, P > 0.30 or 0.40) is not as good as the
fit at lower P values [Rice 1998]. For example, equation (2) describes well the elastic
modulus versus porosity change in ceramics for the porosity range 0 < P < Pc, where Pc is a
critical porosity value [Rice 1998]. For P > Pc, the elastic moduli decrease faster with
increasing P than predicted by equation (2). According to Rice [Rice 1998], the value of Pc

262

1.0

0.9

Relative density, RD
0.8
0.7
0.6

0.4

This study
Selcuk 1997 [31]
Kushi 2009 [33]
Amezawa 2011 [32]

0.36
0.33
Poisson's ratio, 

0.5

0.30
0.27

PG

0.24
0.21
0.18
0.0

0.1

0.2
0.3
0.4
Relative porosity, P

0.5

0.6

Figure 8.6. Poisson’s ratio for GDC10 specimens for the combined data set (the present study,
Selcuk and Atkinson [Selcuk 1997], Amezawa et al. [Amezawa 2011], and Kushi et al.
[Kushi 2009]). The solid curves represents the least-squares fit of the -P data to equation (3)
and the dashed curves represents the least-squares fit of the -P data to equation (2).

263

depends on the pore size and spatial distribution of pores, as well as the state of
agglomeration of the powders, size of initial powders, powder packing parameters and
fabrication techniques.

However, neither Rice [Rice 1998] nor others in the literature give

particular functional relationships for the dependence of Pc on such microstructural or
processing parameters.
Based on empirical observations, for uniaxially hard die pressed and sintered powders,
Pc ~ 0.5 for alumina [Hardy 1995] and Pc ~ 0.5 for hydroxyapatite [Ren 2009].
For the GDC10 data in the present study, a semi-log plot of E versus P for the combined
data set [the present study, Selcuk 1997, Amezawa 2011, Kushi 2009] (Figure 8.5) shows that
in a semi-logarithmic plot, the fit to equation (2) is excellent for 0 < P < 0.4, however,
beginning at approximately P = ~ 0.45, E begins to deviate from the behavior predicted by
equation (2) such that the Young’s modulus drops off faster than the least squares fit to
equation (2). Thus, this behavior is consistent with that described by Rice [1998] for the
drop-off in E for P > Pc. It is important to note that this disagreement is not the result of a
difference in oxygen nonstoichiometry (and hence different lattice parameters and elastic
properties [Wang 2007]) for samples with different porosities/grain sizes. Powder x-ray
diffraction (XRD) analyses of specimen #5 (with a grain size of 1.1 µm) and the raw GDC
powder (with a grain size of ~27 nm) yielded XRD peak positions that were identical within
o

the 0.2 resolution of the XRD; resulting in a lattice parameter of 5.418 ±0.011A for both
samples. The reduced GDC mechanical property study of Wang et al. [Wang 2007] indicates
that the small oxygen vacancy concentration differences indicated by such small lattice
264

Table 8.5. From the literature for the Young’s modulus for partially sintered powder compacts
including the GDC10 specimens in this study, three oxides and titanium metal, the values of
EPNG/E0, where EPNG is the Young’s modulus at P ~ PG and E0 is the Young’s modulus at P
= 0. EPNG/E0 is similar for each material, where EPNG/E0 is independent of the choice of E
versus P model. The degree of densification, is also given.
Ref.

Material

The
present
study

GDC10

Fan et al.

HA

P

E
E
EPNG
measurement
(GPa) /E0
method

0.595 9.5

0.04

0.55

8.3

0.07

Hardy et al. Al2O3

0.45

36

0.09

Nanjangud
et al.

Al2O3

0.415 20

0.05

Deng et
al.

YSZ

0.51

21

0.09

Oh et al.

Ti

0.356

9

0.08

Processing
technique

Hard die
Resonant
pressed at
ultrasound
27.3 MPa,
spectroscopy sintered at
825oC for 5 h
Uniaxially
Resonant cold pressed
ultrasound at 33 MPa,
spectroscopy sintered at
550oC for 2 h
Uniaxially
Ultrasonic
pressed at 45
velocity
MPa, sintered
technique
at 800oC for 2h
Ultrasonic
Extruded,
velocity
sintered at
technique 800oC for 2 h
Single ended
Pulse-echo pressing at 75
method MPa, sintered
at 1100oC for
30 min
Pressed at 70
MPa for 0.6
Compression
ks, sintered at
test
1573 K for
7.2 ks
265



Green
porosity

0.04

0.62

0.11

0.62

0.01

0.46

0.01

0.42

0.09

0.56

0.01

0.36

parameter differences would not cause observable elastic modulus changes.
3.3 Poisson’s ratio as a function of P
The porosity dependence of Poisson’s ratio, is more complicated than that for E, G
and B. In the literature, is relatively insensitive to P for alumina [Phani 2008] and LAST
(Lead–antimony–silver–tellurium) [Ni 2009]. However, based on empirical observations
Boccaccini [Boccaccini 1994] suggested that the changes in  as a function of P is related to
the values of Poisson’s ratio at P = 0 for a given material. For literature from fifteen
different ceramics, including both oxides and non-oxides, for   tended to decrease
as P increased and for  tended to increase as P increased [Boccaccini 1994].
For the GDC10 specimens in the present study, Poisson’s ratio decreased with
increasing porosity (Figure 8.6). For P = 0.07,  is approximately 0.29 and for 0.5 < P < 0.6,
dropped to approximately 0.21. Thus, the  versus P behavior for GDC10 specimens is
broadly consistent with the trends identified by Boccaccini [Boccaccini 1994]. Also, although
Amezawa [Amezawa 2011] and Kushi [Kushi 2009] did not report the  values, the value of
 can be computed from Amezawa et al.’s and Kushi et al.’s E
and G data via [Wachtman 2009]
 = E/2G - 1

(8)

For the combined data (with data from the present study, and references [Amezawa 2011,
Kushi 2009, Selcuk 1997]), a least-squares fit of Poisson’s ratio versus P using both equation
(3) (linear) and equation (2) (exponential) yielded 0 = 0.33, with coefficients of

266

2

determination, R , of 0.819 and 0.844, respectively. Although the fit to equations (2) and (3)
is relatively poor, this is likely due in part to the inherent scatter in the  versus P data.
3.4 Elastic modulus-porosity for other materials in the literature
The literature also shows that porosity dependence of the Young’s modulus-porosity, for
example, can be described by equation (2) for a number of oxides such as Al2O3 [Rem 2009],
Ca10(PO4)6(OH)2 [Ren 2009], 3Y-TZP [Luo 1999], (Bi-Pb)SrCaCuO [Oduleye 1998], CoO
[Petrak 1975], Co0.5Mg0.5O [Petrak 1975] and CoAl2O4 [Petrak 1975] (Table 8.4). In
addition, bE values from the literature for the seven oxides listed in Table 8.4 bracket the bE
of 4.7 for GDC10 in the present study, namely the bE ranges from 5.39 for CoAl2O4 [Petrak
1975] to 2.77 for (Bi-Pb)SrCaCuO [Oduleye 1998].
For the GDC10 specimens in the present study, the Young’s modulus dropped from ~
160 GPa to ~ 10 GPa as P increased from 0.07 to 0.60. This dramatic drop in elastic modulus
with increasing P also is consistent with the literature for a wide range of materials including
the oxides (Ca10(PO4)6(OH)2 [Fan 2012], Al2O3 [Hardy 1995, Nanjangud 1995], YSZ
[Deng 2002] and metal Ti [Oh 2003] (Table 8.5). The degree of densification,
P/PGwas included in Table 8.5 to facilitate direct comparison among studies
with differing values of the green porosity, PG (the present study, [Fan 2012a, Fan 2012b,
Hardy 1995, Nanjangud 1995, Deng 2002, Oh 2003]). If we consider EPNG, the Young’s
modulus at P ~ PG and normalize EPNG by E0, then EPNG/E0 is roughly 0.04 to 0.09 for a
number of materials, including GDC10 in the present study, as well as (Ca10(PO4)6(OH)2

267

[Fan 2012], Al2O3 [Hardy 1995, Nanjangud 1995], YSZ [Deng 2002] and metal Ti [Oh 2003]
(Table 8.5).
4. Summary and Conclusions
Twenty-one Ce0.9Gd0.1O1.95 (GDC10) specimens were hard-die pressed and partially
sintered in air at temperatures from 825oC to 1475oC for 5 hours to produce specimens with
volume fraction porosities, P, from 0.07 to 0.60. Room temperature Young’s, E, shear, G. and
bulk, B, moduli were measured as a function of P using resonant ultrasound spectroscopy.
The least-squares fit of E, G. and B to equations (2) through (5) showed that the exponential
equation (equation (2)) best described the observed porosity dependence. The values of E0,
G0 and B0, obtained by extrapolating E, G and B to zero porosity are useful since E0, G0 and
B0 can be compared to (1) aggregated average moduli from single crystal data or (2) elastic
moduli calculated from density functional theory.
The decrease in elastic moduli with increasing P was very dramatic, with the E, G and B
values at P = 0.60 being only on the order of roughly 5 % of the E0, G0 and B0 values,
respectively. As summarized in Table 8.5, for specimens with very high porosity levels,
similar drastic decreases in elastic moduli for specimens have been observed for a number of
other oxide materials in addition to GDC10.
In solid oxide fuel cell applications, porous GDC structure will undergo thermal and
mechanical stresses arising from manufacturing, external loading and differences in thermal
expansion coefficients between cell layers. The knowledge of elastic moduli of GDC is
essential to model the response of the SOFCs to internal or external stresses and to design
268

more reliable SOFCs with longer service life. Thus, it is of great importance to know the
elastic moduli at different porosity levels since without using the appropriate moduli in
analytical and numerical models, the predicted stresses and strains will exhibit very large
errors.

5. Acknowledgements
This work was supported, in part, through a Michigan State University Faculty Startup
Package.

269

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CHAPTER 9
ROOM TEMPERATURE HARDNESS OF GADOLINIA-DOPED CERIA AS A
FUNCTION OF POROSITY
X. Fan, E. D. Case, Qing Yang and Jason D. Nicholas
Department of Chemical Engineering and Materials Science, Michigan State University,
East Lansing, MI, 48824
Submitted to: J Materials Science.

Abstract
Porous gadolina doped ceria (Ce0.9Gd0.1O1.95 , GDC10) is commonly used as a
functional electrode support in solid oxide fuel cells, gas sensors, and gas separators. In
addition, dense GDC10 is commonly used as a solid oxide fuel cell electrolyte. Although
porosity affects a wide range of electrical, thermal and mechanical properties of solids, this
study focuses on (i) the Vickers indentation hardness, H, as a function of volume fraction
porosity, P, ranging from 0.08 to 0.60 and (ii) the load dependence of H for Vickers
indentation loads of 0.98 N to 9.8 N. For the thirteen GDC10 included in this study, the
decrease in H with increasing P is approximated by the empirical relationship H =
H0exp(-bHP), where H0 = 5.844 GPa and bH = 6.68. In addition, H was independent of the
applied indentation load.

277

1. Introduction
Porous ceramics have a wide range of uses, including as filters [Yang 2007a, Yang
2007b, Khattab 2012], sensors [Dawicke 1986, Kale 2009, Mahabole 2005], catalytic
supports [Twigg 2007], drug delivery platforms [Son 2011], engineered bone tissue growth
scaffolds [Baumann 2007, Ren 2005], and solid oxide fuel cell (SOFC) electrodes [Nicholas
2010, Nicholas 2012, Vohs 2009, Yang 2012, Sarikaya 2012, Liang 2012]. However, the
porosity that enables the various sensor and electrode functions also affects a broad range of
physical properties, including electrical conductivity [Geis 2002, Shimizi 2009, Sulistyo
2010], dielectric constant [Geis 2002, Sulistyo 2010, Hoepfner 2002, Case 2006, Yang
2010], dielectric breakdown strength [Geis 2002, Ewais 2011], thermal conductivity [Hu
2010], thermal diffusivity [Luo 1997], fracture strength [Fan 2012, Nanjangud 1995, He
2008, Pramanik 2007, Yao 2005, Hu 2010], Weibull modulus [Fan 2012b, Fan 2013],
fracture surface energy [Case 1981, Vandeperre 2004], Young’s modulus [Ren 2009,
Boccaccini 1994, Selcuk 197], shear modulus [Ni 2010, Ni 2009, Schmidt 2010], and
hardness [Hoepfner 2003, Li 2012].
In particular, hardness, H, is related to a material’s wear resistance [Zeng 2005,
Krakhmalev 2006] and machinability [Kamboj 2003, Wang 2002], which are two essential
factors for fabrication and mechanical stability in application.

In addition, for dense, fine

grained materials, a material’s compressive strength is roughly H/3 [Rice 2002]. As is the
case with many material properties, microstructure can affect hardness. Although the
quantity, size, and distribution of inclusions [Canakci 2011] and/or precipitates [Liao 2011]
are the most commonly discussed microstructural features affecting hardness, porosity is also
a critical microstructural feature because it exponentially lowers H.

278

This study focuses on the hardness of gadolina doped ceria (Ce0.9Gd0.1O1.95 , GDC10)
as a function of (i) porosity and (ii) the applied Vickers indentation load. “Dense” gadolinia
doped ceria (GDC), i.e. that with a volume fraction porosity, P, typically ranging from 0 to
0.08, is commonly used as an intermediate temperature (500-700oC) SOFC electrolyte [Minh
1995]. Porous GDC is commonly used as a anode catalyst [Sauvet 2001, Trovarelli 1996],
mixed ionic electronic conductor (MIEC) [Goodenough 2007], oxygen storage material
[Trovarelli 1996], and mechanical support [Liu 2002] in SOFC anodes, gas sensors, and gas
separation membranes [Gellings2007, Chen 2011, Muthukkumaran 2008]. In addition,
porous GDC is also used as an ionic conductor and electro-catalyst support in
high-performance, low-temperature SOFC nano-composite cathodes [Nicholas 2012,
Nicholas 2010]. The P of GDC in these electrode applications typically ranges from 0.2 to
0.55, with the lower limit set by the need for an interpenetrating gas network and the upper
limit set by the mechanical strength of the component and the need to prevent ionic current
focusing [Nicholas 2009].
In this study, measurements of the hardness, H, for 13 partially sintered GDC10
specimens were performed by Vickers indentation. The resulting H decreased roughly
exponentially as a function of increasing porosity over the volume fraction porosity, P, range
from 0.08 to 0.60 where P = 0.60 corresponds to the green state porosity for the specimens in
this study. For fixed values of P, H was approximately independent of Vickers indentation
load over the range from 0.98 N to 9.8 N.
2. Experimental procedure
2.1 Specimen preparation

279

Each of the thirteen specimens in this study was included in a previous study of the elastic
modulus of GDC10 [Fan 2013]. In both this study and the Fan et al. study [Fan 2013],
approximately 2.7 grams of 99.9% pure Ce0.9Gd0.1O1.95 (GDC10) nano-powders (Rhodia,
Inc. Cranbury, NJ) with ~27 nm mean particle size were uniaxially cold pressed at 27.3 MPa in
a 19 mm diameter steel die. The specimens were then fired in air for 5 hours in an electrical
resistance furnace at temperatures from 825oC to 1450oC to produce disc-shaped specimens
with volume fraction porosities, P, ranging from 0.08 and 0.60. The porosity was calculated
from the specimen mass (measured with an electronic balance) and specimen dimensions
(measured by digital calipers). Although Fan et al. [Fan 2013] measured the elastic moduli for
the specimens in the as-sintered state, prior to the hardness testing, each specimen was
polished via an automatic polishing machine (Leco Vari/Pol VP-50, Leco Corporation, St.
Joseph, MI) using a series of diamond pastes with grit sizes ranging from 90 m to 1m.
2.2 Microstructural examination procedure
An Auriga Dual Column focused ion beam-scanning electron microscope (FIB-SEM,
Carl Zeiss Microscopy, LLC, Thornwood, NY) was used to examine the polished and
fractured surfaces of selected GDC10 specimens. An accelerating voltage of 15 kV and a
working distance of 5 mm were used. The average grain size was calculated using linear
intercept technique with a stereographic projection factor of 1.5 [Underwood 1969]. Prior to
SEM examination, a gold coating approximately 15 nm thick was sputtered on the specimen
surfaces to enhance their electrical conductivity.
X-ray diffraction on the sintered specimens and the as-received GDC10 powders was
performed using a Rigaku MiniFlex II diffractometer with Cu Kα radiation using a 2 range
of 20o to 80o, a scan rate of 0.5o min-1 and a 0.05o step size.
280

2.3 Hardness measurements
Using a Vickers indenter (Shimazdu HMV 2000, Kyoto, Japan), ten indentations were
placed on each GDC10 specimen. For every specimen except for the specimens with P =
0.55 and P = 0.60, the indentations were performed using a 4.9 N load and a 10 second
loading time. However, at a load of 4.9 N, the indentation impressions for the most porous
specimens (P = 0.55 and P = 0.60), were greater than 350 m across so that their images
filled the field of view of the optical microscope mounted on the indenter. For specimens
with P = 0.55 and P = 0.60, the Vickers indentation loads therefore were reduced to 2.94 N
and 0.98 N, respectively, which reduced the dimensions of the indentation impressions
sufficiently to allow them to be measured. The hardness measurements performed by the
Vickers indenter were calibrated using a HV 790 steel standard calibration block (Yamamoto
Scientific Tools Lab Co. LTD, Chiba, Japan). For each load used in this study (0.98 N, 2.94
N, 4.9 N and 9.8 N), the calibration factors,  were determined from the average hardness
value obtained from ten indentations of the calibration block using a loading time of 10
seconds. For the entire range of loads included in this study, the calibration factor,  ranged
from 0.95 to 0.97.
The hardness values, H, for the GDC specimens were calculated from the average of 10
indentations per specimen using the following equation [Lawn 2012]
H = 

1.8544F
(1)

( 2 a )2

where the dimensionless constant, is the calibration factor obtained from the indentation of
the calibration block, F is the applied indentation load and a is half of the diagonal length of
the indentation impression.

281

(a)

(b)

Figure 9.1. SEM micrographs of (a) polished surface for the GDC10 specimen with P = 0.08,
(b) fractured surface for the GDC10 specimen with P = 0.10.

282

In addition, the load dependence of H was studied using three of the GDC10 specimens,
namely with P = 0.08, P = 0.23 and P = 0.38. Each of the three specimens were indented at
four values of load (0.98 N, 2.94 N, 4.9 N and 9.8 N), again with ten indentations per load
value on each specimen.
3. Results and Discussion
3.1 Microstructural and phase analysis
SEM micrographs of the polished surface for the specimen with P = 0.08 (Figure 9.1a)
showed irregularly-shaped pores ranging from roughly 1m to 3 m across, with occasional
pore clusters. The fractured surface for the specimen with P = 0.10 (Figure 9.1b) displayed
equiaxed grains and a predominately intergranular fracture mode with pores also in 1 to 3 m
range. No microcracks or macrocracks were observed on either the polished surfaces (Figure
9.1a) or on the fractured surfaces (Figure 9.1b) of the specimens.
In a recent elasticity study, Fan et al. [Fan 2013] found that the grain size-density
trajectory for the GDC10 specimens included in the present study followed the trend typical
of a sintered ceramic compact [Berry 1986, Barsoum 1997]. In particular, as P decreased
from about 0.6 to 0.2 during sintering, there was a relatively small increase in grain size from
about 0.1 m to 0.3 m.

As further densification occurred, the grain growth accelerated,

reaching a maximum grain size in this study of 1.1 m for the minimum porosity of P = 0.08.
All peaks in the XRD analysis (Figure 9.2) corresponded to GDC10, which is
consistent with the 99.9% purity of the starting powders (Section 2.1).
3.2 Hardness as a function of porosity
The H versus P behavior for brittle materials is often described by the empirical
exponential relationship (equation 2) [Rice 1998]

283

Figure 9.2. XRD patterns of the as-received GDC10 powder and GDC10 specimen sintered
at 1400oC (P = 0.10).

284

0.9

Relative density
0.8
0.7
0.6

6

0.4

This study
Mangalaraja et al.

5

Hardness (GPa)

0.5

4
3
2
1
0
0.0

0.1

0.2

0.3
0.4
Relative porosity

0.5

0.6

Figure 9.3. Hardness versus porosity for a combined GDC10 data set (this study and
Mangalaraja et al. [Mangalaraja 2009]), where the curve represents a least-squares fit to
equation 2. Note that in this study the indentation load was 4.9 N for all the specimens except
two with the highest porosities and in Mangalaraja’s study [Mangalaraja 2009] the
indentation load was 5 N. For P > 0.2, the error bars on the H data for this study are smaller
than the symbol size.

285

H = H0exp(-bHP)

(2)

where H0 is the hardness value at P = 0, bH is a dimensionless, material-dependent constant
that is related to the rate of H change as a function of P. A least-squares fit of the H versus P
for the thirteen GDC10 specimens in this study (Figure 9.3) to the equation (2) yielded H0 =
2

5.84 ± 0.57 GPa, bH = 6.68 ± 0.77 with a coefficient of determination, R , of 0.955.
In this study, the maximum P value is P = 0.60, which is nearly equal to the unfired
(green) porosity of PG = 0.62 for the GDC10 specimens. However, few studies in the
literature include values of H for brittle materials for P greater than 0.35. An exception is a
Vickers indentation study of the porosity dependence of H for Ti doped Al2O3- Cr2O3 solid
solutions by Cho [69] with 0.05 < P < 0.45, however in Cho’s study both the composition (Ti
addition and the Al2O3/Cr2O3 ratio) and P varied simultaneously. Also, Ramadass [Ramadass
1983] studied the Vickers hardness of Y2O3 partially stabilized zirconia for 0 < P < 0.45, but
once again both P and the specimen chemistry changed in tandem (with the Y2O3 additions
ranging from 0 to 7.5 mol%). Unfortunately, neither the Cho [Cho 1990] nor the Ramadass
[Ramadass 1983] studies included a sufficient number of specimens to clearly separate the
compositional effects from the P dependence of H. Thus, the current study is one of very few
in the literature to study the porosity dependence of H over a large range of P (0.08 < P <
0.60).
In addition, studies of the porosity dependence of H for GDC10 are also very limited
even for a restricted P range. For example, Mangalaraja [Mangalaraja 2009] fabricated
GDC10 specimens using a nitrate-fuel combustion method with four different organic fuels:
286

urea, citric acid, glycine and poly ethylene glycol. However, the Mangalaraja study included
a very limited porosity range, with P ranging only from about 0.02 to 0.08 [Mangalaraja
2009]. Both the indentation load (4.9 N [Mangalaraja 2009] and 5.0 N [this study]) and the
grain size range (0.4 to 1.0 m [Mangalaraja 2009] and 0.1 to 1.0 m [this study]) were
comparable between the two studies.
This similarity between the indentation load and grain size of the two H studies of
GDC10 [Mangalaraja 2009 and this study] motivated the least-squares fit to equation (2) of
the 13 H values from this study along with four hardness values from Mangalaraja’s study
[Mangalaraja 2009] which yielded the fitting parameters H0 = 6.92 ± 0.50 GPa, bH = 8.37 ±
2

1.01 and R = 0.940.
Despite the relatively high value of the coefficient of determination for the H versus P
data included in this study (Figure 9.3), Rice [Rice 1998] has shown that for a critical
porosity PC (which is typically on the order of 0.3 to 0.4), H versus P data often deviates from
the exponential relationship for P > Pc. A semi-log plot of the hardness data of thirteen
GDC10 specimens in this study (Figure 9.4) shows that H decreases more rapidly than
predicted in equation 2 for P > 0.4. A similar behavior was observed for the Young’s
modulus versus P behavior for GDC10 [Fan 2013] where Pc was approximately 0.45 and as
reviewed by Fan et al [Fan 2013] this behavior is commonly observed in for the Young’s
modulus, E, versus P data for highly porous ceramics. Furthermore, the departure from the
empirical exponential relationship (equation 2) for H as a function of P is also commonly
observed for highly porous ceramics [Rice 2000]. It should be emphasized, however, that
equation 2 (as well as the exponential E versus P relationship) is an empirical equation and

287

0.9

0.8

0.1

0.2

Relative density
0.7
0.6

0.5

0.4

0.5

0.6

Hardness (GPa)

1

0.1

0.01

0.3

0.4

Relative porosity

Figure 9.4. Semi-logarithmic plot of the hardness versus porosity for GDC10 specimens in
this study. The solid line represents a least-squares fit of data to equation (2) for the P range
of 0.08 < P < 0.4.

288

this H versus P behavior may only indicate that the exponential relationship is inadequate to
describe the nature of H for high values of P.
In this study, the diagonal length of the Vickers indentation impression, 2a, ranged from
50.3 m for the specimen with P = 0.10 to 341.5 m for the specimen with P = 0.52. The
typical pore sizes obtained from the SEM micrographs were from 0.3 m to 3 m so that the
ratio of indentation impression size to the pore size falls between roughly 15 and 100. In
addition, for the GDC10 specimens in this study, 2a was >> the mean grain sizes, which
ranged from about 0.1 to 1.0 m. Thus, 2a is large compared to the specimens’
microstructural features (pores and grains) so that the H values reported in this study should
represent bulk values.
The hardness versus porosity relationship for GDC10 from this study and other oxides
from the literature including Ca10(PO4)6(OH)2 [Hoepfner 2003], 3Y-TZP [Luo 1999],
Pb0.74Ca0.26[(Co0.5W0.5)0.05Ti0.95]O3 + 1% MnO [Ricote 1994] is described relatively well
by the exponential relationship (equation 2, Table 9.1). The values of bH for GDC10 from
this study and the combined data set [this study, Mangalaraja 2009] are 6.68 and 8.37,
respectively, which are higher than the bH values for the literature data for the three oxides,
namely bH = 6.03 for Ca10(PO4)6(OH)2 [Hoepfner 2003], bH = 5.04 for 3Y-TZP [Luo 1999],
and bH = 3.08 for Pb0.74Ca0.26[(Co0.5W0.5)0.05Ti0.95]O3 + 1% MnO [Ricote 1994] (Table
9.1).
3.3 Hardness as a function of load
For a variety of materials, the value of H has been observed to be a function of the
applied indentation load [Nix 1998, Sangwal 2000, Bull 1989, Pharr 2010]. When H is a
289

Table 9.1. For the GDC10 specimens included in this study and porosity-hardness data for
several oxides from the literature, the values of bH obtained from a least-squares fit of the
hardness, H, versus porosity, P, data to equation 2 (H = H0exp(-bHP)). The number of data
points, ND, included in the least-squares fit and the relative porosity range of the data is also
specified.
Ref.

Material

This
study

Ce0.9Gd0.1O1.95
(GDC10)

bH
6.68 ±
0.77

H
range
(GPa)
0.05
to
3.55

H meas.
method

Hard die
pressed at
27.3 MPa,
sintered
from 825oC
to 1450oC
for 5 h
Hoepfner Ca (PO ) (OH) 6.03
0.91 Vickers, Uniaxially
10
46
2
and Case
to
load =
cold pressed
2003
5.54 2.94, 4.9 at 6.55
and 9.81 MPa,
N
sintered
from 1050
o
C to 1400
o
C
Luo and 3Y-TZP
5.04
~ 1.5 Vickers Single
Stevens
to
action die
1999
~ 11.8
pressing
followed by
CIP.
Sintered at
1150 –
1450oC
Ricote et Pb
3.08
~ 2.0 Vickers Sintered.
0.74Ca0.26[(Co
a
to
Sintering
al. 1994
0.5W0.5)0.05Ti0.9
~ 3.3
conditions
not
5]O3 + 1% MnO
specified.
a
Data taken from the paper by Datathief and replotted by the authors

290

Vickers,
load =
4.9 N

Processing
technique

Relative
porosity
range
0.08 –
0.60

ND

0.02 –
0.31

42

0 – 0.38

13

0.04 –
0.18

6

13

function of load, the load dependence is typically discussed in terms of either an indentation
size effect (ISE) [Nix 1998] or a reverse indentation size effect (RISE) [Sangwal 2000]. For
ISE, the measured H decreases with increasing applied load, which has been observed for
both ceramics [Bull 1989, Sangwal 2009] and metals [Pharr 2010]. ISE has been attributed to
elastic recovery and working hardening during indentation or surface dislocation pinning in
metals [Gong 1999]. For RISE [Sangwal 2000], the H values decrease with decreasing load,
where stress relaxation during unloading [Sangwal 2000] has been proposed as a mechanism
for RISE [Sangwal 2000].
In this study, three GDC10 specimens with P = 0.08, P = 0.23 and P = 0.38 were
indented under a series of loads (0.98 N, 2.94 N, 4.9 N and 9.8 N), with 10 indentations
performed at each load for each specimen. In this study, the hardness values for each
specimen were relatively constant over the entire load range (Figure 9.5). The average
hardness values (indicated by the horizontal lines in Figure 9.5) were 3.38 ± 0.47 GPa, 1.24 ±
0.15 GPa and 0.38 ± 0.04 GPa for specimens with porosities P = 0.08, P = 0.23 and P = 0.38,
respectively. Thus, for the GDC10 specimens included in this study, no significant ISE or
RISE is observed for the range of indentation loads included in this study.
A recent study Ce1-XGdXO2 (for x = 0 to 2) by Korobko et al. [Korobko 2012] reported
-1

an H value of about 6.8 GPa under a “slow” loading rate of 3 mNs and 8.1 GPa with a
-1

“fast” loading rate of 15 mNs . However the differences in the specimens and experimental
procedures between the Korobko et al. [Korobko 2012] study and the current study makes
comparison of the H data very difficult. For example, Korobko et al. [Korobko 2012] used a
nanoindenter with a Berkovich indenter with a triangular-based indenter tip at a fixed load of

291

Vickers hardness for the specimen with P = 0.38
Vickers hardness for the specimen with P = 0.23
Vickers hardness for the specimen with P = 0.08

5.0
4.5
4.0

Hardness (GPa)

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

0

2

4

6
Load (N)

8

10

Figure 9.5. The hardness versus applied indentation load for three selected GDC10
specimens in this study. The horizontal lines represent the average of the H values as a
function of load, namely the solid line for P = 0.38, the dashed line for P= 0.23 and the dotted
line for P = 0.08.

292

0.15 N while this study employed a Vickers indenter with a square-based pyramidal indenter
tip at loads of 0.98 to 9.8 N. Also, while the instrumented nanoindenter used by Korobko et
al. [Korobko 2012] is capable of varying the load rate, the typical Vickers indenter such as
the instrument used in this study does not have that capability. In addition, the Korobko et al.
[Korobko 2012] stated that for the specimens in their study “grain size exceeded 1.5 m for
all samples” and that the “density of all pellets exceeded 95%”. This uncertainity in the
microstructural parameters makes comparison difficult, especially given the steep slope of
the H versus P curve for P values less than roughly 0.1 (Figure 9.3 and equation 2). Also, it is
widely observed for oxides and other brittle materials [Kolemen 2006, Ozturk 2013,
Daguano 2012, Machaka 2011] that an ISE becomes significant at loads less than roughly 1
to 2 N, such that as the indentation load approaches the 0.15 N employed in the Korobko et
al. [Korobko 2012] nanoindentation study, H typically increases dramatically. Thus, the
differences in microstructure, the experimental apparatus, the experimental technique along
with the possible ISE at the 0.15 N loads make it difficult to make a definitive comparison
between the H values found in this study those reported by Korobko et al. [Korobko 2012].
Mangalaraja [Mangalaraja 2009] also investigated the H versus Vickers indentation load
behavior for GDC10. For GDC10 specimens processed with citric acid and glycine (P ~ 0.02
in both cases), the measured H increased approximately linearly from about 5 GPa to
approximately 9 GPa as the applied load increased from 1 N to 20 N [Mangalaraja 2009].
However, except for an increase from about 4.6 GPa at 1 N load to roughly 5.3 GPa at 5 N
load for GDC10 processed with urea, H was relatively independent of load over the range of
1 to 20 N for specimens processed with urea and poly ethylene glycol fuels. The difference in
the load dependent behavior of H among the processing routes used by Mangalaraja

293

[Mangalaraja 2009] may be related at least in part to processing-related differences in
specimen chemistry since the authors state that “the fuels released high carbonaceous
residues during combustion”. Thus, the amount of residual carbon in the specimens may
differ. Also, the exothermic peak heights are roughly 50% higher for GDC10 specimens
processed with citric acid and glycine than for GDC10 specimens processed with urea and
poly ethylene glycol (Figure 9.3 in [Mangalaraja 2009]). Thus, although the authors
[Mangalaraja 2009] did not discuss it, the differences in the exothermic reactions coincides
with the differences in hardness-load dependence behavior, with marked RISE behavior for
GDC10 specimens processed with citric acid and glycine and little or no RISE behavior for
GDC10 specimens processed with urea and polyethylene glycol.
Studies that include both the load dependence and porosity dependence of H are sparse
in the literature. One study by Milman [Milman 1999] measured the hardness versus load for
silicon carbide specimens with P of 0, 0.05, 0.16 and 0.20 at loads of 1 N, 10 N and 50 N.
For the four porosity groups, a load-dependent H was observed with H decreasing with
increasing load more sharply at higher P than lower P. The hardness values at 50 N load
decreased 18%, 20%, 60% and 70% compared to the H values obtained at 1 N load for
specimens with P = 0, 0.05, 0.16 and 0.20, respectively. In this study, no significant load
dependence of H was observed for GDC10 specimens with P = 0.08, 0.23 and 0.38,
indicating that for the porosity range from 0.08 to 0.38, H for GDC10 is relatively constant
over the load range.
3.4 Hardness versus grain size behavior
In this study, we did not measure H as a function of G, but we will discuss here the
difficulties with H versus G behavior of ceramics in general and the grain size independence

294

of H observed for GDC in particular. A Hall-Petch type relationship between H and grain
size, G, that is available in the literature is [Rice 2000]

H  H0  kG1/ 2

(3)

where H0 is considered as the H value in the single crystal limit and k is called the Petch
parameter. While Hall-Petch type relationships are commonly observed for the strength of
both metals and ceramics [Lawn 2012], for hardness Hall-Petch behavior is very common in
metals [Lawn 2012] but the nature of the H versus GS behavior is uncertain for many
ceramics [Rice 2000, Armstrong 2011]. In an extensive review of the G dependence of H in
ceramics, Rice [Rice 2000] notes that Hall-Petch behavior is common for relatively soft
ceramics such as alkaline halides, but there is a great deal of scatter in the H versus GS data
scatter for most hard ceramics [Rice 2000, Armstrong 2011]. Furthermore, Rice notes [Rice
2000] that H0 obtained from H versus G data is not always a good predictor for the H of
single crystals. In addition, for some ceramics, such as ZrO2 and AlMg2O4, H is independent
of grain size [Rice 2000]. A further complication for using equation 3 is that minima in H
versus G data are often observed for intermediate grain sizes (G on the order of several
microns) and thus the Petch parameter k frequently takes on both negative and positive
values (with unequal magnitudes) for a given set of data [Rice 2000, Armstrong 2011]
whether it suggested that the observed minima may be related to surface or subsurface
cracking near the indentation site. Nevertheless, there is not an alternative to equation 3 in
the literature and thus in general the H versus G behavior for brittle materials is not clear
from the standpoint of the available data or the available functional relationship (equation 3).

295

For gadolina-doped ceria materials in particular, Mangalaraja et al. [Mangalaraja 2009]
noted that for each of the GDC10 specimens included in their study (which represents mean
grain sizes from 0.4 to 1.0 m and P ~ 0.02) “all the samples showed grain size independent
hardness”.

For GDC20 (Ce0.8Gd0.2O2- δ) with a mean grain sizes of 0.5 m to 9.5 m

and P from about 0.02 to 0.01, Zhang et al. [Zhang 2004] observed that H “increased
slightly” as G increased from the smallest to largest grain size. Thus from the literature, and
the dramatic decrease in H with increasing P observed in the present study (Figure 9.3), it
appears that H for GDC is relatively independent of grain size for grains sizes between about
0.4 and 9.5 m. However, a study beyond the scope of the present work aimed at
understanding the full relationship between hardness, porosity, and grain size is needed
before this can be concluded for certain.
4. Summary and Conclusions
This paper reports room temperature GDC10 hardness data over a wide range of
porosities for the first time. Specifically, a total of 13 GDC10 specimens with volume
fraction porosities, P, from 0.08 to 0.60 were cold pressed and partially sintered. Vickers
hardness values were measured (i) as a function of P for a fixed load of 4.9 N (Figure 9.3)
and (ii) as a function of indentation load from 0.98 to 9.8 N for fixed P values of 0.08, 0.23
and 0.38 (Figure 9.5).
The least-squares fit of the hardness data to the exponential H versus P relation
(equation 2) described the hardness versus porosity data relatively well, yielding H0 = 5.84
GPa, bH = 6.68 and a R2 of 0.955. However, for Pc > 0.4 in this case, the hardness decreased
faster with increasing P than is predicted by the exponential relation given in equation 2

296

(Figure 9.4). In contrast to the Mangalaraja study [Mangalaraja 2009], over the range of
loads and porosities included this study, no significant load dependence on the hardness was
observed for the GDC10 specimens (Figure 9.5).
5. Acknowledgements
Authors Qing Yang and Jason D. Nicholas acknowledge support of Michigan State
University Faculty Startup Package.

297

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298

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303

CHAPTER 10
ROOM TEMPERATURE MECHANICAL PROPERTIES OF NATURAL MINERAL
BASED THERMOELECTRICS
X. Fan1, E. D. Case1, X. Lu2, D. T. Morelli1,2
1
Department of Chemical Engineering and Materials Science, Michigan State
University, East Lansing, MI, 48824
2
Department of Physics & Astronomy, Michigan State University, East Lansing, MI,
48824
Submitted to: Journal of Materials Science

Abstract
Low-cost, highly efficient thermoelectric materials for waste heat recovery applications
can be made by combining the naturally-occurring thermoelectric mineral tetrahedrite
(Cu10Zn2As4S13) and the synthetic compound (Cu12Sb4S13). To better utilize this material
in waste harvesting applications, it is essential to characterize the material’s mechanical
properties including elastic modulus, hardness and fracture toughness. In this study, powders
of Cu10Zn2As4S13 were mixed with varying amounts of Cu12Sb4S13 and then densified by
hot pressing. The room temperature mechanical properties were investigated as a function of
(i) composition and (ii) ball milling time. Elastic moduli were measured using resonant
ultrasound spectroscopy (RUS). Hardness and fracture toughness were determined by
Vickers indentation technique.

304

1. Introduction
Because of their potential ability to convert heat to electricity, thermoelectric materials
are under intense scrutiny by both experimental and theoretical researchers. Through a
variety of new approaches, significant improvements in thermoelectric figure of merit have
been achieved over the last decade. One example are the filled skutterudite compounds, first
investigated in the mid-1990’s [Morelli 1995, Sales 1996], that have recently been shown to
exhibit dimensionless figure of merit exceeding 1.5 at 800 K [Shi 2011]; another example are
hierarchically-structured PbTe –PbS solid solutions, that combine point defects, grainboundaries, and precipitated second phases on the nanoscale to achieve ZT values in excess
of two [Biswas 2012].
While these improved ZT values are exciting, some concerns remain regarding their
implementation in commercial devices, mainly due to their use of low abundance elements
such as rare earths and tellurium. For this reason, work continues to focus on the
development of materials that not only can exhibit high thermoelectric performance, but also
possess the potential to be synthesized economically on a large scale; examples include
oxide-based and silicon-based thermoelectrics. As is typical of new high-performance
materials in many applications, there is a trade-off between high figure of merit and cost, and
one can make a good case that materials that do not necessarily exhibit the highest
performance but that can be produced inexpensively will have wide-scale implementation in
heat-recovery and power generation schemes. Another good example of such compounds,
recently reported by Lu, et al., [Lu 2013a] are semiconductors based on tetrahedrite,
Cu12Sb4S13. They showed that compounds of composition Cu12-x(Fe,Zn)xSb4S13 can reach
figure of merit close to unity at 700 K. Unlike typical semiconductor thermoelectrics, in

305

which the doping level needs to be fairly carefully controlled to optimize ZT, the figure of
merit in these compounds is relatively insensitive to the amount of metal impurity x. Lu, et
al. noted that the range of x spanned the range determining the composition of natural
mineral tetrahedrites, and for this reason they further suggested that one might be able to use
the natural mineral itself as a source thermoelectric material. In a subsequent very recent
work, Lu and Morelli showed that tetrahedrite samples containing as much as 50 % natural
mineral tetrahedrite did indeed possess similar values for ZT [Lu 2013b]. Since tetrahedrites
are the most widespread sulfosalts on Earth, this presents the potential for an earth-abundant,
inexpensive source of thermoelectric materials.
In addition to ZT and the transport properties, mechanical properties are crucial to the
successful fabrication and application of thermoelectric generators. For example, waste heat
recovery applications will impose a variety of thermal and mechanical stresses on
thermoelectric materials originating from the heat up and cool down of the waste heat source.
The thermal stress, T, can be induced by the thermal gradients or thermal transients
[Kaliakin 2002, Martin 1973] during waste heat recovery. In order to simulate the thermal
stress by either analytic or finite element method, the Young’s modulus and Poisson’s ratio
are required [Kaliakin 2002, Zienkiewicz 2005], such that
T 

E(T)(T)T
1  (T) 

where E(T) is the temperature-dependent Young’s modulus, (T) is the temperaturedependent Poisson’s ratio, (T) is the temperature-dependent thermal expansion, and T is
the temperature difference.

306

However, the literature on the mechanical properties of thermoelectric materials is
limited. In particular, for both naturally-occurring and synthetic tetrahedrite compositions
such as Cu10Zn2As4S13, mechanical property data are entirely absent in the literature.
2. Experimental procedure
2.1 Specimen preparation
The natural thermoelectric mineral used in this study had the chemical composition
Cu9.7Zn1.9Fe0.4As4S13 (Stefano Fine Minerals, Ann Arbor, Michigan, USA). Two groups
of specimens were processed, one group for a composition-property study and a second
group for a milling time-property study. For the composition-property study, the synthetic
thermoelectric (TE) material Cu12Sb4S13 was first synthesized by direct solid state reaction
of the elements Cu (99.99% purity), Sb (99.9999% purity), and S (99.999%) from AlfaAesar, which were loaded in stoichiometric ratios into quartz ampoules. The ampoules were
-5

then evacuated to <10 Torr, sealed, loaded into a vertical furnace and heated at a ramp rate
of 0.3 ºC min-1 to 650ºC, and then held at that temperature for 12 hours. The reacted
-1

powders were then cooled to room temperature at the rate of 0.4 ºC min . The Cu12Sb4S13
and the natural mineral powders were mixed by ball milling for 30 min in a SPEX
Mixer/Mill using a stainless steel mill jar with stainless steel grinding media. In a glove box
with flowing argon gas, the specimens were hot pressed at 723 K for 20 minutes at 70 MPa
pressure using a 10 mm graphite die.
For the milling time-property study, the elemental Cu powders (10 micron mean particle
size, 99.9% purity), Sb shot (99.999% purity) and S chunks (99.999% purity) from AlfaAesar were mixed in the stoichiometric ratio needed to form Cu12Sb4S13 using a tungsten
307

carbide milling jar with tungsten carbide grinding media. The natural mineral powders of
composition Cu9.7Zn1.9Fe0.4As4S13 were added to the milling jar in a mass ratio of 1:1 with
the elemental powders. The powders were then milled in the Spex mill for 30 minutes. The
specimens were then densified by hot pressing, using the same time-temperature-pressure
profile used to densify the specimens in the composition-property study.
2.2 Microstructural and phase characterization
The microstructure of the specimens included in this study was analyzed with a JEOL
7500F scanning electron microscope (SEM) using a working distance of 8mm and an
accelerating voltage of 15 kV. Mean grain sizes were determined using linear intercept
technique with a minimum of 150 intercepts per micrograph [Wurst 1972].
X-ray diffraction analysis was performed on each of the densified specimens using a
Rigaku Miniflex II bench-top X-ray diffractometer (Cu Kα radiation). The XRD results were
analyzed using a commercial software package (Jade).
2.3. Elasticity measurements
The Young’s modulus, E, shear modulus, G, Poisson’s ratio, ν, and bulk modulus, B,
were measured via a non-destructive method, namely, the resonant ultrasound spectroscopy
[Migliori 1997] (RUS, RUSpec, Quasar International, Albuquerque, NM, USA). The RUS
apparatus excited, detected and recorded resonance modes of the specimens over a frequency
range from 20 to 600 kHz. The elastic moduli were calculated from the resonant frequencies
along with the mass, shape, and the dimensions of the specimens using a commercial
software (RPModel package, Quasar International, Albuquerque, NM, USA). Additional
details of the RUS measurement technique are available elsewhere [Ren 2008, Ren 2009,
Schmidt 2013].
308

2.4 Hardness and fracture toughness measurements
Hardness, H, and fracture toughness, Kc, were evaluated using Vickers indentation.
Prior to the indentation, specimens were polished with diamond paste with diamond grit sizes
from 90 m to 1 m using an automatic polishing machine (Leco Vari/Pol VP-50, Leco
Corporation, St. Joseph, MI).
For specimens NM II and 45% (Table 10.1), an indentation load, F, of 4.9 N and a load
time of 5 s were used. For the other specimens in this study, spalling occurred near the
indentation site so F was lowered to 2.94 N but the 5 s loading time was retained. Ten
Vickers indentations (Shimazdu HMV 2000, Kyoto, Japan) were placed on each specimen.
Hardness was calculated using
H=

1.8544F

(1)

(2a ) 2

where 2a is the diagonal length of the indentation impression [Wachtman 2009]. Fracture
toughness was calculated using equation (2)

Kc 

(E / H) 0.5 F

(2)

c1.5

where is a dimensionless calibration constant equal to 0.016 [Anstis 1981], E is the
Young’s modulus measured by RUS, H is the hardness value and c is half of the radial crack
length.
3. Results and discussion
3.1 Microstructural and phase characterization
In each of the specimens included in this study, the grains were approximately equiaxed,
with quasi-spherical pores along the grain boundaries (Figure 10.1). From SEM micrographs

309

Figure 10.1. SEM micrographs of fracture surfaces of specimens included in both the
composition and milling time studies. For the composition study, the microstructures are
shown in figures (a) – (c) for the weight fraction of synthetic phase, x, where (a) x = 0, (b) x
= 0.5, (c) x = 1. For the milling time study, the microstructures correspond to (d) 1 h milling
time, (e) 9 h milling time.
310

Figure 10.1 (cont’d). SEM micrographs of fracture surfaces of specimens included in both
the composition and milling time studies. For the composition study, the microstructures are
shown in figures (a) – (c) for the weight fraction of synthetic phase, x, where (a) x = 0, (b) x
= 0.5, (c) x = 1. For the milling time study, the microstructures correspond to (d) 1 h milling
time, (e) 9 h milling time.

311

0.55

Grain size (m)

0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.0

0.2
0.4
0.6
0.8
1.0
Weight fraction of the synthetic phase

0.35

Grain size (m)

0.30
0.25
0.20
0.15
0.10
0

2

4
6
Milling time (hour)

8

10

Figure 10.2. Grain size as a function of (a) the weight fraction of the synthetic phase; (b)the
milling time.

312

of fracture surfaces, the mean grain sizes of the composition-property group of specimens
range from about 0.23 to 0.54 microns (Figure 10.2a). For the milling time-properties study,
the grain size decreased exponentially with milling time from about 0.33 m for specimens
fabricated from powders milled for one hour, reaching a steady-state mean grain size of
approximately 0.12m after 4 hours of milling (Figure 10.2b).
The fracture mode appears to be a function of both composition and milling time. For
example, in terms of the weight fraction of synthetic phase, x, the fracture mode changes
from mostly transgranular, to predominantly intergranular to mostly transgranular to a
mixture of transgranular and intergranular for x = 0, 0.25, 0.5, 0.75 and 1.0. As a function of
milling time, at 1 hour the fracture surfaces are intergranular, then mostly intergranular at 3
hours, a mixture of transgranular and intergranular at 4.5 hours, and then mostly intergranular
at milling times of 6 and 9 hours. The fracture mode can be a function of several parameters,
including grain size and inclusions, where one likely source of inclusions is the
contamination from the milling process. For example, in a study on lead zirconate titanate
(PZT) ceramics, the fracture mode was found to be predominantly intergranular at grain size
of approximately 9.5 m and entirely transgranular at the large grain size of about 18 m
[Kim 1990].
In addition, the XRD analysis did not detect the presence of minor phases in any of the
specimens included in this study.
3.2 Elastic moduli, hardness and fracture toughness as a function of composition
For the solid state compositions that result from combinations of the synthetic TE
material Cu12Sb4S13 and the natural TE mineral Cu9.7Zn1.9Fe0.4As4S13, the room
temperature Young’s modulus, E, shear modulus, G and bulk modulus, B (Table 10.1), the

313

Table 10.1. For the specimens, the number of RUS resonant peaks measured, N, the RMS
error, the Young’s modulus, E, shear modulus, G, Poisson’s ratio, , bulk modulus, B, mass
density, .
Specimen
Label
0 Syn-disk
0 Syn-bar

a

0.45 Syn-bar
0.25 Syn-disk
0.50 Syn-disk
0.75 Syn-disk
1 Syn-disk
0.50 Syn-1hdisk
0.50 Syn-3hdisk
0.50 Syn4.5h- disk
0.50 Syn-6hbar
0.50 Syn-9hdisk

E (GPa)

G (GPa)



35
19

RMS
Error
0.40%
0.58%

B
(GPa)
56.67 ± 0.12 21.97 ± 0.03 0.290± 0.002 44.9
58.55 ± 0.41 22.98 ± 0.08 0.274± 0.008 43.2


(g/cc)
4.30
4.26

33
25
31
13
26
19

0.32%
0.44%
0.51%
0.63%
0.15%
0.40%

59.98 ± 0.18
61.27 ± 0.12
55.45 ± 0.14
51.03 ± 0.14
33.54 ± 0.02
50.85 ± 0.09

22.56 ± 0.05
23.23 ± 0.03
21.2 ± 0.04
19.08 ± 0.04
12.5 ± 0.06
19.49 ± 0.03

0.329± 0.003
0.317± 0.002
0.308± 0.003
0.337± 0.003
0.342± 0.001
0.304± 0.002

58.5
56.3
48.1
52.2
35.3
43.3

4.70
4.48
4.65
4.72
4.62
4.40

19

0.11%

55.99 ± 0.03 21.56 ± 0.01 0.298± 0.001 46.3

4.32

26

0.12%

4.5

21

0.49%

52.92 ± 0.03 20.06 ±
0.319± 0.001 48.7
0.008
55.23 ± 0.10 21.04 ± 0.02 0.312± 0.002 49.1

13

0.11%

55.01 ± 0.03 20.99 ±
0.008

4.51

N

a

0.310± 0.001 48.3

4.42

The geometry of the bar is not ideal for RUS measurements since the opposing surfaces of
the bar are not parallel.

314

hardness, H, (Table 10.1) and the Poisson’s ratio, , are functions of composition (Figure
10.3). For the weight fraction of the synthetic phase Cu12Sb4S13, x (0 ≤ x ≤1), the
composition dependence of property, A (where A is E, G, B, H, or ), can be described
relatively well by the equation
A = A0 + (A1 – A0)x + k x (1-x)

(3)

where A0 = the value of A at x = 0, A1 = the value of A at x = 1, and k is a parameter
(sometimes called the “bowing parameter”) which has the same unit as A (Figure 10.3).
When k = 0, equation (3) reduces to a rule of mixtures relationship. For E, G, and H, the
2

2

coefficient of determination, R , is between 0.973 and 0.982, although for the B data, R is
only 0.801 and thus equation (3) did not fit the B data as well (Table 10.2). In addition, for
2

Poisson's ratio, , 0 = 0.284 +/- 0.008, 1= 0.341 +/- 0.011, k = 0.047 +/- 0.041, R = 0.851
and xmax = 1.106, where xmax is the x value when  reaches the highest value on the least
squares fitting curve.
For solid solution systems in the literature, equation (3) also describes the compositional
dependence of E, G [Ravinder 2001, Ren 2007] and H [Schenk 1998, Ren 2008]. It should
be noted that while E and G are measures of a material’s resistance to elastic deformation, H
is a measure of the resistance to plastic deformation. Thus, although the elastic moduli and H
represent fundamentally different physical entities, when equation (3) is normalized by A0,
the changes in the normalized E, G, and H values are nearly identical (Figure 10.4).

315

Table 10.2. For the composition-property study, the results of the least-squares fit the elastic
moduli, E, G, and B, and hardness, H, data to equation 3.
2

A0 (GPa)

A1 (GPa)

K (GPa)

R

xAmax

E

57.59 ± 1.25

34.16 ± 1.71

48.68 ± 6.41

0.975

0.233

G

22.44 ± 0.43

12.75 ±0.59

17.26 ± 2.20

0.982

0.259

B

44.46 ± 3.01

35.90 ± 4.11

60.20 ± 15.41

0.801

0.429

H

3.26 ± 0.12

1.81 ± 0.12

2.72 ±0.05

0.973

0.233

Young's, shear, bulk modulus (GPa)
Poisson's ratio

Property

Young's modulus
Shear modulus
Bulk modulus
Poisson's ratio

70
60
50
40
30
20
10
0.4
0.3
0.2
0.0

0.2
0.4
0.6
0.8
1.0
Weight fraction of the synthetic phase

Figure 10.3. Young’s, shear, bulk modulus and Poisson’s ratio as a function of the weight
fraction of the synthetic phase for the mineral TE materials in this study. The solid, dashed,
dotted and dash-dotted curves represent respectively the least squares fit of equation (3) to
the Young’s modulus, shear modulus, bulk modulus and Poisson’s ratio data.
316

E
G
H

Normalized E, G and H

1.2

1.0

0.8

0.6

0.4
0.0
0.2
0.4
0.6
0.8
1.0
Weight fraction of the synthetic phase

Figure 10.4. Normalized Young’s, shear modulus and hardness as a function of the weight
fraction of the synthetic phase (in each case, the data were normalized by dividing the each
coefficient in equation (3) by A0). The solid curve represents a least squares fit of all the
normalized E, G and H data to equation (3).

317

1.0
Hardness
Fracture toughness

3.5

0.9
0.8
0.7

3.0

0.6
2.5

0.5
0.4

2.0

0.3
1.5

0.2
0.0

Fracture toughness (MPa m0.5)

Hardness (GPa)

4.0

0.2
0.4
0.6
0.8
1.0
Weight fraction of the synthetic phase

Figure 10.5. Hardness and fracture toughness as a function of the weight fraction of the
synthetic phase for the mineral TE materials in this study. The solid curve represents a least
squares fit of the hardness data to equation (3); the dashed line represents the mean value of
Kc over the entire weight fraction of the synthetic phase.

318

The fracture toughness, Kc, was essentially independent of composition for the
specimens included in this study, with an average Kc value of 0.47 ± 0.04 MPa•m

0.5

(Figure

10.5). A Kc value that is independent of composition for a solid solution system has been
observed by Shibata [Shibata 1997]. For the Cr2O3-Al2O3 solid solution system, the H and
Kc values were essentially constant ~23.5 GPa and ~4 MPa•m

0.5

, respectively over a range

of Al2O3 from 0 to 25 mol % [Shibata 1997]. However, the literature on Kc as a function of
composition for solid solution systems is relatively limited. There are a number of papers on
KC versus composition for zirconia systems, however the extent of transformation
toughening is a sensitive function of the Zr content [Fu 2009]. Also, a number of researchers
studied solid solution matrices that include nano/micro scale inclusions such that it is
difficult to separate the contributions to KC from solid solution composition and the
inclusions [Charitidis 2007].
The elastic moduli and Poisson’s ratio reported in this study are global properties of the
specimen since they are determined by RUS from the spectrum of mechanical resonances for
the bulk specimen. However, the H and Kc values are determined via Vickers indentation
measurements and thus it is important to consider whether H and Kc are “local” or “global”
values. The hardness values are functions of the diagonal length of the Vickers pyramidal
indentation impression, 2a, (equation 2), where average value of 2a in this study was 40 μm.
In comparison, the grain sizes of every specimen included in this study were less than 1 μm.
Thus, the hardness values represent an average over thousands of individual grains. Also, for
the indentation measurements of Kc, the average crack length, c, ranged from about 45 to 65
319

Young's, shear, bulk modulus (GPa)
Poisson's ratio

Young's modulus
Shear modulus
Bulk modulus
Poisson's ratio

70
60
50
40
30
20
10
0.4
0.3
0.2
1

2

3
4
5
6
7
Milling time (hours)

8

9

Figure 10.6. Young’s, shear, bulk modulus and Poisson’s ratio as a function of milling time.

4.0

1.0

Hardness (GPa)

0.9

3.5

0.8
0.7

3.0

0.6
0.5

2.5

0.4
0.3

2.0

0.2

0

1

2

3
4
5
6
7
Milling time (hour)

8

9

10

Figure 10.7. Hardness and fracture toughness as a function of milling time.

320

Fracture toughness (MPa m0.5)

Hardness
Fracture toughness

μm so that the fracture toughness measurements represent the interaction of the indentationinduced crack with more than a hundred grains. Thus, the H and Kc values determined
should be relatively insensitive to localized fluctuations in microstructure.
3.3 Elastic moduli, hardness and fracture toughness as a function of milling time
During ball milling of brittle materials [Suryanarayana 2001, Hall 2009, Pilchak 2007],
including thermoelectrics [Hall 2009, Pilchak 2007], the powder particle size initially
decreases rapidly, reaching a steady state (grindability limit) typically within a few hours.
The current study shows that ball milling for times longer than 4 hours does not result in a
smaller as-sintered grain size for the solid solution formed by mixing then sintering the
natural mineral and synthetic tetrahedrite powders. This result is significant since ball
milling (using a variety of mill jar and milling media) can result in significant contamination
that generally increases as the milling time increases [Singh 2009].
The Young’s shear and bulk moduli along with Poisson’s ratio were approximately
constant as a function of milling time (Figure 10.6). In contrast, the hardness and fracture
toughness values (Figure 10.7) apparently increase and then either go through a maximum or
reach a plateau for milling times of 4 hours or more. However, the scatter in the H and KC
data make it difficult to clearly identify the H and Kc versus milling time trends.
4. Summary and conclusions
The composition dependencies of the Young’s modulus, E, shear modulus, G (Figure
10.3) and hardness, H (Figure 10.5), are described well by equation 3, in agreement with the
trends observed for the elastic moduli and hardness of a number of solid solution systems in
the literature. When normalized by the A0, value of the property in equation 3 for zero mass

321

fraction of the synthetic tedrahedrite, the changes in E, G and H are remarkably similar
(Figure 10.4). However, the fracture toughness, Kc, is essentially independent of the
composition over the entire solid solution range.
For the milling time-property study, the elastic moduli and Poisson’s ratio are
essentially independent of milling time. The scatter in the H and Kc make it difficult to
discern a trend as a function of milling time (Figure 10.7).
5. Acknowledgements
The authors acknowledge the financial support of the Department of Energy,
“Revolutionary Materials for Solid State Energy Conversion Center,” an Energy Frontiers
Research Center funded by the U.S. Department of Energy, Office of Science, Office of
Basic energy Sciences under award number DE-SC0001054.

322

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323

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325

CHAPTER 11
ROOM TEMPERATURE MECHANICAL PROPERTIES OF THERMOELECTRIC
INTERMETALLIC MATERIAL ZrNiSn
X. Fan1, L. G. GUERRERO2, E. D. Case1, H. Sun1, D. T. Morelli1
1
Department of Chemical Engineering and Materials Science, Michigan State
University, East Lansing, MI, 48824
2
Technical University of Madrid, Madrid, Spain, 28040

Abstract
The intermetallic half-Heusler compound ZrNiSn is of interest as a thermoelectric
material for waste heat recovery applications. For this study, ten polycrystalline ZrNiSn
specimens were prepared from arc melted ingots that were subsequently powder processed
and then densified by pulsed electric current sintering. Using resonant ultrasound
spectroscopy (RUS), the room temperature Young’s modulus, shear modulus and Poisson’s
ratio were determined. Also, the room temperature hardness was measured by Vickers
indentation. Using a range of sintering temperatures between 800oC and 1025oC, the volume
fraction porosity, P, of the specimens was intentionally varied from 0.01 to 0.24. Both
porosity dependence of the elastic moduli and the hardness can be described well by the
empirical exponential relationship A(P) = A0exp(-bAP) where A is the porosity-dependent
mechanical property, A0 is the value of A at P = 0, and bA is a measure of the rate of decrease
in A with increasing P.

326

1. Introduction
Thermoelectric (TE) materials are capable of converting waste heat into electricity, and
have attracted wide interest recently. Among the various types of the TE materials, the class
of the intermetallic compounds known as the half-Heusler (HH) have received extensive
attention [Zou 2013, Kong 2012, Populoh 2012, Fu 2013]. Half-Heusler compounds have
high figure of merit (ZT near 1) at high temperatures [Yu 2009]. Half-Heusler compounds
have chemical formula of XYZ, for example, MNiSn (M = Hf, Zr, Ti) – based HH have been
widely studied as n-type TE materials [Populoh 2012, Shen 2001].
However, most studies of HH compounds have focused on the improvement of transport
properties [Zou 2013, Kong 2012, Populoh 2012, Yu 2009], while the literature on
mechanical property measurements of HH compounds are very limited [Verges 2011].
Nevertheless, designers need the elastic moduli of a material in order to model its stress and
strain behavior under applied thermal and/or mechanical stresses. For example, the finite
element analysis requires the Young’s modulus and Poisson’s ratio in order to construct the
stiffness matrix [Martin 1973]. In addition, the elastic moduli can be used to monitor the
level of microcrack damage in a brittle material [Fan 2012a]. Moreover, hardness is related to
the machinabily [Kamboj 2003] and wear resistance [Krakhmalev 2006] of the material,
which are directly related to the fabrication and stability of TE materials.
In general, porosity affects mechanical, electrical and thermal properties of materials.
For example, both the thermal conductivity, κ and electrical conductivity, σe, of ceramics
[Rice 1998], metals and semiconductors decrease with increasing porosity. In particular, for
327

thermoelectric materials such as CoSi [Kim 2002], Ca2.76Cu0.24Co4O9 [Kim 2013],
Al-doped ZnO [Ohtaki 2011] and the intermetallic half-Heusler compound FeVSb [Fu 2013],
the observed decreases in κ and σe with increasing P has been attributed to the scattering of
phonons and charge carriers by pores [Ohtaki 2011, Fu 2013]. The efficiency of TE materials
is characterized by a dimensionless figure of merit (ZT, ZT = S2σeT/κ), where S is the Seebeck
coefficient, σe is the electrical conductivity, T is the absolute temperature, and κ is the thermal
conductivity. Both κ and σe decrease with increasing porosity in manners such that κ = κ0
exp(-bκ P) and σe =σe0 exp(-bσe P), where

κ0 and σe0 are κ and σe at zero porosity, P,

respectively, bκ and bσe are material-dependent constants. The values of bκ and bσe were
empirically found to be in the range of 1.5 to 2.0 and 1.5 to 3.5, respectively [Case 2012].
Seebeck coefficient has been found enhanced by porosity [Kallel 2012, Lee 2010]. Thus, the
ZT could be increased, remained constant, or decreased in the presence of porosity.
A wide range of mechanical properties are also porosity dependent, including fracture
strength [Fan 2012b], fracture surface energy [Vandeperre 2004], elastic moduli [Fan 2012b]
and hardness [Hoepfner 2003].
In this paper, the room temperature elastic moduli and hardness of half-Heusler
compound ZrNiSn as a function of porosity were studied. It is of great importance to study
porosity in thermoelectric materials for the following reasons: (i) porosity has the potential to
increase ZT in TE materials; (ii) porosity may improve the thermal fatigue resistance in TE
materials in analogy to thermal barrier coatings [Case 2012]; (iii) the obtained
modulus-porosity relation allows the comparison of data with different porosity levels by
328

different researchers; (iv) the extrapolation value of the modulus-porosity relation allows the
comparison of experimental data with porosity to theoretical calculations without porosity.
2. Experimental procedure
2.1 Specimen preparation
The starting materials for the ZnNiSn ingots were obtained from Alfa Aesar, namely Zr
lump material (roughly 3-6mm in diameter, 99.8% pure, stock#36253), Ni slugs (6.35mm dia
x 12.7mm length, 99.995%, purity, stock# 42330) and Sn shot (3mm diameter, 99.999% pure,
stock# 11010). An arc melting furnace (Centorr, Model 5SA) was used to melt the elements
in stoichiometric proportions to form the ZrNiSn phase on a copper hearth without
water-cooling and under flowing argon gas.
After casting, the ingot surfaces with lightly sanded with 1200 grit sandpaper and were
then cleaned for 5 minutes in a detergent bath in an ultrasonic cleaner. The specimens were
then rinsed with DI water, cleaned in an ethanol bath and then rinsed with DI water.
The cleaned ingots were then crushed, ground, sieved and reground (CGSR) (Retsch
RM200, Retsch GmbH, Haan, Germany) in a glovebox with flowing argon until all powders
passed through a 53 m sieve. The CGSR powder was then either (i) densified directly using
pulsed electric current sintering (PECS) or (ii) dry milled prior to PECS densification via a
planetary mill (PM100, Retsch, Newtown, PA) using a 250 cc alumina-lined milling jar and
spherical alumina grinding media [Pilchak 2007].
2.2 Pulsed electric current sintering (PECS)
For both the CGSR powders (without planetary milling) and the ball milled ZnNiSn
329

powders, approximately 2 grams of powder per specimen were PECS processed (Model 10,
Series 4 by Thermal Technologies LLC, Santa Rosa, CA) in a 12.7 mm diameter graphite die
with grafoil covering each of the punch faces. The sintering temperatures, peak pressures and
times are summarized in Table 11.1.
For each of the PECS processed specimens, the heating and cooling ramp rates were
10oC/min and the pressure ramp rate was 22.5 MPa/min. The specimens fabricated from the
CGSR powders (without planetary milling) were labeled as HH-X (X = the sintering
temperature), and the specimens fabricated from ball milled powders were labeled as
HH-BM-Y (Y = the sintering temperature).
2.3 Microstructural analysis
The CGSR powders, the ball milled powders, and fractured surfaces of specimens
HH-1000, 800 and HH-BM-1000 were observed using a scanning electron microscope (SEM,
Carl Zeiss Microscopy, LLC, Thornwood, NY) with an accelerating voltage of 20 kV and a
working distance of 5 mm.
Crystallographic phase analysis was performed using a Rigaku Ultima III diffractometer
with CuKα radiation with 2θ between 20o and 70o. X-ray diffraction (XRD) was performed
on specimens HH-900 and HH-950.
The PECS processed disk-shaped specimens were then polished using diamond paste
with grit sizes ranging from 90 m to 1 m. The specimens were then cut into bars (Table
11.1) for microstructural analysis, as well as elastic modulus and hardness measurements.
The mean grain sizes of the densified ZrNiSn specimens were determined using a linear
330

intercept method with 150 to 200 intercepts per SEM micrograph and a stereographic
projection factor of 1.5 [Underwood 1968, Case 1981].
2.4. Elastic modulus measurement
The room temperature elastic moduli of the ZrNiSn specimens included in this study
were measured using the resonant ultrasound spectroscopy (RUS) technique [Miglori 1997].
A commercial resonant ultrasound spectroscopy device (RUSpec, Magnaflux Quasar,
Albuquerque, NM) was used in which the specimen is placed on a tripod of piezoelectric
transducers. One transducer excites mechanical vibrations in the test specimen while the
other two transducers detect the vibrations. In this study, the frequency for the driving
transducer was swept from 20 kHz to 600 kHz in 29,999 steps. The elastic moduli are then
calculated based on the mechanical resonance frequencies, the mass, dimensions and
geometry of the specimens [Miglori 1997] using commercial software (RPModel, Magnaflux
Quasar, Albuquerque, NM).
2.5. Hardness measurements
Prior to the room temperature hardness measurements on the ZrNiSn specimens, the
Vickers indenter (Shimazdu HMV 2000, Kyoto, Japan) was calibrated using a standard
hardness calibration block (HV 790 steel standard calibration block, Yamamoto Scientific
Tools Lab Co. LTD, Chiba, Japan). The hardness, H, was calculated from the average of at
least 10 indentations per specimen using [Wachtman 2009]
H=

1.8544F

(1)

(2a ) 2

331

Table 11.1. For each cut and polished specimen included in this study, the PECS temperature,
T, and the relative porosity, P, the mass and dimensions. For each specimen, the PECS
pressure was 50 MPa, with a holding time of 20 minutes except for specimen HH-900, whose
holding time was 10 minutes.
Specimen
label

PECS T (oC)

Mass (g)

Dimensions (cm
x cm x cm)

HH-800

800

0.704

0.81 x 0.60 x 0.24 0.24

HH-850

850

0.656

0.84 x 0.62 x 0.20 0.23

HH-900

900

0.856

0.83 x 0.62 x 0.25 0.15

HH-950

950

0.742

0.80 x 0.61 x 0.23 0.14

HH-975

975

0.685

0.82 x 0.59 x 0.20 0.09

HH-1000

1000

0.542

0.81 x 0.59 x 0.14 0.01

HH-1025

1025

0.634

0.81 x 0.61 x 0.17 0.04

HH-BM-800

800

0.829

0.91 x 0.60 x 0.22 0.13

HH-BM-850

850

0.796

0.90 x 0.65 x 0.19 0.09

HH-BM-900

900

0.837

0.85 x 0.64 x 0.20 0.04

0.699

0.90 x 0.60 x 0.17 0.04

HH-BM-1000 1000

332

Relative
porosity, P

where the dimensionless constant,  is the calibration factor obtained from the indentation of
the calibration block, F is the applied indentation load and a is half of the diagonal length of
the indentation impression.
3. Results and Discussion
3.1 Microstructure and phase analysis
Micrographs of CGSR powder showed a wide distribution of particle size, from
sub-micron, a few microns to roughly 40 to 50 m (Figures 11.1 a). After ball milling, the
powders were reduced to particle sizes from sub-micron to roughly 20 m (Figures 11.1b ).
On the fractured surface of HH-800 (Figure 11.2a), a matrix of small grains with
distributed large grains was observed. The average grain size of the small matrix grains
ranged from roughly 1 μm to 5 μm, with large grains in the 10 m to 30 m range (Figure
11.2a). Roughly half of the area of the fractured surface is occupied by large grains(Figure
11.2a). For the specimen HH-BM-800 (Figure 11.2b), about one third of the fractured surface
are occupied by large grains at the size of 10 – 15 m, and no grains larger than ~ 20 m
were observed.
For the densified HH specimens, the volume fraction porosity, P, for the specimens
ranged from 0.24 to 0.01, with the highest porosity for specimen HH-800 and lowest porosity
for specimen HH-1000 (Table 11.1). For specimens densified using the same PECS condition
but from different powders, namely the CGSR powders and the ball milled powders, the final
porosities are different. For example, the final porosity for HH-800 was 0.24, while for
HH-BM-800 was 0.13. The lower final porosity for
333

Figure 11.1. SEM micrographs of powder processed ZrNiSn, for (a) CGSR powders, (b) ball
milled powders.

334

Figure 11.2. SEM micrographs of fractured surface for (a) HH-800, and (b) HH-BM-1000.

335

HH-BM-800 is due to smaller starting powder particle size and the relatively narrow particle
size distribution [Barsoum 1997].
An XRD analysis showed that the specimen was relatively pure ZrNiSn (Figure 11.3).
However, a faint peak indicated by the arrow (Figure 11.3) is not yet identified.
3.2 Elastic modulus measurement
Elastic moduli were measured on the polished, bar-shaped specimens. Details on the
number of resonant peaks, rms error and elastic moduli are listed in Table 11.2.
The porosity dependence of the elastic moduli (Figure 11.4) is described well by the
empirical relation [Rice 1998, Fan 2012b]
A = A0exp (-bAP)

(2)

where A represents the elastic moduli (Young’s modulus, E, shear modulus, G and Poisson’s
ratio, ), A0 is the elastic moduli at P = 0 and bA is a unitless, material-dependent constant
that is proportional to the rate of decrease in A with increasing P.
As reviewed by Rice [Rice 1998], for volume fraction porosities, P, greater than a critical
porosity, PC, the experimentally-determined elastic modulus sometimes decrease much faster
with increasing P than is predicted by equation 2. Often PC is from roughly 0.3 to 0.4 [Rice
1998, Fan 2013]. For the range of P included in this study (0.01 < P < 0.24), equation 2 does
describe the modulus-porosity relationship quite well. For example, the least-squares fits of
the E, G, and  data to equation (2) yielded E0 = 214.0 ± 6.8 GPa, bE = 4.1 ± 0.3; G0 = 85.1
± 2.9 GPa, bG = 4.0 ± 0.3;  0 = 0.262 ± 0.004, b = 0.87 ± 0.10, with coefficients of

336

30000

HH-900
HH-950

Intensity (A.U.)

25000
20000
15000
10000
5000
0
20

30

40

50
60
2 (deg)

70

80

90

Figure 11.3. XRD spectra for specimens HH-900 and HH-950. The faint peak indicated by
the blue arrow is not yet identified.

337

Table 11.2. For RUS analysis for each specimen, the number of resonant peaks, N, the rms
error, the Young’s modulus, E, the shear modulus, G, the bulk modulus, B, and Poisson’s ratio,
.
Specimen label

N

rms error

E (GPa) G (Gpa)

B (Gpa)



HH-800

19

0.49%

76.7

31.5

45.1

0.216

HH-850

27

0.45%

73.8

30.2

44.0

0.220

HH-900

26

0.47%

121.9

49.8

73.5

0.224

HH-950

24

0.36%

115.5

47.2

69.8

0.224

HH-975

28

0.48%

154.5

61.8

102.9

0.250

HH-1000

22

0.40%

191.6

75.9

134.4

0.262

HH-1025

23

0.21%

188.0

74.9

128.5

0.256

HH-BM-800

25

0.39%

116.5

47.1

73.7

0.236

HH-BM-850

22

0.39%

150.32

60.56

96.7

0.241

HH-BM-900

22

0.37%

189.3

76.4

120.4

0.238

HH-BM-1000

21

0.49%

187.4

75.8

125.7

0.251

338

2

determination, R , of 0.970, 0.970 and 0.912, respectively.
3.3 Hardness measurement
The Vickers hardness, H, was measured on the polished surfaces using five different
loads, namely 0.98 N, 2.94 N, 4.9 N, 9.8 N and 19.6 N (Figure 11.5). The H values were
nearly independent of load ranging from 2.94 N to 19.6 N for selected HH specimens. The
slightly higher hardness values obtained for the indentation load of 0.98 N may be due to the
avoidance of indentation on porous areas. For lower loads such as 0.98 N, the indentation
impression is too small to read if was placed in the porous areas.
Hardness-porosity relations were investigated indented at load of 4.9 N.
Hardness-porosity data were fit to the empirical relationship
H = H0exp (-bHP)

(3)

where H is hardness, H0 is the hardness at zero porosity and bH is a unitless constant (Figure
11.6). Least-squares fits of H values for all the specimens to equation (3) yielded H0 = 9.7 ±
2

2

0.9 GPa, bH = 5.6 ± 1.1 and R = 0.812. The low R value indicated that equation (3) did not
fit the hardness data as a function of porosity well.
3.4 Comparison to literature values for elastic moduli and hardness
In the literature (Table 11.3), there are some theoretical simulations of half-Heusler
compounds such as NiTiSn [Hichour 2012], CoVSn [Hichour 2012], IrMnAl [Hamidani
2009], IrMnSn [Hamidani 2009], IrMnSb [Hamidani 2009], and intermetallic compound of
Fe-Al-Ti [Krein 2010]. The calculated Young’s modulus of NiTiSn [Hichour 2012], CoVSn
[Hichour 2012], IrMnAl [Hamidani 2009] and intermetallic compound of Fe-Al-Ti [Krein
339

150
120

0.35

90

0.30

210
180

Poisson's ratio

Young's and shear modulus (GPa)

0.50

Young's modulus (CGSR)
Shear modulus (CGSR)
0.45
Poisson's ratio (CGSR)
Young's modulus (BM)
Shear modulus (BM)
0.40
Poisson's ratio (BM)

240

60
0.25

30
0

0.20
0.00

0.05

0.10
0.15
Porosity

0.20

0.25

Figure 11.4. Young’s modulus, shear modulus and Poisson’s ratio versus porosity for
specimens fabricated from CGSR powders (solid symbols) and specimen fabricated from ball
milled powders (open symbols). The solid curves for each data set represent the least-squares
fit to equation 2.

340

HH-BM-800
HH-900
HH-950
HH-1000
HH-1025

Hardness (GPa)

12
11
10
9
8
7
6
5
4
3
2
1
0

5

10
Load (N)

15

20

Figure 11.5. Hardness as a function of load for specimens fabricated from CGSR powders
and ball milled powders.

10

Specimen from ball milled powders
Specimens from CGSR powders

9

Hardness (GPa)

8
7
6
5
4
3
2
1
0.00

0.05

0.10

0.15

0.20

0.25

Porosity

Figure 11.6. Hardness vs. P for specimens fabricated from CGSR powders (open symbol) and
specimen fabricated from ball milled powders (solid symbol) indented at load = 4.9N.
341

2010] are comparable to the E value for the ZrNiSn in this study, except for IrMnSn
[Hamidani 2009], IrMnSb [Hamidani 2009] which have lower E values. Germond [2010]
obtained elastic moduli of ZrNiSn using indentation technique. The Young’s modulus of
223.5 GPa and 241.1 GPa [Germond 2010] are somewhat higher than the value for the
specimens in this study (E0 = 215 GPa). However, indentation method is a static technique,
and measures the Young’s modulus of a localized area, while RUS used in this study is a
dynamic method that measures the elastic response of the bulk specimen. Elastic modulus for
Fe-Al-Ti compound measured by the indentation method [Prakash 2001] showed comparable
E values to the HH specimens in this study.
Table 11.4 gives hardness data from the literature for two half-Heusler compounds
[Verges 2011, Germond 2010] along with the specimen fabrication technique, relative density
and grain size. The specimens from the study by Verges [2011] had a relative density of 95%.
Using the hardness values obtained from HH specimen with a relative density of 96%, it is
found that the hardness value in this study (H = 8.54 ± 0.37 GPa) is comparable to that in
the study by Verges (H = 9.2 ± 0.4 GPa) [Verges 2011] However, the grain size of the
specimens in Verges’s study [Verges 2011] was not stated, and the composition was different
from this study. For the specimens from Germond’s study [Germond 2010], the composition
is the same as in this study. The hardness values were ~14 GPa [Germond 2010], which are
higher than the values in this study. However, the relative density of the specimens were not
given [Germond 2010]. The hardness value (~14 GPa) measured by Germond [Germond
2010] is even higher than the extrapolation value obtained from the hardness-porosity plot
342

Table 11.3. Comparison of the Young’s modulus, shear modulus and Poisson’s ratio data
from this study to data for intermetallic compounds from the literature.
References Material

This study ZrNiSn
Germond
2010

Young’s
modulus
(GPa)
187.4

ZrNiSn PECS
ZrNiSn –
Hot
pressed
NiTiSn

223.5

Shear Poisson’s P
modulus ratio
(GPa)
75.8
0.251
0.01
NS

a

NS

NS
NS

241.1

NS

NS

219.96

87.84

0.25

CoVSn

243.24

95.69

0.27

Hamidani IrMnAl
2009
IrMnSn

231.15

92.66

0.247

79.19

28.61

0.331

IrMnSb

135.65

50.94

0.383

Fe-(10-27.
5)
at.%Al-(1
0-30)
at.%Ti
Fe-25Al

“just
above
200”

NS

NS

178

NS

NS

Fe-25Al-6
Ti
Fe-25Al-2
5Ti
Fe-25Al-2
0Ti-4Cr

200

NS

NS

265

NS

NS

240

NS

NS

Hichour
2012

Prakash
2001

Krein
2010

a
b

Not stated by the authors.
Theoretical calculation assume dense materials (P = 0).
343

b

0

b

0

b

0

b

0

Measurement
/calculation method
Resonant ultrasound
spectroscopy
Depth sensing
indentation
techniques

Theoretical calculation
by full-potential linear
muffin-tin orbital
(FP-LMTO) method
Theoretical calculation
by full potential
linearized augmented
plane wave method

b

0

NS

b

0

Indentation depth as a
function of applied load
during loading and
unloading of Vickers
indentation
Ab initio calculation

b

0

b

0

NS

Dynamic measurement
device

Table 11.4. Comparison of the hardness data from this study to data for half-Huesler
compounds from the literature.
Reference and material

Powder preparation Load Vickers
Relative Grain
& specimen
(N) Hardness
density size
fabrication
(GPa)
o
This study
PECS at 900 C and 0.98 8.54 ± 0.37 96 %
Need
ZrNiSn
50 MPa for 10 min
to give
a grain
size
Verges 2011
Hot pressed at
1.96 9.2 ± 0.4
95%
a
NS
100 MPa and
Zr0.5Hf0.5NiSn0.99Sb0.01
820oC for 1 h
Germond 2010
Mechanical
0.49 14.15
N/A
43 nm
ZrNiSn
alloying
Hot pressed at
850 oC for 60 min,
100 MPa for 180
min
Mechanical
0.49 13.69
N/A
39 nm
alloying
SPS at 825oC for 7
min
a

Not stated by the authors.

344

(Figure 11.6). It is important to note that the grain size for Germond’s HH specimens is in the
nano regime, from 39 nm to 43 nm [Germond 2010]. For the specimens in this study, the
grain sizes are much larger (with grain size from roughly 1 – 30 m) than Germond’s HH
specimens [Germond 2010]. As indicated by the Hall-Petch relationship [Rice 2000], the
hardness values increase with decreasing grain size, for grain size greater than the grain size,
usually around 10 nm. Hall-Petch relationship can be expressed as [Rice 2000]
H = Hc + k (GS)

-0.5

where Hc is the H value in the single crystal limit, k is the Petch parameter and GS is the
grain size.
4. Summary and Conclusions
Arc melted ZrNiSn ingots were powder processed and densified by PECS over a
temperature range from 800oC to 1025oC in order to intentionally induce a range of porosity,
P, from about 0.01 to 0.24 in the specimens. The porosity-dependent elastic moduli (Young’s
modulus, shear modulus, Poisson’s ratio) were measured via RUS and the hardness was
determined by Vickers indentation.
Over the range of porosity included in this study, the Young’s and shear modulus versus
porosity as well as the hardness versus porosity were relatively well described by the
empirical exponential relationship A(P) = A0exp(-bAP), which is also the porosity
dependence that been reported in the literature for a wide range of brittle materials. In
addition, the elastic modulus and hardness values compare favorably with the corresponding
literature values.
345

Acknowledgements
The authors acknowledge the financial support of the Department of Energy, “Revolutionary
Materials for Solid State Energy Conversion Center,” an Energy Frontiers Research Center
funded by the U.S. Department of Energy, Office of Science, Office of Basic energy Sciences
under award number DE-SC0001054.

346

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350

CHAPTER 12
SUMMARY AND CONCLUSIONS

In this study, the mechanical properties of hydroxyapatite (HA), gadolina doped ceria
(GDC10) and thermoelectric materials (Cu10Zn2As4S13 - Cu12Sb4S13and ZrNiSn) were
characterized as a function of microstructure and/or composition.
For HA specimens, unimodal porosity ranging from 0.08 to 0.62 was induced by partial
sintering. A total of more than 500 specimens were fabricated, whose elastic moduli were
measured by resonant ultrasound spectroscopy, and then fractured by biaxial flexure testing.
The porosity, P, dependence of the mean fracture strength, <f>, and the Young’s modulus, E,
for the HA specimens are power law functions of the degree of densification, , where  = 1 P/PG and PG is the green porosity (Chapter 5). The Weibull modulus, m, was studied along
with data from the literature that represent eight different materials and more than 1500
specimens. The m versus P plot including all the data set showed a “U - shaped” trend with a
wide band of m values for P < 0.1 and P > 0.55, and a narrower band in the intermediate
porosity region of 0.1 < P < 0.55 (Chapter 4).
The limited range of Weibull modulus values ( 4 < m < 11) for all the data set in the
porosity range of 0.1 < P < 0.55 regardless of the composition, grain size, testing techniques
or surface finish of the specimens has important implications since porosity of the majority of
the applications of porous brittle materials fall within this range (Chapter 4). A medium to
high scatter in the fracture strength is to be expected, thus requiring the designers of such

351

components to include high safety factors in their designs. The fracture strength and Young’s
modulus are crucial to the design of brittle components too. The power law functions of the
degree of densification for fracture strength and Young’s modulus can be applied to many
partially sintered brittle materials (Chapter 5).
As a follow-up study, about 300 HA specimens with high porosity from 0.59 to 0.62
were fabricated at sintering T from 350oC to 1025oC. The fracture strength decreased
monotonically with decreasing Tsinter, from 4.8 MPa for specimens sintered at 1025oC to
0.66 MPa for specimens sintered at 350oC (Chapter 7). The decrease in fracture strength with
limited porosity increase is due to the neck growth during initial sintering stage. The Weibull
modulus remained surprisingly high, ranging from 6.6 to 15.5 (Chapter 7).
This study (Chapter 7) is the first research on Weibull modulus for highly porous
materials (near green density). Despite their low strength, the surprisingly high Weibull
modulus implied high mechanical reliability, which may indicate to designers that new,
practical applications are possible for highly porous materials.
The effect of microcracking on the reductions of the Young’s modulus, E, and Poisson’s
ratio, , was studied by inducing an array of Vickers indentation microcracks on the surfaces
of HA specimens (Chapter 6). Both E and decreased approximately linearly with increasing
microcrack damage. The results agreed with the theoretical predictions on the
microcracking-elastic moduli relations.
This study (Chapter 6) validated the theoretical work by direct measurements of crack
number density and crack length. The agreement of this study and a study on alumina [Kim
352

1993] supported the conclusion that the linear elastic modulus – microcrack damage trend
may apply to a wide variety of brittle materials.
For the gadolinia doped ceria (Ce0.9Gd0.1O1.95 or GDC10), the room temperature
elastic properties (Chapter 8) and hardness (Chapter 9) were measured as a function of
porosity. GDC10 specimens with porosities from 0.07 to 0.60 were produced by hard die
pressing and partial sintering. The room temperature Young’s modulus, shear modulus, bulk
modulus and Poisson’s ratio decreased exponentially with increasing porosity (Chapter 8).
The Vickers hardness also decreased with increasing P exponentially in porosity range of
0.08 < P < 0.60 (Chapter 9). For Vickers indentation loads of 0.98 N to 9.8 N, hardness
showed no dependence with regard to the applied load.
The knowledge of elastic moduli of GDC10 at different porosity levels is essential to
model the response of the SOFCs to internal or external stresses and to design more reliable
SOFCs with longer service life (Chapter 8). In addition, hardness, H, is related to a material’s
wear resistance and machinability, which are two essential factors for fabrication and
mechanical stability in application (Chapter 9).
For thermoelectric solid solution system Cu10Zn2As4S13 - Cu12Sb4S13, the room
temperature elastic moduli, hardness and fracture toughness were investigated as a function
of (i) composition and (ii) ball milling time. The composition dependence of the Young’s
modulus, E, shear modulus, G and hardness, H were described well by the parabolic
relationship (Chapter 10), in agreement with the trends observed for the elastic moduli and
hardness of a number of solid solution systems in the literature. The fracture toughness, Kc,
353

is independent of the composition change. For the study on ball milling time, the grain size
decreased with increasing milling time, and reached to a steady state after 4 hours of milling.
The elastic moduli and Poisson’s ratio are essentially independent of milling time. The scatter
in the H and Kc make it difficult to discern a trend as a function of milling time (Chapter 10).
For the half-Heusler compound ZrNiSn prepared from arc melted ingots followed by
powder processing and densified by pulsed electric current sintering, the room temperature
Young’s modulus, shear modulus, Poisson’s ratio and hardness were studied (Chapter 11).
Both porosity dependence of the elastic moduli and the hardness can be described well by the
empirical exponential relationship A(P) = A0exp(-bAP) where A is the porosity-dependent
mechanical property, A0 is the value of A at P = 0, and bA is a measure of the rate of decrease
in A with increasing P (Chapter 11).
In waste heat recovery applications, thermoelectrics will undergo thermal and
mechanical stresses arising from thermal gradient, thermal transient and mechanical loading.
The elastic moduli and Poisson’s ratio are required for the analysis of stress and strain by
analytical or numerical simulations. The knowledge of hardness and fracture toughness is
also crucial to successful design and application of such components.
Mechanical properties reflect how a component subjected to various thermal or
mechanical stresses will respond. Studying the mechanical property – microstructure
relationships is essential for a better understanding of the material’s response under different
circumstances. The knowledge of the mechanical properties will guide us to design and
fabricate more reliable components with longer service life and higher safety factor.
354

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356

CHAPTER 13
FUTURE WORK
In this study, the effect of total volume fraction porosity on elastic moduli, fracture
strength and Weibull modulus of hydroxyapatite was explored. However, the effect of pore
size, pore morphology and pore size distribution on the mechanical properties is not clear.
Bimodal pore distribution is of great importance for hydroxyapatite application as bone
scaffolds since the bimodal pores are biologically and physiologically essential for bone
ingrowth and blood/nutrient circulations. Strategies of incorporating bimodal pores include
burning out sacrificial fugitives (corn starch, polymer beads, etc.) and direct forming of gas
bubbles. Future studies should focus on fabricating specimens with bimodal pore size
distribution, and studying how the bimodal pores affect the mechanical properties. In
addition, studying the porosity effect on other brittle materials could also be done.
In the current study, the room temperature elastic modulus, hardness and fracture
toughness were studied for thermoelectric (TE) materials and solid oxide fuel cell (SOFC)
materials. In commercial applications, these materials will be operated under cyclic
temperatures and possible mechanical loading. Knowing the Young’s modulus and Poisson’s
ratio at elevated temperature would be critical to model the thermal stress and strain using
analytical or numerical simulations. Thus, future work needs to be done to measure the
elasticity as a function of temperature for the TE and SOFC materials. Moreover, the
mechanical integrity of these materials is essential in applications. Thermal fatigue testing is
capable to evaluate whether the material could withstand a number of thermal cycles. In the
future, thermal fatigue testing can be done on the TE and SOFC materials. Path towards
improving thermal fatigue resistance for these materials needs to be considered. Porosity has

357

been proved to enhance thermal fatigue resistance for thermal barrier coatings. Future studies
aiming to improve thermal fatigue resistance for the TE and SOFC materials can incorporate
an engineered porosity ranging from 0.1 to 0.2. Besides working on the materials, future
studies could also be done to investigate mechanical integrity of the entire thermoelectric
module and the whole SOFC system.

358