EXPLOITING DESIGN PRINCIPLES IN NATURE TO SYNTHETIC COMPOSITES
By
Abhishek Dutta

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering—Doctor of Philosophy
2013

ABSTRACT
EXPLOITING DESIGN PRINCIPLES IN NATURE TO SYNTHETIC COMPOSITES
By
Abhishek Dutta
The current need of technology demands the generation of light-weight, durable and
energy efficient structural materials. In order to attain this goal, over the last few decades there
has been an increasing trend towards understanding design guidelines existing in natural
composites and to implement the same in the development of synthetic composites. Of the
innumerable structural biological composites existing in Nature, the nacreous layer in the
seashells has received significant attention owing to its unique architecture and exceptional
mechanical properties. Over the past decade, researchers all over the globe have addressed the
reasons which are responsible for this attribute, and also attempted to synthesize them in the
laboratory. However from the perspective of structural application, it is imperative to
understand the suitability/applicability of these type of composites, should they be subjected to
a variety of loading rates. In our investigations, we have attempted to address as to why bioinspired nacre-mimetic composites are suited for impact loading conditions. We have also tried
to understand the mechanics of these type of composites when subjected to impact loading
and unearth the parametric attributes that are responsible for superior resistance. Additionally,
we have also tried to interpret what does it take for the nacreous layer to promote not only
significant fracture resistance but at the same time attain light-weight as well. Excerpts from
our investigations provide beneficial guidelines in designing tough bio-inspired composites at
any length scale.

ACKNOWLEDGMENTS

The author would like to express his sincere gratitude to his advisor Dr. Srinivasan Arjun
Tekalur for his persistent support, guidance, and critique throughout the duration of this
research work. The author would also like to express his gratitude to Dr. Alfred C. Loos, Dr.
Lawrence T. Drzal, and Dr. Gary J. Blanchard for all their valuable suggestions and associated
help in each and every part associated with this research.
The author would also like to thank Dr. Parimal Maity for his continued encouragement.
In addition to that, the author is also thankful to Wei Zhang, Oishik Sen, Andy VanderKlok,
David Gonzalez and Aiswarya Venkadachalam for providing an amicable environment in the
laboratory. The author also acknowledges the support of Gail Berry, Mike McClean, Adam and
Todd for their help throughout the course of the research.
Last but not the least, the author would like to thank his parents, his elder brother &
sister and his wife for their unabated belief in the author's capabilities and perennial support.

iii

TABLE OF CONTENTS

LIST OF TABLES

vi

LIST OF FIGURES

vii

KEY TO SYMBOLS

xii

1.

Synthetic Staggered Architecture Composites
1.1. Introduction
1.2. Materials and Methods
1.3. Results and Discussion
1.3.1. Effect on composite stiffness and strength
1.3.2. Effect on composite toughness
1.4. Conclusion
REFERENCES

2.

High Strain Rate Mechanical Behavior of Seashell-Mimetic Composites: 28
Analytical Model Formulation & Validation
2.1. Introduction
29
2.2. Mathematical Formulation
34
2.3. Materials and Methods
42
2.3. Results and Discussion
45
2.4. Conclusion
54
REFERENCES

3.

Optimal Overlap Length in Staggered Architecture Composites Under Dynamic
Loading Conditions
3.1. Introduction
3.2. Mathematical Formulation
3.3. Results and Discussion
3.3.1. Limitations of the analytical model and future directions
3.4. Conclusion
REFERENCES

61

Enhancing Resistance and Load Transfer Via Choice of Optimal Overlap Length
4.1. Introduction
4.2. Analytical Model
4.2.1. Solution scheme for elemental displacements
4.2.2. Computation of crack driving force (CDF)

106
107
110
116
119

4.

iv

1
2
6
8
12
16
21

62
65
80
96
100

4.2.3. Limitations associated with the analytical model
4.3. Results and Discussion
4.3.1. Validation by experimental data
4.3.2. An illustrative example
4.4. Conclusion
REFERENCES
5.

Conclusions
5.1. Design Guideline
5.2. Future Scope of Work

121
123
135
137
139
145
146
147

v

LIST OF TABLES

Table 1.1.

Influence of starting powder size & metal (base) plate on ceramic 10
content and aspect ratios of the synthesized silica-SC 15 composites.

Table 2.1.

Results of dynamic compression experiments on SiO 2 SC-15 composites.

Table 2.2.

Results of dynamic compression experiments on Al 2O3 SC-15 52
composites.

Table 3.1.

Comparison of experimentally measured joint strength against those 88
predicted by analytical model for varying overlap area and loading rates.

Table 4.1.

Recent advances on modeling staggered architecture biological 108
composites by various researchers.

Table 4.2.

Recent advances on investigating existence of characteristic overlap 109
length in staggered architecture biological composites by various
researchers.

vi

51

LIST OF FIGURES

Figure 1.1.

Brick and mortar type structural arrangement observed in seashells.

2

Figure 1.2.

Microstructure of the synthesized composites (c, d) exhibiting staggered 9
arrangement (a, b)-shown as blown-out view to obtain aspect ratios
associated with various samples. ρ and Φ w indicate the aspect ratio and
ceramic content respectively in the synthesized composites.

Figure 1.3.

TGA curves of the samples identifying varying ceramic content (Φ w) in 10
the synthesized composites (a); blown-out view another sample to
obtain aspect ratios associated with it. ρ indicates aspect ratio.

Figure 1.4.

Compressive stress-strain plots (a) for different sample ID's as indicated 13
in Table 1.1; variation of Young's modulus of elasticity (b) and maximum
compressive strength (c) of silica-SC 15 epoxy staggered composites for
varying ceramic content and aspect ratio; (d) sample used for
compressive experiments (specimen dimensions mentioned in Materials
and Methods); failure of the specimens along the interface (e).

Figure 1.5.

Dependence of stiffness of a staggered architecture composite as a 15
function of aspect ratio and ceramic content as investigated theoretically
under quasi-static [Gao et al., 2003] and dynamic rates of loading [Dutta
et al., 2013] (plot adapted from [Dutta et al., 2013]).

Figure 1.6.

Representative load-displacement curves obtained upon fracture testing 17
of samples in accordance with ASTM C1421 (a), variation of fracture
toughness of silica-SC 15 brick and mortar composites with varying
ceramic content and aspect ratio and their comparison against the
fracture toughness of silica (ceramic) and SC 15 (polymer) alone (b).

Figure 2.1.

Dependence of stiffness of biocomposites (exhibiting brick and mortar 31
type of architecture at its most elementary level) as a function of aspect
ratio of ceramic bricks and volume fraction of ceramic as laid down by
Gao et al., 2003. Inset shows the brick and mortar type microstructural
arrangement observed in seashells, and microstructure of bone.

vii

Figure 2.2.

(a) Schematic of the brick and mortar type structural arrangement 34
subjected to external loading; (b) load acting on the ceramic and transfer
of load between ceramic platelets by shearing of the intermediate
polymer layers; (c) variation of shear stress and displacement profile
along the ceramic thickness for the outer ceramic; (d) variation of shear
stress and displacement profile along the ceramic thickness for the inner
ceramic.

Figure 2.3.

(a) Typical specimen and its representative microstructure (b, c); (d) 44
schematic showing distribution of ceramic platelets in polymeric matrix
in brick and mortar type microstructural composites; (e) region under
consideration highlighting the overlap region; (f, g) deformed platelets
and intermediate polymer upon tensile and compressive loading
respectively; (h) specimen for quasi-static experiments; (i) fractured
specimen under quasi-static loading.

Figure 2.4.

(a) Schematic of a Split Hopkinson Pressure Bar (SHPB); (b) plot of pulses 48
traced in the oscilloscope based on the pulses recorded by the strain
gages in the incident bar (solid pink) and transmission bar (solid yellow);
(c) specimen for SHPB experiments; (d) fractured specimen upon impact
loading; (e) ratio of force on the transmission face to the incident face
over time; (f) plot of strain-rate versus time.

Figure 2.5.

Dynamic stress strain curves obtained for SiO 2 SC-15 ceramic polymer 50
composite (a) and Al2O3 SC-15 ceramic polymer composite (b)
corresponding to varying ceramic contents and strain-rates.

Figure 2.6.

Plot of (a) experimental stress, and (b) strains (at peak stress) for SiO 2 SC- 52
15 ceramic polymer composite corresponding to varying ceramic
contents at different strain-rates.

Figure 2.7.

Plot of (a) experimental stress, and (b) strains (at peak stress) for Al 2O3 53
SC-15 ceramic polymer composite corresponding to varying ceramic
contents at different strain-rates.

viii

Figure 3.1.

(a) Schematic of brick-and-mortar microstructure identifying the overlap 65
region; Representative loading pulses obtained from SHPB experiments:
(b) triangular pulse, and (c) half-sinusoidal pulse.

Figure 3.2.

Shear stress distribution profile over the overlap length at varying times 81
under the application of triangular loading pulse.

Figure 3.3.

Variation of shear stress at the extremities of the joint against overlap 83
length for nacre (a), and its dependence on loading rate (b).

Figure 3.4.

Variation of shear stress at the extremities of the joint against overlap 85
length for spider-silk (a), and its dependence on loading rate (b).

Figure 3.5.

Variation of shear stress at the extremities of the joint against overlap 86
length for collagen in tendon/bone (a), and its dependence on loading
rate (b).

Figure 3.6.

Variation of elastic strain energy density against varying overlap length at 91
different intervals of time under the application of triangular pulse
loading (a), and dependence of elastic strain energy density on loading
rate, under the application of triangular pulse loading, shown via (1)
varying max stress (keeping time at maximum stress fixed)-shown in red
and (2) varying time at max stress (keeping maximum stress fixed)shown in black; dependence of w eff is markedly pronounced towards
increasing stress-levels than for time under the application of triangular
pulse loading.

Figure 3.7.

Plot showing the dependence of maximum shear stress (at the ends of 93
overlap) as a function of aspect ratio and volume fraction of the ceramic
bricks for: aspect ratio varying between 4 to 200 (a) and, 4 to 800 (b)
under the application of triangular pulse loading, corresponding to a
given loading-rate.

Figure 3.8.

Plot showing the dependence of the ratio of composite modulus to 95
Young's modulus of the ceramic bricks for varying aspect ratio and
volume fraction of the ceramic bricks: correlation between dynamic (a)
and quasi-static (b) regime, corresponding to a given loading rate.

ix

Figure 3.9.

Variation of shear stress at the interface as a function of overlap length 99
in a material (in the current case, nacreous layer in seashells) under (a)
static, and (b) dynamic rates of loading.

Figure 4.1.

Toughening mechanisms associated with nacre and region under
consideration.

Figure 4.2.

(a) Schematic of staggered architecture of calcium carbonate bricks 113
(shown in green) in biopolymer matrix (shown in yellow); (b) expanded
view of the 2-dimensional unit cell structure (with plane strain

111

infinitesimal deformation) showing the pre-cracked length (L1) and
instantaneous overlap length (L2) and coordinate systems; (c) expanded
view of the 2-dimensional unit cell structure (with plane strain
infinitesimal deformation) highlighting the elemental blocks and the
original overlap length (LO); (d) splitting of parent crack tip for crack
driving force calculations; (e) schematic identifying shear and normal
force resultants NC and QC respectively.
Figure 4.3.

Variation of interfacial shear stress (at the extremities of the overlap) as 124
a function of varying overlap length and pre-crack. The figure in the inset
shows the expanded view for lower shear levels in the unit cell.

Figure 4.4.

Plot of crack driving force as a function of overlap length in nacre for
varying pre-crack lengths.

Figure 4.5.

Variation of crack driving force against applied stress for overlap length 128
on either end of the characteristic value in the nacreous layer.

Figure 4.6.

Mode mixity parameter as a function of overlap length for varying pre- 129
crack lengths.

Figure 4.7.

Variation of crack driving force upon overlap length for varying pre-crack 131
length associated with the primary crack. The figure in the inset identifies
the angle (θ) at which crack driving force of the kinked crack attains a
maximum value for varying pre-crack length (0.09-1.29μm).

x

127

Figure 4.8.

Variation of crack driving force upon overlap length for varying pre-crack 132
length associated with the primary crack. The figure in the inset identifies
the angle (θ) at which crack driving force of the kinked crack attains a
maximum value for a pre-crack length of 1.49 μm.

Figure 4.9.

(a) Schematic of staggered architecture of calcium carbonate bricks 133
(shown in green) in biopolymer matrix (shown in yellow); (b) expanded
view of the 2-dimensional unit cell structure (with plane strain
infinitesimal deformation) highlighting the elemental blocks and the
original overlap length (Lo); (c) crack trajectory through the overlap
length and crack jump/deflection from one unit cell to another.

Figure 4.10.

Existence of optimal length scale, as predicted by the analytical model, 136
for an Al99.5- Loctite 3421 single lap adhesive bonded joint.

Figure 4.11.

Existence of optimal overlap lengths for varying widths of composite 138
adherend (10 mm, 40 mm and 100 mm) and adhesive thickness (0.50
mm, 1.00 mm, 2.00 mm, 3.00 mm and 5.00 mm). The hollow and solid
circles are representative of characteristic length with respect to
minimization of crack driving force (a,c,e) and shear stress (b,d,f)
respectively.

Figure 5.

Design guideline for synthetic composite showing the interplay of 146
optimization scheme and manufacturing technique.

xi

KEY TO SYMBOLS

Φ

Volume fraction of ceramic.

W

Inter-ceramic spacing.

V

Freeze front velocity.

Μm

Microns.

Φw

Weight fraction of ceramic.

E

Young's modulus of the composite (in Chapter 1)/ of the ceramic
(in Chapter 3).

Em

Young's modulus of the mineral/ceramic.

Sp

Strength of the polymer (or, polymer-ceramic interface).

Sm

Compressive strength of the ceramic bricks.

Ρ

Aspect ratio (in Chapter 1)/density (in Chapter 3) of ceramic
bricks.

KIc

Fracture toughness (MPa m

a/W

Ratio of crack length to the width of specimen.

Pmax

Maximum load.

S

Span.

B

Specimen width.

xii

0.5

).

J

Fracture energy.

Θp

Effective strain to which the polymer can deform before failure.

Sp

Yield strength of the polymer.

Sint

Ceramic polymer interface strength.

τp

Polymeric shear strength (in Chapter 1)/shear stress (in Chapter
2).

p

Strain in the polymer.

Gp

Protein (polymer) shear modulus.

To

Axial stress resultant in the outer ceramic.

Ti

Axial stress resultant in the inner ceramic.

ao

Average acceleration of the cross-section of the outer ceramic.

ai

Average acceleration of the cross-section of the inner ceramic.

H

Thickness of the ceramic (in Chapter 2)/Polymer thickness (in
Chapter 3).

uo

Horizontal displacement of the outer ceramic brick.

ui

Horizontal displacement of the inner ceramic brick.

o

Shear stress in the outer ceramic.

i

Shear stress in the inner ceramic.

xiii

Gm

Shear modulus of the ceramic.

o

Shear strain in the outer ceramic.

i

Shear strain in the inner ceramic.

z'

Location of origin for the outer ceramic.

z''

Location of origin for the inner ceramic.

uop

Displacement of the ceramic at the interface between the outer
ceramic block and the intermediate polymer.

uip

Displacement of the ceramic at the interface between the inner
ceramic block and the intermediate polymer.

p

Shear strain in the polymer.

tp

Thickness of the polymer.

2L*

Width of the ceramic layer.

cc

Wave velocity through the ceramic.

Α

A parameter associated with structural and geometrical
properties.

F(t)

Prescribed loading at the inner and outer ceramic.

p

Approximate shear strain.

N1(x), N2(x)

First order interpolation functions and correspond to weighted
residuals in Galerkin method.

 e 
K 



Elemental stiffness matrix.

xiv

 e 
M 



Elemental mass matrix.

C

Wave velocity in the incident bar (in Chapter 2)/through the
ceramic (in Chapter 3).

eff
o

Effective strain in the outer ceramic.

predicted

Predicted strain.

predicted

Predicted stress.

Eb

Young's modulus of the transmission bar.

Ab

Cross-sectional area of the transmission bar.

As

Instantaneous cross-sectional area of the specimen.

ls

Thickness of the specimen for SHPB testing.

J

Polar moment of inertia of the cross-section of the specimen.

εi

Incident strain record.

εr

Reflected strain record.

εt

Transmission strain record.

ἑ

Strain rate.

εexp

Strain value obtained experimentally.

εcal

Calculated strain value.

xv

1  1  x,t 

Normal stress developed in the upper ceramic brick.

2  2  x,t 

Normal stress developed in the lower ceramic bricks.

 or, ζ

Shear stress developed in the polymer.

u1  u1  x,t 

In-plane displacement of the upper ceramic brick.

u2  u2  x,t 

In-plane displacement of the lower ceramic brick.

G

Shear modulus of the polymer.

B

Half the width of ceramic bricks.

T

Time.

u4  u4  x,t 

A variable dependent upon terms u1  u1  x,t  and u2  u2  x,t  .

w  x,t 

Smoothing function satisfying boundary conditions.

v  x,t 

A variable leading to solution for u4  x,t  .

 t

External acting stress on the unit-cell structure.

K

A parameter dependent upon E, G, b and h.

p04 ,pn4

Variables for computing another variable p4 .

L

Overlap length.

n4

Eigen values for v  x,t  , n  1 .

Xn4

Eigen functions for v  x,t  , n  1 .

T04 ,Tn4

Variables for computing temporal variation T4 corresponding to
v  x,t  .
xvi

an4 ,en4

Variables for computing Tn4 .

a04 ,b04 ,d04

Variables for computing T04 .

u3  u3  x,t 

Another

s  x,t 

Smoothing function satisfying boundary conditions.

r  x,t 

A variable leading to solution for u4  x,t  .

p03 ,pn3

Variables for computing another variable p3 .

n3

Eigen values for r  x,t  , n  1 .

Xn3

Eigen functions for r  x,t  , n  1 .

T03 ,Tn3

Variables for computing temporal variation T3 corresponding to
r  x,t  .

an3 ,en3

Variables for computing Tn3 .

a03 ,b03

Variables for computing T03 .

max or, 0

Maximum value of externally applied stress.

ta or t f

Time at which max occurs.

Μs

Micro-seconds.

P

Loading rate (stress per unit time).

Nm

Nanometer.

ct

Wave velocity in the transmission bar.

variable dependent upon terms

u2  u2  x,t  .

xvii

u1  u1  x,t  and

Als

Area of the adhesive lap joint.

u1 avg

Average acceleration.

weff

Elastic strain energy density.

eff

Effective stress.

eff

Effective strain.


s

Strain rate of the organic matrix.

L

Characteristic length.

Loptimal

Optimal overlap length.

L1

Pre-cracked length in the unit cell.

L2

Instantaneous overlap length in the unit cell.

Lo

Original overlap length.

Nc

Shear force resultant.

Qc

Normal force resultant.



Thickness of the bio-polymer in nacre.

Z

Thickness direction of the ceramic bricks in nacre.

x, x1, and x2

Horizontal axes with different origins.

a0

Void length.

da

Length of pre-existing incremental crack.

xviii

uix

Displacement along the horizontal direction corresponding to
block i (=1,4).

ui
z

Displacement along the thickness direction corresponding to block
i (=1,4).

ixx

Axial stress corresponding to block i (=1,4).

i
zz

Peeling stress corresponding to block i (=1,4).

ixz

Shear stress corresponding to block i (=1,4).

p
zz

Axial strain in the biopolymer.

p
zz

Normal (peel) strain in the biopolymer.

p
xz

Shear strain in the biopolymer.

p
zz

Normal (peel) stress in the biopolymer.

p
xz

Shear stress in the biopolymer.

c c
C11 ,C22 ,Cc
44

Orthotropic properties (stiffness) of ceramic bricks.

Cp ,Cp ,Cp
22 33 44

Material properties (stiffness) of the bio-polymeric matrix.

M1 ,M2 ,M3

c c
Variables computed from C11 ,C22 ,Cc , Cp ,Cp ,Cp .
44 22 33 44

A
ux

Displacement along the horizontal direction corresponding to
point A.

A
uz

Displacement along the thickness direction corresponding to point
A.

xix

uB
x

Displacement along the horizontal direction corresponding to
point B.

uB
z

Displacement along the thickness direction corresponding to point
B.

W

Work required to close the virtual crack.

GI

Mode I crack driving force.

GII

Mode II crack driving force.

E1 ,E2 ,E3

Young's modulus along orthogonal directions for the mineral
component in the nacreous layer.

G12 ,G23 ,G13

Shear modulus values for the mineral component in the nacreous
layer.

Ep ,Ga , a

Properties of biopolymer layer in nacre.

Lcharacteristic   min 

Characteristic length on minimization of maximum shear stress.

Lcharacteristic   CDFmin 

Characteristic length on minimization of crack driving force (CDF).

xx

CHAPTER 1: SYNTHETIC STAGGERED ARCHITECTURE COMPOSITES

1

1.1. Introduction
Biological materials are composites structures synthesized by nature via bottom-up
route and exhibit complex hierarchical design over several length scales. Nacre, or the motherof-pearl, is one of the most widely studied biocomposites since it is inherently light weight and
strong (high stiffness and specifically, superior toughness) [Jackson et al., 1988; Sarikaya et al.,
1990; Menig et al., 2000; Kamat et al., 2000; Barthelat and Espinosa, 2007]. Nacre is
composed of polygonal mineral tablets (width=5-10 µm & thickness=0.4-0.5 µm) of high weight
fraction (Φ=0.95) which are cemented together with very small amount of polymeric matrix
(thickness=20-30 nm) which is composed of proteins and polysaccharides. Irrespective of the
degree of hierarchical complexity exhibited by the biocomposites, at the most elementary level,
they exhibit a generic microstructure comprising of nanometer sized inorganic crystals
embedded in a soft organic matrix (biopolymers) in the form of a staggered architecture [Gao
et al., 2003], otherwise known as a brick-and-mortar type structural arrangement (Figure 1.1).

Figure 1.1. Brick and mortar type structural arrangement observed in seashells. For
interpretation of the references to the color in this and all other figures, the reader is referred
to the electronic version of this dissertation.
2

Nature cleverly uses aspect ratio and ceramic content for fine tuning properties for
functions. For instance, bone is primarily designed for withstanding time-dependent or, impact
loading (for example, withstanding stresses while running, jumping, accidental fall) and hence
the microstructure is designed primarily from enhanced toughness point of view and moderate
stiffness. On the contrary, seashells (which lie in the abyss of oceans) apart from providing
defense to the organism (it is hosting) against predatory attacks, is also subjected to quasistatic loading (for example, enormous hydrostatic pressure). Hence, emphasis on the shell
structure evolution is directed primarily towards providing enhanced stiffness. Such an amazing
capability of tuning properties based on a function has rarely been seen in synthetic
composites. Although bone and nacre differ from each other in varying aspects, deformation
and toughening mechanisms in both these materials are similar. Thus, in order to synthesize
composites having brick and mortar architecture, freeze-casting technique can be used to first
synthesize the ceramic relic. It consists of four principal steps: (1) ceramic slurry preparation, (2)
freezing the ceramic-solvent system, (3) sublimation of the frozen solvent, and (4) sintering of
the porous ceramic relic. A short summary of the recent work in the literature of freeze-casting
is given as follows. Porous alumina (Al2O3) and silicon nitride (Si 3N4) ceramics were produced
using water as the carrier where growth of dendritic shaped ice crystals was observed upon
freezing which produced elliptical pores (dendritic channel) having a lengths of major and minor
axis equal to 500 μm and 30 μm respectively [Fukusawa et al., 2002]. Simultaneously,
camphene (C10H16) was also being used as a freezing vehicle because of its environment

3

friendliness and its ability to cast at room temperatures dense Al 2O3 ceramics [Sofie and
Dogan, 2001]. Araki and Halloran, 2005 produced Al2O3 ceramics by using an amine derivative
of a fatty acid condensation polymer as a dispersant in addition to the Al 2O3-C10H16 slurry
which was cast on polyurethane molds and subsequently sintered at 1600 °C for 4 hours and
obtained circular cross-sectional channels. Porous silica (SiO 2) ceramics were produced using
polyvinyl alcohol (PVA) as the binder by casting the slurry onto glass tubes which was
subsequently immersed in liquid nitrogen at a controlled speed [Zhang et al., 2005]. Porous
hydroxyapatite (HAP) scaffold were produced using glycerol as the carrier, PVA as a binder (1.5
wt%) and Dynol 604 as the dispersant (0.75 wt%) by casting the solutions on PVC tubes which
was covered with polyurethane foam [Fu et al., 2008]. Similarly, porous HAP ceramic scaffold,
titanium foam and HAP/tricalcium phosphate (TCP) ceramic scaffold respectively using
camphene as the freezing vehicle [Chino and Dunand, 2008; Macchettaa et al., 2009; Yoon et
al., 2007]. Pore channels generated were greater than 100 μm which enabled their use for
bone-tissue engineering applications. Porous HAP have been produced which were modified by
silica [Blindow et al., 2009], with functionally graded core-shell structure (using camphene as
the carrier) [Soon et al., 2011], and using distilled water as the carrier, PVA as a binder and
ammonium polyacrylate as the dispersant [Zuo et al., 2011]. Titanium foams [Li and Dunand,
2011] and porous Si3N4 ceramics [Ye et al., 2010] have been produced using distilled water as
the media and using a variety of compounds as an addition agent to observe subsequent pore

4

structure and geometry (for example, ammonium polymethacrylate anionic dispersant,
polyacrylamide dispersion agent, etc). Macroporous alumina ceramics [Jing et al., 2010], Al2O3ZrO2 ceramic (20:80 with a 40-80 wt% solid loading) [Liu et al., 2010], and porous alumina
ceramic using Al2O3 sol [Yoon et al., 2010] as a substitute for water and as a medium for
making ceramic slurry have also been produced. The authors [Yoon et al., 2010] also discussed
the use of PVA both as a binder as well as an emulsifying agent to stabilize air bubbles during
magnetic stirring. Following the manufacturing of the ceramic backbone, the ceramic-polymer
composites can be synthesized via infusion of polymer into the porous ceramic. Deville and coauthors [Deville et al., 2006] were the first to produce Alumina epoxy lamellar composite by
filling the porous IT scaffolds by epoxy. Al2O3-PMMA brick and mortar and lamellar composites
[Launey et al., 2009] have been synthesized by introducing a polymeric phase (by free radical
polymerization of methylmethacrylate (MMA) initiated by 2,2 ’-azobisisobutyronitrile (AIBN))
into the porous Al2O3 freeze cast component. By grafting the Al 2O3 with another polymeric
component [3-(trimethoxysilyl)propyl methacrylate (γ-MPS)], significant improvement in
mechanical properties have been obtained in comparison to the non-grafted, Al2O3 and PMMA
counterparts alone.
All the above mentioned researches have attempted to mimic lamellar/brick and mortar
microstructure in its entirety. In our current investigation, we are not trying to mimic brick and
mortar microstructure of nacre; rather, what we are interested is in drawing inspiration from
5

staggered architecture in biological composites at their elementary level (such as nacre: 0.900.95 ceramic content [Jackson et al., 1988], bone: 0.50 ceramic content [Currey, 2002], dentin:
0.43 ceramic content [Currey, 2002], etc.), we want to (a) synthesize synthetic ceramic-polymer
composites with varying ceramic content and aspect ratio, and (b) experimentally understand
the structure-property correlationship of these composites when subjected to external loading.
Excerpts drawn from this investigation would thereby help in custom-design manufacture of
hybrid bio-inspired composite materials.
1.2. Materials and Methods
Sodium dodecyl sulfate (C 12H25NaO4S) was obtained from Sigma Aldrich (St. Louis, MO,
USA). Poly (vinyl alcohol) [-CH2CH(OH)-]n 98% hydrolyzed having an average molecular weight
Mw 13,000-23,000 was obtained from Aldrich Chemical Company, Inc. (Milwaukee, WI, USA).
Sucrose (C12H22O11) crystals were obtained from Roche Diagnostics Corporation (IN, USA). Two
types of ceramic powders were purchased: Silica (SiO 2) spheres of 8µm and 1µm diameter were
obtained from Fiber Optic Center Inc (New Bedford, MA, USA). SC-15 epoxy resin (toughened
two phase) was obtained from Applied Poleramic Inc. (Benicia, CA, USA).
The ceramic scaffolds were prepared following the sequence of steps as have been
carried out by authors [Deville et al., 2006; Launey et al., 2009] and for brevity purposes not
described in detail. Suspension of micrometer sized ceramic powders were prepared by
dispersing ceramic powders of different concentrations (30 wt%, 40 wt%, and 50 wt%) into a
solution composed of 2 wt% polyvinyl alcohol (PVA), 4 wt% sucrose and 5 wt% sodium dodecyl
6

sulfate (SDS) in order to produce ceramic scaffolds having varying porosity. The colloidal
solutions were sonicated for 10 minutes duration (Fisherbrand, FB 11021) at 35W. The ceramic
suspensions were casted in teflon (PTFE) molds on top of a metal plate which was cooled by
liquid nitrogen. The frozen samples were subsequently transferred to a freeze dryer (Freeze
Dryer 1.0, Labconco, Kansas City, MO) for a period of 48 hours which in turn promotes
sublimation of ice and thereby, we are left behind with the ceramic green body. Densification of
the ceramic was by obtained by sintering at 1550 °C for 2 hours. The epoxy is prepared by
mixing 100wt% SC-15 Part A (resin) and 30wt% of SC-15 Part B (hardener). The mixture is
stirred gently for approximately 10 minutes followed by degassing for another 10 minutes.
Finally, the scaffolds were infiltrated with the two-phase epoxy followed by curing of the epoxy
at 60°C for 2 hours and 94°C for 4 hours.
Morphological features of the samples were acquired using a JEOL JSM-6400V scanning
electron microscope (JEOL Ltd., Tokyo, Japan). Samples thermogravimetric analysis (TGA Q500
-1

V20.10 Build 36, TA Instruments) were conducted at a heating rate of 20 °C min

up at

temperatures up to 550°C. ASTM D695 was followed for the determination of compressive
strength of these ceramic polymer composites under quasi-static rates of loading. The
specimen dimension chosen for testing was 5mm × 5mm × 10 mm and was performed in a MTS
machine (10kN load cell) at a crosshead speed of 1mm/min. ASTM C1421 was followed for the
determination of fracture toughness of the synthesized composites. The specimen dimension

7

chosen for testing was 4mm × 4mm × 16 mm (with an initial crack length of 1.40 mm), and was
performed in a MTS machine (10kN load cell) at a crosshead speed of 0.01 mm/s.
1.3. Results and Discussion
The first task in the current investigation involves synthesizing composites with varying
ceramic content and aspect ratio. As mentioned earlier, freeze-casting technique followed by
infusion of polymer (into the porous ceramic) has been employed for synthesis of ceramicpolymer composites. The most critical stage of the entire freeze-casting process is the
controlled solidification of the slurry. Formation of the structure and future porosity are
determined in this step. The cooling rates, for freezing the water to ice, typically will range from
-1

-1

0.1°C min to 10°C min . Freezing front velocity, which is directly proportional to cooling rate,
plays an important role in the final microstructure development. With reference to the basic
crystallography of ice, water solidifies into anisotropic hexagonal ice crystals during freeze
casting. Empirically, the inter-ceramic spacing w has been found to vary with freezing-front
velocity v [Deville, 2008] as follows: w 

1
. In order to control the freezing rate and thereby,
vn

the porosity of the ceramic, we employ a simple and fundamental approach. The ceramic
suspensions are casted in PTFE molds on top of a metal plate which in-turn is cooled by liquid
nitrogen. The freezing velocity or the solidification rate of the solution can be tuned by
changing the base metal plate. In current experiments, we have employed two types of metal
plate: aluminum and copper. Since, thermal conductivity of copper (401 W/m-K) is greater than
the thermal conductivity of aluminum (167 W/m-K), for the same ceramic concentration in

8

solution, denser ceramic backbone (or, lower porosity ceramics) will be obtained which in-turn
implies, higher weight fraction of ceramic content in synthesized ceramic-polymer composites.
Additionally, the starting particle size is seen to have an effect on the solidification kinetics, and
subsequently on the structural wavelength (or rather, the final thickness of the ceramic bricks).
Hence, we employed two different sized powders (particle diameters of 8µm and 1µm) to
control the structural wavelength and thereby, the aspect ratio of the ceramic blocks in the
synthesized composite. Thus following this simple approach, both ceramic content and aspect
ratio were controlled in the synthesized silica-SC 15 composites. Figure 1.2(a,c) and Figure
1.3(b) shows the staggered microstructure of a representative sample.

Figure 1.2. Microstructure of the synthesized composites (c,d) exhibiting staggered
arrangement (a, b)-shown as blown-out view to obtain aspect ratios associated with various
9

Figure 1.2. (cont'd)
samples. ρ and Φw indicate the aspect ratio and ceramic content respectively in the synthesized
composites.

Figure 1.3. TGA curves of the samples identifying varying ceramic content (Φ w) in the
synthesized composites (a); blown-out view another sample to obtain aspect ratios associated
with it. ρ indicates aspect ratio.
Table 1.1. Influence of starting powder size & metal (base) plate on ceramic content and aspect
ratios of the synthesized silica-SC 15 composites.
Ceramic Content

Initial
Particle
Size

in composite (Φw)

Aspect
Ratio (ρ)

*From TGA

*From SEM

Aluminum

1-μm

35%

15

Φw=0.35/ρ=15

30%

Copper

1-μm

40%

15

Φw=0.40/ρ=15

40%

Aluminum

8-μm

55%

10

Φw=0.55/ρ=10

40%

Copper

8-μm

58%

10

Φw=0.58/ρ=10

Ceramic
Conc. in
Solution

Plate
Type

30%

10

ID

Table 1.1. (cont'd)
45%

Aluminum

8-μm

60%

10

Φw=0.60/ρ=10

50%

Aluminum

8-μm

65%

7

Φw=0.65/ρ=7

50%

Copper

8-μm

70%

6

Φw=0.70/ρ=6

Table 1.1 lists the details of the above set of processes for different starting ceramic
concentrations & powder size. Figure 1.3(a) demonstrates the TGA curves of different samples
for subsequent evaluation of ceramic content in the samples, the values of which are tabulated
in Table 1.1. Aspect ratio of the ceramic platelets is defined as the ratio of length to the width
of the ceramic platelet. Corresponding to each sample ID, SEM was done to obtain the aspect
ratio of the ceramic platelets. Figure 1.2(a-d) shows the microstructure of the synthesized
samples which helps in identifying the aspect ratio of the ceramic platelets in the composite.
For a given particle size as a precursor, as the ceramic content in the composite increases, the
aspect ratio of the platelets decreases which is intuitive as the thickness of the platelets
remains constant; however, the length of the platelets decreases owing to higher particle
distribution in the interdendritic space [Deville, 2008]. Nevertheless, the aspect ratio of the
ceramic platelets varied from 6-10 & 15-16 for 8µm and 1µm ceramic spheres respectively.
Ceramic concentration in solution greater than 30wt% & 50 wt% for 1-µm and 8-µm were not
probed into as it was difficult to maintain suspension of ceramic particles in solution under
those conditions.
An integral part of understanding predator-prey interactions is to first understand the
mechanisms operating in food web dynamics. For instance in a marine food chain, otters and
crabs can be considered as the predators who use their molar tooth and claws respectively to
11

prey the protein-rich flesh of the organism (acting as the prey) hosted by seashells. Similarly,
bone has to be durable enough to withstand high stresses associated with biting of tooth,
sudden fall, etc. Predatory attacks involve application of compressive loads to the structural
component associated with the prey that has to be not only durable but fracture-resistant as
well. However, it can be pointed out that mechanical behavior of structural biocomposites is
tuned predominantly for the function that it is specifically designed for. For instance, bone is
primarily designed for withstanding time-dependent or, impact loading (for example,
withstanding stresses while running, jumping, accidental fall) and hence the microstructure is
designed primarily from enhanced toughness point of view and moderate stiffness. On the
contrary, seashells (which lie in the abyss of oceans) apart from providing defense to the
organism it is hosting against predatory attacks, is also subjected to quasi-static loading (for
example, enormous hydrostatic pressure). Hence, emphasis on the shell structure evolution is
directed primarily towards providing enhanced stiffness. In order to observe whether the
synthesized composites also exhibit this variance in mechanical behavior for varying ceramic
content and aspect ratio, quasi-static compression and fracture experiments were performed
to better understand this aspect.
1.3.1. Effect on composite stiffness and strength
Figure 1.4(a) shows the representative stress-strain plots obtained upon quasi-static
compressive loading of the synthesized composites. It has been shown theoretically, as shown
in Figure 1.5(a,b), that the stiffness of a staggered architecture composite can be tuned via

12

controlling its aspect ratio for a given ceramic content under both static [Gao et al., 2003] and
dynamic rates [Dutta et al., 2013] of loading.

Figure 1.4. Compressive stress-strain plots (a) for different sample ID's as indicated in Table 1.1;
variation of Young's modulus of elasticity (b) and maximum compressive strength (c) of silica-SC
15 epoxy staggered composites for varying ceramic content and aspect ratio;
13

Figure 1.4. (cont'd)
(d) sample used for compressive experiments (specimen dimensions mentioned in Materials
and Methods); failure of the specimens along the interface (e).
Indeed similar observations have been observed from experimentally obtained data as
shown in Figure 1.4(b,c) where as the aspect ratio increases, the stiffness of the composite
increases as well even if high polymeric content is present in the composite.
For a ceramic polymer composite having a staggered microstructure, the ceramic forms
the load-bearing component and load transfer from one platelet to another is accomplished via
shear-deformation of the polymer. Larger is the aspect ratio, larger will be the force that would
be transmitted from one ceramic brick to another via the polymer matrix. Failure of the
interfacial matrix is initiated at the extremities of the polymeric layer joint via nucleation and
growth of plastic zone from either ends of the overlap length towards the center [Apalak and
Gunes, 2007; Ouinas et al., 2010; da Costa et al., 2012; Sayman et al., 2013]. Indeed failure of
the synthesized composite has been observed along the interfaces as shown in Figure 1.4(e).
Hence, aspect ratio of the ceramic platelets plays a superior role in its contribution to the
composite's stiffness.
It might be speculated that the shear model might not be applicable under compressive
loading scenario for staggered architecture composites, because microbuckling has been
observed as the failure mode operational under quasi-static and dynamic rates of loading in
seashells [Menig et al., 2000]. This aspect, however, can be addressed as follows [Dutta et al.,
2012]:

14

Figure 1.5. Dependence of stiffness of a staggered architecture composite as a function of
aspect ratio and ceramic content as investigated theoretically under quasi-static [Gao et al.,
2003] and dynamic rates of loading [Dutta et al., 2013] (plot adapted from [Dutta et al., 2013]).
The ceramic backbone in a ceramic polymer composite is brittle in nature. Hence, in
these composites failure initiates at the micro-scales in the form of interfacial matrix failure
starting at the joint extremities. The nacreous layer in seashells is a exhibits a hierarchical
architecture (2-3 levels) where the staggered architecture is exhibited only at the level of microscales; however, the meso-scale is representative of a lamellar architecture (equivalent to a
fiber-reinforced composite structure) comprising of 300 μm layers between nearly 20 μm layers
of viscoplastic material [Menig et al., 2000]. Plastic microbuckling is the dominant failure
mechanism in fiber-reinforced composites under the action of compressive loads, and thereby
based on this analogy, microbuckling was hence observed in seashells under compressive
loading conditions. Similar inferences have been laid down [Menig et al., 2000] where it has
been observed in the lamellar structures of seashells that cracks initiates first in the viscoplastic
layers separating the mesolayers. Additionally, with reference to the microstructure of the
15

synthesized composites, ceramic bridges present amidst the ceramic layers (an attribute
associated with nacre at micro-scales) also cripples Euler buckling of the ceramic layers at
microscopic scales [Deville et al., 2006]. Similarly, buckling is restrained at micro-scales, under
rapid-compression ballistic conditions, via synchronized deformation twinning of the nano-scale
particles [Huang et al., 2011].
Similarly, for a staggered ceramic-polymer composite, the strength [Gao et al., 2003]
can be expressed as follows:

 Sp Sm 
S  min 
,

2 
 2

(1.1)

where, S p is the strength of the polymer (or, polymer-ceramic interface) and Sm is the
compressive strength of the ceramic bricks. Based on the experimentally obtained data and the
theoretical foundation [Gao et al., 2003; Ji and Gao, 2010], it can be clearly seen from Figure
1.4(c) that aspect ratio (ρ) plays a superior role than ceramic content in its contribution towards
the strength of the staggered ceramic-polymer composite and confirms that fine tuning of
properties can be obtained in synthetic composites by varying ceramic content and aspect
ratio. Thus, from the point-of-view of stiffness and strength, the aspect ratio of mineral crystals
significantly affects the mechanical properties of the composite materials in a number of
aspects (for instance, microstructural load transfer) as is evident from established theoretical
models [Ji and Gao, 2010] and current experimental observations.
1.3.2. Effect on composite toughness

16

Figure 1.6. Representative load-displacement curves obtained upon fracture testing of samples
in accordance with ASTM C1421 (a), variation of fracture toughness of silica-SC 15 brick and
17

Figure 1.6. (cont'd)
mortar composites with varying ceramic content and aspect ratio and their comparison against
the fracture toughness of silica (ceramic) and SC 15 (polymer) alone (b).
What makes bioinspired staggered architecture very promising from structural point of
view is its unique structural arrangement which confers it superior toughness in comparison to
its ceramic and polymer counterparts alone. Whether similar attributes exist in the synthesized
composites, fracture toughness of the composites were measured to observe this effect and
the results are shown in Figure 1.6(a,b).
ASTM C1421 was followed for the determination of fracture toughness of the
synthesized composites. Figure 1.6(a) shows the representative load-displacement curves
obtained during fracture testing of the samples in a three-point bending set up. As per ASTM
C1421, determination of K Ic, as shown in Figure 1.6(b) is as follows:



 Pmax S   3a / W 1/2 
KIc  g 

 BW3/2   2 1  a / W 3/2 



 

where,
 a
g  g  
W

(1.2)

2
1.99  a / W 1  a / W  2.15  3.93a / W   2.7 a / W  




1  2a / W 

As it can be seen from Figure 1.6(b), the measured fracture toughness of all the
synthesized composites is significantly greater than the fracture toughness of the ceramic
(silica: 0.64 MPa m

0.5

) and the polymer (SC 15: 1.54 MPa m

0.5

) [Robinette et al., 2007]. Once

again, the effect of both aspect ratio and ceramic content of the ceramic can be seen to
18

contribute a superior role to the composite's fracture toughness even if high polymeric content
is present in the composite. This can be explained as follows.
Dugdale's cohesive strip model [Gao et al., 2003; Ji and Gao, 2010] is given by the
following equation,



J  1   L  pdp  1   Lp min Sp ,Sint , p



(1.3)

where, J is the fracture energy, Θ p denotes the effective strain to which the polymer can
deform before failure, Sp denotes the yield strength of the polymer, Sint denotes the ceramicpolymer interface strength, and τ p denotes shear strength of the polymer. With reference to
equation (1.3), the expression under the integral is representative of the dissipation energy by
the polymer per unit volume and thus, Θ p is a key parameter which contributes to fracture
energy. The polymeric matrix is thereby essential for enhancing toughness via energy
dissipation due to its viscoelastic nature that contributes to large deformation. Also it is
discernible that inelasticity is a key material parameter that renders a material notch insensitive
by enabling a material to eliminate stress concentration at strain concentration locales.
However, the aspect ratio also plays an important role which is explained as follows. The
staggered microstructure provides a strain amplification mechanism [Ji and Gao, 2010] and
contributes to full utilization of large deformation capability of the polymer.

p 

 


    m   
2 1    
L 

19

(1.4)

As it can be seen from equation (1.4), the capability of the polymeric matrix to be
strained is magnified over composite strain by the aspect ratio of the ceramic platelets, thereby
allowing the polymer to deform and dissipate energy at the microscopic level without giving
rise to large deformation on the composite level. For composites having high ceramic content,
fracture is often triggered before plastic instability inception via abrupt percolation of damage
across the material [Mortensen and Llorca, 2010]. This explains, that why toughness decreases
in composites with high ceramic content and lower aspect ratios. As is evident from Figure
1.4(c) and Figure 1.6(b), higher toughness comes at a cost of decreasing strength; this is in
agreement with analytical foundation [Zhang et al., 2010] where this behavior can be
attributed to increasing difficulty in driving the polymeric matrix deformation unless smaller
aspect ratios are used. Another way to enhance toughness of the composites is by grafting the
ceramic platelets with polymeric chains. This in turn will lead to increase in S int which in current
non-grafted synthesized composites is one of the most vulnerable entities under contemporary
loading conditions.
From structural engineering application view point, a knowledge and an understanding
of the synthesis-structure-property relationship in these composites is vital for the
development of advanced ceramic-polymer composites with enhanced mechanical stiffness,
strength and toughness. Based on the information obtained from experimental data, we can
summarize our observations as follows. If high stiffness and strength is desired, it would be
advantageous to have composites synthesized having low aspect ratios and high ceramic
content. Similarly, if high toughness is desired, it would be advantageous to have composites
20

synthesized having high aspect ratios and low ceramic content. This is in agreement with
Wilbrink and co-authors [Wilbrink et al., 2010] where they demonstrated, via development of a
unit-cell model, that as the aspect ratio of the bricks increases, a trade-off exists between
increasing strength and decreasing ductility. Indeed, the same set of phenomena can be
observed in biological composites as well. As mentioned earlier, bone comprises of plate-like
crystals (having an aspect ratio varying from 25-50) embedded in a collagen-rich protein matrix
(~0.40) resulting in variation of elastic modulus from ∼2−25 GPa and a strength varying from
∼150−200 MPa. Similarly, nacre is made of enormously high ceramic content (~0.90-0.95) of
plate-like inorganic tablets (having an aspect ratio from 10-20) resulting in variation of elastic
modulus from ∼40−70 GPa and a strength varying from ∼20−120 MPa [Espinosa et al., 2009].
From literature, it can be recalled that toughening rate of bone is higher than that of nacre;
corresponding to a crack extension of 0.6 mm, resistance increases to 30 kJm
only to 1.5 kJm

−2

−2

in bone but

in nacre [Barthelat and Espinosa, 2007; Wang and Gupta, 2011]. Similarly, elk

antler falls in the realm of tougher side of the bone family materials, exhibits a toughness
increase of 60 kJm

−2

at a crack extension of 0.6 mm [Wang and Gupta, 2011].

1.4. Conclusion
'Mother Nature' cleverly uses the aspect ratio and content of inorganic matter to
synthesize composites having varying degrees of stiffness, strength and toughness. Such an
amazing capability of fine-tuning the properties for functions has rarely been demonstrated in
synthetic composites. In this article, an attempt has been made to synthesize staggered
21

ceramic-polymer composites having varying ceramic content and aspect ratio using freezecasting technique. Inferences obtained from experimental investigation provides us useful
information as to their effects on the mechanical properties of synthesized composites. This will
in-turn help us in custom-design manufacture of hybrid bio-inspired composite materials.

22

REFERENCES

23

REFERENCES

Araki, K., Halloran, J.W. 2005. Porous ceramic bodies with interconnected pore
channels by a novel freeze casting technique. J. Am. Ceram. Soc. 88, 1108–1114.
Apalak, M.K., Gunes, R., 2007. Elastic flexural behavior of an adhesively bonded single
lap joint with functionally graded adherends. Mater. Design 28(5), 1597-1617.
Barthelat, F., Espinosa, H.D. 2007. An Experimental Investigation of Deformation and
Fracture of Nacre-Mother of Pearl. Exp. Mech. 47, 311–324.
Blindow, S., Pulkin, M., Koch, D., Grathwohl, G., Rezwan, K. 2009. Hydroxyapatite/SiO2
composites via freeze casting for bone tissue engineering. Adv. Eng. Mater. 11(11), 875-84.
Currey, J.D. 2002. Bones: Structure and Mechanics, Princeton University Press,
Princeton.
Chino, Y., Dunand, D. 2008. Directionally freeze-cast titanium foam with aligned,
elongated pores. Acta Mater. 56, 105–113.
Deville, S., Saiza, E., Nalla, R.K., Tomsiaa, A.P. 2006. Freezing as a path to build complex
composites. Science 311, 515–518.
Deville, S. 2008. Freeze-casting of porous ceramics: A review of current achievements
and issues. Adv. Eng. Mater. 10(3), 155-69.
Dutta, A., Tekalur, S.A., Miklavcic, M. 2013. Optimal overlap length in staggered
architecture composites under dynamic loading conditions. J. Mech. Phys. Solids. 61(1), 145160.
Dutta, A., Vanderklok, A., Tekalur, S.A., 2012. High strain rate mechanical behavior of
seashell-mimetic composites: Analytical model formulation and validation. Mech. Mater. 55,
102-111.
da Costa Mattos, H.S., Monteiro, A.H., Palazzetti, R., 2012. Failure analysis of
adhesively bonded joints in composite materials. Mater. Design 33, 242-247.
Espinosa, H.D., Rim, J.E., Barthelat, F., Buehler, M.J. 2009. Merger of structure and
material in nacre and bone–Perspectives on de novo biomimetic materials. Prog. Mater. Sci. 54,
1059–1100.
24

Fukasawa, T., Deng, Z.-Y., Ando, M., Ohji, T., Kanzaki, S. 2002. Synthesis of porous
silicon nitride with unidirectionally aligned channels using freeze-drying process. J. Am. Ceram
Soc. 85, 2151–2155.
Fu, Q., Rahaman, M.N., Dogan, F., Bal, B.S. 2008. Freeze casting of porous
hydroxyapatite scaffolds. I. processing and general microstructure. J. Biomed. Mater. Res. B
86(1), 125-135.
Gao, H., Ji, B., J¨ager, I.L., Arzt, E., Fratzl, P. 2003. Materials become insensitive to flaws
at nanoscale: lessons from nature. P. Natl. Acad. Sci. USA 100, 5597–5600.
Huang, Z., Li, H., Pan, Z., Wei, Q., Chao, Y.J., Li, X. 2011. Uncovering high-strain rate
protection mechanism in nacre. Sci. Rep. 1:148, 1-5.
Jackson, A.P., Vincent, J.F.V., Turner, R.M. 1988. The mechanical design of nacre. P. R.
Soc. London B. 234, 415–440.
Jing, L., Zuo, K.H., Zhang, F.Q., Chun, X., Fu, Y.F., Jiang, D.L., Zeng, Y.P. 2010. The
controllable microstructure of porous Al 2O3 ceramics prepared via a novel freeze casting route.
Ceram. Int. 36(8), 2499-2503.
Ji, B., Gao, H. 2010. Mechanical principles of biological nanocomposites. Annu. Rev.
Mater. Res. 40, 77-100.
Kamat, S., Su, X., Ballarini, R., Heuer, A.H. 2000. Structural basis for the fracture
toughness of the shell of the conch Strombus gigas. Nature 405, 1036–1040.
Launey, M.E., Munch, E., Alsem, D.H., Barth, H.B., Saiz, E., Tomsia, A.P., Ritchie, R.O.
2009. Designing Highly Toughened Hybrid Composites Through Nature-Inspired Hierarchical
Complexity. Acta Mater. 57, 2919-2932.
Liu, G., Zhang, D., Meggs, C., Button, T.W. 2010. Porous Al2O3-ZrO 2 composites
fabricated by an ice template method. Scripta Mater. 62(7), 466-468.
Li, J.C., Dunand, D.C. 2011. Mechanical properties of directionally freeze-cast titanium
foams. Acta Mater. 59(1), 146-58.
Menig, R., Meyers, M.H., Meyers, M.A., Vecchio, K.S. 2000. Quasi-static and dynamic
mechanical response of Haliotis rufescens (abalone) shells. Acta Mater. 48, 2383–2398.

25

Macchettaa, A., Turner, I.G., Bowena, C.R. 2009. Fabrication of HA/TCP scaffolds with a
graded and porous structure using a camphene-based freeze-casting method. Acta Biomater. 5,
1319–27.
Mortensen, A., Llorca, J. 2010. Metal matrix composites. Annu. Rev. Mater. Res. 40,
243-270.
Ouinas, D., Bouiadjra, B.B., Achour, T., Benderdouche, N., 2010. Influence of disbond
on notch crack behavior in single bonded lap joints. Mater. Design 31(9), 4356-4362.
Robinette, J., Bujanda, A., DeSchepper, D., Dibelka, J., Costanzo, P., Jensen, R.,
McKnight, S. 2007. Nanosilica modification of elastomer-modified VARTM epoxy resins for
improved resin & composite toughness. Aberdeen Proving Ground, MD : Army Research
Laboratory,
Report
Number:
ARL-TR-4084
(http://www.dtic.mil/cgibin/GetTRDoc?AD=ADA466485) [Access Date: August 17, 2012].
Sarikaya, M., Gunnison, K.E., Yasrebi, M., Aksay, J.A., 1990. Mechanical
property-microstructural relationships in abalone shell. Mater. Res. Soc. 174, 109-116.
Sofie, S.W., Dogan, F. 2001. Freeze casting of aqueous alumina slurries with glycerol. J.
Am. Ceram. Soc. 84(7) 1459–1464.
Soon, Y.M., Shin, K.H., Koh, Y.H., Lee, J.H., Choi, W.Y., Kim, H.E. 2011. Fabrication and
compressive strength of porous hydroxyapatite scaffolds with a functionally graded core/shell
structure. J. Eur. Ceram. Soc. 31(1-2), 13-18.
Sayman, O., Ozen, M., Korkmaz, B., 2013. Elasto-plastic stress distributions in
adhesively bonded double lap joints. Mater. Design 45, 31-35.
Wilbrink, D.V., Utz, M., Ritchie, R.O., Begley, M.R. 2010. Scaling of strength and
ductility in bioinspired brick and mortar composites. Applied Physics Letters 97, 193701.
Wang, R., Gupta, H.S. 2011. Deformation and fracture mechanisms of bone and nacre.
Annu. Rev. Mater. Res. 41, 41-73.
Yoon, B.H., Koh, Y.H., Park, C.S., Kim, H.E. 2007. Generation of large pore channels for
bone tissue engineering using camphene-based freeze casting. J. Am. Ceram. Soc. 90, 1744–
1752.
Ye, F., Zhang, J.Y., Zhang, H.J., Liu, L.M. 2010. Pore structure and mechanical properties
in freeze cast porous Si3N4 composites using polyacrylamide as an addition agent. J. Alloy.
Compd. 506(1), 423-27.
26

Yoon, H.J., Kim, U.C., Kim, J.H., Koh, Y.H., Choi, W.Y., Kim, H.E. 2010. Macroporous
alumina ceramics with aligned microporous walls by unidirectionally freezing foamed aqueous
ceramic suspensions. J. Am. Ceram. Soc. 93(6), 1580-1582.
Zhang, H., Hussain, I., Brust, M., Butler, M.F., Rannard, S.P., Cooper, A.I. 2005. Aligned
two- and three-dimensional structures by directional freezing of polymers and nanoparticles.
Nature Mater. 4, 787–93.
Zhang, Z., Zhang, Y.-W., Gao, H. 2010. On optimal hierarchy of load-bearing biological
materials. Proc. R. Soc. B 278, 519-525.
Zuo, K.H., Zhang, Y.A., Zeng, Y.P., Jiang, D.L. 2011. Pore-forming agent induced
microstructure evolution of freeze casted hydroxyapatite. Ceram. Int. 37(1), 407-10.

27

CHAPTER 2: HIGH STRAIN RATE MECHANICAL BEHAVIOR OF SEASHELL-MIMETIC
COMPOSITES: ANALYTICAL MODEL FORMULATION & VALIDATION

28

2.1. Introduction
Superior strength and light weightiness are two discernible facets in biological
composites, like bone and nacre, and has caught substantial attention of the scientific
community. Biological composites have been tried, tested and designed by the theory of
evolution over millions of years.

Nacre and bone exhibit high stiffness, strength and

specifically, superior toughness [Barthelat and Espinosa, 2007; Jackson et al., 1988; Kessler et
al., 1996; Kamat et al., 2000; Menig et al., 2000, 2001; Sarikaya et al., 1990] which is orders of
magnitude higher with respect to its inorganic (brittle minerals) and organic (ductile polymers)
components. The brittle minerals tend to have different structures and compositions. With
specific reference to mollusks, the shell material is typically composed of aragonite and/or
calcite where the orientation of the inorganic component may be prismatic, nacre, crossedlamellar or foliated in nature. Nacre is composed of mineral tablets (of high weight fraction
Φ~0.95) and soft polymeric matrix, composed of proteins and polysaccharides, laid in a brickand-mortar structural arrangement. Significant macroscopic deformations have been observed
in nacre in the form of high values of failure strains of the order of nearly 5%. Quasi-static
-1

compression tests [Menig et al., 2000], at loading rates of 1–500 MPas , were performed on
nacre under two loading configurations: (a) load applied perpendicular to the lamellar bricks
and (b) load applied along the plane of bricks. Probability estimation using Weibull function was
employed which showed that there is a 50% chance of failure at stress levels of 450 MPa and
200 MPa for loading configurations (a) and (b) respectively. Similar observations (for static case
only) were inferred in the works of [Barthelat et al., 2006; Barthelat and Espinosa, 2007;
29

Jackson et al., 1988; Sarikaya et al., 1990]. In nacre, a number of factors contribute to
toughening mechanisms, namely (a) pull–out of tablets [Jackson et al., 1988], (b) crack
deflection by the organic layers [Menig et al., 2000], (c) platelet interlocks [Katti et al., 2005],
and (d) presence of nanoasperities & tablet waviness and sliding which results in high inelastic
strains and hence whitening [Barthelat and Espinosa, 2007], which in-turn is responsible for its
superior fracture resistance.
An underlying course in the evolutionary process of shells is shell-crushing predation by
crustaceans [Kitching and Lockwood, 1974; Vermeij, 1976, 1977, 1978; Zipser and Vermeij,
1978]. Shell breaking methods is recognizable in other tropical and subtropical crab species
[Rossi and Parisi, 1973; Shoup, 1968; Vermeij, 1976, 1978; Zipser and Vermeij, 1978 ].
Predatory attack by fishes on larger gastropods, living in the open, is also pretty common
[Bertness and Cunningham, 1981]. Similarly, bone suffers fracture and failure due to impact
loading from falls, accidents, ballistic impact and blast loading. Note that a number of the
studies in literature are limited to quasi-static loading scenario. This is intriguing given the fact
that both this material is fundamentally used as a basic structural material for synthesizing and
in orchestrating complex load bearing/shielding structures against predatory attack.
Understanding the mechanics of failure of the biological composites under such a wide range of
impact loading will provide us design strategies for fabrication of hybrid bio-inspired
composites.
Biological composites are synthesized by nature via bottom-up route exhibiting complex
hierarchical design over nano-micro-meso-macro scales. For instance, seashells and bone

30

(vertebral) exhibit 2-3 [Currey, 1977; Jackson et al., 1988; Menig et al., 2000, 2001] and 7 levels
[Currey, 1984; Landis, 1995; Rho et al., 1998; Weiner and Wagner, 1998] of hierarchy
respectively. Biocomposites, notwithstanding the degree of hierarchical complexity, at the most
elementary level, exhibit an analogous microstructure comprising of nanometer sized inorganic
crystals reinforced in a soft biopolymer (organic) matrix in the form of a staggered architecture
[Fratzl et al., 2004; Gao et al., 2003; Jager and Fratzl, 2000], otherwise known as a brick-andmortar type structural arrangement (shown in the inset in Figure 2.1).

Figure 2.1. Dependence of stiffness of biocomposites (exhibiting brick and mortar type of
architecture at its most elementary level) as a function of aspect ratio of ceramic bricks and
volume fraction of ceramic as laid down by Gao et al., 2003. Inset shows the brick and mortar
type microstructural arrangement observed in seashells, and microstructure of bone.
Crystalline plate-like (2–4nm thick and up to 100nm long) reinforcements (Φ~0.50-0.67)
in collagen-rich protein matrix [Landis, 1995; Landis et al., 1996; Roschger et al., 2001] are
31

present in bone and dentin. As discussed earlier, nacre is made of enormously high volume
fraction (0.90-0.95) of plate-like inorganic tablets (0.4-0.5 μm thick and 5-10 μm long) with a
small amount of soft matrix (thickness of around 20-30 nm) in between the platelets [Currey,
1977; Jackson et al., 1988; Kamat et al., 2000; Menig et al., 2000; Wang et al., 2001]. In the
literature, there have been attempts to analyze the stiffness of the staggered nanostructure of
bone [Jager and Fratzl, 2000; Kotha et al., 2000]. A mechanical model was proposed by Jager
and Fratzl (2000), to estimate the maximum stress and strain of the composite, by advancing
previously established models for the mechanical properties of mineralized collagen fibrils
[Wagner and Weiner, 1992] by setting forth inorganic crystals in a staggered arrangement in
agreement with the distribution of gaps in the collagen fibril [Hulmes et al., 1995; Landis, 1995;
Veis and Sabsay, 1987]. Micromechanical model developed by Kotha et al. (2000) was based
on shear-lag theory for deriving axial and shear stress distribution in platelets on the
assumption of a fundamental basis that the load carried by the platelets remains constant and
inter-platelet load transfer occurs via shear deformation of the matrix. Tension-shear chain
(TSC) model developed by Gao et al., (2003) and Ji and Gao (2004), as shown in Figure 2.1. and
given by equation (2.1) below, unraveled the reason for high stiffness of the biocomposites.

1 4 1   
1


22 Em
E G 
p

(2.1)

On the basis of the mathematical expression as given by equation (1), the composite
stiffness estimate (E) is dependent on factors, namely, mineral Young's modulus (E m), protein

32

shear modulus (G p), mineral volume fraction (Ф), and aspect ratio of the mineral crystals (ρ).
Additionally, beyond a certain value of aspect ratio, the composite stiffness (E) becomes nearly
constant. Also, the TSC model predicts the stiffness with a reasonable good accuracy. However,
all the different theoretical models developed earlier predict material behavior under quasistatic loading conditions.
Several questions still remain unanswered when we shift our focus to the dynamic
regime, such as (1) Is the material behavior better under dynamic loading conditions than its
static counterpart? (2) what are the parameters that govern the material behavior of
particulate composites under dynamic loading conditions? Menig et al. (2000, 2001) carried out
dynamic compression test on abalone and conch shells in a Split Hopkinson Pressure Bar (SHPB)
and found out that dynamic compressive strength of abalone is approximately 50% higher than
static compressive strength. However, a theoretical analysis would provide guidance on the
development of nanostructured materials by tailor-designing materials from a scale of
nanometers and up. Currently we barely have any theoretical basis on how to design a
hierarchical material to achieve a particular set of macroscopic properties. Recently, Wei et al.
(2012) developed a continuum model, under static loading conditions, which communicates the
existence of characteristic overlap length (a function of cross-linking chemistry of the matrix
with its reinforcement and mechanical properties of reinforcement) which contributes to
optimization of both strength and toughness in nacre, collagen molecules and spider-silk fibers.
In order to understand the mechanical response of staggered and lamellar composites under
dynamic rates of loading, in the current research we attempt to use the fundamental aspect of
33

shear-lag theory i.e., ceramic scaffold serves as the load-bearing member and the polymer
contributes to transfer of load via shear, however applying it under compressive loading
scenario en-route to the development of a theoretical model followed by subsequent
experimental investigation in a SHPB.
2.2. Mathematical Formulation

Figure 2.2. (a) Schematic of the brick and mortar type structural arrangement subjected to
external loading; (b) load acting on the ceramic and transfer of load between ceramic platelets
by shearing of the intermediate polymer layers; (c) variation of shear stress and displacement
34

Figure 2.2. (cont'd)
profile along the ceramic thickness for the outer ceramic; (d) variation of shear stress and
displacement profile along the ceramic thickness for the inner ceramic.
Figure 2.2. shows the schematic of the representative element of a ceramic polymer
composite having a brick and mortar structural arrangement at its elementary level. Following
assumptions have been taken into consideration:
(a) ceramic blocks are linear elastic,
(b) constant axial stress distribution throughout the ceramic thickness,
(c) constant shear stress distribution throughout the polymer thickness, and
(d) inconsequential peeling thickness.
Performing force equilibrium on the outer ceramic and inner ceramic, we obtain

 To  To   To  2pdx  c hdx  ao
To
2uo
 2p  chao  ch
x
t2

(2.3)

avg

 Ti  Ti   Ti  2pdx  c hdx  ai
Ti
2ui
 2a  chai  ch
x
t2

(2.2)

(2.4)

(2.5)

avg

where, subscripts 'c', 'p', 'o, and' 'i' denotes the ceramic, polymer, outer (ceramic) and inner
(ceramic) respectively. T represents the axial stress resultant (positive in tension) in the
ceramic, τp represents the shear stress in the polymer, a is the average acceleration of the

35

cross-section of the ceramic, h represents the thickness of the ceramic and u represents the
horizontal displacement of the ceramic brick. Let τ o and τi denote the shear stress in the outer
and the inner ceramic respectively with z' and z'' as its origins. Assuming linear variation of
shear stress through the ceramic thickness, we obtain

o 

2p
h

z'

 2z'' 
i  p  1 

h 


(2.6)

(2.7)

If Gc denotes the shear modulus of the ceramic, then the shear strains in the outer (γ o)
and inner ceramic (γi) can be represented as

o 

i 

2p
Gch

z'

p  2z'' 
1 

Gc 
h 

(2.8)

(2.9)

Integrating shear strains given by (2.8) and (2.9) over the thickness of the ceramic, we
obtain the following expressions for horizontal displacement of the outer ceramic (u o) and
inner ceramic (ui), given by

uo  x,z'   uop  x  

p  x  z'2 p  x h

Gch
4Gc

(2.10)

p  x  
z''2 
 z''

Gc 
h 



(2.11)

ui  x,z''   uip  x  

36

where uop and uip represent the displacement of the ceramic at the interface between the
outer ceramic block and the intermediate polymer and between the polymer and the inner
ceramic respectively. The axial forces in the outer (T o) and inner ceramic (Ti) is given by

h
2 uo
 duop
h dp 
To  Ec 
dz'  Ech 


6Gc dx 
 dx
h x
z'
2

(2.12)

h u
i dz''  E h  duip  h dp 
Ti  Ec 

c 
 dx 6Gc dx 
z''0 x

(2.13)

Assuming polymeric shear thickness to be constant throughout the thickness of the
polymer layer, the shear strain in the polymer (γ p) may be expressed as follows

p 

uip  uop
tp

(2.14)

where, tp represents the thickness of the polymer. Thus, it can be shown that

2 p 1  2uip 2uop 

 

2 tp  x2
2 
x
x 


(2.15)

Rearrangement of the terms in equation (2.12) and (2.13) yields

duop To
h dp


dx
hEc 6Gc dx

(2.16)

duip
T
h dp
 i 
dx hEc 6Gc dx

(2.17)

37

Substitution of equations (2.16) and (2.17) in equation (2.15) followed by
rearrangement of the terms leads to the following expression

hGp  2 p

Ti To

 hEc  tp 

x x
3Gc  x2


(2.18)

where Gp denotes the shear modulus of the polymer. The average accelerations of the outer
and inner ceramic layers is given by

h
*
L
2 2uo  x,y,z',t 
dz'dy


t2
*
h
y L z'
2uop hGp 2 p
2uo
2
ao  x,t  



h
6Gm t2
t2
t2
*
avg
L
2

 dz'dy
*
y L z' h
2

(2.19)

L*
h 2u  x,y,z',t 
i
dz'dy


2u
t2
* z''0
2uip hGp 2 p
 i
y L
ai  x,t  



6Gm t2
t2 avg
L*
h
t2

 dz'dy
y L* z''0

(2.20)

where, 2L* represents the width of the ceramic layer. Combining equations (2.19) and (2.20)
together, taking into account (2.14) and rearranging the terms, we get

hGp  2 p

ai  ao   tp 

3Gc  t2

Subtracting equation (2.3) from (2.5), we get

38

(2.21)

 Ti To 
 x  x   4p  ch ai  ao 



(2.22)

Replacement of bracketed terms in the right-hand side and left-hand side of equation
(2.22) by (2.18) and (2.21) respectively followed by rearrangement of the terms leads us to the
generation of the governing differential equation of the system, given by

2 p 1 2 p 1

 
2 c2 t2  p
x
c
where cc denotes the wave velocity through the ceramic cc 

12GpGc
1

 E 3G ht  h2G
c
c p
p



(2.23)

Ec
and, α is given by
c



(2.24)

Combining equations (2.14), (2.16) and (2.17), we get

Gp  p



 Ti  To 
1 

 3Gc 1     x h2Ec 1   

(2.25)

The initial conditions and the boundary conditions of the problem, given by equations
(2.26-2.29), can be represented as

 t 0  0

(2.26)


0
t t 0

(2.27)

At x = 0, Ti = 0 and To = -F(t)

(2.28)

At x = L, Ti = -F(t) and To = 0

(2.29)
39

where, F(t) is the prescribed loading at the inner and outer ceramic and the negative sign is
representative sign of compressive load (as in a SHPB experiment) and L denotes the overlap
length. Note that the prescribed loading is equal on both the ends which in turn ensures that
the specimen is in a state of dynamic equilibrium with minimal wave propagation effects.
Solution of the governing equation is carried out using Galerkin method of weighted
residuals [Cook et al., 2007] as follows:
For a finite element x1 ≤ x ≤ x2, the residual equation for the element is


x2      1 2 
p
p 1 
Ni  

 p x  0 , i = 1, 2

 2 2

 
x1  x  x  c t


(2.30)

where p represents the approximate shear strain, given in equation (2.31) as

p  N1  x  p  N2  x  p
1
2

(2.31)

N1(x) and N2(x) are first order interpolation functions and correspond to weighted
residuals in Galerkin Method. Substituting i =1,2 in equation (2.30) followed by rearrangement
of the terms leads to the following equation, given as

40

x2 N N
 x2  N N

1
1
   1 1  N1N1  x   1 2  N1N2  x 


 
 x  x x 


 p 
x1  x x 
1

 1 


x2 N N
x2  N N
  p2 
1
1

   2 1  N2N1  x   2 2  N2N2  x  

 x x 
 
 x  x x 


x1 
 1

x2 1
 x2 1

 p

 
N1N1x 
N1N2x 


2
 x cc2
  p   x x 
x1 cc
1
1

 1   

 

x2 1
 x2 1
  p2   p



 
N2N1x 
N2N2x 
 x

2
 x cc2

x2 



x1 cc
 1


(2.32)

which is equivalent to

 

 

  e   (e)   e   (e)   e  


K  p  M  p  f 





(2.33)

where (2.32) can be simplified to obtain

 1 1  x2  x1  2 1
1
 e 
K 

  6 1 2


  x2  x1   1 1 



(2.34)

  e    x2  x1  1 0 


M  


2c2 0 1 

(2.35)

The elemental mass matrix given by equation (2.32-2.33) is consistent mass matrix and
has been converted into diagonally lumped mass matrix, as shown in equation (2.35), using HRZ
lumping scheme.
For the representative element, total displacement is the sum of deformation
corresponding to the ceramic element and the deformation corresponding to the shear
deformation of the protein, and thus strain can be expressed as given by equation (2.36).

41

eff 
o

 uo

avg h 1    p

x

x

(2.36)

x L

Based on the condition of velocity compatibility on the specimen incident bar interface,
it can be laid down that particle velocity of the specimen and that of the incident bar are equal.
Thus, the predicted strain εpredicted, as given by equation (2.37), can be computed as follows

predictedc  eff cc
o

(2.37)

where c represents stress-wave velocity in the incident bar. Taking into consideration our initial
assumption that the load is carried by the ceramic skeleton, the stress in the specimen,
σpredicted, can be computed by taking into consideration, on the basis of similar grounds, strain
continuity between adjacent elements and is given by equation (2.38) as

 E A  c  T
predicted   b b   c  i
 As   c  hEc

(2.38)

where Eb, Ab and As correspond to Young's modulus of the transmission bar, cross-sectional
area of the transmission bar and cross-sectional area of the specimen respectively. Lamellar
microstructure can be visualized as an extension of brick and mortar microstructure where
inter-ceramic brick spacing in a given zero is essentially equal to zero (can be imagined as
ceramic bricks in a given row are fused together and forming a rod). Note that this
mathematical formulation is thereby equally applicable for composites having lamellar
architecture as its elementary microstructure.
2.3. Materials and Methods

42

Sodium dodecyl sulfate (C 12H25NaO4S) was obtained from Sigma Aldrich (St. Louis, MO,
USA). Poly (vinyl alcohol) [-CH2CH(OH)-]n 98% hydrolyzed having an average molecular weight
Mw 13,000-23,000 was obtained from Aldrich Chemical Company, Inc. (Milwaukee, WI, USA).
Sucrose (C12H22O11) crystals were obtained from Roche Diagnostics Corporation (IN, USA). Two
types of ceramic powders were purchased for synthesizing two types of ceramic polymer
composite. Silica (SiO2) spheres having Ф 8µm were obtained from Fiber Optic Center Inc (New
Bedford, MA, USA). Aluminum oxide (Al 2O 3) spheres having Ф 10µm were obtained from Sigma
Aldrich (Milwaukee, WI, USA). SC-15 epoxy resin (toughened two phase) was obtained from
Applied Poleramic Inc. (Benicia, CA, USA).
Ceramic-polymer composites were prepared by freeze-casting technique [Deville et al.,
2006; Launey et al., 2009], as shown in Figure 2.3(a,b,c). Samples thermogravimetric analysis
-1

(TGA Q500 V20.10 Build 36, TA Instruments) were conducted at a heating rate of 20 °C min up
at temperatures up to 600 °C. ASTM D695-10 was followed for the determination of
compressive strength of these ceramic polymer composites under static rates of loading. The
specimen dimension chosen for testing was 5mm × 5mm × 10 mm and was performed in a MTS
machine (10 kN load cell) at a crosshead speed of 1mm/min. For dynamic compression
behavior, material properties have been investigated using SHPB technique. A conventional
SHPB or the Kolsky Bar [Kolsky, 1963, 1964)] consists of a striker bar, an incident bar and a
transmitter bar, as illustrated in Figure 2.4(a). The specimen is sandwiched between the
43

incident and transmitter bar (made of high strength aluminum alloy (alloy 7075), each of which
was 6ft (1.8 m) long and 5/8 inch (15.9 mm) in diameter. The specimen dimension chosen for
testing purpose was 11.22mm × 11.22mm × 9.161 mm. Choice of this dimension has been
explained in the Discussion section. The responses were measured with electrical resistance
strain gages, bonded to the middle of the incident and transmitter bars, connected to Vishay
Signal Conditioning Amplifiers (Model 2310B), and the response from the strain gages were
recorded using a LeCroy digital oscilloscope (Model354A). The specimen can be subjected to a
wide range of strain rates by employing striker bars of various lengths. Performing one
dimensional wave analysis on the strain signal obtained from the incident and transmitter bars,
and coupling an important assumption that wave propagation effects within a short specimen
may be neglected, we can obtain the stress-strain profile in the specimen as given below.

Figure 2.3. (a) Typical specimen and its representative microstructure (b, c); (d) schematic
showing distribution of ceramic platelets in polymeric matrix in brick and mortar type
44

Figure 2.3. (cont'd)
microstructural composites; (e) region under consideration highlighting the overlap region; (f,
g) deformed platelets and intermediate polymer upon tensile and compressive loading
respectively; (h) specimen for quasi-static experiments; (i) fractured specimen under quasistatic loading.
The specimen strain is calculated from the reflected pulse which is as follows:

s  t  

2c t
 r  t  dt
ls 0

(2.39)

where, c and ls denote the wave speed in the bar and the thickness of the specimen
respectively. The average stress in the specimen is given by

A
s  Eb b t
As

(2.40)

where, A s is the instantaneous cross sectional area of the specimen.
2.4. Results and Discussion
Typically, cylindrical samples are employed for determination of mechanical behavior of
specimens under dynamic rates of loading. Specimen design in a cylindrical specimen is
governed by choice of an appropriate slenderness ratio (which is the ratio of the length to
diameter of the sample). However, for non-cylindrical specimens, corresponding slendernessratio has not been defined. In recent studies [Sen, 2011], a general design criterion for noncylindrical samples, as given by equation (2.41), has been proposed by suggesting an

45

appropriate cross-sectional dimension and a slenderness ratio for a specimen with arbitrary
cross-sectional geometry.

1.4 

ls
 2.8
J
As

(2.41)

where J denotes the polar moment of inertia of the cross-section of the specimen. In our
current investigation, we chose a ratio of 2 and cross-sectional dimension of 11.22 mm×
11.22mm, which in turn based on the design criterion, gives l s = 9.161mm.
A composite having a brick and mortar type of architecture at its most elementary level
can be visualized as an adherend (the ceramic) held together by an adhesive (the polymer). In
our model, we have combined the effects of both inelastic deformation of the polymer and
shear deformation of ceramic by including the effects of shear strain acceleration of the
polymer and out-of-plane thickness of the ceramic. Figure 2.3(d) shows the schematic of the
distribution of ceramic platelets in the polymeric matrix as observed in staggered architecture
composites. The overlap region, as shown in Figure 2.3(e), is responsible for inter-platelet load
transfer via shear deformation of the polymer under the action of tension (Figure 2.3(f)) and
compressive (Figure 2.3(g)) loading respectively. One might speculate micro-buckling of the
platelets to be the dominating deformation mechanism in compression based on compressive
experiments on nacre by Menig et al., 2000. It is to be noted that the basic load transfer
mechanism at the micro-scales still remains the same as is depicted in Figure 2.3(g). The
inorganic matter associated with nacreous layer (or, ceramic in a ceramic polymer composite) is
brittle and inflexible in nature. Hence, failure initiates in these types of composites in the
46

interfacial matrix at the extremities of the joint at micro-scales; the coordinated sliding of layer
segments of the same approximate length produces an overall rotation of the specimen in the
region with a decrease in length and hence, it appears to be buckled at meso-scales. The
following points illustrate this aspect further: (a) seashell is a hierarchical material which
exhibits brick and mortar construction of ceramic and biopolymer is exhibited in the microscale, whereas a lamellar architecture is exhibited in the meso-scales as shown in the figure
below (comprising of 300 μm thick layers, spaced apart by layers of viscoplastic material of
thickness of about 20 μm); (b) under compressive loading conditions, microbuckling governs
failure mechanism of fiber-reinforced composites [Espinosa et al., 1998] when loading is in a
direction parallel to reinforcements, and hence based on the similarity of mesostructure in
seashells against that of fiber-reinforced composites, microbuckling was thereby observed
under compressive loading conditions in nacre. In the lamellar structures associated with the
mesostructure in seashells, cracks initiate first in the at the viscoplastic material layers
separating the mesolayers [Menig et al., 2000; Su et al., 2012]. Presence of ceramic bridges in
between the ceramic layers (an attribute associated with nacre at micro-scales) also hinders
Euler buckling of the ceramic layers at micro-scales [Deville, 2006]. Similarly, as was postulated
by Huang et al., 2011, under rapid-compression ballistic conditions, buckling of the material is
restrained at micro-scales via deformation twinning of the coordinated nano-scale particles.
Under the application of an external loading, hence the failure of the specimen is pronounced
along the interface for both static, as shown in Fig. 3(i), and under dynamic loading conditions

47

via nucleation and growth of plastic zone from either ends of the overlap length towards the
center.

Figure 2.4. (a) Schematic of a Split Hopkinson Pressure Bar (SHPB); (b) plot of pulses traced in
the oscilloscope based on the pulses recorded by the strain gages in the incident bar (solid pink)
and transmission bar (solid yellow); (c) specimen for SHPB experiments; (d) fractured specimen
upon impact loading; (e) ratio of force on the transmission face to the incident face over time;
(f) plot of strain-rate versus time.
With reference to studies [Gao et al., 2003; Ji and Gao, 2004], for a brick and mortar
type of composite, the strength of the staggered composite is dependent on the strength of the
48

ceramic-polymer interface, yield strength of the polymer, shear strength of the polymer and
maximum compressive strength of the ceramic bricks. Typically, in a ceramic-polymeric
composite, the most vulnerable entity under loading conditions (both, static and dynamic) is
the interfacial ceramic-polymer strength which in turn is thus representative of the ceramicpolymer joint strength. Note that, the ceramic in the current synthesized ceramic-polymer
composites were not grafted. Thus, dynamic compression experiments on these type of
composites would provide us ceramic-polymer interfacial strength of the composite under high
rates of loading.
Two types of ceramic-polymer composites were synthesized: (a) SiO2-SC 15 epoxy resin,
and (b) Al2O3-SC 15 epoxy resin. Furthermore, corresponding to each type, (a) & (b), ceramicpolymer composites were synthesized having varying weight fractions (Φw) of the ceramic
content in them. Figure 2.4(b) shows a typical plot of pulses obtained during dynamic
compressive loading on the ceramic-polymer composites. The solid pink line represents the
incident record (εi) and reflected record (εr). The solid yellow line represents transmitted
record (εt). Note that, a large opposite unloading wave can be observed in the later part of the
reflected wave. This is a very special phenomenon and this can be attributed to occurrence of
an elastic hysteresis effect, which may be caused by viscoelastic response of this kind of
polymer [Hao et al., 2005]. Figure 2.4(e,f) shows a typical force-equilibrium and strain rate vs.
time plot obtained under these strain rates. As it can be seen from Figure 2.4(e), reasonable
equilibrium was attained. Figure 2.4(f) shows a plot of strain-rate versus time. Constant strain
49

rate was achieved, however for a short period of time. Corresponding value was thereby
chosen as the representative strain-rate corresponding to a given loading condition.

Figure 2.5. Dynamic stress strain curves obtained for SiO2 SC-15 ceramic polymer composite (a)
and Al2O3 SC-15 ceramic polymer composite (b) corresponding to varying ceramic contents and
strain-rates.
50

The dynamic stress-strain curves for SiO2 SC-15 & Al2O3 SC-15 composite systems are
shown in Figure 2.5(a,b) respectively. Post peak stress, the stress-strain data is representative
of the unloading behavior. Figure 2.4(c) shows the specimen used for testing in SHPB. Failure of
the specimens, as shown in Figure 2.4(d) initiates at the peak stress. Table 2.1 and Table 2.2
lists the results obtained from dynamic compression experiments using SHPB on SiO 2 SC-15 &
Al2O3 SC-15 composite systems respectively. Static compressive strengths of SiO 2 SC-15
composite system having ceramic weight fractions of 0.45, 0.55 & 0.60 (as determined from
TGA) were found out to be 134 MPa, 170 MPa and 213 MPa respectively. Similarly, Static
compressive strengths of Al 2O3 SC-15 composite system having ceramic weight fractions of
0.45, 0.55, 0.60 & 0.70 (as determined from TGA) were found out to be 105 MPa, 108 MPa, 115
MPa and 130 MPa respectively.
Table 2.1. Results of dynamic compression experiments on SiO 2 SC-15 composites.

Φw
0.45
0.45
0.45
0.55
0.55
0.55
0.60
0.60
0.60

ἑmax
-1

(s )
1310
1580
1640
1200
1330
1500
1140
1160
1250

σpredicted σexperiment
(MPa)
(MPa)
165.11
167.02
176.06
189.79
175.99
182.21
191.51
198.27
198.23
208.99
195.78
202.47
197.91
203.57
203.20
211.64
210.24
219.71

51

Error
(%)

εcal

εexp

Error
(%)

1.15
7.23
3.42
3.41
5.15
3.30
2.78
3.99
4.31

0.0344
0.0388
0.0409
0.0291
0.0354
0.0372
0.0263
0.0308
0.0290

0.0354
0.0425
0.0431
0.0310
0.0371
0.0412
0.0292
0.0321
0.0308

2.70
8.61
5.01
6.22
4.61
9.81
9.90
4.01
5.88

Table 2.2. Results of dynamic compression experiments on Al 2O3 SC-15 composites.

Φw
0.45
0.45
0.45
0.55
0.55
0.55
0.60
0.60
0.60
0.70
0.70
0.70

ἑmax
-1

(s )
1170
1230
1490
860
890
1120
1160
1180
1190
960
1070
1100

σpredicted σexperiment
(MPa)
(MPa)
165.30
169.40
171.23
175.77
174.84
172.10
162.81
173.39
162.93
168.61
195.73
196.97
182.13
190.15
184.98
189.01
195.63
201.90
217.63
232.01
213.77
230.34
210.17
224.07

Error
(%)

εcal

εexp

Error
(%)

2.42
2.58
1.59
6.10
3.37
0.63
4.22
2.13
3.11
6.20
7.19
6.20

0.0345
0.0364
0.0379
0.0229
0.0230
0.0266
0.0294
0.0277
0.0251
0.0228
0.0257
0.0165

0.0373
0.0402
0.0406
0.0245
0.0256
0.0287
0.0319
0.0297
0.0276
0.0240
0.0267
0.0186

7.41
9.54
6.64
6.59
10.21
7.38
7.76
6.60
8.96
4.98
3.58
11.06

Figure 2.6. Plot of (a) experimental stress, and (b) strains (at peak stress) for SiO 2 SC-15 ceramic
polymer composite corresponding to varying ceramic contents at different strain-rates.

52

Figure 2.7. Plot of (a) experimental stress, and (b) strains (at peak stress) for Al 2O3 SC-15
ceramic polymer composite corresponding to varying ceramic contents at different strain-rates.
As it can be seen from Table 2.1. and Table 2.2., the peak stresses and strains (εcal - at
peak stress) as predicted by the analytical model are in reasonable agreement with the
experimentally obtained stresses and strains. Dynamic compressive strength of ceramicpolymer composites is significantly higher than its static counterpart (especially, for Al2O3 SC15 composite) and this large discrepancy in fracture strength values is indicative of a strong
strain-rate dependence of the mechanical properties of this type of ceramic polymer
composites. However maximum strains, as it can be seen from Figure 2.5(a,b), in dynamic
regime (~5%-6%) are approximately the same with respect to its static counterpart (~6%-6.5%).
From microstructural point of view, this could possibly be attributed to the occurrence of two
deformation mechanisms, under high strain-rate loading, in the ceramic bricks: (a) partial
dislocation emission, and (b) deformation twinning as it can be inferred from the recently
53

conducted studies [Huang et al., 2011] on their outlook towards uncovering high-strain rate
protection mechanism in nacre. In fact, an additional observation can be noted from Table 2.1
and Table 2.2. It can be seen that for a fixed ceramic weight fraction strain levels increase,
though minimal, with increasing strain-rate (shown in Figure 2.6(b) and Figure 2.7(b) for SiO2
SC-15 & Al2O3 SC-15 composite system respectively), which again follows the reasoning stated
above. Also, it can be seen that ceramic-polymer composites having 0.55-0.60 weight fraction
of ceramic provide the optimal performance in terms of stresses and strains (at peak stress).
The current model predicts failure strength within ±10%. Nevertheless, future studies will be
undertaken towards understanding the effect of kinking in these type of composites to
determine if it plays a major role in strength determination.
Typically, ceramics exhibit rate-insensitive or even negative rate dependence behavior.
If compared against the mechanical behavior obtained by brick-and-mortar ceramic-polymer
composites, these composites exhibit dynamic self-stiffening behavior [Huang et al., 2011],
which is similar to that observed in nacre. This kind of self-stiffening behavior under dynamic
rates of loading will find particular relevance in application of these types of composites on
their usage as protective materials when subjected to ballistic impact. The outcome of the
current research will provide highlights on the applicability of hybrid bio-inspired composites as
body armored materials.
2.5. Conclusion
Investigation on the mechanical behavior of ceramic-polymer composites, having brick
and mortar structure as its most elementary level of architecture, under dynamic rates of
54

loading have been performed in a SHPB. An analytical model has been formulated, for dynamic
rates of loading, based on the mechanism that mineral platelets serve as the principle load
bearing member & intermediate polymeric layer serve to transfer load between the platelets
following shear deformation) and considering the fact that a composite having a brick and
mortar type of architecture can be visualized as an adherend (the bricks/ceramic) held together
by an adhesive (the mortar/polymer). In a ceramic-polymeric composite, the interfacial
ceramic-polymer strength is representative of the ceramic-polymer joint strength and the
analytical model attempts to predict the same. The results obtained from the analytical model
are in reasonable agreement with the experimentally obtained stresses and strains. It was
found that dynamic compressive strength of ceramic-polymer composites is significantly higher
than its static counterpart and this can be attributed to strong strain-rate dependence of the
mechanical properties of this type of ceramic polymer composites. From microstructural point
of view, this could possibly be attributed to the occurrence of two deformation mechanisms,
under high strain-rate loading, in the ceramic bricks: (a) partial dislocation emission, and (b)
deformation twinning. Additionally, it was seen that for a fixed ceramic weight fraction, strain
levels increase with increasing strain-rate which follows the justification stated above. Thus,
these composites exhibit dynamic self-stiffening behavior which will find particular relevance in
application of these types of composites as protective material when subjected to ballistic
impact.

55

REFERENCES

56

REFERENCES

ASTM D695-10: Standard Test Method for Compressive Properties of Rigid Plastics,
ASTM International.
Bertness, M.D., Cunningham, C. 1981. Crab shell-crushing predation and gastropod
architectural defense. J. Exp. Mar. Biol. Ecol. 50, 213-230.
Barthelat, F., Li, C-M, Comi, C., Espinosa, H.D. 2006. Mechanical properties of nacre
constituents and their impact on mechanical performance. J. Mater. Res. 21, 1977-1986.
Barthelat, F., Espinosa, H.D. 2007. An Experimental Investigation of Deformation and
Fracture of Nacre-Mother of Pearl. Exp. Mech. 47, 311–324.
Currey, J.D. 1977. Mechanical properties of pearl in tension. P. R. Soc. London B 196,
443-463.
Currey, J.D. 1984. The Mechanical Adaptations of Bones. Princeton University Press,
Princeton, NJ, pp 24–37.
Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, J.R. 2007. Concepts and applications of
finite element analysis. 4th edition. India:Wiley.
Deville, S., Saiza, E., Nalla, R.K., Tomsiaa, A.P. 2006. Freezing as a path to build complex
composites. Science 311, 515–518.
Espinosa, H.D., Lu, H.C., Xu, Y. 1998. A Novel Technique for Penetrator Velocity
Measurement and Damage Identification in Ballistic Penetration Experiments. J. Comp. Mater.
32(8), 722-743.
Fratzl, P., Gupta, H.S., Paschalis, E.P., Roschger, P. 2004. Structure and mechanical
quality of the collagen–mineral nano-composite in bone. J. Mater. Chem. 14, 2115–2123.
Gao, H., Ji, B., J¨ager, I.L., Arzt, E., Fratzl, P. 2003. Materials become insensitive to flaws
at nanoscale: lessons from nature. P. Natl. Acad. Sci. USA 100, 5597–5600.
Hulmes, D.J.S., Wess, T.J., Prockop, D.J., Fratzl, P. 1995. Radial packing, order and
disorder in collagen fibrils. Bioph. J. 68, 1661–1670.

57

Hao, X., Gai, G., Lu, F., Zhao, X., Zhang, Y., Liu, J., Yang, Y., Gui, D., Nan, C-W 2005.
Dynamic mechanical properties of whisker/PA66 composites at high strain rates. Polymer 46,
3528-3534.
Huang, Z., Li, H., Pan, Z., Wei, Q., Chao, Y.J., Li, X. 2011. Uncovering high-strain rate
protection mechanism in nacre. Sci. Rep. 1:148, 1-5.
Jackson, A.P., Vincent, J.F.V., Turner, R.M. 1988. The mechanical design of nacre. P. R.
Soc. London B. 234, 415–440.
Jager, I., Fratzl, P. 2000. Mineralized collagen Fibrils: a mechanical model with a
staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746.
Ji, B., Gao, H. 2004. Mechanical properties of nanostructure of biological materials. J.
Mech. Phys. Solids 52, 1963–1990.
Kolsky, H. 1963. Stress Waves in Solids. Publisher: Dover Publications, Inc., New York,
N.Y.
Kolsky, H. 1964. Stress Waves in Solids. J. Sound Vib. 1, 88-110.
Kitching, J.A., Lockwood, J. 1974. Observations on shell form and its ecological
significance in thaid gastropods of the genus Lepsiella in New Zealand, Marine Biology, 28, 131141.
Kessler, H., Ballarini, R., Mullen, R.L., Kuhn, L.T., Heuer, A.H. 1996. A biomimetic
example of brittle toughening: (I) steady state multiple cracking. Comp. Mater. Sci. 5, 157–166.
Kamat, S., Su, X., Ballarini, R., Heuer, A.H. 2000. Structural basis for the fracture
toughness of the shell of the conch Strombus gigas. Nature 405, 1036–1040.
Kotha, S.P., Kotha, S., Guzelsu, N. 2000. A shear-lag model to account for interaction
effects between inclusions in composites reinforced with rectangular platelets. Compos. Sci.
Technol. 60, 2147–2158.
Katti, K.S., Katti, D.R., Pradhan, S.M., Bhosle, A. 2005. Platelet interlocks are the key to
toughness and strength in nacre. J. Mater. Res. 20, 1097-1110.
Landis, W.J. 1995. The strength of a calcified tissue depends in part on the molecular
structure and organization of its constituent mineral crystals in their organic matrix. Bone 16,
533–544.

58

Landis, W.J., Hodgens, K.J., Song, M.J., Arena, J., Kiyonaga, S., Marko, M., Owen, C.,
McEwen, B.F. 1996. Mineralization of collagen may occur on fibril surfaces: evidence from
conventional and high voltage electron microscopy and three dimensional imaging. J. Struct.
Biol. 117, 24–35.
Launey, M.E., Munch, E., Alsem, D.H., Barth, H.B., Saiz, E., Tomsia, A.P., Ritchie, R.O.
2009. Designing Highly Toughened Hybrid Composites Through Nature-Inspired Hierarchical
Complexity. Acta Mater. 57, 2919-2932.
Menig, R., Meyers, M.H., Meyers, M.A. & Vecchio, K.S. 2000. Quasi-static and dynamic
mechanical response of Haliotis rufescens (abalone) shells. Acta Mater. 48, 2383–2398.
Menig, R., Meyers, M.H., Meyers, M.A. & Vecchio, K.S. 2001. Quasi-static and dynamic
mechanical response of Strombus gigas (conch) shells. Mater. Sci. Eng. A 297, 203–211.
Rossi, A.C., Parisi, V. 1973. Experimental studies of predation by the crab Eriphia
verrucosa on both snail and hermit crab occupants of conspecific gastropod shells. Bollettino di
Zoologia 40, 117-135.
Rho, J.Y., Kuhn-Spearing, L., Zioupos, P. 1998. Mechanical properties and the
hierarchical structure of bone. Med. Eng. Phys. 20, 92–102.
Roschger, P., Grabner, B.M., Rinnerthaler, S., Tesch, W., Kneissel, M., Berzlanovich, A.,
Klaushofer, K., Fratzl, P. 2001. Structural development of the mineralized tissue in the human
L4 vertebral body. J. Struct. Biol. 136, 126–136.
Shoup, J.B. 1968. Shell opening by crabs of the genus Calappa. Science 160, 887-888.
Sarikaya, M., Gunnison, K.E., Yasrebi, M., Aksay, J.A., 1990. Mechanical
property-microstructural relationships in abalone shell. Mater. Res. Soc. 174, 109-116.
Sen, O., Tekalur, S.A., Jilek, C. 2011. The determination of dynamic strength of single lap
joints using the split Hopkinson pressure bar. Int. J. Adhes. Adhes. 31, 541-549.
Sen, O. 2011. Radial Inertia in Non-cylindrical Specimens in a Kolsky Bar. Conference
Proceedings of the Society for Experimental Mechanics Series 17, 169-170.
Vermeij, G.J. 1976. Interoceanic differences in vulnerability of shelled prey to crab
predation. Nature 260, 135-136.
Vermeij G.J. 1977. Patterns in crab claw size: the geography of crushing. Syst. Zool. 26,
138-152.
59

Vermeij, G.J. 1978. Biogeography and adaptation: patterns of marine life, Harvard
University Press. Cambridge, Page: 416.
Veis, A., Sabsay, B. 1987. The collagen of mineralized matrices. In Bone and Mineral
Research/5, W. A. Peck, editor. Elsevier Science Publications, New York, Amsterdam, Tokyo. pp:
1-63.
Wagner, H.D., Weiner, S. 1992. On the relationship between the microstructure of bone
and its mechanical stiffness. J. Biomech. 25, 1311–1320.
Weiner, S., Wagner, H.D. 1998. The material bone: structure–mechanical function
relations. Annu. Rev. Mater. Sci. 28, 271–298.
Wang, R.Z., Suo, Z., Evans, A.G., Yao, N., Aksay, I.A. 2001. Deformation mechanisms in
nacre. J. Mater. Res. 16, 2485–2493.
Wei, X., Naraghi, M., Espinosa, H. 2012. Optimal Length Scales Emerging from Shear
Load Transfer in Natural Materials--Application to Carbon-Based Nanocomposite Design, ACS
Nano 6(3), 2333–2344.
Zipser, E., Vermeij, G.J. 1978. Crushing behavior of tropical and temperate crabs. J. Exp.
Mar. Biol. Ecol. 3I, 155-172.

60

CHAPTER 3. OPTIMAL OVERLAP LENGTH IN STAGGERED ARCHITECTURE COMPOSITES UNDER
DYNAMIC LOADING CONDITIONS

61

3.1. Introduction
Nacre refers to the inner shining layer in seashells. Microstructure of nacre reveals the
existence of mineral tablets in a soft polymeric matrix in a brick-and-mortar structural
arrangement [Jackson et al., 1988; Sarikaya et al., 1990]. Two of the most discerning aspects of
nacre are exhibition of superior strength and synergistic toughness [Barthelat and Espinosa,
2007; Jackson et al., 1988; Kessler et al., 1996; Kamat et al., 2000; Menig et al., 2000, 2001;
Sarikaya et al., 1990]. Multifarious factors contribute to toughening mechanisms, namely tablet
pull-out [Jackson et al., 1988], crack deflection by the polymeric matrix [Menig et al., 2000],
platelet interlocks [Katti et al., 2005], presence of nanoasperities [Evans et al., 2000; Wang et
al., 2001], diffusive tablet sliding [Barthelat and Espinosa, 2007] arising from tablet waviness,
aragonite bridge reinforcements at interface [Song et al., 2003], and re-locking of tablets
[Meyers et al., 2008] due to persistent contact of broken aragonite bridges, all contribute to
interfacial strengthening and thereby, superior fracture resistance.
So far, literature studies have addressed the interdependence of material properties
and geometrical parameters on mechanical behavior of the staggered architecture materials
under quasi-static loading conditions. Analytical model proposed by Jaeger and Fratzl (2000)
estimated the maximum stress and strain of the composite, by advancing previously
established models for the mechanical properties of mineralized collagen fibrils [Wagner and
Weiner, 1992] by setting forth inorganic crystals in a staggered arrangement in agreement with
the distribution of gaps in the collagen fibril [Hulmes et al., 1995; Landis, 1995; Veis and
Sabsay, 1987]. Micromechanical model developed by Kotha et al. (2000) derived axial and

62

shear stress distribution in platelets on the assumption of a fundamental basis that the load
carried by the platelets remains constant and inter-platelet load transfer occurs by shear.
Tension-shear-chain (TSC) model postulated by Gao et al. (2003) established the relationship
between stiffness, aspect ratio of minerals, mineral volume fraction and material properties.
Additionally, a critical overlap length, using Griffith’s fracture criterion, was developed which
attributed to the existence of an optimal aspect ratio of mineral crystals in biological
composites. Shuchun and Yueguang (2007) used shear-lag model to study the interdependence
of the overall elastic modulus and number of hierarchical levels in bone-like materials, and
compared their results against TSC model and finite element simulations. Investigations by
Chen et al. (2009) were directed towards understanding characteristic length for efficient stress
transfer in staggered biocomposites via derivation of an analytical model followed by numerical
simulations. Zhang et al. (2010) employed homogenization method to investigate the effect of
platelet distribution (regular, stairwise, random), in staggered architecture materials, on the
stiffness, strength, failure strain and energy storage capacity. Zhang et al. (2011) developed a
quasi-self-similar hierarchical model to investigate the cause responsible for determination and
existence of optimal number of hierarchical levels in staggered architecture materials, for
varying mineral content, to obtain maximal toughness. Liu et al. (2011) used perturbation
method to obtain analytical expressions for displacement and stress fields in the staggered
nanocomposite structure, under quasi-static loading conditions, when subjected to uniaxial
tension. Recently, Wei et al. (2012) laid down a criterion which reveals the existence of a

63

unique overlap length in biological composites that contributes to an optimization on both,
strength and toughness fronts.
Resistance to crushing against crustacean predators is an integral part of shell evolution.
Shell breaking methods by predators is representative of a dynamic loading event. It has been
observed that under high strain-rates (using a Split-Hopkinson Pressure Bar(SHPB)), nacre
exhibits superior damage tolerance and elevated fracture strength [Huang et al., 2011]. This
behavior has been attributed to, based on electron microscopic analyses, on the existence of
phenomenon's like partial dislocation emission and the onset of deformation twinning within
the mineral crystals. However, minimal literature exists which addresses analytically the factors
responsible for this behavior under dynamic rates of loading.
The following sections attempt to investigate and answer the following questions with
regards to composites having staggered microstructure as its most elementary level of
architecture under impact loading conditions. (1) Does there exist a structure-property
correlationship for these materials under dynamic events? (2) Critical overlap length exists
under quasi-static loading conditions which optimizes both strength and stiffness; is there any
associated critical overlap length involved in load transfer mechanism between the adjacent
reinforcements in dynamic regime? (3) As had been postulated by Gao et al. (2003), building
blocks in biological composites are at nanometer scales under quasi-static loading conditions; is
the same argument true under impact loading? In order to address these questions, we based
our analyses in the following manner: (a) Analytical model development for staggered
microstructural materials under time dependent loading considering linear elastic behavior, (b)

64

identification of the critical overlap length criterion involved in these events, (c) predicting
overlap lengths for other biocomposites exhibiting staggered microstructure as its most
elementary level of architecture, for example nacre, spider-silk and collagen in bone/tendon,
(d) predicting & validating the joint strength, using the analytical model, of aluminum-Loctite
lap joint under impact loading using SHPB [Sen et al., 2011], (e) provide an explanation, based
on the analytical model, for superior damage tolerance and elevated dynamic compressive
strength of nacre, and (f) insights into the effect of loading rate.
3.2. Mathematical Formulation
We begin our analysis by taking into consideration the simplified 2D unit cell, as shown
in Figure 3.1(a), and by employing 'shear-lag theory' [Volkersen, 1938].

Figure. 3.1. (a) Schematic of brick-and-mortar microstructure identifying the overlap region;
Representative loading pulses obtained from SHPB experiments: (b) triangular pulse, and (c)
half-sinusoidal pulse.
65

Applying force equilibrium to the upper and lower ceramic bricks, we obtain

1 
2u1
 
x b
t2

(3.1)

2 
2u2
 
x b
t2

(3.2)

where, 1  1  x,t  and 2  2  x,t  denotes the normal stress developed in the upper and
lower ceramic bricks,  denotes the shear stress developed in the polymer, b denotes half the
ceramic brick width and h, the polymeric thickness. Assuming a state of pure shear in the
polymer, we have

 u u 
  G  1 2  where, u1  u1  x,t  , u2  u2  x,t  denotes in-plane
 h 

displacement of the upper and lower ceramic bricks respectively and G denotes the shear
modulus of the polymer. Assuming linear elastic behavior of the ceramic bricks, i.e.,   E

u
,
x

where E denotes Young's modulus of elasticity of the ceramic, equations (3.1) and (3.2) can we
rewritten as equations (3.3) and (3.4), to obtain the governing differential equation of the
system, given by

2u1 G
1 2u

u1  u2   2 21
x2 Ebh
c t
2u2
G
1 2u

u1  u2   2 22
x2 Ebh
c t
where c 

(3.3)

(3.4)

E
denotes the wave velocity through the ceramic and,  denotes the density of the


ceramic. The boundary conditions of the system are given by
66

 t
u2

x  0,t 
E

(3.5)

1  0,t   0 

u1
0
x  0,t 

(3.6)

2 L,t   0 

u2
0
x L,t 

(3.7)

 t
u1

x L,t 
E

(3.8)

2  0,t     t  

1 L,t     t  

The initial conditions of the system are given by

u1  x,0   0

(3.9)

u2  x,0   0

(3.10)

u1
0
t  x,0 

(3.11)

u2
0
t  x,0 

(3.12)

We thus have coupled partial hyperbolic differential equations subjected to non-homogenous
boundary conditions. Given the nature of the problem, we employed advanced separation of
variables technique coupled with rearrangement of terms to obtain a closed-form solution for
the displacements in the ceramic blocks.
We now introduce a variable, u4  u4  x,t  which is defined as follows:

u4  x,t   u1  x,t   u2  x,t 

67

(3.13)

Additionally, we define u4  u4  x,t  in an alternative manner as follows:

u4  x,t   v  x,t   w  x,t 

(3.14)

where w  x,t  represents a smoothing function satisfying only the boundary conditions, i.e.,

 t
w

x  0,t 
E

(3.15)

 t
w

x L,t 
E

(3.16)

Subtracting equation (3.4) from equation (3.3) and substituting equation (3.14), we obtain
another governing differential equation of the system, given by

2v
1 2v
2w
1 2w
 kv 

 kw 
x2
c2 t2
x2
c2 t2
where, k 

(3.17)

2G
, and is subjected to boundary conditions and initial conditions, given by
Ebh

equations (3.18-3.21) respectively.

v
0
x  0,t 

(3.18)

v
0
x L,t 

(3.19)

v  x,0   w  x,0 

(3.20)

v
w

t  x,0 
t  x,0 

(3.21)

68

Since boundary conditions are to be satisfied, choice of w  x,t  is made on the basis of
Neumann boundary conditions, and is thus represented by

w  x,t  


  t   x2
 x

E  L



(3.22)

Substituting equation (3.22) into right hand side of equation (3.17), we thus have the following
differential equation of the system, given by

2v
1 2v
 nx 
 kv 
 p4  x,t   p04  t    pn4  t  cos 

2
2 t2
 L 
x
c
n1

(3.23)

 2  x2
    t  1  x2
 d2  t 
 x
p4  x,t      k   x  

 L
 E
 dt2
 L
c2E  L






(3.24)

where,

p04  t  

pn4  t  

 12  kL2 
 L  d2  t 
1 L
t  
p4  x,t  dx   


 6EL 
L x 0
 6c2E  dt2



2 L
 nx 
 2LK   1  cos n  
 p4  x,t  cos 
t
 dx  

L x 0
 L 
 E   n22 

 2L  1  cos n   d2  t 

,n  1


 c2E  n22  dt2

(3.25)

(3.26)

Assume, v  x,t   X4  x  T4  t  . The Eigen values and functions are thus given by

2
 n 
n4    ,n  0
 L 

(3.27)

 nx 
Xn4  x   cos 
 ,n  0
 L 

(3.28)

69

Hence the formal solution of the problem is given by formal Fourier series, as

 nx 
 nx 
v  x,t    Tn4  t  cos 
  T04  t    Tn4  t  cos 

 L 
 L 
n0
n1

(29)

Substituting equation (3.29) into left hand side of equation (3.23) and equating both sides, we
get

d2Tn4  2   n 2 


 c k     Tn4  c2pn4  t  ,n  1
   L  
dt2

 

(3.30)

d2T04
 kc2T04  c2p04  t 
2
dt

(3.31)

The initial conditions, equations (3.20, 3.21), can be rewritten as follows:

 nx 
v  x,0   T04  0    Tn4  0  cos 

 L 
n1
 x2
 0
 w  x,0      x 
 L
 E


L  0 
2L  0   1  cos n  
 T04  0  
,Tn4  0   


6E
E  n22 

(3.32)

dT  0 
dT  0 
v
 nx 
 04
  n4
cos 

t  x,0 
dt
 L 
n1 dt

w
1 d  0   x2
nx 
  x   q4  x   q04   qn4 cos 




t  x,0 
E dt  L
 L 
n1


dT  0 
1 L
L d  0 
 04
 q04 
,and
 q4  x  dx 
dt
L x 0
6E dt
dT  0 
2 L
2L  1  cos n   d  0 
 nx 
 n4
 qn4 
dx   
,n  1
 q4  x  cos 


dt
L x 0
E  n22  dt
 L 


70

(3.33)

Equation (3.30) can be rewritten as follows:

2


d2Tn4
2 T  c2a  k  t   1 d   t   ,n  1
 en4 n4
n4
 E
dt2
c2E dt2 



(3.34)

2
 1  cos n  
 n 
where, an4  2L 
, and en4  c k    . Equation (3.34) can be rearranged

 L 
 n22 
further to obtain equation (3.35) as follows:

2
 d2T
a d  t  2 
an4

 n4  n4
e
n4  Tn4  E   t   
2
2 
E dt
 dt




2
c2an4k  t  en4an4   t  an4  nc 2




  t
E
E
E  L 

(3.35)

Equation (3.35) can be written in a simplified manner to obtain equation (3.36) as follows:

d2y4  t  2

 en4 y4  t   g4  t 
dt2

(3.36)

a
y4  t   Tn4  n4   t  ,and
E
a  nc 2
g4  t   n 
  t
E  L 

(3.37)

where,

We use Laplace Transform to solve equation (3.36). Let us denote L y4  t   Y4  s  , and

L g4  t   G4  s  . Thus, we obtain

71

dy  0  2
s2Y4  s   sy4  0   4
 en4 Y4  s   G4  s 
dt



 dy 0 

s
1
1

 y 0  
 4  
 G s 
 Y4  s  
 s2  e2  4
 s2  e2  dt
 s2  e2  4

n4 

n4 

n4 
 y4  t   y 4  0  cos  en4 t  

(3.38)

t 1
dy 4  0  sin  en4 t 
 
sin  en4  t     g 4   d
dt
en4
en4
0

Note that,

a
y4  0   Tn4  0   n4   0  
E
2L  0   1  cos n   2L  0   1  cos n  




  0,and
E  n22 
E  n22 
dy 4  0  dTn4  0  an4 d  0 



dt
dt
E
dt
2L  1  cos n   d  0  2L  1  cos n   d  0 
 
 
0


E  n22  dt
E  n22  dt

(3.39)

and thus, using equations (3.37-3.39), we finally obtain

a
a  nc 2 t
Tn4  t    n4   t   n4 
      sin  en4  t     d,n  1
E
Een4  L  0

(3.40)

Lets us now solve equation (3.31). We proceed as follows. Equation (3.31) can be rewritten as
follows:


d2  t  
d2T04

 d2 T04  c2  a04   t   b04
04

dt2
dt2 



(3.41)

 12  kL2 
 L 
 , b04   
where, a04   
, and d04  c k . Equation (3.41) can be rearranged
2E 
 6EL 
 6c 


further to obtain equation (3.42) as follows:
72

 d2T
d2  t   2 
 04  c2b04
d
T  c2b04   t   

2
2  04  04


 dt
dt 

 2
2a  t  d2 c2b  t   2c   t
c 04   04 04  

 EL 



(3.42)

Equation (3.42) can be written in a simplified manner to obtain equation (3.43) as follows:

d2m4  t  2

 d04m4  t   f4  t 
dt2

(3.43)

m4  t   T04  c2b04   t  ,and
 2c2 
  t
f4  t   
 EL 



(3.44)

where,

We use Laplace Transform to solve equation (3.43). Let us denote L m4  t   M4  s  , and

L f4  t   F4  s  . Thus, we obtain
dm4  0  2
s2M4  s   sm4  0  
 d04M4  s   F4  s 
dt



 dm 0 

s
1
1

m  0   
 4  
F s 
(3.45)
 M4  s  
 s2  d2  4
 s2  d2  dt
 s2  d2  4
04 
04 
04 



t 1
dm4  0  sin  d04 t 
 m4  t   m4  0  cos  d04 t  
 
sin  d04  t     f4   d
dt
d04
d04
0
Note that,

 L 
 L 
m4  0   T04  0   c2b04   0       0       0   0,and
 6E 
 6E 
dm4  0  dT04  0  2
d   0   L  d  0   L  d   0 

 c b04
 
 
0
dt
dt
dt
 6E  dt
 6E  dt
73

(3.46)

and thus, using equations (3.44-3.46), we finally obtain

2c2 t
T04  t   c2b04  t  
     sin  d04  t     d
ELd04 0

(3.47)

Substituting the expressions Tn4 ,T04 obtained from equations (3.40, 3.47) into equation (3.29)
and finally into equation (3.14), we finally get

2

 nx    t   x
 x
u4  x,t   u1  x,t   u2  x,t   T04  t    Tn4  t  cos 



E  L
 L 
n1



(3.48)

We now introduce another variable, u3  u3  x,t  which is defined as follows:

u3  x,t   u1  x,t   u2  x,t 

(3.49)

Additionally, we define u3  u3  x,t  in an alternative manner as follows:

u3  x,t   r  x,t   s  x,t 

(3.50)

where s  x,t  represents a smoothing function satisfying only the boundary conditions, i.e.,

 t
s

x  0,t 
E

(3.51)

 t
s

x L,t 
E

(3.52)

Adding equations (3.3) and (3.4), and substituting equation (3.50), we obtain another governing
differential equation of the system, given by

2r 1 2r
2s 1 2s



x2 c2 t2
x2 c2 t2

74

(3.53)

subjected to boundary conditions and initial conditions, given by equations (3.54-3.57)
respectively.

r
0
x  0,t 

(3.54)

r
0
x L,t 

(3.55)

r  x,0   s  x,0 

(3.56)

r
s

t  x,0 
t  x,0 

(3.57)

Since boundary conditions are to be satisfied, choice of s  x,t  is made on the basis of
Neumann boundary conditions, and is thus represented by

s  x,t  

x  t 
E

(3.58)

Substituting equation (3.58) into right hand side of equation (3.53), we thus have the following
differential equation of the system, given by

2r 1 2r
 nx 

 p3  x,t   p03  t    pn3  t  cos 

2 c2 t2
 L 
x
n1

(3.59)

2
x d  t
p3  x,t  
c2E dt2

(3.60)

where,

p03  t  

 L  d2  t 
1 L
p3  x,t  dx  


L x 0
 2c2E  dt2
75

(3.61)

 2L  cos n   1  d2  t 
2 L
 nx 
pn3  t  
 p3  x,t  cos 

 dx   2 
L x 0
 L 
 c E  n22  dt2

(3.62)

Assume, r  x,t   X3  x  T3  t  . The Eigen values and functions are thus given by

2
 n 
n3    ,n  0
 L 

(3.63)

 nx 
Xn3  x   cos 
 ,n  0
 L 

(3.64)

Hence the formal solution of the problem is given by formal Fourier series, as

 nx 
 nx 
r  x,t    Tn3  t  cos 
  T03  t    Tn3  t  cos 

 L 
 L 
n0
n1

(3.65)

Substituting equation (3.65) into left hand side of equation (3.59) and equating both sides, we
get

d2Tn3  nc 2

T  c2pn3  t  ,n  1
2  L  n3


dt

(3.66)

d2T03
 c2p03  t 
2
dt

(3.67)

The initial conditions, equations (3.56, 3.57), can be rewritten as follows:

 nx 
r  x,0   T03  0    Tn3  0  cos 

 L 
n1
x  0 
 s  x,0   
E
L  0 
2L  0   cos n   1 
 T03  0   
,Tn3  0   


2E
E  n22 

76

(3.68)

dT  0 
dT  0 
r
 nx 
 03
  n3
cos 

t  x,0 
dt
 L 
n1 dt


s
x d  0 
 nx 

 q3  x   q03   qn3 cos 

t  x,0 
E dt
 L 
n1

dT  0 
1 L
L d  0 
 03
 q03 
,and
 q3  x  dx  
dt
L x 0
2E dt


(3.69)

dTn3  0 
2 L
2L  cos n   1  d  0 
 nx 
 qn3 
,n  1
 q3  x  cos 

 dx   
dt
L x 0
E  2n2  dt
 L 

Equation (3.66) can be rewritten as follows:

2
d2Tn3 2
2a d   t  ,n  1
 en3Tn3  c n3
dt2
dt2

(3.70)

 2L  cos n   1 
 nc 
where, an3  
, and en3  
 . Equation (3.70) can be rearranged
2E  n22 
 L 
 c 

further to obtain equation (3.71) as follows:

 d2T
d2  t   2 
2
 n3  c2an3
 e
T  c2an3  t    en3c2an3  t  ,n  1

2
2  n3  n3


 dt
dt 


(3.71)

Equation (3.71) can be written in a simplified manner to obtain equation (3.72) as follows:



d2y3  t  2
 en3y3  t   g3  t 
dt2

(3.72)

where,

y3  t   Tn3  c2an3  t  ,and
2
g3  t   en3c2an3  t 

77

(3.73)

We use Laplace Transform to solve equation (3.72). Let us denote L y3  t   Y3  s  , and

L g3  t   G3  s  . Thus, we obtain
dy  0  2
s2Y3  s   sy3  0   3
 en3Y3  s   G3  s 
dt



 dy 0 

s
1
1
 y 0  
 3  
 G s 
(3.74)
 Y3  s   
 s2  e2  3
 s2  e2  dt
 s2  e2  3
n3 
n3 
n3 



t 1
dy  0  sin  en3t 
 y3  t   y3  0  cos  en3t   3
 
sin  en3  t     g3   d
dt
en3
en4
0
Note that,

y3  0   Tn3  0   c2an3  0  


2L  0   cos n   1  2L  0   cos n   1 



  0,and
E  n22 
E  n22 

dy3  0  dTn3  0  2
d  0 

 c an3

dt
dt
dt
2L  cos n   1  d  0  2L  cos n   1  d  0 
 
 
0


E  n22  dt
E  n22  dt

(3.75)

and thus, using equations (3.73-3.75), we finally obtain

t
Tn3  t   c2an3  t   en3c2an3      sin  en3  t     d,n  1
0

(3.76)

Lets us now solve equation (3.67). We proceed as follows. Equation (3.67) can be rewritten as
follows:

d2  t 
d2T03
 c2a03
dt2
dt2

78

(3.77)

 L 
where, a03  
 . Equation (3.77) can be rearranged further to obtain equation (3.78) as
2c2E 

follows:

 d2T
d2  t  
 03  c2a03
0
2 
 dt2
dt 


(3.78)

Equation (3.78) can be written in a simplified manner to obtain equation (3.79) as follows:

d2m3  t 

0
dt2

(3.79)

m3  t   T03  c2b03  t 

(3.80)

where,

We use Laplace Transform to solve equation (3.79). Let us denote L m3  t   M3  s  . Thus, we
obtain

dm3  0 
s2M3  s   sm3  0  
0
dt
 1  dm  0 
1
 M3  s     m3  0     3
s
 s2  dt
 m3  t   m3  0   t

(3.81)

dm3  0 
dt

Note that,

L 
 L 
m3  0   T03  0   c2a03  0        0       0   0,and
 2E 
 2E 
dm3  0  dT03  0  2
d  0 
 L  d  0   L  d  0 

 c a03
  
 
0
dt
dt
dt
 2E  dt
 2E  dt
and thus, using equations (80-82), we finally obtain
79

(3.82)

T03  t   c2a03  t 

(3.83)

Substituting the expressions Tn3 ,T03 obtained from equations (3.76, 3.83) into equation (3.65)
and then into equation (3.50), we finally get

 nx  x  t 
u3  x,t   u1  x,t   u2  x,t   T03  t    Tn3  t  cos 

E
 L 
n1

(3.84)

u  x,t   u4  x,t 
u1  x,t   3
,and
2
u  x,t   u4  x,t 
u2  x,t   3
2

(3.85)

Thus, we have

3.3. Results and Discussion
As mentioned earlier, the current formulation has been derived with respect to any
arbitrary pulse loading. In SHPB experiments, obtained loading pulses typically exhibit either

 t 

triangular   t   max t or, half-sinusoidal   t   max sin
 waveforms. Both type of
ta
 2ta 
loading pulses were applied to the staggered composite. As per the formulation carried out by
Wei et al. (2012), under quasi-static loading conditions, shearing stress at the ends of the
overlap ( max ) is given in terms of the maximum applied stress ( 0 ) as follows:

 L 2G 
 b 2G
max   x 0   x L  0
coth
 2 Ebh 

2 Ebh



(3.86)

Substitution of the geometrical and material properties of nacre [Barthelat et al., 2005;
Espinosa et al., 2011, Meyers et al., 2008, Jackson et al., 1988] into equation (3.85), results in

80

0 =133 MPa. In order to visualize the variation of shear strength in the joint throughout the
overlap length and over time, a maximum stress ( max ) of 133 MPa is hence applied and the
time taken to attain this stress level (ta) is chosen to be 133 μs. Shear stress distribution profiles
obtained were almost identical corresponding to both types of loading. Figure 3.2 shows the
shear stress distribution profile. As it can be seen from Figure 3.2, the shear stresses are
maximum at the either ends of the overlap. With progressing time, sliding between the ceramic
bricks continues to increase and thereby, shear stress at the ends continues to increase along
with increasing stress levels toward the center of the joint. Indeed similar behavior has been
observed under quasi-static loading conditions [Wei et al., 2012] as well as under finite element
(FE) simulations [Sen et al., 2011] which justifies the efficacy of the analytical model.

Figure 3.2. Shear stress distribution profile over the overlap length at varying times under the
application of triangular loading pulse.
81

Biological composite structures such as nacre, bone, etc. exhibit a generic
microstructure (brick and mortar) at their most elementary level of architecture [Gao et al.,
2003] where inorganic reinforcements serve as the load bearing member and the organic
matrix mediates the load between adjacent ceramic bricks via shear. Under impact loading
conditions, the total time for the event is in the order of tens of microseconds. Experimental
studies [Sen et al., 2011] on adhesively bonded lap-joints account shearing failure of the
adhesive as the predominant mode of failure under high-strain rate loading. Therefore,
optimization of overlap length to provide maximum shearing resistance (at the ends) under
dynamic loading conditions is a decisive parameter in the evolution of shells. Once the
inorganic and organic constituents are fixed, nature cleverly chooses a characteristic overlap
length such that shearing stress is a minimum and is thus, below or at par with the shear
strength of the organic matrix. In order to prove this hypotheses, three natural materials were
chosen (nacreous layer in abalone shell, spider silk and collagen in tendon/bone) and their
characteristic overlap length scales were predicted based on this hypothesis. Figure 3.3-3.5.
identifies those critical overlap length scales for nacreous layer in abalone shell, spider silk and
collagen in tendon/bone respectively.

82

Figure 3.3. Variation of shear stress at the extremities of the joint against overlap length for
nacre (a), and its dependence on loading rate (b).
In nacre, microscopic observations [Meyers et al., 2008] reveals the existence of 0.5μm
(≡ 2b) thick aragonite bricks bonded together by 25 nm (≡ h) thick organic matrix and, an
average overlap length of 1.6 μm [Barthelat et al., 2005; Espinosa et al., 2011]. The Young's
modulus of elasticity (≡ E) of the aragonite bricks is chosen to be 105.39 GPa and the shear
modulus of the biopolymer matrix (≡ G) is approximately 1.4 GPa [Wei et al., 2012]. As it can be
seen from Figure 3.3(a), shear stress at the ends is minimum for an overlap length of ~2.5 μm.
Note that the predicted value is reasonably close to the value observed in nature and that
predicted by critical length scales [Wei et al., 2012] (of approximately 1.69 μm) under quasistatic loading scenario. Additionally, this critical overlap length remains constant even under
increasing rates of loading as shown in Figure 3.3(b) and clearly justifies why this particular
overlap length is chosen by nature once the material parameters (E, G) and geometrical
parameters (b, h) are fixed. Also, note that shear stress distribution profile, as shown in Figure

83

3.2, was developed for an overlap length of 1.6 μm and interestingly, the maximum shear stress
at the extremities at the end of the event matches exactly with the average shear strength of
the polymer (≡ 37 MPa) [Jackson et al., 1988], which justifies the choice of this overlap length
by nacreous layer corresponding to time-dependent loading event.
Next the analytical model is applied to identify the critical overlap length between βsheets in spider silk, the constituents of which are in nanometer scale levels. The building
blocks for spider silk fibrils are β-sheets. Molecular dynamic simulations [Keten and Buehler,
2008; Keten et al., 2010] report an elastic modulus (≡ E) of approximately 22.6 GPa, and shear
modulus (≡ G), which is representative of the crosslinking between b-sheets, of 4.6 GPa. β-sheet
has a thickness (≡ 2b) of about 1 nm, and the distance between β-sheets (equal to the length of
H-bond), is approximately 0.3 nm (≡ h). Experimental studies have revealed β-sheet length in
the range from 2 to 8 nm [Penel et al., 2003]. Once again, substitution of these values predicts
the critical overlap length of ~5 nm (shown in Figure 3.4) for dynamic loading event, which is in
good agreement with experimental data. Additionally, the observed value is also in close
agreement with those obtained by Wei et al. (≈ 5.7 nm) [Wei et al., 2012] in their analyses
under quasi-static loading conditions.
Finally, we apply the analytical model to identify the critical overlap length of collagen
fibrils in tendon/bone. Experiments on tropocollagen molecules [Sun et al., 2002] reveal a wide
distribution of elastic modulus ranging from 0.35-18 GPa has been attributed probably to varied
strain rates used in the studies. An elastic moduli (≡ E) of approximately 1 GPa was chosen for
the tropocollagen molecule corresponding to the current scenario. For wet environmental

84

conditions, over the overlap length, crosslinking shear modulus (≡ G) in collagen fibrils was
chosen to be approximately 3.4 MPa following measurements by micromechanical bending
[Yang et al., 2008].

Figure 3.4. Variation of shear stress at the extremities of the joint against overlap length
for spider-silk (a), and its dependence on loading rate (b).
Tropocollagen molecule has a diameter (≡ 2b) of about 1.23 nm, and intra-tropocollagen
molecular gap (≡ h) is about 0.24 nm [Landis et al., 1993]. Electron microscopy 3-D
reconstruction technique identifies an overlap length of 27 nm [Landis et al., 1993].
85

Substitution of the above mechanical and geometrical parameters into the analytical model
predicts the critical overlap length of ~36 nm (shown in Figure 3.5) for dynamic loading event,
which once again is in close agreement with experimental data and also in close agreement
with those obtained for characteristic length scales (≈ 31 nm) [Wei et al., 2012] under quasistatic loading conditions.

Figure 3.5. Variation of shear stress at the extremities of the joint against overlap length for
collagen in tendon/bone (a), and its dependence on loading rate (b).

86

Next, we utilize our analytical model to predict the joint strength of an aluminumLoctite adhesively bonded lap joint under impact loading conditions. Experiments, using SHPB,
were carried out under high rates of loading for these lap joint specimens [Sen et al., 2011]. In
order to predict the joint strength we proceeded as follows:

x 

u1
x L,t

f

(3.87)

v x L,tf   v t
 c 
  t    x
 ct 

(3.88)

A E 
 predicted  b b t
Als
where, c, ct represent stress-wave velocity in the aluminum, and wave velocity through the
transmission bar (Aluminum alloy 7075) respectively. Transmitted strain is calculated on the
basis of velocity compatibility condition at the specimen transmission bar interface. A b, Als
represent the cross-sectional area of the transmission bar, and area of the adhesive lap joint
respectively. E b, tf denotes the Young's modulus of elasticity of the transmission bar and failure
time of adhesively bonded lap joint. Table 3.1 lists the predicted joint strength against
experimentally measured data and they are found to be in reasonably close agreement with
each other. Based on this observation, the efficacy of our analytical model can be validated.

87

Table 3.1. Comparison of experimentally measured joint strength against those predicted by
analytical model for varying overlap area and loading rates.
Loading Rate
(kN/μs)
0.64
1.08
0.87

0.65

1.08
1.52

σMaximum
(MPa)

tf
(μs)

27.68
37.77
40.51
38.22
41.08
38.22
38.22
21.22
25.67
30.76
43.59
57.07
41.94
72.42
66.04

12
24
16
18
16
12
18
12
18
20
13
18
20
16
16

Overlap
Length
(mm)

Overlap
2

Area (mm )

10

158.75

10

158.75

10

158.75

15

238.5

15

238.5

15

238.5

σExperimental
(MPa)

σPredicted
(MPa)

34.62
47.24
50.67
47.80
36.29
47.51
34.91
17.67
21.37
25.61
36.29
47.51
34.91
60.29
54.98

33.78
40.44
49.44
46.64
35.38
46.32
34.03
17.22
20.83
24.96
35.38
46.32
34.03
58.77
53.59

E = 71.7 GPa, G = 1.2 GPa, h = 0.5 mm, 2b = 15.9 mm
Thus, once the material properties and geometrical parameters are fixed, choice of
overlap length can be based upon the essence of minimization of shear stress at the extremities
to promote maximum load transfer by polymer via shear and to withstand the entire dynamic
event. Too far a deviation from the optimal length will raise shear stress at the ends
appreciably, which will in-turn will result in premature failure of the joint (will not be able to
withstand the whole incident pulse) and thereby, reduce joint efficiency. Thus, perspectives
drawn from this point-of-view can be utilized as a vital design guideline in tailoring adhesively
bonded single lap composite joints (at micro/meso/macro scales). Note that, the current
observations are based on triangular pulse loading. However, similar set of observations are
88

encountered if a half-sinusoidal pulse loading is applied, and for brevity purposes are not
discussed here.
Based on fracture mechanics concepts, Gao et al. (2003) postulated the adoption of
nanometer sized inorganic reinforcements in order to achieve a state of flaw-intolerance.
Presence of nanometer sized inorganic crystals can be visualized from another point-of-view.
Shear transfer efficiency of staggered architectural composites can be qualitatively interpreted
by consideration of elastic strain-energy density via shear transfer mechanism. The effective
strain, eff , and the effective stress, eff , can be written as

2u1
eff  lim
tta L L,t 
eff  x,t   lim
tta

b  t   b L  x u1 avg

2b  h

(3.89)

(3.90)

and the elastic strain energy density, weff , is thus given by

1
weff   d  eff eff
2

(91)

for triangular loading pulse loading. Similar to our previous analyses, once the material
properties and geometrical parameters are fixed, w eff becomes a function of overlap length (L).
A plot of elastic strain energy density variation against overlap length is shown in Figure 3.6(a).
As it can be seen from Figure 3.6(a), as the length of the overlap decreases and reaches
nanoscale dimensions, elastic strain energy density of the staggered composite increases. Note
that, the plot in Figure 3.6(a) corresponds to dimensions and properties for nacreous layer in

89

seashell. In fact, this phenomena has been observed experimentally (for macro-scale
specimens) where for a given loading rate, joint-strength increases with decreasing overlapping
area [Sen et al., 2011]. It has been observed that the toughening rate (tearing modulus) in bone
is higher than in nacre [Wang and Gupta, 2011], which can be inferred from Figure 3.6(a). and
attributed to the presence of nanometer sized building blocks in its staggered architecture. T his
explains why nature chooses nano-sub-micrometer scale overlapping feature for biocomposites
structural integrity.
One of the other interesting outcomes of the current analytical model is that as the
loading rate increases, the elastic strain energy density (weff) of the staggered composite
increases as well. As it can be seen from Figure 3.6(b), dependence of weff is markedly
pronounced towards increasing stress-levels than for time rise of the loading pulse. Thus,
staggered composites showed increasing energy absorption behavior with increasing loadingrates and thereby, exhibit dynamic self-stiffening behavior [Huang et al., 2011]. This aspect of
self-stiffening behavior under impact loading conditions will find particular relevance in
application of these types of composites as protective materials under impact.

90

Figure 3.6. Variation of elastic strain energy density against varying overlap length at different
intervals of time under the application of triangular pulse loading (a), and dependence of elastic
strain energy density on loading rate, under the application of triangular pulse loading, shown
via (1) varying max stress (keeping time at maximum stress fixed)-shown in red and (2) varying
time at max stress (keeping maximum stress fixed)-shown in black; dependence of w eff is
markedly pronounced towards increasing stress-levels than for time under the application of
triangular pulse loading.
Note that, based on the current model, we can also explain the reason as to why the
dynamic compressive strength of abalone nacre is nearly 50% higher than its static counterpart,
as observed from SHPB experiments by Menig et al. (2000) and Huang et al. (2011). As was
postulated by Gao and coworkers [Gao et al., 2003; Gao, 2006], under quasi-static conditions,
nature uses, particularly, aspect ratio of the mineral crystals to promote strain amplification
mechanism and thereby obtain an optimal balance for deformation in organic and inorganic
91

components to obtain maximum potential. de Gennes and Okumura (2000) and Okumura and
de Gennes (2003) presented an analytical solution for a layered system, based on laminar
architecture of the seashells, and accounted the existence of weaker stress concentration
ahead of the crack tip in these materials in comparison to traditional isotropic elastic materials,
under quasi-static loading conditions. In our current investigation, we have attempted to
establish a link between superior strength of nacre, under impact loading conditions for a given
loading-rate, and its microstructural dependence, via the existence of an optimum overlap
length during shear-transfer. The loading rates applied for dynamic compression testing [Menig
-1

et al. (2000)] of abalone nacre varied from 15-25 GPas . Corresponding to loading rates of 15
-1

GPas , 20 GPas

-1

-1

and, 25 GPas , the current analytical model predicts an effective strain of

0.0070, 0.0090, and 0.0117 respectively, which are in reasonable agreement with the
experimentally observed values of 0.008, 0.0095, and 0.0145 respectively, obtained by Menig
et al. (2000). Additionally, the predicted maximum compressive strength is approximately 450
MPa which is again in reasonable agreement with experimentally obtained strengths of nearly
500 MPa [Menig et al., 2000; Huang et al., 2011]. Predicted shear strains corresponding to
-1

loading rates of 15 GPas , 20 GPas

-1

and, 25 GPas

-1

were 0.178, 0.237, and 0.297 respectively

which is less than experimentally obtained maximum shear strains of approximately 0.45
[Menig et al., 2000]. This accounts for the reason as to why staggered architecture composites
exhibit elevated compressive strength under time-dependent loading conditions. Additionally,
under the current model, the elastic strain energy density (w eff) will thereby turn out to be
92

higher for dynamic loading than for quasi-static loading conditions, which can explain the
reason for superior damage tolerance of staggered architecture composites. The outcome of
the current research will thus provide beneficial guidelines in custom-design manufacture of
hybrid bio-inspired protective composite materials.

(a)

(b)
Figure 3.7. Plot showing the dependence of maximum shear stress (at the ends of overlap) as a
function of aspect ratio and volume fraction of the ceramic bricks for: aspect ratio varying
93

Figure 3.7. (cont'd)
between 4 to 200 (a) and, 4 to 800 (b) under the application of triangular pulse loading,
corresponding to a given loading-rate.
Ji and Gao (2004, 2010) and Ji (2008) had analyzed the staggered arrangement of the
biological structure previously under quasi-static loading conditions. They postulated that
under quasi-static conditions, two parameters play a significant role: (a) presence of
nanometer-sized crystals to obtain a state of maximum flaw tolerance and, (b) presence of an
optimum aspect ratio of the mineral crystals to promote strain amplification mechanism and
thereby obtain an optimal balance for deformation in organic and inorganic components to
obtain maximum potential. Apart from visualizing the design in staggered architecture
materials from optimal overlap length point of view, the current model can also be visualized
from aspect ratio and volume fraction of the ceramic point of view, for a given thickness of the
ceramic brick.
Under time-dependent loading conditions, optimum aspect ratios exist for varying
volume fractions of ceramic in staggered architecture materials, as shown in Figure 3.7(a,b).
For lower volume fractions of the ceramic, a wide range of optimal aspect ratio exists which
would correspond to maximum load transfer capability of the joint via attainment of a state of
minimum shear. For example, bone and dentin comprise of plate-like crystals having aspect
ratio varying from 25-50 and a volume fraction of 0.40-0.55 [Ji and Gao (2004)]. For very high
volume fraction of the ceramic, the range of aspect ratio available to obtain a state of minimum

94

shear is narrow, as shown in Figure 3.7(a). Similarly, nacre comprises of ceramic bricks having
an aspect ratio varying from 10-16 and a volume fraction of nearly 0.95 [Ji and Gao (2004)].

(a)

(b)
Figure 3.8. Plot showing the dependence of the ratio of composite modulus to Young's modulus
of the ceramic bricks for varying aspect ratio and volume fraction of the ceramic bricks:
correlation between dynamic (a) and quasi-static (b) regime, corresponding to a given loading
95

Figure 3.8. (cont'd)
rate.
In order to obtain a more clearer understanding of the interdependence of aspect ratio,
volume fraction of the ceramic, and constituent material properties under impact loading
conditions, a plot of ratio of composite modulus to the ceramic Young's modulus as a function
of aspect ratio and volume fraction is shown in Figure 3.8(a). Figure 3.8(b) shows the
dependence obtained under quasi-static conditions using the Tension-Shear-Chain (TSC) model
as was postulated by Ji and Gao (2004).
It should be noted that extensive literature exists quantifying role of microstructure and
material parameters on the behavior of staggered architecture, however under quasi-static
loading conditions. Note that, in our current investigations, what we are trying to establish is
that staggered architecture in biological composites is designed in such a unique manner that
the structure would be able to provide maximum resistance both under quasi-static loading
conditions, and when subjected to severe dynamic loads when attacked by predators (as
resistance to shell crushing is an integral part of shell evolution process).
3.3.1. Limitations of the analytical model and future directions
Formulation of the model is laid on the foundation of linear elastic material behavior
only, i.e. the stresses are linearly dependent on strain only (via material properties E,G which
are considered constant); however, the organic matrix being a visco-elastic material, under


dynamic loading conditions, strain-rate ( s ) of the organic matrix is expected to play a key role

96


in load transfer via shear. Incorporation of strain-rate ( s ) dependent terms and large-scale
deformation aspects would help in understanding mechanical behavior of these composites
under very high-strain rate loading; for example under blast loading conditions. Additionally,
incorporation of plastic behavior of the intermediate matrix is also expected to contribut e to
the cause. This could possibly account for the difference obtained between experimentally
observed and theoretically predicted overlap length values in nacre as shown previously. Work
is in progress in this direction. To further and better understand this aspect, the model can be
extended to include strain-rate dependence and large-scale deformation of the intermediate
polymer, effect of alignment (regular, stairwise, random), number of hierarchical levels, etc.
These can be identified as future directions of the current work. Additionally, the current model
is restricted to the scenario where adherends are made of same materials. Work is in progress
to understand the behavior of staggered architecture composites where adherends are made
of dissimilar materials.
Under static loading conditions, characteristic lengths predicted were essentially based

2Ebh
on (a) stress transfer point of view, L 
, as was postulated by Chen et al. (2009) and,
G
Ebh
(b) an optimization on both strength and toughness fronts, L  2.318
, as postulated by
G
Wei et al. (2012). Under dynamic rates of loading, the shear stress developed at the interface is
also a function of the overlap length (L). If a joint has to withstand the entire dynamic event,
critical overlap length exists based on the essence of minimization of shear stress at the

97

extremities to promote maximum load transfer capability by polymer via shear (once the
material properties and geometrical parameters are fixed). Thereby, in order to obtain
optimum joint efficiency, an optimal overlap length is evaluated as follows:

G

Loptimal  Minimize    x,t    Minimize   u4  x,t   

L
h
L

 t L

2 t

    2c

    sin  d04  t     d

 6E

ELd04 
0





 2L  1  cos n  



 

  t 

E  n22 







 

 
G 
2
 t

1  cos n 
 n 

 Minimize      2c 

    sin  c k     t     d 



 h  n1  E 
2 2 2
 L 

 
  kL  n   0

 






 nx 

cos 




 L 





 2


   t   x  x 


 E  L






L (3.92)



It is to be noted that there does not exist a direct relationship between dynamic overlap length
and static overlap length. This can be better addressed with reference to the figure shown
below.

98

Figure 3.9. Variation of shear stress at the interface as a function of overlap length in a material
(in the current case, nacreous layer in seashells) under (a) static, and (b) dynamic rates of
loading.
Under static loading conditions, the shear stress (at the extremities of the interface)
[Wei et al. (2012)] as given by the equation below, decreases till a critical value is attained and
remains constant thereafter.

b
2G
 L 
static  L  
cosh  ;  
x  0, 
Ebh
 L 
 2 
 2  2sinh  
 2 

(3.93)

On the contrary, it can be clearly seen (Figure 3.9) that critical overlap length exists under
dynamic rates of loading, and thus addresses the vitality of optimal overlap length under
dynamic rates of loading.
Under quasi-static loading conditions, choice of critical overlap length in staggered
biocomposites attempts to optimize mechanical performance on both strength and toughness
fronts. However under impact loading conditions, since shear transfer mechanism is the most
99

versatile parameter, choice of critical overlap length is mainly directed towards minimization of
peak shear stresses at the extremities of the joint. However, nature simultaneously addresses
the nano-sub-micro overlap length scales in an attempt to improve on toughness perspective as
well. This feature can be adapted as an important guideline in the design of adhesively bonded
lap joints, gradient microstructural composites, etc. under dynamic rates of loading. The
outcome of the current research will thus provide beneficial guidelines in custom-design
manufacture of hybrid bio-inspired protective composite materials.
3.4. Conclusions
In our current investigation, we have addressed the following aspects with regards to
materials exhibiting staggered microstructure as its most elementary level of architecture
under impact loading conditions, by considering a triangular loading pulse: (1) Identifying the
structure-property correlationship for these materials, (2) existence of a critical overlap length
which optimizes load transfer mechanism between the adjacent reinforcements, (3) ratify the
choice of nanometer building blocks in biological composites under time-dependent loading, (4)
generalizing the application of the concept to the case of macro-scale adhesively bonded lap
joints, (5) explaining the reason behind superior damage tolerance and elevated dynamic
compressive strength of nacre, and (6) theoretically justify the applicability of staggered
microstructural composites under impact loading conditions. The outcome of the current
research will provide beneficial guidelines in custom-design manufacture of hybrid bio-inspired
protective composite materials.

100

REFERENCES

101

REFERENCES

Barthelat, F., Espinosa, H.D., 2007. An Experimental Investigation of Deformation and
Fracture of Nacre-Mother of Pearl. Exp. Mech. 47, 311–324.
Barthelat, F., Tang, H., Zavattieri, P., Li, C.M., Espinosa, H., 2007. On the Mechanics of
Mother-of-Pearl: A Key Feature in the Material Hierarchical Structure. J. Mech. Phys. Solids 55,
306-337.
Chen, B., Wu, P.D., Gao, H., 2009. A characteristic length for stress transfer in the
nanostructure of biological composites. Compos. Sci. Technol. 69, 1160-1164.
de Gennes, P.G., Okumura, K., 2000. On the toughness of biocomposites. C. R. Acad. Sci.
Paris, t.1, Ser. IV, 257-261.
Evans, A., Suo, Z., Wang, R., Aksay, I., He, M., Hutchinson, J., 2001. Model for the
Robust Mechanical Behavior of Nacre. J. Mater. Res. 16, 2475-2484.
Espinosa, H.D., Juster, A.L., Latourte, F.J., Loh, O.Y., Gregoire, D., Zavattieri, P.D., 2011.
Tablet-Level Origin of Toughening in Abalone Shells and Translation to Synthetic Composite
Materials. Nature Comm. 2, 173.
Gao, H., Ji, B., Jäger, I., Arzt, E., Fratzl, P., 2003. Materials Become Insensitive to Flaws
at Nanoscale: Lessons from Nature. Proc. Natl. Acad. Sci. U.S.A. 100, 5597-5600.
Gao, H., 2006. Application of fracture mechanics concepts to hierarchical biomechanics
of bone and bone-like materials. Int. J. Fracture, 138, 107-137.
Hulmes, D.J.S., Wess, T.J., Prockop, D.J., Fratzl, P., 1995. Radial packing, order and
disorder in collagen fibrils. Bioph. J. 68, 1661–1670.
Huang, Z., Li, H., Pan, Z., Wei, Q., Chao, Y.J., Li, X., 2011. Uncovering high-strain rate
protection mechanism in nacre. Sci. Rep. 1, 148; DOI:10.1038/srep00148.
Jackson, A.P., Vincent, J.F.V., Turner, R.M., 1988. The mechanical design of nacre. P. R.
Soc. London B. 234, 415–440.
Jager, I., Fratzl, P., 2000. Mineralized collagen Fibrils: a mechanical model with a
staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746.

102

Ji, H., Gao, H., 2004. Mechanical properties of nanostructure of biological materials. J.
Mech. Phys. Solids 52, 1963-1990.
Ji, B.H., 2008. A study of the interface strength between protein and mineral in
biological materials. J. Biomech. 41, 259–266.
Ji, B., Gao, H., 2010. Mechanical principles of biological nanocomposites. Annu. Rev.
Mater. Res. 40, 77-100.
Kessler, H., Ballarini, R., Mullen, R.L., Kuhn, L.T., Heuer, A.H., 1996. A biomimetic
example of brittle toughening: (I) steady state multiple cracking. Comp Mater Sci. 5, 157–166.
Kamat, S., Su, X., Ballarini, R., Heuer, A.H., 2000. Structural basis for the fracture
toughness of the shell of the conch Strombus gigas. Nature, 405, 1036–1040.
Kotha, S.P., Kotha, S., Guzelsu, N., 2000. A shear-lag model to account for interaction
effects between inclusions in composites reinforced with rectangular platelets. Compos. Sci.
Technol. 60, 2147–2158.
Katti, K.S., Katti, D.R., Pradhan, S.M., Bhosle, A., 2005. Platelet interlocks are the key to
toughness and strength in nacre. J. Mater. Res. 20, 1097-1110.
Keten, S., Buehler, M.J., 2008. Asymptotic Strength Limit of Hydrogen-Bond Assemblies
in Proteins at Vanishing Pulling Rates. Phys. Rev. Lett. 100, 198301.
Keten, S., Xu, Z., Ihle, B., Buehler, M.J., 2010. Nanoconfinement Controls Stiffness,
Strength and Mechanical Toughness of -Sheet Crystals in Silk. Nature Materials 9, 359-367.
Landis, W., Song, M., Leith, A., McEwen, L., McEwen, B., 1993. Mineral and Organic
Matrix Interaction in Normally Calcifying Tendon Visualized in Three Dimensions by HighVoltage Electron Microscopic Tomography and Graphic Image Reconstruction. J. Struct. Biol.
110, 39-54.
Landis, W.J., 1995. The strength of a calcified tissue depends in part on the molecular
structure and organization of its constituent mineral crystals in their organic matrix. Bone 16,
533–544.
Liu, G., Ji, B.H., Hwang, K.C., Khoo, B.C., 2011. Analytical solutions of the displacement
and stress fields of the nanocomposite structure of biological materials. Compos. Sci. Technol.
71, 1190-1195.
Menig, R., Meyers, M.H., Meyers, M.A., Vecchio, K.S., 2000. Quasi-static and dynamic
mechanical response of Haliotis rufescens (abalone) shells. Acta Mater. 48, 2383–2398.
103

Menig, R., Meyers, M.H., Meyers, M.A., Vecchio, K.S., 2001. Quasi-static and dynamic
mechanical response of Strombus gigas (conch) shells. Mater. Sci. Eng. A 297, 203–211.
Meyers, M., Lin, A., Chen, P., Muyco, J., 2008. Mechanical Strength of Abalone Nacre:
Role of the Soft Organic Layer. J. Mech. Behav. Biomed. Mater. 1, 76-85.
Okumura, K., de Gennes, P.-G., 2001. Why is nacre strong? : Elastic theory and fracture
mechanics for biocomposites with stratified structures. Eur. Phys. J. E 4, 121-127.
Penel, S., Morrison, R.G., Dobson, P.D., Mortishire Smith, R.J., Doig, A.J., 2003. Length
Preferences and Periodicity in Strands. Antiparallel Edge Sheets Are More Likely to Finish in Non
Hydrogen Bonded Rings. Protein Eng. 16, 957.
Sarikaya, M., Gunnison, K.E., Yasrebi, M., Aksay, J.A., 1990. Mechanical
property-microstructural relationships in abalone shell. Mater. Res. Soc. 174, 109-116.
Sun, Y.L., Luo, Z.P., Fertala, A., An, K.N., 2002. Direct quantification of the flexibility of
type I collagen monomer. Biochemical and Biophysical Research Communications 295, 382-386.
Song, F., Soh, A., Bai, Y., 2003. Structural and mechanical properties of the organic
matrix layers of nacre. Biomaterials 24, 3623-3631.
Shuchun, Z., Yueguang, W., 2007. Effective elastic modulus of bone-like hierarchical
materials. Acta Mech. Solida Sin. 20, 198-205.
Sen, O., Tekalur, S.A., Jilek, C., 2011. The determination of dynamic strength of single
lap joints using the split Hopkinson pressure bar. Int. J. Adhes. Adhes. 31, 541-549.
Volkersen, O., 1938. Die Nietkraftverteilung in zugbeanspruchten Nietverbindungen mit
konstanten Laschenquerschnitten, Luftfahrtforsch 15, 41.
Veis, A., Sabsay, B., 1987. The collagen of mineralized matrices. In Bone and Mineral
Research/5, W. A. Peck, editor. (Elsevier Science Publications, New York, Amsterdam, Tokyo).
pp. 1-63.
Wagner, H.D., Weiner, S., 1992. On the relationship between the microstructure of
bone and its mechanical stiffness. J. Biomech. 25, 1311–1320.
Wang, R., Suo, Z., Evans, A., Yao, N., Aksay, I., 2001. Deformation mechanisms in nacre.
J. Mater. Res.16, 2485-2493.
Wang, R., Gupta, H.S., 2011. Deformation and fracture mechanisms of bone and nacre,
Annu. Rev. Mater. Res. 41, 41-73.
104

Wei, X., Naraghi, M., Espinosa, H.D., 2012. Optimal length scales emerging from shear
load transfer in natural materials: application to carbon-based nanocomposite design. ACS
Nano 6(3), 2333-2344.
Yang, L., Van Der Werf, K.O., Fitié, C.F.C., Bennink, M.L., Dijkstra, P.J., Feijen, J., 2008.
Mechanical Properties of Native and Cross-Linked Type I Collagen Fibrils. Biophys. J. 94, 22042211.
Zhang, Z.Q., Liu, B., Huang, Y., Hwang, K.C., Gao, H., 2010. Mechanical properties of
unidirectional nanocomposites with non-uniformly/randomly staggered platelet distribution. J.
Mech. Phys. Solids 58, 1646-1660.
Zhang, Z., Zhang, Y.-W, Gao, H., 2011. On optimal hierarchy of load-bearing biological
materials. Proc. R. Soc. B, 278, 519-525.

105

CHAPTER 4. ENHANCING RESISTANCE AND LOAD TRANSFER VIA CHOICE OF OPTIMAL
OVERLAP LENGTH

106

4.1. Introduction
Over the past decade, increasing attention has been focused towards producing
composites that are both light-weight and durable via extracting design principles from
structural biological composites for example, nacre. The inner shining layer in seashells is
referred to as the nacreous layer and exhibits a staggered architecture in the form of an
overlapping arrangement of mineral tablets in a soft polymeric matrix [Jackson et al., 1988;
Sarikaya et al., 1990]. What confers nacre an ideal model for biomimetic inspiration are two
promising aspects: light weight and superior toughness [Barthelat and Espinosa, 2007; Jackson
et al., 1988; Kessler et al., 1996; Kamat et al., 2000; Sarikaya et al., 1990]. Light weight is
accomplished via embedding a polymeric matrix in a network of ceramic/mineral bricks,
whereas toughness has been attributed to existence of a multitude of toughening mechanisms,
namely tablet pull-out [Jackson et al., 1988], crack deflection through the biopolymer [Menig
et al., 2000], platelet interlocks [Katti et al., 2005], presence of nanoasperities [Evans et al.,
2000; Wang et al., 2001], diffusive tablet sliding [Barthelat et al., 2007] arising from tablet
waviness, aragonite bridge reinforcements at interface [Song et al., 2003], re-locking of tablets
[Meyers et al., 2008] due to persistent contact of broken aragonite bridges, and synchronized
deformation twinning of the nano-scale particles in ceramic bricks [Huang et al., 2011]. All the
aforementioned factors contribute to both interfacial strengthening and fracture resistance in
varying degrees of resistance.
Till date, a number of literature studies have attempted to address the behavior of the
staggered architecture materials on the basis of interdependence of material properties and

107

geometrical parameters. Table 4.1 provides the references and the approach adopted by
various researchers with reference to modeling of biological composites, in a brief manner.
Table 4.1. Recent advances on modeling staggered architecture biological composites by
various researchers.
Authors
Description
Jaegar and Fratzl, Predicted modulus of staggered architecture composites along the
2000; Kotha et longitudinal direction of the inclusions as a function of aspect ratio and
al., 2001
concentration of the inclusions.
Developed an analytical model based on laminar architecture of seashells
de Gennes and
to account for the existence of weaker stress concentration ahead of the
Okumura, 2000;
crack tip in these materials in comparison to traditional isotropic elastic
Okumura and de
materials along with an examination of the effect of differential stiffness
Geness, 2003
of inclusions and interfaces.
Developed the Tension-shear-chain (TSC) model to describe the
Gao et al., 2003
mechanics of staggered architecture composites and study the
mechanical properties of these type of biological composites.
Studied the interdependence of the elastic modulus of the composite
Shuchun
and
and number of hierarchical levels in bone-like materials via comparison
Yueguang, 2007
of their results against TSC model and FE (finite element) simulations.
Investigated the effect of platelet distribution (regular, stairwise,
Zhang et al., 2010 random) on stiffness, strength, failure strain and energy absorption
capability in materials exhibiting staggered architecture.
Used a quasi-self-similar hierarchical model to comprehend the existence
Zhang et al., 2011
of an optimal number of hierarchical levels in biological composites.
Provided analytical expressions for displacement and stress fields in
Liu et al., 2011
staggered nanocomposite structures under static loading conditions
under uniaxial tension.
Barthelat
and Developed a micromechanical model taking into consideration the effect
Rabiei, 2011
of toughening associated with the process zone.
The nacreous layer can be visualized as translation of unit cell structure in 2-dimensions
comprising of overlapping mineral tablets having distinct overlap and core regions. The overlap
region is responsible for inter-mineral load transfer via shear deformation of the polymer. One
might consider what could possibly be the reason associated with the existence of specific

108

overlap length in the nacreous layer in seashells. Analytical models and numerical simulations
have been developed and certain criterion been proposed to address this aspect as listed in
Table 4.2 below.
Table 4.2. Recent advances on investigating existence of characteristic overlap length in
staggered architecture biological composites by various researchers.
Authors
Chen et al., 2009

Wei et al., 2013

Dutta et al., 2013

Description
Under static rates of loading, the existence of characteristic length was
attributed to attainment of efficient stress transfer in staggered
biocomposites.
Under static rates of loading, the existence of unique overlap length in
biological composites was attributed to an attempt to optimize both
strength and toughness frontiers in staggered architecture biological
composites. Note that, the toughness in their investigations was
defined in terms of elastic strain energy density.
From the perspective of impact rates of loading, existence of optimal
overlap length was attributed to an attempt to obtain maximum shear
transfer efficiency.

What contributes to the superior load-transfer and energy dissipative capabilities of the
nacreous layer is the meandering nature of the crack path throughout its architecture. Higher is
the level of tortuousness associated with the crack path, higher will be the energy required to
drive the crack throughout the individual microscopic unit cells and thereby, superior will be
the toughness of the nacreous layer on a macroscopic level. Note that, the current scenario
under consideration is representative of an interface fracture problem. On a conservative
approach, it is opportune to consider that an interface is as much fracture resistant as the least
tough participating component in the system. As per Griffith criterion, cracks will initiate and
delamination or cohesive failure will occur in the unit-cell once the crack driving force (rate of
energy input) ahead of the crack tip exceeds the fracture resistance (rate of energy dissipated)
109

of either the biopolymer itself, or that of the interface. The objective thus is to find out how
structural overlap length is responsible for minimizing this event to the maximum extent
possible. Under given set of loading conditions, if overlap length is adopted such that crack tip
driving forces required for crack initiation can be kept to a minimum, higher will be load that
further needs to be applied to elevate the energy required to attain the fracture resistance of
the barrier(s) to promote delamination (shown later in Figure 4.5) in the unit-cell. This would
in-turn raise the toughness associated with the individual unit-cells which would contribute to
global toughness of the composite. The main outcome of the developed analytical model in the
current study is to obtain a simplified expression for crack driving force in terms of local finite
stress concentrations in the biopolymer layer of nacre. This would in-turn provide an insight
into the severity of load carried and transferred by the joint. Finally, the article concludes with
validation of the hypothesis against previously published experimental data followed by
proposing a design guideline to guide staggered architecture composite development to obtain
optimized performance in terms of stiffness, load-transfer and toughness.
4.2. Analytical model
The remarkable macroscopic toughness of nacre is as a result of all the microstructural
mechanisms taking place ahead of the crack tip (otherwise referred to as intrinsic toughening),
behind the crack tip (otherwise referred to as extrinsic toughening), around the crack tip and at
the crack tip [Launey and Ritchie, 2009] as shown in Figure 4.1. Extrinsic toughening is
associated with mechanisms such as crack-bridging and in-situ phase transformations. On the
contrary, intrinsic toughening is associated with the role of type, size, distribution of

110

reinforcement and it's interfacial properties with the binding matrix. Crack growth ensues from
extrinsic toughening whereas crack initiation is primarily dependent upon intrinsic toughening.
So, the question is how does the nacreous layer allay crack initiation in its micro-architecture?
In the current study, we will try to better understand the design principle in the nacreous layer
from a fracture mechanics perspective from the point-of-view of intrinsic toughening.

Figure 4.1. Toughening mechanisms associated with nacre and region under consideration.
In order to address this, we develop our analytical model by taking into consideration a
unit cell with plane strain infinitesimal deformations, as shown in Figure 4.2. Following are the
assumptions associated with the analytical model:
(1). One-dimensional analysis.
(2). Participating members exhibit linear elastic behavior (ceramic bricks, bio-polymer).
(3). Normal stress distribution in the ceramic bricks (no bending).
111

(4). Shear stress distribution in the bio-polymer.
(5). Since the bio-polymer is very thin in comparison to the brick thickness, the stress
distribution is considered constant throughout the joint thickness [Luo and Tong, 2004].
(6). There is no rotation of the transverse normal about the y-axis, which is directed into the
plane of the paper.
The unit cell has been discretized into four elemental blocks (labeled 1 to 4). The total
width of the mineral tablets is 2b and thickness of the biopolymer is given by  . Local
rectangular Cartesian coordinate system for elemental blocks 1-4 have been defined as shown
in Figure 4.2. It has also been assumed that there is no rotation of the transverse normal about
the y-axis.

112

Figure 4.2. (a) Schematic of staggered architecture of calcium carbonate bricks (shown in green)
in biopolymer matrix (shown in yellow); (b) expanded view of the 2-dimensional unit cell
structure (with plane strain infinitesimal deformation) showing the pre-cracked length (L1) and
instantaneous overlap length (L2) and coordinate systems; (c) expanded view of the 2dimensional unit cell structure (with plane strain infinitesimal deformation) highlighting the
elemental blocks and the original overlap length (L O); (d) splitting of parent crack tip for crack
driving force calculations; (e) schematic identifying shear and normal force resultants N C and
QC respectively.
113

With reference to Figure 4.2, x, x1, and x2 represents horizontal axes with different
origins (along the longitudinal direction of the bricks) and the z direction is associated in the
thickness direction. With reference to Figure 4.2(b), as a starting point, it has been considered
that there is a pre-existing void (of length, a0), an incremental crack (of length, da) in the cell.
Subsequent analyses are based on the value of that instantaneous crack length. The
displacements are denoted by u and are associated with a subscript and superscript. The
subscript corresponds to direction of displacement (either x, x 1, x2, or z) whereas the
superscript corresponds to elemental block number (labeled 1 through 4). We assume that in
all the elemental blocks, the variation of displacements is a function of either x, x1, or x2. The
elemental strains are thus defined as follows:
Elemental block #1:

du1
1  x
xx
dx1

(4.1)

1  0
zz

(4.2)

du1
1  z
xz
dx1

(4.3)

du2
2  x
xx
dx2

(4.4)

2  0
zz

(4.5)

Elemental block #2:

114

2
2  duz
 xz
dx2

(4.6)

du3
3  x
xx
dx

(4.7)

3  0
zz

(4.8)

3
3  duz
 xz
dx

(4.9)

du4
x
dx

(4.10)

Elemental block #3:

Elemental block #4:

4 
xx

4
zz  0

(4.11)

4
4  duz
 xz
dx

(4.12)

The stresses in the elemental blocks (i = 1..4) have been computed as follows:

c
c
i  C11i  C22i
xx
xx
zz

(4.13)

c
i  C22i  Cc i
zz
xx
33 zz

(4.14)

i  Cc i
xz
44 xz

(4.15)

The strains and the stresses in the biopolymer are obtained as follows:

115

1  du3 du4 
p   x  x 
xx
2  dx
dx 



(4.16)

1 4
p  ux  u3
xz
x






(4.17)





(4.18)

1 3 4
p  uz  uz
zz


p  Cp p  Cp p
zz
22 xx
33 zz

(4.19)

p  Cp p
xz
44 xz

(4.20)

where, p and p denotes the normal (peel) stress and shear stress in the polymer
zz
xz
respectively.
4.2.1. Solution scheme for elemental displacements
Force equilibrium on the elemental blocks leads to the development of following
governing differential equations of the system, given by

d2u1
x 0
2
dx1

(4.21)

d2u1
z 0
2
dx1

(4.22)

d2u2
x 0
2
dx2

(4.23)

d2u2
z 0
2
dx2

(4.24)

116

d2u3
x  M u3  u4  0
1 x x
dx2





(4.25)





(4.26)

d2u4
x  M u3  u4  0
1 x x
dx2

4
 3
d2u3
z  M  dux  dux   M u3  u4  0
2
3 z z
dx
dx 
dx2



(4.27)

4
 du3 du4 
d2uz
3 4
 M2  x  x   M3 uz  uz  0
2
 dx
dx 
dx



(4.28)









p
p
Cp
44 , M  C22 , and M  C33 . The boundary conditions of the
where M1 
3
2
c
bCc
bC11
2bCc
44
44
system are given by

u1
z
u2
x

x1 0

x2 0

du1
x
dx1
du1
z
dx1

4
duz
dx

0

(4.29)

0

(4.30)

0

(4.31)

0

(4.32)

0

(4.33)

x1 0

x1 0

x L2
117

du3
x
dx

du2
x
dx2




c
C11

(4.34)




c
C11

(4.35)

0

(4.36)

0

(4.37)

0

(4.38)

x L2

x2 0

du4
x
dx

du2
z
dx2
du3
z
dx

x L2

x2  0

x L2

where, L1  a0 and L2  Lo  a0 . The continuity equations for forces across the element
boundaries are thus given by

du1
x
dx1
du1
z
dx1
du2
x
dx1

x1 L1

du3
 x
dx


x1 L1

x1 L1

3
duz
dx

4
dux
dx

118

(4.39)

x 0

(4.40)

x 0

(4.41)

x 0

du2
z
dx1


x1 L1

4
duz
dx

(4.42)

x 0

The continuity equations for displacements across the element boundaries are defined
as follows:

u1
x

u1
z
u2
x
u2
z

x1 L1

x1 L1

x2 L1

x2 L1

 u3
x

3
 uz
4
 ux
4
 uz

x 0

x 0

x 0

x 0

(4.43)

(4.44)

(4.45)

(4.46)

We thus have coupled second-order linear differential equations subjected to
homogenous boundary conditions. The governing equations corresponding to each elemental
blocks are solved first which results in 16 constants of integration. The unknown constants of
integration are determined from the boundary conditions and continuity conditions of the
system to finally obtain the generalized displacements associated with each elemental block.
4.2.2. Computation of crack driving force (CDF)
As per linear elastic fracture mechanics, energy released from the propagation of a crack
is equivalent to work needed to close that small crack propagation. The energy release rate (or
rather, rate of energy input) is representative of the severity of load acting on the infinitesimal
crack in the biopolymer layer. It is also representative of energy-flux to crack tip which acts a

119

driving force for quasi-static advancement of crack and subsequently fracture. Thus in our
present analyses energy release rate ahead of the crack tip will be henceforth perceived as
crack driving force.
As mentioned previously, considering plane strain conditions, we assume a virtual crack
tip displacement by an amount da . The parent crack tip at point C splits into points A and B,
and translates to point C'. Crack driving force ahead of a crack-tip due to an incremental
increase in crack length is equivalent to work (or rather, energy) required to close that
infinitesimal increment and is thus computed as follows [Yang et al., 2008]:

G



 



W
1 
A

N uB  uA  Qc uB  uz 
z
 c x x


da 2da 

(4.47)

where, W represents the work required to close the virtual crack. As shown in Figure
4.2(d,e), shear and normal force resultants Nc and Q c respectively can be computed as follows:

da
Nc    p dx
xz
x 0

(4.48)

da
Q c   p dx
zz
x 0

(4.49)

p and p are obtained from the solution of the problem prior to virtual crack-tip
zz
xz
A A
extension. ux , uz , uB and uB are found out by solving the problem by substitution of overlap
x
z
length L2  Lo  a0 by L2  Lo  a0  da and by using the following relations:

A
ux  u2
x

x2 a0 da

120

(4.50)

A
uz  u2
z

uB  u1
x
x
uB  u1
z
z

(4.51)

x2 a0 da

(4.52)

x1 a0 da

(4.53)

x1 a0 da

Practically speaking the unit-cell structure in the nacreous layer of seashells is
representative of an adhesive joint and in order to better understand the crack initiation and
growth process, it is imperative to extract the crack driving force contributions associated with
both mode-I and mode-II, and thus we introduce the mode-mixity parameter. In order to
investigate the effect of components of crack driving force that contributes to crack initiation,
Equation (4.47) can alternatively be written as





(4.54)





(4.55)

GII 

1 
N uB  uA 
 c x x 

2da 

GI 

1 
A
Q c uB  uz 
z



2da 

G 
MMP  tan1  II 
 GI 

(4.56)

where, GII , GI and MMP denotes mode I crack driving force, mode II crack driving force
and mode mixity parameter respectively. For pure mode I failure, MMP=0 and for pure mode II
failure, MMP =


.
2

4.2.3. Limitations associated with the analytical model

121

In our current study, the microstructural resistance to crack initiation has been
associated from a local perspective in terms of crack driving force. Lower the value of crack
driving force, higher is the microstructural resistance associated with the unit-cells to crack
initiation. At this stage, we would like to point out that applied stress levels responsible for
crack initiation could possibly be higher than the strength of the nacreous layer. Since the
foundation of the model is based upon linear elastic deformation conditions, specific levels of
external energy needed to be applied to generate appropriate level of rate of energy input
(CDF) to surpass corresponding fracture resistance might possibly be over predicted. As we
know, inelastic energy dissipation associated in the process zone [Barthelat and Espinosa,
2007], viscoplastic deformation and microvoid coalescence mechanisms do contribute to a
significant extent on the rate of energy dissipation in the forward zone associated with the
crack-tip (as was shown in Figure 4.1).
However, note that under quasi-static rates of loading, the precursor to inelastic
deformations is tablet sliding where the tablets start to slide upon one another once the shear
strength of the material has been reached and is subsequently accompanied by interface
strengthening. Additionally, under high strain-rate loading, the bio-polymer in the nacreous
layer is expected to behave in a hard and brittle manner [Zachary and Burger, 1980] which
would lead to conditions favoring brittle fracture and thereby, linear elastic conditions can be
assumed without significant loss of fidelity.
The question that can be posed at this stage is under either of those scenarios what is
that optimal overlap length which would delay this event (crack initiation following tablet

122

sliding) and enhance interface strengthening to the maximum extent possible? It can be
envisaged that a crack will advance only if the crack driving force (rate of energy input)
associated with the local crack tip exceeds the fracture toughness (rate of energy dissipated) of
(a) the bio-polymer, and/or (b) the interface. Thus, in our analyses we have focused our
attention to the abovementioned perspective only.
4.3. Results and Discussion
It has already been reported in the past by Jackson et al., 1988 that approximately a
third of the toughness of wet nacre samples has been associated with tablet pull-out. As per
the TSC model postulated by Gao et al., 2003 the inter-platelet load transfer occurs via shearing
deformation of the polymer in its existing overlap length. The question is whether this length
scale is just an arbitrary random number chosen by Nature or, is it associated with optimizing
the structure-property correlationship? If the latter one, what are those mechano-biological
principles that are operating underneath?
In order to test the validity of the analytical model, the variation of shear stress at
mineral biopolymer interface is plotted for varying overlap lengths and pre-crack in the unit cell
as shown in Figure 4.3. The properties of the mineral component in the nacreous layer
[Barthelat et al., 2006] are: E1=144 GPa, E2 =76 GPa, E3 =82 GPa, G12=47.2 GPa, G13 =25.6 GPa,
G23=41.3 GPa, ν12=0.44, ν13=-0.06, ν23=0.18, 2b=0.5 μm. The properties of the biopolymer
layer in nacre [Xu et al., 2011] are: Ep=10.57 GPa, Ga=1.4 GPa, νa =0.45, η=20 nm. The observed
overlap length in the nacreous layer is reported to be 1.67μm [Espinosa et al., 2011].

123

Figure 4.3. Variation of interfacial shear stress (at the extremities of the overlap) as a function
of varying overlap length and pre-crack. The figure in the inset shows the expanded view for
lower shear levels in the unit cell.
When an external load is applied, interfacial shear stress attains a maximum value at the
extremities of its overlap. As it can be seen from Figure 4.3, the maximum interfacial shear
stress (at the ends of the joint) attains a minimum value for overlap length exceeding 2-2.5 μm
and reaches the shear strength of the biopolymer ≡ 37 MPa [Jackson et al., 1988]. This
observation is in direct agreement with those obtained by Dutta et al., 2013 in their estimation
of characteristic overlap length in nacre under dynamic rates of loading respectively. The region
to the right-hand side of 2-2.5 μm might be considered as the preferred zone as it contributes

124

to a minimization of shear stress at the interface and optimizes intertablet load transfer via
shear deformation of the intermediate biopolymer.
Resistance to crack advancement is an attribute associated with extrinsic toughening
that takes place in the wake of the crack, and thus is responsible for R - curve. On the contrary,
microstructural damage mechanisms in the cohesive zone and in bounded material are
responsible for crack initiation which can be attributed to intrinsic toughening. Intrinsic
toughening is dependent upon material properties and independent of length of crack or
geometry of test specimen (except plane stress/plane strain). This brings into forefront that
crack driving force to initiate fracture is a result of dissipation mechanism associated with
intrinsic toughening. The current scenario under consideration is representative of an interface
fracture problem. Note that an interface is as much fracture resistant as the least tough
participating component in the system. As per Griffith criterion, cracks will initiate and
delamination or cohesive failure will occur in the unit-cells once the crack driving force (rate of
energy input) ahead of the crack tip exceeds the fracture resistance (rate of energy dissipated)
of either the biopolymer itself, or that of the interface. The objective at present is to find out
how structural overlap length is responsible for minimizing delamination in the unit-cells to the
maximum extent possible.
2

Fracture toughness of the organic matrix varies between 20-160 mJ/m [Song et al.,
2003] whereas intrinsic fracture toughness of the organic interface has an upper limit varying
2

between 400-1000 mJ/m [Rabiei et al., 2012]. This value is comparable to the mineral
toughness in the nacreous layer. This wide variation can be associated with viscoelastic nature
125

of the biopolymer, presence of nanoasperities, mineral bridges or a combination of all the
modes. Once the crack driving force ahead of the crack-tip reaches fracture resistance of the
biopolymer and/or the interface, the crack will initiate and start to grow in the unit-cell.
Resistance to crack advancement is an attribute associated with extrinsic toughening
that takes place in the wake of the crack, and thus is responsible for R - curve. On the contrary,
microstructural damage mechanisms in the cohesive zone and in bounded material are
responsible for crack initiation which can be attributed to intrinsic toughening. Intrinsic
toughening is dependent upon material properties and independent of the length of the crack
or geometry of the test specimen (except plane stress/plane strain). This brings into forefront
that crack driving force to initiate fracture is a result of dissipation mechanism associated with
intrinsic toughening. Under a given set of loading conditions, if overlap length is adopted such
that crack tip driving forces required for crack initiation can be kept to a minimum, higher will
be load that further needs to be applied to elevate the energy required attain the fracture
resistance of the barrier(s) to promote delamination in the unit-cell. This would in-turn raise the
toughness associated with the individual unit-cells which would contribute to global toughness
of the composite. Figure 4.4 shows the variation of crack driving force against overlap length
for varying pre-crack lengths in the nacreous layer.

126

Figure 4.4. Plot of crack driving force as a function of overlap length in nacre for varying precrack lengths.
As it can be seen from Figure 4.4, under the application of a given stress, crack driving
force is limited to a minimum value for an overlap length of approximately 1.6-2.5μm. In fact,
the crack driving force value corresponding to this overlap length scale range is less than or
comparable to the fracture resistance of the biopolymer. Thus it can now be clearly understood
as to why nacre adopts 1.69μm as its characteristic overlap length, not only to maximize its
load-transfer capability but at the same time provide resistance to crack initiation to the
maximum extent possible via minimization of crack driving force.

127

Figure 4.5. Variation of crack driving force against applied stress for overlap length on either
end of the characteristic value in the nacreous layer.
In order to better understand the influence of overlap length on the response
associated with the unit cells, Figure 4.5 highlights the effect of stress levels needed to
generate appreciable crack driving force for crack initiation corresponding to values of overlap
length on either side of the characteristic length in the nacreous layer. Clearly it can be seen
that if overlap lengths are smaller than characteristic length, significantly lower stress levels are
sufficient to reach crack driving force values to surpass biopolymer fracture resistance
(horizontal dotted blue line). Similarly, if the overlap lengths are larger than characteristic
length, it does not contribute to any additional property enhancement; it simply increases the
weight of the entire bonded unit. That explains the reason behind choice of characteristic
128

length scales by Nature in the nacreous layer. Additionally, at this stage we would like to
highlight the fact the stress levels that are needed to be actually applied would be less than
those reported above. This can be explained as follows. Energy input to initiate and drive the
crack is obtained from the energy that is applied to the system externally. Since linear elastic
deformation was a key assumption en route to model development, corresponding stress
values are thereby very large. In reality, the same energy is input to the system via elasticplastic deformation (shown in the inset) and thereby stress levels required for crack initiation
would be less than those predicted above.

Figure 4.6. Mode mixity parameter as a function of overlap length for varying pre-crack lengths.
129

As mentioned previously, the unit-cell structure in the nacreous layer of seashells is
representative of an adhesive joint and in order to better understand the crack initiation and
growth process, it is imperative to extract the crack driving force contributions associated with
both mode-I and mode-II, and thus we introduce the mode-mixity parameter. Clearly it can be
seen with reference to Figure 4.6 that the relative magnitude associated with mode-I and
mode-II are dependent upon the overlap length in a premeditated manner. For pure mode-I
failure, MMP=0° and for pure mode-II failure, MMP=90°.
With reference to Figure 4.6, substantial contributions from mode-II component are
likely to ensue crack initiation and subsequent growth via interfacial failure within the unit cell.
The inset in Figure 4.6 shows the strength of the nacreous layer under different loading
configurations [Menig et al., 2000]. Of all the loading configurations, it can be seen that the
nacreous layer is extremely weak under peeling. In order to provide maximum resistance
against peeling, choice of overlap length is kept at a characteristic value (≡ 1.6-2.5μm) in order
to promote maximum resistance against peeling and thereby ensuing initiation and propagation
via shear within the unit cells.
Existing literature in the experimental work on the nacreous layer associates toughness
with significant contribution of inelastic deformations [Barthelat and Espinosa, 2007] and
occurrence of 'process zone' [Barthelat and Rabiei, 2011]. Nonetheless, these events are
associated with sliding of ceramic bricks in a large region around the vicinity of cracks. The crack
trajectory is extremely meandering. Upon the application of external loading, numerous microcracks/voids are generated in all the unit-cells associated with the forward region in the
130

damage zone as shown in Figure 4.1. With progressive loading, these micro-voids coalesce
together thereby rendering the crack path through the damage zone extremely tortuous. The
question posed at this juncture how do these dispersed cracks in the unit-cells join together?

Figure 4.7. Variation of crack driving force upon overlap length for varying pre-crack length
associated with the primary crack. The figure in the inset identifies the angle (θ) at which crack
driving force of the kinked crack attains a maximum value for varying pre-crack length (0.091.29μm).
The current problem at hand represents mixed-mode crack propagation. Taking into
consideration Griffith's energy theory as the valid criteria which explains crack growth, kinking
of the crack will take place and the crack will grow in the direction (θ) along which the elastic
131

energy release per unit crack extension attains a maximum value. As it can be seen from Figure
4.7 and Figure 4.8, the initiation and propagation of the crack in the unit-cells is primarily shear
dominated throughout a majority of the overlap length. As the overlap length narrows down
appreciably, the crack kinks at an angle through the biopolymer at an angle of approximately
75°. This probably accounts for the reason as to why the crack does not run through completely
in a given row.

Figure 4.8. Variation of crack driving force upon overlap length for varying pre-crack length
associated with the primary crack. The figure in the inset identifies the angle (θ) at which crack
driving force of the kinked crack attains a maximum value for a pre-crack length of 1.49 μm.

132

By the time crack driving force attains a value which is comparable to the fracture
resistance associated with the mineral or tablet, it has almost reached the very edge of the
joint. The sharp kinking angle thereby is responsible for connecting associated cracks in unitcells of other columns, and thereby rendering the overall trajectory extremely tortuous as
shown in Figure 4.9.

Figure 4.9. (a) Schematic of staggered architecture of calcium carbonate bricks (shown in green)
in biopolymer matrix (shown in yellow); (b) expanded view of the unit cell structure (with plane
strain infinitesimal deformation) highlighting the elemental blocks and the original overlap
length (Lo); (c) crack trajectory through the overlap length and kinking of the crack from one
unit cell to another.
The current analytical model is only an approximate model for the nanocomposite
structure. It should be noted that crack propagation process is inelastic in nature (associated
133

with molecule bond breaking process); however in a simplified manner, growth of bondline
crack can be perceived as alteration of geometry associated with the overlap to account for
changes in the load path and thereby, this approach was followed to get a preliminary
understanding of the crack propagation process.
Under static loading conditions, characteristic lengths predicted were essentially based
on (a) stress transfer point of view, L 

2Ebh
, as was postulated by Chen et al., 2009 and,
G

(b) maximizing elastic strain energy density, L  2.318

Ebh
, as postulated by Wei et al.,
G

2012. Similarly under dynamic rates of loading, characteristic length was predicted based on
maximum shear

transfer efficiency point of view, Loptimal  Minimize    x,t  , as was

L

postulated by Dutta et al., 2013. In this article, we have shown using fracture mechanics
concepts (under quasi-static rates of loading) that characteristic length exists in the nacreous
layer in seashells which is based upon the existence of upper and lower bounds as given by
equation (4.57) below:

Lcharacteristic   CDF   Loptimal  Lcharacteristic   min 
min

(4.57)

Too far a deviation of optimal length on the right hand bound of equation (57) would
provide no additional structural property enhancement, but would only add to weight of the
adhesively bonded unit. Similarly, too far a deviation of optimal length on the left hand bound
of equation (4.57) might produce a further light weight design; however, it would at the same
time reduce the load withstanding capability rendering it more prone to fracture via increasing

134

crack driving force ahead of the crack tip and simultaneously raising the shear stress in adhesive
beyond its allowable limit/strength. Thus, the optimal length scale criterion, as given by
equation (4.57), serves as an important design guideline in the not only in synthesizing
adhesively-bonded structures but staggered architecture composites which would not only be
light-weight but tough as well. In order to validate our claim, in the subsequent paragraphs we
illustrate the unanimity that exists between our model predictions against previously published
experimental data followed by additional illustrative example.
4.3.1. Validation by experimental data
Lucic et al., 2006 investigated the existence of optimal overlap length in single lap
adhesive bonded joint comprising of Al99.5 as the adherend material and Loctite 3421 as the
(two-component) structural adhesive. The thickness and width of the adherend was 30 mm and
1.95 mm respectively. The thickness of the adhesive was 0.15 mm. The overlap length was
varied from 15 - 60 mm to explore optimal overlap length which corresponds to maximizing the
joint strength. Using the same geometrical and material parameters as were used in their
investigation, in our current analysis we tried to explore whether our model predictions, as per
our design criterion as given by equation (4.57), gave similar results against their experimental
observations. As it can be seen from Figure 4.10, optimal length scale predicted by the model
lies in the range of 30-35 mm.

135

(a)

(b)
Figure 4.10. Existence of optimal length scale, as predicted by the analytical model, for an
Al99.5-Loctite 3421 single lap adhesive bonded joint.
136

Optimal length observed experimentally by Lucic et al., 2006 lie in the range of 35-50
mm and thus, model predictions are in reasonable agreement with previously reported
experimental data and thus, justifies the efficacy of our proposed design guideline. Note that,
the unit for change in shear stress and CDF in Figure 4.10 and Figure 4.11 is MPa and J/m

2

respectively.
4.3.2. An illustrative example
Consider the adherend material in single-lap adhesive joint to be made of plain weave
S2-glass/SC-15 epoxy laminate (Deka et al., 2007: E1=24.85 GPa, E2=24.85 GPa, E3=11.04 GPa,
G12=2.67 GPa, G13=1.97 GPa, G23=1.97 GPa, ν12=0.11, ν13 =0.18, ν23=0.18) and adhesive to be
SC-15 epoxy resin (Applied Poleramic Inc.: Ep=2.00 GPa, Ga =1.40 GPa, νa=0.35). Fracture
toughness of cured SC-15 epoxy resin, S2-glass/SC-15 epoxy laminate for initiation and
2

2

2

propagation are 0.989 kJ/m , 0.688 kJ/m and 1.104 kJ/m respectively. Figure 4.11 shows the
variation of optimal overlap lengths for varying widths of adherend (10 mm, 40 mm and 100
mm) and adhesive thickness (0.50 mm, 1.00 mm, 2.00 mm, 3.00 mm and 5.00 mm) where the
hollow and solid circles are representative of characteristic length with respect to minimizati on
of crack driving force, as shown in Figure 4.11(a,c,e), and shear stress, as shown in Figure
4.11(b,d,f) respectively. With increasing adhesive thickness, crack driving force increases,
interfacial shear stress decreases and bounds of characteristic overlap length shift toward
higher numbers.

137

Figure 4.11. Existence of optimal overlap lengths for varying widths of composite adherend (10
mm, 40 mm and 100 mm) and adhesive thickness (0.50 mm, 1.00 mm, 2.00 mm, 3.00 mm and
138

Figure 4.11. (cont'd)
5.00 mm). The hollow and solid circles are representative of characteristic length with respect
to minimization of crack driving force (a,c,e) and shear stress (b,d,f) respectively.
Clearly, optimal overlap lengths exists in adhesively bonded structures which are
governed by characteristic overlap length as given by equation (57). Note that in Figure 4.11,
the hollow and solid circles are representative of characteristic length with respect to
minimization of crack driving force and shear stress respectively. Thus under quasi-static
loading conditions, choice of characteristic overlap length ensures optimal combinations both
from the perspective of weight reduction, maximum interfacial shear resistance, resistance to
crack initiation, and consequently superior toughness. In our current investigation, nacre has
been chosen as the structural biological composite for illustrative purposes. Feature drawn
from current analysis can be adapted as an important guideline for designing adhesive bonded
structures and staggered architecture materials at any level of scale. The outcome of the
current research will thereby provide beneficial guidelines in designing hybrid bio-inspired
protective composite materials [Dutta et al., 2012].
4.4. Conclusions
In our current investigation, we showed the influence of structural overlap length on
minimizing delamination in the unit-cells to the maximum extent possible via development of
an analytical model. The main outcome of the developed analytical model was to provide a
simplified expression for crack driving force in terms of local finite stress concentrations in the
biopolymer layer of nacre. Based on the above analysis, we have attempted to address that the

139

'well-made' structure of the nacreous layer is built with respect to two bounds: a lower-bound
based on an energy criteria (to allay crack-initiation) and an upper-bound based on stress
criterion (to improve load-transfer/shear-transfer efficiency). Excerpts drawn from the current
study would provide beneficial guidelines in designing light-weight and tough adhesively
bonded structures and staggered architecture composites.

140

REFERENCES

141

REFERENCES

Barthelat, F., Li, C.-M., Comi, C., Espinosa, H.D. 2006. Mechanical properties of nacre
constituents and their impact on mechanical performance. J. Mater. Res. 21(8), 1977-1986.
Barthelat, F., Espinosa, H.D., 2007. An Experimental Investigation of Deformation and
Fracture of Nacre-Mother of Pearl. Exp. Mech. 47, 311–324.
Barthelat, F., Tang, H., Zavattieri, P., Li, C.M., Espinosa, H., 2007. On the Mechanics of
Mother-of-Pearl: A Key Feature in the Material Hierarchical Structure. J. Mech. Phys. Solids 55,
306-337.
Barthelat. F., Rabiei, R., 2011. Toughness amplification in natural composites. J. Mech.
Phys. Solids 59, 829-840.
Chen, B., Wu, P.D., Gao, H., 2009. A characteristic length for stress transfer in the
nanostructure of biological composites. Compos. Sci. Technol. 69, 1160-1164.
de Gennes, P.G., Okumura, K., 2000. On the toughness of biocomposites. C. R. Acad. Sci.
Paris, t.1, Ser. IV, 257-261.
Deka, L.J., Bartus, S.D., Vaidya, U.K., 2007. Numerical modeling of simultaneous and
sequential multi-site impact response of S-2 glass/epoxy composite laminates. In: Proceedings
of Composites and Polycon. American Composites Manufacturers Association, Tampa, Florida
USA, October 17-19.
Dutta, A., Vanderklok, A., Tekalur, S.A., 2012. High strain rate mechanical behavior of
seashell-mimetic composites: Analytical model formulation & validation, Mech. Mater. 55, 102111.
Dutta, A., Tekalur, S.A., Miklavcic, M., 2013. Optimal length scales in staggered
architecture composites under dynamic loading conditions, J. Mech. Phys. Solids 61(1), 145-160.
Evans, A., Suo, Z., Wang, R., Aksay, I., He, M., Hutchinson, J., 2001. Model for the
Robust Mechanical Behavior of Nacre. J. Mater. Res. 16, 2475-2484.
Espinosa, H.D., Juster, A.L., Latourte, F.J., Loh, O.Y., Gregoire, D., Zavattieri, P.D., 2011.
Tablet-Level Origin of Toughening in Abalone Shells and Translation to Synthetic Composite
Materials. Nature Comm. 2, 173.

142

Gao, H., Ji, B., Jäger, I., Arzt, E., Fratzl, P., 2003. Materials Become Insensitive to Flaws
at Nanoscale: Lessons from Nature. Proc. Natl. Acad. Sci. U.S.A. 100, 5597-5600.
Huang, Z., Li, H., Pan, Z., Wei, Q., Chao, Y.J., Li, X., 2011. Uncovering high-strain rate
protection mechanism in nacre. Sci. Rep. 1, 148; DOI:10.1038/srep00148.
Jackson, A.P., Vincent, J.F.V., Turner, R.M., 1988. The mechanical design of nacre. P. R.
Soc. London B. 234, 415–440.
Jager, I., Fratzl, P., 2000. Mineralized collagen Fibrils: a mechanical model with a
staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746.
Kessler, H., Ballarini, R., Mullen, R.L., Kuhn, L.T., Heuer, A.H., 1996. A biomimetic
example of brittle toughening: (I) steady state multiple cracking. Comp Mater Sci. 5, 157–166.
Kamat, S., Su, X., Ballarini, R., Heuer, A.H., 2000. Structural basis for the fracture
toughness of the shell of the conch Strombus gigas. Nature, 405, 1036–1040.
Kotha, S.P., Kotha, S., Guzelsu, N., 2000. A shear-lag model to account for interaction
effects between inclusions in composites reinforced with rectangular platelets. Compos. Sci.
Technol. 60, 2147–2158.
Katti, K.S., Katti, D.R., Pradhan, S.M., Bhosle, A., 2005. Platelet interlocks are the key to
toughness and strength in nacre. J. Mater. Res. 20, 1097-1110.
Luo, Q., Tong, L., 2004. Linear and higher order displacement theories for adhesively
bonded lap joints. Int. J. Solids Struct. 41(22-23), 6351-6381.
Launey, M.E., Ritchie, R.O. 2009. On the fracture toughness of advanced materials. Adv.
Mater. 21, 2103-2110.
Liu, G., Ji, B.H., Hwang, K.C., Khoo, B.C., 2011. Analytical solutions of the displacement
and stress fields of the nanocomposite structure of biological materials. Compos. Sci. Technol.
71, 1190-1195.
Menig, R., Meyers, M.H., Meyers, M.A., Vecchio, K.S., 2000. Quasi-static and dynamic
mechanical response of Haliotis rufescens (abalone) shells. Acta Mater. 48, 2383–2398.
Meyers, M., Lin, A., Chen, P., Muyco, J., 2008. Mechanical Strength of Abalone Nacre:
Role of the Soft Organic Layer. J. Mech. Behav. Biomed. Mater. 1, 76-85.
Okumura, K., de Gennes, P.-G., 2001. Why is nacre strong? : Elastic theory and fracture
mechanics for biocomposites with stratified structures. Eur. Phys. J. E 4, 121-127.
143

Sarikaya, M., Gunnison, K.E., Yasrebi, M., Aksay, J.A., 1990. Mechanical
property-microstructural relationships in abalone shell. Mater. Res. Soc. 174, 109-116.
Song, F., Soh, A., Bai, Y., 2003. Structural and mechanical properties of the organic
matrix layers of nacre. Biomaterials 24, 3623-3631.
Shuchun, Z., Yueguang, W., 2007. Effective elastic modulus of bone-like hierarchical
materials. Acta Mech. Solida Sin. 20, 198-205.
Wang, R., Suo, Z., Evans, A., Yao, N., Aksay, I., 2001. Deformation mechanisms in nacre.
J. Mater. Res.16, 2485-2493.
Wei, X., Naraghi, M., Espinosa, H.D., 2012. Optimal length scales emerging from shear
load transfer in natural materials: application to carbon-based nanocomposite design. ACS
Nano 6(3), 2333-2344.
Xu, Z.-H., Yang, Y., Huang, Z., Li, X. 2011. Elastic modulus of biopolymer matrix in nacre
measured using coupled atomic force microscopy bending and inverse finite element
techniques. Mat. Sci. Eng. C 31(8), 1852-1856.
Yang, C., Chadegani, A., Tomblin, J.S., 2008. Strain energy release rate determination of
prescribed cracks in adhesively-bonded single-lap composite joints with thick bondlines.
Composites Part B: Engineering 39(5), 863-873.
Zhang, Z.Q., Liu, B., Huang, Y., Hwang, K.C., Gao, H., 2010. Mechanical properties of
unidirectional nanocomposites with non-uniformly/randomly staggered platelet distribution. J.
Mech. Phys. Solids 58, 1646-1660.
Zhang, Z., Zhang, Y.-W, Gao, H., 2011. On optimal hierarchy of load-bearing biological
materials. Proc. R. Soc. B, 278, 519-525.

144

CHAPTER 5: CONCLUSIONS

145

5.1. Design Guideline
In the current study, an attempt was made to understand the mechano-biological
principles existing in structural biological composites in Nature exhibiting staggered
architecture. As mentioned previously, Nature optimizes stiffness, strength and toughness by
fine tuning the ceramic content and aspect ratio of ceramic platelets.

Figure 5. Design guideline for synthetic composite showing the interplay of optimization
scheme and manufacturing technique.
If the same strategy were to be applied to synthetic composite design, choice of
participating materials and manufacturing technique need to go hand-in-hand so that an

146

optimized performance can be attainable in synthetic composites. Figure 5 shows the design
guideline that can be adopted from the current study with the help of an associated example.
Assuming that epoxy matrix, reinforcement material, starting particle size and volume fraction
are known (fixed) beforehand, based on structural property enhancement desired, different
optimized aspect ratios exist based on either stiffness, load-transfer and toughness via
maximization of Young's modulus, minimization of maximum shear stress and minimization of
crack driving force respectively. Based on a judicial choice of aspect ratio, the manufacturing
technique (in the current case, freeze casting) technique can be adjusted accordingly to vary
the cooling rate and/or freezing time to regulate the aspect ratio in the final composites and
thereby, obtain optimized performance.
5.2. Future Scope of Work
Optimization scheme with respect to toughness carried in the current study addressed it
from a quasi-static point of view. As a extension of the current work, it would be opportune to
unravel the corresponding parameter and associated criterion for toughness optimization
under dynamic rates of loading. In the current study, the ceramic material was same
throughout and had the same thickness. As a future work, it would be interesting to investigate
material response and establish a design criterion for gradient architecture (dissimilar materials
and/or dissimilar thickness) materials. The manufacturing technique needs to be tuned
accordingly to custom-design manufacture functional gradient architecture materials.

147