SURFACE FORM MEMORY BY
INDENTATION-PLANARIZATION TRAINING OF NITI SHAPE
MEMORY ALLOYS

By

Xueling Fei

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Chemical Engineering and Materials Science

2011

iv

ABSTRACT
SURFACE FORM MEMORY BY INDENTATION-PLANARIZATION
TRAINING OF NITI SHAPE MEMORY ALLOYS
By
Xueling Fei

Shape memory alloys respond to deformation by indentation with strong
two-way cyclic displacements on heating and cooling. Shallow indents can vanish on
heating, and deeper indents will change depth when thermally cycled. In the present
work, following indentation by an added step of surface planarization can furthermore
produce what has been termed “Surface Form Memory” or SFM. The term SFM
describes an effect that causes one surface form to reversibly transform to another,
under thermal excitation. The term surface form is meant to distinguish between say,
flat and “bumpy” surfaces, on a scale much larger than the intrinsic surface roughness.
Surface protrusions, or „exdents‟ may reach an amplitude near 20% of the initial
indent depth, and may be created on nano to macroscopic size scales. The surface
form memory effect may have special applications into novel optical,
microelectromechanical, and tribological systems.

Surface deformation of a NiTi shape memory alloy (SMA) was
accomplished using spherical, cylindrical, pyramidal, and flat punch indenters. Single
indents and indent arrays have been made. The amplitude of SFM after planarization

v

was found to be related to the size of the subsurface deformation zone and can be
maximized by thermal cycling, multiple indentations, by changing indent depth and
spacing, and by varying indentation temperature parameters. The spatial extent of the
subsurface deformation zone was studied by both experimental methods and finite
element modeling. Methods such as successive front and back thinning probed the
size of the deformation zone directly, while finite element modeling provided
stress-strain information supporting the conclusion that the deformation zone
responsible for the two-way effect was a slip zone that had plastic strains larger than
7%.

NiTi SFM, when studied in constrained recovery experiments, showed an
energy density up to 10 MJ/m3, which is about the same as that of a conventional NiTi
actuator. This proves SFM‟s potential to be used in high energy density actuation
devices.

Finally, Laser Shock Peening (LSP) was demonstrated to make indents in
bulk NiTi alloys efficiently. These dynamically made indents have larger two-way
response ratio than the quasi-static indents made by conventional indenters.
Combining the high energy density and high efficiency of the LSP process, LSP SFM
may see special applications into thin film actuators and MEMS devices.

vi

This dissertation is dedicated
to my dear parents and beloved friends

iv

ACKNOWLEDGEMENTS

I would like to thank my dear advisor and friend, Professor David S.
Grummon, for his guidance in teaching, research, and life. Professor Grummon has
taught me how to conduct research, trained me to be a professional engineer, given
me opportunities to develop my communication and leadership skills through team
management and conference organization.
I would also like to express my gratitude towards Professor Yang-Tse Cheng
at University of Kentucky for his guidance in research and career development. His
encouragement motivated me to pursue further in research and to develop myself as a
competent researcher.
I am also grateful towards my PhD committee members, Professor Martin A.
Crimp, Professor Thomas R. Bieler, from the Department of Chemical Engineering
and Materials Science, and Professor Tom Pence, from the Department of Mechanical
Engineering, for giving valuable comments and suggestions during my PhD study. I
will also thank my collaborators, Chang Ye, Gary J. Cheng at Purdue University for
diversifying my research experience. Last but not least, I will thank my colleagues
Ke-bin Low, John W. Foltz, my undergraduate assistants Corey O‟Connell, Matt Weir,
Neil Murphy, Scott Sutton, Doug Schriner for all the help and laughter in the
laboratory. Without them, a lot of work will not be accomplished.

v

TABLE OF CONTENTS

LIST OF TABLES …………………………………………………………………...ix
LIST OF FIGURES …………………………………………………………………..x
NOMENCLATURES…………………………………………………………….. xviii
CHAPTER 1
BACKGROUND AND LITERATURE REVIEW ..................................................... 1
1.1 Introduction ....................................................................................................... 1
1.2 NiTi Shape Memory Alloys ................................................................................ 4
1.2.1 Historic Development ........................................................................... 4
1.2.2 The Shape Memory Effect ….……………………………………… .... 4
1.2.3 Thermodynamics of Shape Memory Alloys ......................................... 11
1.3 Two-way Shape Memory Effects ..................................................................... 14
1.3.1 Introduction ........................................................................................... 14
1.3.2 Review of TWSME Training Methods ................................................. 15
1.3.3 The Mechanisms of TWSME ............................................................... 25
1.3.4 Indentation in Shape Memory Alloys ................................................... 34
1.3.5 The Indentation Induced TWSME ...................................................... 38
1.4 Contact Mechanics ……………………….……………………………….... .. 40
1.4.1 Elastic Contact ………………………………………………………. 41
1.4.2 Elastic-Plastic Contact ........................................................................ 44
1.4.3 Indentation Related Contact Mechanics ............................................. 47
1.5 Finite Element Modeling the Subsurface Deformation Zone ......................... 51
1.5.1 Introduction ......................................................................................... 51
1.5.2 The 3D Superelastic Model ................................................................ 52
1.6 Summary ......................................................................................................... 55
CHAPTER 2
EXPERIMENTAL METHODOLOGY ................................................................... 57
2.1 Introduction ..................................................................................................... 57
2.2 Materials ......................................................................................................... 58
2.2.1 NiTi Shape Memory Alloys ................................................................ 58
2.2.1.1 Source and Thermal Response ...................................................... 58
2.2.1.2 Stress-strain Response . ................................................................. 59
2.2.1.3 Indentation Response . ………………………………………….. 61
2.2.2 Stainless Steel, Aluminum, and Indenter Materials .. ........................... 62
2.2.3 Laser Shock Peening Materials ........................................................... 63
2.3 Indentation Methods ....................................................................................... 63

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2.4
2.5

2.6

2.7

2.8

2.9

2.3.1 Indenter Materials and Properties . ....................................................... 63
2.3.2 Indentation Loading . ............................................................................ 64
2.3.3 Post Indentation Thermal Cycling . ...................................................... 64
2.3.4 Laser Shock Peening (LSP). ................................................................. 65
Indent and Displacement Characterization ..................................................... 66
Obtaining SFM by Indentation/Planarization ................................................. 67
2.5.1 Thermo-mechanical Preparation . ……………………………………. 67
2.5.2 Grinding Techniques ........................................................................... 68
2.5.2.1 Cold Planarization ....................................................................... 68
2.5.2.2 Post-recovery A-planarization ..................................................... 70
Maximizing Indentation-induced TWSME .................................................... 71
2.6.1 Effect of a/r Ratio ............................................................................... 71
2.6.2 Effect of Thermal Cycling .................................................................. 71
2.6.3 Effect of Indentation Spacing ............................................................. 72
2.6.4 Effect of Multiple Indentations ........................................................... 73
2.6.5 Effect of Indentation Temperature ...................................................... 74
2.6.6 Effect of Prior Cold Rolling ................................................................ 74
Determining the Size of the Active Zone ....................................................... 75
2.7.1 Front-Thinning Experiments ............................................................... 76
2.7.2 Back-Thinning Experiments ............................................................... 76
2.7.3 Direct Observation of the Indent Deformation Zone .......................... 77
Indent Replication and Energy Estimates ....................................................... 78
2.8.1 Introduction ......................................................................................... 78
2.8.2 One Shot Replica Indentation ............................................................. 78
2.8.3 Reversible Replica Indentation by SFM ............................................. 80
Finite Element Modeling ................................................................................ 83
2.9.1 The 3D Constitutive Model ................................................................ 83
2.9.2 Finite Element Program Implementation ............................................ 84

CHAPTER 3
RESULTS and DISCUSSION ................................................................................. 86
3.1 Introduction ..................................................................................................... 87
3.2 Developing and Maximizing the Indentation-Induced TWSME .................... 87
3.2.1 Indent Recovery by the Indentation-Induced TWSME ...................... 87
3.2.2 Maximizing Indent Recovery ............................................................. 88
3.2.2.1 Effect of a/r Ratio ........................................................................ 88
3.2.2.2 Effect of Thermal Cycling ........................................................... 94
3.2.2.3 Effect of Indentation Spacing ...................................................... 98
3.2.2.4 Effect of Multiple Indentations ................................................. 100
3.2.2.5 Effect of Indentation Temperature ............................................ 102
3.2.2.6 Effect of Prior Cold Rolling ...................................................... 106
3.2.2.7 Maximum Two-way Response .................................................. 108
3.3 Planarization for Surface Form Memory ...................................................... 111
3.3.1 Introduction to the Indentation-Planarization Method ...................... 111

vii

3.4

3.5

3.6

3.7

3.3.2 Post-recovery M-planarization .......................................................... 111
3.3.3 Pre-recovery M-planarization ........................................................... 117
3.3.4 Post-recovery A-planarization .......................................................... 118
Characterization of the Active Deformation Zone ........................................ 122
3.4.1 Front Thinning Experiments ............................................................. 122
3.4.2 Back Thinning Experiments ............................................................. 128
3.4.3 Two-way Strain in the Active Subsurface Zone ............................... 132
3.4.4 Direct Observation of the Indentation Deformation Zone ................ 141
Indent Replication ......................................................................................... 145
3.5.1 Introduction ....................................................................................... 145
3.5.2 One Shot Spherical Replica Indentation ........................................... 146
3.5.3 Reversible Spherical Replica Indentation ......................................... 149
3.5.3.1 Spherical Exdent and Replica Indent ........................................ 149
3.5.3.2 Effect of Preload Stress ............................................................. 152
3.5.3.3 Low Cycle Fatigue Test ............................................................ 155
3.5.3.4 Factors Affecting the Constrained Recovery ............................ 157
3.5.3.5 Work Output and Energy Density Calculation .......................... 160
SFM from Dynamic Indentation (LSP) ........................................................ 163
3.6.1 Indent Recovery and SFM ................................................................ 163
3.6.2 Mechanism of SFM in Dynamic Indents .......................................... 172
Modeling Indent Recovery and SFM ............................................................ 180
3.7.1 Finite Element Modeling of the Deformation Zone ......................... 180
3.7.1.1 Spherical Indentation and the Deformation Zone ....................... 180
3.7.1.2 Cylindrical Indentation and the Deformation Zone .................... 183

CHAPTER 4
CONCLUSIONS AND FUTURE WORK ............................................................ 187
4.1 Conclusions ................................................................................................... 187
4.2 Future Development ...................................................................................... 189
APPENDIX ............................................................................................................ 191
BIBLIOGRAPHY .................................................................................................. 193

viii

LIST OF TABLES

Table 2.1

Transformation Temperatures measured by DSC and SQUID ........ 58

Table 2.2

Parameters of the Initial Spherical Indents in Martensitic NiTi ....... 79

Table 3.1

Curve-fitting model for two-way indent depth change for spherical and
cylindrical indents ............................................................................ 94

Table 3.2

Thermal cycling parameters and indent depth changes for spherical
indents with a/r ratios of 0.4 and 0.65 ............................................. 95

Table 3.3

The parameters and their effects on maximizing the two-way responses
of the spherical indents .................................................................. 109

Table 3.4

The indent responses of spherical and cylindrical indents at different a/r
ratios ............................................................................................... 127

Table 3.5

Summarize the a/r ratios and D* measured in the front and back
thinning experiments ...................................................................... 133

Table 3.6

Replica Indent and Residual Exdent Dimensions ........................... 148

Table 3.7

Path dependency effect on the replica indent depth at the preload of
250 MPa ......................................................................................... 160

Table 3.8

Calibration Indentation Data for 440 SS, 304 SS, and Al 6061-T6 at
433 K .............................................................................................. 160

Table 3.9

Calculation of Energy Density for Replica Indents made under 250
MPa ................................................................................................ 163

Table 3.10

Comparison of Quasistatic and LSP Indent with same contact
radius ................................................................................................ 171

Table 3.11

Comparison of Quasistatic and LSP Indent Parameters with same
two-way indent depth change ........................................................ 172

Table 3.12

Data and regression coefficients used in estimating the dynamic yield
stress for slip deformation in the martensite for the present work... 177

ix

LIST OF FIGURES

Figure 1.1

Recovery of spherical indents made in martensitic NiTi upon heating..2

Figure 1.2

Flat-to-bump transition or Surface Form Memory (SFM) of the GM
logo made by the indentation-planarization method in a NiTi SMA .. 2

Figure 1.3

Schematic illustration of austenite and martensite phase with invariant
plane as a habit plane ......................................................................... 6

Figure 1.4

The lattice change from B2 parent (a) to B19‟ martensite (b). i, j, and k
refer to parent lattice, and i‟, j‟, and k‟ refer to martensite phase. a, b, c
refer to lattice parameters,  refers to the angle between [-1 0 0] and
[0 -1 -1]. [Otsuka and Ren, 2005] ...................................................... 7

Figure 1.5

Seterographic projection of 24 habit plane variants for positive shear
based on WLR theory for <011>m Type II twinning [Otsuka et al.
2005] .................................................................................................. 10

Figure 1.6

TEM observation and the analysis (right-hand side) of a) typical
triangular morphology of B19‟ martensite, b) selected area diffraction
pattern [Madangopal et al. 1997] ..................................................... 10

Figure 1.7

(a) Microstructure of un-deformed martensite twins; (b) microstructure
of martensite twins after 4% deformation strain in tension; (c)
microstructure of deformation bands inside martensite twins after 4%
deformation strain in compression [Liu et al. 1998] ........................ 11

Figure 1.8

Stress-strain curve of a NiTi specimen under uniaxial tension and
compression [Liu et al. 1998] .......................................................... 11

Figure 1.9

Schematic drawing of superelasticity in temperature-stress space.... 14

Figure 1.10

Schematic drawing of the thermomechanical training of NiTi two-way
shape memory spring [Yu et al. 2006]. The specimen was annealed at
500 oC................................................................................................. 17

Figure 1.11

TWSME strain tw as a function of the number of thermal cycles and
training strain t [Lahoz et al. 2002] ................................................ 17

x

Figure 1.12

Plot of the two-way displacement ratio (or two-way strain) as a
function of the number of training cycles.  T is the total elongation
due to training [Perkins et al. 1984] ................................................. 18

Figure 1.13

Demonstration of “all-round shape memory effect” in Ti–51Ni alloy
from (a) to (e) when cooled down to 77 K [Nashida et al. 1984] ...... 19

Figure 1.14

Mechanism of “all-round shape memory effect” [Kainuma et al. 1987]
............................................................................................................ 20

Figure 1.15

Effect of the number of training cycles on the two-way shape memory
strain of the Ti36Ni49Hf15 alloy with different training strains [Meng et
al. 2004] ........................................................................................... 22

Figure 1.16

Effect of compressive prestrain on the development of two-way shape
memory effect in NiTi shape memory alloys [Kim et al. 2009] ........ 22

Figure 1.17

The TWSME obtained after a single deformation of 6.2% strain in an
equiatomic NiTi alloy [Liu et al. 1999] ........................................... 24

Figure 1.18

Effect of prestraining in tension on the development of two-way shape
memory effect in an equiatomic NiTi alloy [Liu et al. 1999] .......... 24

Figure 1.19

Dark Streaks in the austenite phase after several training cycles. (a) to
(d) 3 cycles, (e) to (f) 5 cycles [Perkins et al. 1984] .......................... 27

Figure 1.20

MD simulations show martensite nucteation and growth, (a)-(c) for a
pair dislocation, (e)-(g) for two pair dislocations. (d) and (h) show the
stress fields of one pair and two pair dislocations [Zhang et al. 2003]28

Figure 1.21

Transmission electron micrograph showing a “Comb-shaped”
arrangement of dislocations observed in the thermomechanical cycled
Cu-Zn-Al alloy. (a) zone axis G=[-1-11]. (b) Same zone as in (a) but
under a B=[001]1 direction of observation. Here the projection of
both dislocations in a pair are coincident: the plane containing a pair is
(1-10) which corresponds to their glide plane [Rios-Jara et al. 1987]
............................................................................................................ 29

Figure 1.22

a) and b) Shear stress-strain curves obtained at T=285 K during
isothermal shearing test in the first training and second training
[Manach et al. 1993] .......................................................................... 31

Figure 1.23

Shear stress-strain curve obtained during a super-elastic cycle with
strain of ±5% at T=313 K after first shearing cycling. The dash line in

xi

the hysteresis indicates the estimated reversible stress contribution to
the hysteresis.  irr is the irreversible stress contribution induced by
training [Manach et al. 1993] ............................................................. 33
Figure 1.24

Shear stress-strain curve obtained during a superelastic simple shear
cycling with strain of ±5% at T=313 K. The dash line in the hysteresis
indicates the estimated reversible stress contribution to the hysteresis
[Manach et al. 1993] .......................................................................... 33

Figure 1.25

Plots of the recovery ratio, true stress against the representative strain
(0.2 a/r). Complete recovery occurs when the representative strain is
less than 5%, beyond which slip plasticity occurs [Ni et al. 2002] ... 36

Figure 1.26

a) Indent depth recovery upon heating for pseudoelastic, shape
memory alloys and [111] Ni with max load up to 16000 N; b) Indent
depth recovery upon heating for pseudoelastic and shape memory
alloys with max load up to 2000 N [Frick et al. 2006] .................... 37

Figure 1.27

Two-way indent depth change of a deep spherical indent upon heating
and cooling [Zhang et al. 2006] ......................................................... 38

Figure 1.28

Three dimensional profiles of reversible surface exdents: (a) a 3x3
array of spherical exdents on the surface of the austenite phase of NiTi
at high temperature which disappears when the sample is cooled to the
martensite phase. (b) A protruding line on the surface of the austenite
phase that nearly disappears in the martensite phase. (c) A nanoscale
reversible protrusion on the surface a NiTi film. [Zhang et al, 2006]
......................................................................................................... ...39

Figure 1.29

Representative strain vs. mean stress in spherical indentation, and its
correlation with equivalent true stress-true strain behavior in uniaxial
tension (Solid line). Metal A is Steel and metal B is Copper. [Tabor
1948, 1996] ........................................................................................ 46

Figure 1.30

Finite element modeling of indentation of an elastic-plastic half space
by a rigid sphere: development of the plastic zone. Broken line:
contour of J2. PY is the normal load needed to first initiate plastic flow
[Hardy et al. 1971] ............................................................................. 46

Figure 1.31

Flow patterns exhibited in a material that piles up [Mata et al. 2002]
.......................................................................................................... ..47

Figure 1.32

Comparison of the plastic zone size between conical and spherical
indentations at different values of a/r for (a) highly elasto-plastic solid

xii

(E = 200 GPa, s = 1000 MPa, and n = 0.4), (b) a solid with reduced
elasto-plastic character (E =200 GPa, s = 400 MPa, and n = 0.2), and
(c) a fully plastic solid (E = 200 GPa, s = 50 MPa, and n = 0.1) [Mata
et al. 2006] ......................................................................................... 48
Figure 2.1

a) DSC, b) SQUID measurements of the transformation temperatures
of the equi-atomic NiTi shape memory alloys ................................... 59

Figure 2.2

Compressive stress strain behavior for the martensitic NiTi used in the
study. 1 is the stress for slip deformation, 2 is the stress for
martensite detwinning ........................................................................ 60

Figure 2.3

Nanoindentation in NiTi by MTS XP Nanoindenter with an
indentation depth of 1.5 m ............................................................. 61

Figure 2.4

a) Load-Displacement curves of 304 Stainless Steel (Upper), b)
6061-T6 (Lower) used in the constrained recovery experiment ........ 62

Figure 2.5

Schematic illustration of the laser shock peening process on NiTi
shape memory alloys.......................................................................... 65

Figure 2.6

Optical image of the laser shock peened NiTi discs. The left has indent
made with beam size of 0.5 mm and the right with beam size of 1 mm
............................................................................................................ 66

Figure 2.7

Comparison of post-recovery M-planarization after first recovery and
pre-recovery M-planarization without first recovery ......................... 69

Figure 2.8

Schematic illustration of the setup of the constrained recovery
experiment. 304 stainless steel (SS) is used as an example of the
replica material................................................................................... 82

Figure 3.1

Schematic illustration of initial indent depth recovery of the spherical
indent made with a 0.79 mm diameter tungsten carbide ball under
700 N in the martensitic state............................................................. 88

Figure 3.2

Plot of the indent depth change (normalized to the indenter radius) as a
function of the a/r ratio for spherical indents .................................... 90

Figure 3.3

Plot of the two-way response, normalized to ho as a function of a/r
ratio based on the experimental results and calculations based on Eq.
(3.2) .................................................................................................... 91

Figure 3.4

Two-way indent depth changes for cylindrical indents, normalized to
the indenter radius, plotted as a function of the c/r ratio ................... 93

xiii

Figure 3.5

Change of the indent depth upon thermal cycling for spherical indents
with a/r ratios of 0.4 and 0.65 ............................................................ 96

Figure 3.6

Change of the indent depth during heating and cooling for spherical
indents with a/r ratios of 0.4 and 0.65 ............................................... 96

Figure 3.7

SQUID measurements of the as-indented specimen before and after
thermal cycling and the as-received NiTi specimen .......................... 98

Figure 3.8

Plot of the indent depth change against the normalized indentation
spacing d/a for 500 N spherical indents ............................................. 99

Figure 3.9

Change of indent depth as a function of repeated indentation times for
500 N spherical indents .................................................................... 101

Figure 3.10

Plot of the initial indent depth as a function of repeated indentation
times for 500 N spherical indents .................................................... 102

Figure 3.11

Plot of initial indent depth as a function of indentation temperature for
500 N spherical indents .................................................................... 104

Figure 3.12

Plot of indent depth change as a function of indentation temperature
for 500 N spherical indents .............................................................. 105

Figure 3.13

Plot of the indent depth change ratio (normalized to initial indent
depth) as a function of indentation temperature for 500 N spherical
indents .............................................................................................. 105

Figure 3.14

Plot of the initial indent depth as a function of the percentage of cold
rolling for 500 N spherical indents .................................................. 107

Figure 3.15

Plot of indent depth change as a function of the percentage of cold
rolling for 500 N spherical indents .................................................. 108

Figure 3.16

Comparison of the two-way response due to thermal cycling for
spherical indents made by double indentation at 333 K and single
indentation at 293 K with similar a/r ratios ..................................... 111

Figure 3.17

Indent training (a-c) and planarization (d) to obtain Surface Form
Memory (SFM) „exdents‟ in NiTi. Thermally cycling between (b) 
(c) and (d)  (e) is indefinitely repeatable. Indentation results in three
deformation zones under the indent: a: Plastic Slip Zone, b: Detwinned
Martensite Zone, c: Elastic Deformation Zone (Semi infinite volume)
.......................................................................................................... 113

xiv

Figure 3.18

Cyclic exdent-to-flat surface transition after planarization of a
spherical indent ................................................................................ 115

Figure 3.19

Profiles of various exdent shapes measured after
indentation-planarization in a series of hot-cold-hot: a) cylindrical
bumpy-to-flat transition; b) punch bumpy-to-flat transition; c) mesh
bumpy-to-flat transition; d) pyramidal bumpy-to-flat transition; e)
spherical exdent is compared to pyramidal exdent. (The vertical scale
is adjusted for better visualization) .................................................. 115

Figure 3.20

Surface profiles of the initial 500N spherical indent, exdent upon
heating and cooling after pre-recovery M-planarization ................. 119

Figure 3.21

Six possible variations of Surface Form Memory in NiTi SMAs ... 120

Figure 3.22

Surface Profiles of 200 N spherical indent, indent upon cooling after
post-recovery A-planarization, and subsequent heating .................. 124

Figure 3.23

Subsequent thermal cycling of 200 N spherical indentation after
post-recovery A-planarization to ensure the reversibility of surface
feature transition .............................................................................. 125

Figure 3.24

Schematic illustration of the front thinning experiment .................. 125

Figure 3.25

Results of the front-thinning experiments. The stable cyclic exdent
height is plotted as a function of surface removal depth for spherical
indents with a/r ratios between 0.4 and 0.95 ................................... 126

Figure 3.26

Results of successive planarization experiments. The stable cyclic
exdent height is plotted as a function of surface removal depth for
cylindrical indents with a/r ratios between 0.4 and 0.95 ................. 127

Figure 3.27

Schematic illustration of the back thinning experiment .................. 129

Figure 3.28

Plot of indent depth change as a function of the remained material,
normalized over the contact radius a ............................................... 130

Figure 3.29

Plot of the indent in martensite and austenite, normalized over the
contact radius a, as a function of the remained material, normalized
over the contact radius a .................................................................. 131

Figure 3.30

Plot of D* based on front and back thinning experiments with
spherical indentation deformation zone modeled by ABAQUS ...... 131

xv

Figure 3.31

Exdent heights for both spherical and cylindrical indents with various
a/r ratios between 0.4 and 0.95 plotted as a function of the size of the
subsurface active zone, D*. The slope of the linear curve fit gives the
average strain in the volume defined by D* .................................... 134

Figure 3.32

Stable cyclic exdent height for spherical indents as a function of the
depth of the subsurface active zone, D*, in the front and back thinning
experiments ...................................................................................... 134

Figure 3.33

Data for spherical indents for all a/r ratios studied normalized to the
experimentally determined depth of the subsurface active zone, D*.
The data may be adequately fitted to eq. (3.7) which has a single
adjustable parameter ........................................................................ 135

Figure 3.34

2D illustration of residual spherical cap after a depth of dpl of material
being removed .................................................................................. 136

Figure 3.35

Data for cylindrical indents for all a/r ratios studied normalized to the
experimentally determined depth of the subsurface active zone, D*
........................................................................................................ ..140

Figure 3.36

Schematic illustration of residual area in the cylindrical cap after a
depth of dpl of material away. Afan is the area of the fan with angle ,
Atri is the area of a triangle inside the fan. Ares is double the rest area of
the fan. Atot is the area of the hemisphere. dpl is the depth of material
being removed and D* is the size of the deformation zone ............. 140

Figure 3.37

Schematic drawing of the cross-sectional view of the spherical indent
after cutting ...................................................................................... 142

Figure 3.38

The cross-sectional view of a spherical indent at room temperature
(293 K) after polishing, after heating to 433 K, and after cooling to
243 K and measured at 293 K optically ........................................... 143

Figure 3.39

Schematic drawing of the cylindrical indent after cutting............... 144

Figure 3.40

The cross-sectional view of a cylindrical indent at room temperature
(293 K) after polishing, after heating to 433 K, and after cooling to
243 K and measured at 293 K optically ........................................... 145

Figure 3.41

Residual exdents in NiTi (left) and Replica indents in aluminum (right)
at 293 K made directly following the replication experiments. Images
(a,b), (c,d) and (e,f) correspond to initial indent a/r ratios of 0.42, 0.68
and 1.00 respectively. In each image the field size is 1.9 mm x 2.5 mm.
Vertical scales vary, as shown. Images on the left have been inverted to

xvi

correspond to the relative orientations of the features during their
formation .......................................................................................... 147
Figure 3.42

Depths of replica indents plotted as a function of the post-replication
height of the corresponding NiTi exdents when re-warmed to 423 K
(the austenitic condition).................................................................. 148

Figure 3.43

The optical image of the NiTi exdent and replica indent in 304
stainless steel after replica indentation with a preload of 250 MPa
.......................................................................................................... 150

Figure 3.44

The surface profiles of NiTi exdent heated to 433 K (the upper left
figure) and a replica indent in 304 stainless steel at 293 K (the lower
left figure) after replication under preload of 250 MPa. The dimensions
of the exdent and replica indent are plotted in the right figure ........ 151

Figure 3.45

The relation between replica indent depth and preload stress for
stainless steel and aluminum. No replica indent is assumed at 0 preload
stress ................................................................................................. 152

Figure 3.46

The relation between the replica indent depth and material hardness at
preload stress of 250 MPa ................................................................ 153

Figure 3.47

Plot of replica indent depth against the material hardness at preload
stresses of 125, 200, 250, and 300 MPa........................................... 154

Figure 3.48

Plot of H* against the preload stress. Parabolic curve fitting suggests
480 MPa is the largest preload stress that can be used to make a replica
indent................................................................................................ 155

Figure 3.49

Plot of the preload stress against H*/ preload. Linear curve fitting
suggests 480 MPa is the largest preload stress that can be used to make
a replica indent ................................................................................. 155

Figure 3.50

Low cycle fatigue of NiTi exdent in the constrained recovery
experiments ...................................................................................... 157

Figure 3.51

Path dependency of replica indentation: A) cooling without load; B)
cooling under load............................................................................ 159

Figure 3.52

Schematic illustrations of (a) initial indent depth-recovery, and (b)
exdent-to-flat surface transition during heating and cooling after
planarization of the LSP indent........................................................ 166

xvii

Figure 3.53

Cyclic exdent-to-flat surface transition after planarization of the
dynamic indent ................................................................................. 167

Figure 3.54

Comparison of the initial indent depth between the spherical indent
made under 700 N with r =400 m spherical ball and the dynamic
indent under laser intensity of 2.5 GW/cm2 and beam size of 0.5 mm
.......................................................................................................... 170

Figure 3.55

Comparisons of the initial indent depth, two-way recovery between the
quasi-static spherical indent and the dynamic indent. The dynamic
indent is magnified to have the same vertical length as the spherical
indent................................................................................................ 170

Figure 3.56

Comparisons of initial indent depth and two-way recovery between the
spherical indent and the LSP indent. Vertical scales are identical for
each impression but the base of the quasistatic indent is off the scale
.......................................................................................................... 171

Figure 3.57

Finite element mesh with 15,000 nodes and a refined area as the
contact region and 4 mm spherical indenter .................................... 180

Figure 3.58

Finite element model of the spherical indentation process: a) 10% of
the indentation steps; b) 60% of the indentation steps; c) 100% of the
indentation steps (Indentation step finished) ................................... 182

Figure 3.59

Effective plastic strain developed after spherical indentation with an
a/r of a) 0.4, b) 0.65, and c) 0.95 simulated by ABAQUS finite element
model................................................................................................ 182

Figure 3.60

Effective plastic strain developed after cylindrical indentation with an
a/r of a) 0.41, b) 0.65, and c) 0.95 simulated by ABAQUS finite
element model .................................................................................. 184

Figure a

Schematic illustration of the indentation-planarization process ....... 190

xviii

LIST OF NOMENCLATURES
DSC : Differential Scanning Calorimetry
TEC : Thermal Electric Cooler
SQUID : Super-conducting Quantum Interferometry Device
LSP : Laser Shock Peening
SFM : Surface Form Memory
TWSME : Two-way Shape Memory Effect
SMAs : Shape Memory Alloys
Ms : Martensite Start Temperature
Mf : Martensite Finish Temperature
As : Austenite Start Temperature
Af : Austenite Finish Temperature
Ra : Squar-Root-Mean Roughness
ho : Initial Indent Depth
Hm : Hardness of Martensite
H* : The Hardness Limit in Replica Indentation
hmax : Maximum Exdent Height
a : Contact Radius of a Spherical Indent
c : Half Trench Width of a Cylindrical Indent
r : Indenter Radius
dpl : Depth of Planarization

xix

D* : Depth of the Active Zone
Arepl : Area of the Replica Indent
Prepl : Load of the Replica Indent
Rqs : Radius of Quasistatic Indenter
Cel : Elastic Wave Propagation Speed
Cpl : Plastic Wave Propagation Speed
HEL : Hugoniot Elastic Limit
P : LSP Pressure
Dp : Plastified Depth

c: Two-way Indent Depth Change
he : Two-way Exdent Height

: Stress
 : Strain
 p : Peak Strain
 r : Representative Strain
 p : Pressure Pulse Duration
: Material Density
 : Lame‟s Constant
 : Shear Modulus
r,(sph or cyl,crit) : Critical representative strain (5%) for spherical or cylindrical indents
sph or cyl : Coefficient of representative strain for spherical or cylindrical indents

xx

Chapter 1 BACKGROUND AND LITERATURE REVIEW

1.1

Introduction
Various macro-, micro-, and nano-indentation methods have been used in

shape memory alloys (SMAs) as alternatives to conventional tensile tests for
measuring shape memory and superelastic characteristics of SMAs. For example, as
shown in Figure 1.1, shallow spherical* indents made by a Continuous Stiffness
Machine (CSM) instrumented indenter can completely recover after heating above Af
due to the shape memory mechanism, while deep spherical indents became shallower
but remained after heating due to irreversible slip deformation. These indents
deepened after cooling below Mf, generating a two-way depth change. When a
planarization step was used in the martensitic state after the indent's initial heating and
cooling, a thermally-reversible flat-to-bumpy transition can be observed, as shown in
Figure 1.2. We have termed this surface transition “Surface Form Memory” (SFM).
These transitions are fully reversible and cyclically stable.

It is demonstrated that SFM can be achieved by indenting with a variety of
tools in a martensitic NiTi surface, heating and cooling, and then planarizing in the
cool condition to restore a flat surface.

An inverted image of the tool will repeatedly

appear and vanish as the temperature is varied from Mf to Af and back to Mf.

*It should be noted that only shallow spherical or cylindrical indents can recover
completely.
†

1
Experiments have recorded tens of cycles with apparent convergence on a

Indenting arrays of balls or rods has led to the observation of some of the largest SFM
amplitudes.

Figure 1.1 Recovery of spherical indents made in martensitic NiTi upon heating [Ni et
al. 2002].

Figure 1.2 Flat-to-bump transition or Surface Form Memory (SFM) of the GM logo
made by the indentation-planarization method in a NiTi SMA.

Shot peening such as sand blasting can also give rise to SFM. However, this
stochastic process has no control over indent position and indent depth. Laser shock
peening, on the other hand, allows precise position control and creation of complex
arrays.

2

Indentation methods have been demonstrated as a robust technique to achieve
SFM. The ability to transition from a flat to a bumpy surface may find applications in
a number of areas. SFM in NiTi thin films at very small length scales allows its
application to the fabrication of thermally switchable diffraction gratings or other
optical devices. Other applications include information storage and smart
aerodynamic and hydrodynamic surfaces. SFM transitions generate appreciable
mechanical work during austenitization, which may find application to MEMS
actuators, mold-release surfaces, thermal clutches, mechanical joining, and variable
friction surfaces and interfaces.

This dissertation work concerns the development, understanding, and
applications of SFM. Detailed discussion on experimental results, and possible
mechanisms of SFM, are presented in this dissertation. In this chapter, the NiTi
system will be reviewed with an emphasis on transformation behavior and the
mechanisms of the shape memory effect (SME) and transformational superelasticity.
The development of training methods and the mechanisms of the two-way SME
(TWSME) will be discussed in detail. Recent advances in indentation-induced
TWSME will also be reviewed, associated with the contact mechanics and the
indentation deformation zones. The principles of a three-dimensional finite element
model on the indentation-induced shape memory effect will be discussed at the end.

3

1.2

NiTi Shape Memory Alloys

1.2.1 Historic Development
The discovery of shape memory alloys can be dated back to the early 1950s,
but the engineering significance of shape memory alloys was not well recognized
until the shape memory effect was discovered in NiTi alloys [Buehler et al. 1962].
NiTi shape memory alloys in particular have been favored by industry due to their
higher thermal stability, larger recoverable strain, greater shape memory effect (SME)
and superelasticity (SE). In addition, NiTi alloys have very good corrosion resistance
and biocompatibility, which are critical to applications in the biomedical field
[Morgon et al. 2004]. Due to the desire to better develop SMAs‟ application in
aerospace, biomedical, and consumer sectors, much attention has been given to
develop better understanding of the system and mechanisms of SE and SME, which
are discussed below.

1.2.2 The Shape Memory Effect
Shape memory metals consist of many alloys, of which the iron based alloys,
the copper based alloys, and the nickel based alloys are the technically most important.
For simplification and the relation to the present work, the near-equiatomic NiTi
shape memory alloy will be taken as an example, with a focus on the B2-B19‟
transformation route to illustrate the mechanism of SME.

4

In the NiTi alloy system, the high temperature austenite (A) parent phase with
B2 CsCl type ordered bcc structure can transform displacively to the martensite (M)
phase with monoclinic B19‟ type lattice at low temperature. During phase
transformation, as shown in Figure 1.3, martensite matrix and martensite twin form in
the martensite phase and displace along an invariant plane called habit plane. A single
twinned martensite domain is also called a habit plane variant (h.p.v.). The martensite
twins (LCV1 and LCV2), which are the smallest martensite variants, are called lattice
correspondence variants (L.C.V) and have correspondence with the parent phase.

The lattice structure change from B2 to B19‟ during the martensitic
transformation is shown in Figure 1.4. During cooling, [-1 0 0]p in B2 are contracted
and [0 -1 -1]p are extended. A shear strain causes the angle  to change from 90o to
96.8o and the monoclinic lattice structure is created. Due to the differences in
crystallographic structures in M and A phases, elastic strain is created during
martensitic transformation and dissipated by twinning (the lattice invariant shear).
This shear strain created during the martensitic transformation can only be removed
by the self-accommodation motion where a combination of multiple martensite
variants forms a triangular structure to cancel out shear strain macroscopically.

To see this clearly, we know {110}m has 6 equivalent planes, according to the
WLR theory [Lieberman et al. 1955], in which each plane has two lattice
correspondence variants that are caused shearing at opposite directions, positive shear

5

in (+) and negative shear (-). Since there are two lattice invariant shears ([011]m and
[01-1]m Type II twinning), there are totally 24 habit plane variants in the NiTi system
that cluster around the poles of <011> in the austenite phase according to the
stereographic projection in Figure 1.5 [Otsuka and Ren, 2005]. A particular habit
plane variant in the stereographic projection is represented such as 3‟-4‟(+), where 3‟
represents a martensite matrix, and 4‟ represents a martensite twin. 4‟ is related to 3‟
by [01-1]m Type II twinning in positive shear. Generally, during martensitic
transformation by cooling, 2 or 4 habit plane variants in each pole interact in a
self-accommodation motion to cancel out the transformation strain. This is has also
been analyzed in Cu-Zn-Ga, Cu-Al-Ni, etc by Saburi-Wayman [1979].

Figure 1.3 Schematic illustration of austenite and martensite phase with invariant
plane as a habit plane. (For interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this dissertation.)

6

Figure 1.4 The lattice change from B2 parent (a) to B19‟ martensite (b). i, j, and k
refer to parent lattice, and i‟, j‟, and k‟ refer to martensite phase. a, b, c refer to lattice
parameters,  refers to the angle between [-1 0 0] and [0 -1 -1]. [Otsuka and Ren,
2005]

However, due to the monoclinic structure of B19‟ martensite in NiTi, the
self-accommodation mechanism is rather complex. The triangular morphology was
observed TEM and used to explain the self-accommodation motion successfully
[Madangopal et al. 1997], as shown in Figure 1.6.

7

As a result, the macroscopic

transformation strain was canceled out in the triangular morphology formed by three
habit plane variants.

If the self-accommodated martensite variants are deformed in tension after
cooling, the deformation proceeds through twin boundary motion that essentially
converts one martensite variant to another. The most favorable martensite variant that
gives the largest strain under the external stress grows at the expense of the others
until, in the limiting case, the specimen becomes a single crystal of the most favorable
martensite variant. It is important to note that further plastic deformation would
require dislocation-mediated slip. Upon heating to a temperature above Af (Austenite
finish temperature), martensite variants undergo a reverse transformation to the
original shape in the parent phase due to the lattice correspondence. This effect is
called the Shape Memory Effect (SME).

Of significance to the present work, the deformation mechanism varies if the
specimen is placed in compression, due to tension-compression asymmetry. Liu et al.
[1998] studied the stress-strain curves and the microstructure of NiTi shape memory
alloys in tension and compression, as shown in Figure 1.7. Figure 1.7a shows the
microstructure of the un-deformed self-accommodated martensite. Figure 1.7b shows
the microstructure of martensite twins after 4% strain in tension. Martensite variants
have detwinned and started to align with the maximum shear stress direction. Figure
1.7c shows the microstructure of deformation bands inside martensite twins after 4%

8

strain in compression. The martensite has fully detwinned with large deformation
bands being formed. It was found that the martensitic deformation in compression
was not through the movement of adjacent martensite variants to align with the
maximum stress direction as it was in tension, but through the generation and
rearrangement of lattice defects, especially dislocations. This “tension-compression
asymmetry” arises from the different Schmid factors between tension and
compression directions due to the low symmetry in martensite variants [Patoor et al.
1995]. It was reported by Liu et al. [1998] that NiTi alloys work harden more quickly
in compression than in tension, as shown in Figure 1.8, causing the stress to be larger
in compression than in tension for the same amount of strain. Slip deformation is
hence larger in compression than in tension for the same pre-strain. As a result, the
recoverable strain is smaller in compression than in tension for the same deformation
strain because the lattice defects generated in compression inhibit complete shape
recovery.

9

Figure 1.5 Seterographic projection of 24 habit plane variants for positive shear based
on WLR theory for <011>m Type II twinning [Otsuka et al. 2005].

Figure 1.6. TEM observation and the analysis (right-hand side) of a) typical triangular
morphology of B19‟ martensite, b) selected area diffraction pattern [Madangopal et al.
1997].

10

Figure 1.7 (a) Microstructure of un-deformed martensite twins; (b) microstructure of
martensite twins after 4% deformation strain in tension; (c) microstructure of
deformation bands inside martensite twins after 4% deformation strain in compression
[Liu et al. 1998].

11

Figure 1.8 Stress-strain curve of a NiTi specimen under uniaxial tension and
compression [Liu et al. 1998].
1.2.3 Thermodynamics of Shape Memory Alloys
Since martensitic transformations are diffusionless, shear-like transformations,
stress can be a driving force for martensitic transformation. Therefore, martensite can
be formed by stressing at temperature higher than Ms. The stress that is needed to
stress induce martensite is governed by the modified Clausius-Clapeyron equation
expressed in Eq. (1.1).

dTTRANS

T

 O
d
S
H

Eq. (1.1)

Where ζ is the externally-applied deviatoric stress, TTRANS is the stress-induced
transformation temperature, ΔS is the martensitic transformation entropy per unit
volume, ε is the transformation strain, ΔH is the martensitic transformation enthalpy
per unit volume, TO is the stress-free B2-B19‟ equilibrium temperature.

If the ambient temperature is above Af, Martensitic transformation is reversed
when the applied stress is removed upon unloading. This reversion of stress induced
martensite and recovery of deformation upon unloading is known as Superelasticity.

In section 3.5.3, we will estimate the change of transformation temperature
due to loading in a constrained recovery experiment. Equation 1.1 is modified to be

12

dTTRANS  TO d / H to estimate the change of transformation temperature under
stress.
Figure 1.9 schematically shows the fundamental limits on superelasticity,
plotted in stress- temperature space. It is important to note that the critical stress to
induce martensite is positively (linearly) related to temperature in the
stress-temperature space. This can be explained based on Eq 1.1 where dTTRANS /d is
positive since martensitic transformation is an exothermic process. If NiTi is
subjected to sufficient external stress at temperature Tx above Ms, a value between Af
and Md, as shown in Figure 1.9, the austenite phase becomes thermodynamically
unstable and transforms to the martensite phase. The minimum stress to start inducing
martensite is found to be x. Upon releasing the stress, the martensite spontaneously
reverts to austenite phase due to the instability of martensite in the absence of stress,
given that Tx > Af. The ability to stress induce martensite, however, also depends on
whether the stress level exceeds the temperature dependent critical stress for slip, d
in the austenite. If the applied stress magnitude during deformation is larger than the
critical stress for slip, slip occurs before stress-inducing martensite, resulting in
permanent deformation with partial or even no strain recovery, depending on the
stress level used. Therefore, Md, the temperature at which the critical stress loci cross,
is the maximum temperature for stress inducing martensite with full recovery.

In summary, superelasticity occurs when the work done by stress-inducing
martensite equals the free energy change of the NiTi alloy in forming martensite by

13

cooling, the behavior of which is governed by the modified Clausius-Clayperon
equation.

Figure 1.9 Schematic drawing of superelasticity in temperature-stress space.

1.3

Two-way Shape Memory Effects

1.3.1 Introduction
Different from one-way SME, in which the deformed martensite returns to
the original austenite shape after heating but no shape change upon subsequent
cooling, two-way shape memory effect (TWSME) drives a shape change upon both
heating and cooling. For the shape change to occur on cooling, certain martensite
variants need to be selected to return to the shape in martensite. Due to the nature of
the selective formation of martensite variants, TWSME is a weak effect compared to
one-way SME, in which strong reversibility and lattice correspondence take place.
The selective formation of martensite variants is not an intrinsic property of SMAs,

14

thus methods such as thermo-mechanical cycling, constrained aging, slip deformation,
have often been involved in training for TWSME.
In this section, the most common training methods for TWSME will be
reviewed and the usefulness of the methods will be evaluated. The mechanisms of
TWSME developed in the literature in the past several decades will be discussed.
Understanding the training methods and the mechanisms of TWSME is critical to
understand the indentation induce TWSME and Surface Form Memory.

1.3.2

Review of TWSME Training Methods

1.3.2.1 Thermo-Mechanical Cycling
Bending, extension and tensile deformation are often coupled with
thermo-mechanical cycles to develop TWSME in shape memory alloys. Specifically,
if a NiTi specimen is deformed under bending, tension, or compression in the
martensite phase, then the specimen would be heated to austenite and cooled to
martensite without load to finish one thermo-mechanical cycle and start another one.
If the specimen was deformed in the austenite phase, then the specimen would be
cooled to martensite under load, followed by releasing the load and reheating the
specimen to austenite to start another training cycle.

Yu et al. [2006] used thermo-mechanical training on NiTi springs to develop
TWSME. The training cycle was first extending the NiTi spring in the martensitic
state to various strains, unloading, and then heating and cooling the spring between Af

15

and Mf without stress to measure two-way strain. As a result, the NiTi spring
shortened upon heating and elongated upon cooling (as shown in Figure 1.10). Lahoz
et al. [2002] used two bending modes for training. In method 1, a NiTi wire was bent
in the martensitic state to various strains and heated and cooled without load between
Af and Mf to finish one training cycle. In method 2, a NiTi wire was first bent in the
martensitic state to various strains and then repeatedly thermally cycled between Af
and Mf with load to induce TWSME. It was reported that method 1 rather than method
2 is more effective to increase two-way strain, as shown in Figure 1.11. The two-way
strain, which is defined as the ratio of reversible bending angle change over the
bending angle after unloading, reaches a plateau after around five thermo-mechanical
cycles and the maximum two-way displacement ratio (

l
) was found to be 8%.
lo

Perkins et al. [1984] thermo-mechanically cycled a CuZnAl shape memory
alloy in tension and used TEM to study the dislocations in the trained specimens. The
training method is a little different from above since the author deformed the
specimen in the austenite and cooled to martensite under stress, then unloaded and
heated to austenite to start another thermo-mechanical training cycle. It was reported
that the optimum training for TWSME was at approximately fifteen training cycles, as
shown in Figure 1.12.

16

Figure 1.10 Schematic drawing of the thermomechanical training of NiTi two-way
shape memory spring [Yu et al. 2006]. The specimen was annealed at 500 oC.

Figure 1.11 TWSME strain tw as a function of the number of thermal cycles and
training strain t [Lahoz et al. 2002].

17

Figure 1.12 Plot of the two-way displacement ratio (or two-way strain) as a function
of the number of training cycles.  T is the total elongation due to training [Perkins et
al. 1984].

1.3.2.2 Constrained Aging
Constrained aging can create internal stress fields that lead to TWSME.
Fukuda [Fukuda et al. 2001] aged a single crystal Ti-51at% Ni under a compressive
stress along [111]. The specimen was first solution-treated at 1123 K for 3.6 ks
followed by quenching into ice water. After that, the specimen was aged at 773 K for
6 ks under a compressive stress of 300 MPa applied along the [111] direction. Ni4Ti3
precipitates were formed coherently along {111} plane of the parent phase. The
thermally induced R phase had a single variant and the expansion strain along [111]
direction was about 0.8%.

18

Nishida and Honma [1984] reported “all-round shape memory effect”. The
strips of the Ti-51.at% Ni alloy they used were solution-treated at 1073 K for 7.2 ks.
Then they were subjected to constrained-aging at 773 K for 3.6 ks as a half-circular
form as shown in Figure 1.13a. If the specimen is cooled down to 77 K, the shape
changed from Figure 1.13(a)–(e). Upon heating the exact reverse change of the shape
was observed from Figure 1.13(e)–(a). Figure 1.14 schematically shows the stress
state of the sample. Compressive stress and tensile stress arise in the lower part and
upper part respectively. The bending stress is relieved if the precipitates are formed
normal to the compressive stress which is normal to {111} plane, as shown in Figure
1.14. A TEM study in [Kainuma et al. 1987] confirmed that constrained-aging
induced precipitates are formed in such a way that a special microstructure and stress
field were developed to promote TWSME.

(a)
e
(b)
e
(c)
e
(d)
e
(e)
e
Figure 1.13 Demonstration of “all-round shape memory effect” in Ti–51Ni alloy from
(a) to (e) when cooled down to 77 K [Nashida et al. 1984].

19

Figure 1.14 Mechanism of “all-round shape memory effect” [Kainuma et al. 1987].

Precipitates formed in constrained aging training can be observed using TEM
[Zhang et al. 2003]. It has been shown the oriented strained fields around precipitates
in NiTi can cause martensite variants to align in preferential directions. The stress
fields around the precipitates can be recovered again after reversion to austenite phase.
The mechanism of the precipitate-enabled TWSME is clear. However, the available
two-way shape memory strain was not large (less than 1%), and the increase of Ni to
form Ti3Ni4 precipitates results in the lowering of Ms below ambient temperature,
limiting its applications.

20

1.3.2.3 Plastic Deformation and TWSME
Plastic deformation training methods were used in early studies of TWSME.
Researchers favored severe deformation in NiTi SMAs in order to achieve maximum
two-way response [Liu et al. 1990]. Deformation in the martensitic state was used to
introduce dislocations [Zhang et al. 1986; Otsuka et al. 1986; Kainuma et al. 1987;
Meng et al. 2004; Omori et al. 2003]. The dislocation-related stress fields induced by
martensitic deformation enable the formation of directional martensite variants upon
cooling. For example, Meng et al. [2004] bent a TiNiHf high temperature shape
memory alloy in the martensitic state to create slip deformation in the alloy. Figure
1.15, which plots the two-way shape memory strain as a function of the number of
training cycles, shows that the two-way strain increases with the plastic deformation
strain. The maximum two-way strain was reached around 0.9% after 6 training cycles
at deformation strain of 7.1%.

Kim et al. [2009] studied the effect of compressive loading on the
development of TWSME in NiTi shape memory alloys. The author did compressive
load cycling to preset strains (1-6% in the experiment) in the martensitic state, and
thermally cycled the specimen after unloading to measure the two-way strain. The
two-way strain data, as shown in Figure 1.16, indicated a strong linear relation with
compressive deformation strain. The maximum two-way strain reached around 2.3%
on average for NiTi specimens at a compression of 6%. However, these results could
not generate a whole picture of the relationship between two-way strain and

21

compressive strain. If the compressive strain was increased further, the two-way strain
may further increase or possibly decrease due to heavy slip deformation.

Figure 1.15 Effect of the number of training cycles on the two-way shape memory
strain of the Ti36Ni49Hf15 alloy with different training strains [Meng et al. 2004].

Figure 1.16 Effect of compressive prestrain on the development of two-way shape
memory effect in NiTi shape memory alloys [Kim et al. 2009].

22

Slip-plasticity training has the advantage of requiring fewer training cycles
(often just a single cycle) compared to traditional thermo-mechanical training
methods. Figure 1.17 shows the TWSME obtained from a single uniaxial tensile
deformation training [Liu et al. 1999]. Plastic deformation to a strain of 6.2% led to a
1.44% two-way strain upon subsequent thermal cycling. Tensile pre-strain of 8%
leads to a two-way strain of 2.4%, as shown in Figure 1.18, which agrees with the
2.6% two-way strain developed by tensile deformation to 8% strain reported by
Prader et al. [1997] for a cold-worked and annealed NiTi specimen. However, the
two-way strain (1.44% at 6.2% tensile strain) is lower than what was developed under
compression strain (2.3% at 6% compressive strain), as shown in Figure 1.16. This
difference might be due to tension-compression asymmetry [Patoor et al. 1995]. It
was reported that the critical deformation stress is higher in compression than in
tension, which suggests that there are more dislocations developed in compression
than in tension for the same deformation strain. It is possible that these dislocations
help increase the two-way strain in compression. From Figure 1.18, it can also be seen
that the maximum two-way strain was 4.1% for a tensile strain of 13.3%. This
important evidence suggests the two-way strain in compression can be higher than
2.3% if the deformation strain increases (refer to Figure 1.16). Also shown in Figure
1.18, further increasing the deformation strain decreases the two-way strain because
heavy slip deformation serves as a barrier for the specimen to recover.

23

Figure 1.17 The TWSME obtained after a single deformation of 6.2% strain in an
equiatomic NiTi alloy [Liu et al. 1999].

Figure 1.18 Effect of prestraining in tension on the development of two-way shape
memory effect in an equiatomic NiTi alloy [Liu et al. 1999].

24

1.3.3 The Mechanisms of TWSME
1.3.3.1 Introduction
There are 3 mechanisms proposed for TWSME in the literature: the retained
matensite theory [Saburi et al. 1974], the internal residual stress field theory [Zhang et
al. 2003], and the microstructural anisotropy and thermodynamical anisotropy theory
[Stalmans et al. 1992], the details of which will be reviewed below.

It was speculated by Saburi et al. [1974] that the retained martensite variants
at temperature above Af can preferentially grow and modify the arrangement of other
variants upon cooling. It was also reported by Zhang et al. [2003] that the residual
stress changed thermoelastic martensitic transformation from the formation of 24
self-accommodated martensite variants to certain directional martensite variants upon
cooling. It was also found by Stalmans et al. [1992] that, in the Cu-Zn-Al alloys, a
great number of dislocations in the parent phase could favor the growth of certain
martensite variants out of the 24 possible habit plane variants. Below is the detailed
discussion of these mechanisms of TWSME in the literature.

1.3.3.2 The Retained Martensite Theory
Saburi found a reversible shape memory effect in Cu-Zn-Ga alloys due to slip
deformation in the martensitic phase and proposed the retained martensite theory in
which the retained martensite variants at a temperature above Af can preferentially
grow and modify the arrangement of other variants upon cooling [Saburi et al. 1974;

25

Perkins et al. 1984]. These retained martensite variants can be generated by
overstraining the martensite in the martensitic state or stress-inducing martensite at a
temperature above Af. The overstrained martensite is thought to be stabilized by
dislocation induced strain fields generated under repeated training cycles. These may
then serve as nucleation sites for the formation and preferential growth of martensite
variants upon cooling. However, experimental results also show the retained
martensite can be removed by overheating during martensite-to-austenite
transformation without affecting TWSME. Thus the retained martensite theory is thus
an unlikely mechanism of TWSME.

1.3.3.3 The Residual Stress Field Theory
Shroeder and Wayman [1977] did thermo-mechanical cycling on Cu-Zn
single crystals where the specimens were first deformed in the martensitic state and
then recovered to austenite without load in each training cycling. After training, a
single orientation of martensite variant was formed during cooling, thus showing
TWSME. Delaey and Thienel [1975] documented the changes occurring in the
microstructure of a polycrystalline CuZnAl sample as it was subject to up to ten
training cycles. The conclusion was that nucleation of a particular martensite variant
is responsible for the spontaneous shape change on cooling. Delaey further noted that
plastic deformation developed during training results in a residual stress field that may
influence martensite microstructures during cooling. Perkins et al. [1984] also noted
that the plastically deformed local regions may explain TWSME. As shown in Figure

26

1.19, the thermo- mechanically trained specimen has dark streaks which are thought
to be full of tangled dislocations. Martensite plates once residing in these regions will
form into these regions again upon martensitic transformation. These preferentially
formed martensite variants reoriented the subsequently formed martensite variants,
thus generating TWSME.

1 m

1 m

a
e

1 m

c
e

1 m
e

1 m
b

d

e
e

1 m

e

f

e
e
e
e
Figure 1.19 Dark Streaks in the austenite phase after several training cycles. (a) to (d)
3 cycles, (e) to (f) 5 cycles [Perkins et al. 1984].

Moreover, molecular dynamics simulation [Zhang et al. 2003] suggested that
thermally induced martensite nucleation sites are either above or below the paired
dislocations, where there are tension or compression stress fields of the dislocations,
as shown Figure 1.20. During growth, nuclei of martensite line up in the middle of the
paired dislocations and grow to a single variant via the movement of dislocation by
shear. Therefore, dislocation arrays in the material developed by thermomechanical
cycling selectively grow certain martensite variants to accommodate the

27

e

dislocation-induced stress field. The existence of these directional martensite variants
serves as the basis of dislocation assisted TWSME.

Figure 1.20 MD simulations show martensite nucleation and growth, (a)-(c) for a pair
dislocation, (e)-(g) for two pair dislocations. (d) and (h) show the stress fields of one
pair and two pair dislocations [Zhang et al. 2003].

In addition, Rios-Jara and Guenin [1987] studied in detail the correspondence
between dislocation arrays and repeatedly induced variants in thermo-mechanically
cycled single crystal Cu-Zn-Al alloy. They reported that all of these dislocations in
the single crystal of the 1 parent phase formed by thermo-mechanical cycling of
copper based alloys have the same <110>1 type Burgers vector, as shown in Figure
1.21. The maximum shear in a strain field of this type of dislocation coincided with
the macroscopic shear observed in the martensitic transformation. These dislocations
shared the same <110>1 Burgers vector and accumulated in bands parallel to the
trace given by the habit plane of the induced variant. By studying the isolated
dislocations, it was found that the glide planes of these dislocations correspond to the

28

basal planes of the induced martensite variants, thus achieving minimal free energy in
these variants. However, there are 4 out of 24 crystallographically equivalent
martensite variants that satisfy this condition. Therefore, this theory could not
describe the one-to-one correspondence between martensite variant and austenite, as
required by TWSME.

[a]
e
2 m
e

[b]
e

Figure 1.21 Transmission electron micrograph showing a “Comb-shaped”
arrangement of dislocations observed in the thermomechanical cycled Cu-Zn-Al alloy.
(a) zone axis G=[-1-11]. (b) Same zone as in (a) but under a B=[001]1 direction of
observation. Here the projection of both dislocations in a pair are coincident: the plane
containing a pair is (1-10)1 which corresponds to their glide plane [Rios-Jara et al.
1987].

29

Furthermore, Manach et al. [1993] thermo-mechanically trained a room
temperature austenitic Ti-50.2 at% Ni shape memory alloy on a hydraulic
tension-compression device to develop TWSME. The first training procedure
consisted of 20 cycles of heating to 363 K (Af was 305 K) and cooling to 285 K (Ms
was 276 K) without load, followed by an isothermal loading up to a shear strain of 5%
and unloading to 0% (the stress went to zero before the strain went to zero). The
deformation strain was recovered by heating in the second training cycle. It should be
noted with cyclic training deformation strain accumulated and could not be fully
recovered. A superelastic cycle via shearing was then performed between -5% and 5%
in the austenitic state at T=313 K to observe the stress-strain characteristics of the
trained material. After this, the specimen underwent a second training procedure that
was in the exactly same way as the first procedure, but with a reversed shearing
direction.

Figure 1.22 shows the shear stress-strain curves during the first and second
training cycles. The difference in the shear stress during the first and second training
cycles indicates the existence of internal stress fields. This is more clearly shown in
Figure 1.23 and 1.24 which show stress-strain curves being asymmetric in the
opposite directions. Since the existence of an internal stress field will assist
martensitic transformation in the stressed direction and act as an obstacle in the
opposite direction, the shear stress needed to induce martensitic transformation has to
be lower for the training direction (the same as the stress direction) than for the

30

opposite one. The asymmetry in the hysteresis also suggests that internal stresses
create friction between the martensite variants during phase transformation. Some
variants have to overcome this friction energy, thus making them thermodynamically
unfavorable to form during cooling, which leads to a thermo-dynamical anisotropy
theory that will be discussed in the next section. The thermodynamically favorable
martensite variants will preferentially form during cooling, competing with the
formation of the self-accommodated martensite variants.

Figure 1.22 a) and b) Shear stress-strain curves obtained at T=285 K during
isothermal shearing test in the first training and second training [Manach et al. 1993].

1.3.3.4 Thermodynamic Anisotropy Theory
Stalmans et al. [1992(a) and (b)] studied TWSME induced by
thermo-mechanical training of Cu-based SMAs and speculated that dislocations
developed during training induced a microstructural anisotropy. The microstructural
anisotropy leads to martensite variants having different free energies, thus causing a
thermodynamic anisotropy.

31

It was found that TWSME reached a maximum after about 30 training cycles
and decreased thereafter due to an increasingly imperfect one way shape memory
recovery caused by dislocation accumulation. They also calculated the free energies
of the martensite variants developed after two-way training and compared them to the
case of the untrained thermoelastic martensite variants (The training stress dependent
free energy,

 G which is derived from Clausius-Claperyon equation equals the

product of training stress () and transformation strain (  TR (m) ) of martensite
variant (m) resolved along the direction of external stress:

 G     TR (m) ).

The martensite variants formed after two-way training were found to have lower free
energies than the untrained thermo-elastic martensite variants, thus being
thermodynamically favored and preferentially formed during cooling. It is the
dislocation arrays in the martensite variants that lower the free energy and provide
suitable sites for martensite variants to form. Therefore, plastic deformation to
generate dislocation arrays is assumed to be a prerequisite to develop TWSME.

32

Figure 1.23 Shear stress-strain curve obtained during a super-elastic cycle with strain
of ±5% at T=313 K after first shearing cycling. The dash line in the hysteresis
indicates the estimated reversible stress contribution to the hysteresis.  irr is the
irreversible stress contribution induced by training [Manach et al. 1993].

Figure 1.24. Shear stress-strain curve obtained during a superelastic simple shear
cycling with strain of ±5% at T=313 K. The dash line in the hysteresis indicates the
estimated reversible stress contribution to the hysteresis [Manach et al. 1993].

33

To conclude, it is clear that retained martensite is not the origin of TWSME
since the retained martensite can be removed by overheating without greatly affecting
TWSME. Dislocations developed by plastic deformation are important for the
TWSME observed. However, over deformation negatively affects TWSME because
experimental results suggest that a large number of dislocations developed by over
deformation can resist M-A-M phase transformations. Therefore, the training induced
microstructural anisotropy and thus thermodynamic anisotropy maybe the most likely
mechanism of TWSME, because martensite variants are found to be different
thermodynamically after training. Even though it is difficult to directly observe how
dislocation arrays preferentially select martensite variants, dislocation arrays are
calculated to have lowest energy in the martensite variants where they were originally
formed. Therefore, the existence of these dislocation arrays helps form these
thermodynamically favorable martensite variants. However, there is limited study on
the origin of TWSME in literature. Many researchers tend to agree with the
thermodynamic anisotropy theory proposed by Stalmans et al. [1992].

1.3.4

Indentation in Shape Memory Alloys
As stated in the introduction to this chapter, indentation methods have

historically provided an alternative to conventional tensile tests to characterize
mechanical properties. Compared to other mechanical testing techniques, the
indentation method has several advantages, such as the simplicity of sample
preparation, characterization ability under wet environments, and the quantitative

34

characterization at the nanometer scales. Furthermore, as is well known, indentation
can induce just the sort of dislocation substructures that drive TWSME.

Indentation has also been used to probe shape memory and superelastic
effects at multiple length scales and under complex loading conditions. Ni et al. [2003]
used both micro-indentation and nano-indentation on bulk NiTi shape memory alloys
and observed the indentation-induced shape memory and super-elastic characteristics.
As was shown in Figure 1.1, shallow spherical indents made by an instrumented
indenter can completely recover after heating above Af due to the shape memory
mechanism. Deeper spherical indents became shallower on heating but were not
completely removed.

Figure 1.25 shows that a complete shape memory recovery occurs up to the
point at which the representative strain is larger than 5%. The representative strain, r,
equals 0.2a/r for spherical indents, where a is the indent contact radius and r is the
indenter radius. r correlates with the true strain in the uniaxial tensile test. (A detailed
discussion on representative strain will be given in section 1.4.2.) Above a
representative strain of 5%, slip plasticity occurs in martensite and dislocations
remain after heating above Af, generating a residual indent in austenite.

35

Figure 1.25 Plots of the recovery ratio, true stress against the representative strain
(0.2a/r). Complete recovery occurs when the representative strain is less than 5%,
beyond which slip plasticity occurs [Ni et al. 2002].

Frick et al. [2006] used a Berkovich nanoindenter to characterize the
stress-induced martensitic transformation and shape memory response at the nano
scale. Figure 1.26 shows the indent depth recovery upon heating for pseudoelastic and
shape memory NiTi alloys, and [111] Ni. Full depth recovery was achieved in the
shape memory alloy at 100 N. This is probably because the blunting of the
Berkovich indenter tip leads to spherical- like indentation under low load such as 100
N. However, when the load is increased, the depth recovery decreased due to severe
deformation under the sharp indenter.

Apart from indenting bulk material, Shaw et al. [2003] indented shape
memory thin films and characterized shape memory and super-elastic responses in
thin films. Several other groups have also investigated shape memory and superelastic

36

effects using micro- and nano-indentation techniques [Liu et al. 1999; Cheng et al.
2001; Gall et al. 2001; Ma et al. 2003, 2004; Qian et al. 2004; Liu et al. 2005; Shaw et
al. 2005; Frick et al. 2005; Huang et al. 2005; Komvopoulos et al. 2005; Zhang et al.
2006; Wood et al. 2006; Su et al. 2007; Crone et al. 2007; Yan et al. 2007; Arciniegas
et al. 2008]. Indenter type, indentation depth, and indentation temperature play an
important role in the shape memory and superelastic responses after indentation.
These parameters were used to maximize the indentation-induced TWSME in the
present study.

Figure 1.26 a) Indent depth recovery upon heating for pseudoelastic, shape memory
alloys and [111] Ni with max load up to 16000 N; b) Indent depth recovery upon
heating for pseudoelastic and shape memory alloys with max load up to 2000 N
[Frick et al. 2006].

37

1.3.5

The Indentation Induced TWSME
Plastic deformation can generate dislocations in martensite that strongly affect

subsequent shape-memory mechanisms [Fernandez et al. 2003; Stalmans et al. 1992;
Liu et al. 1998; Wang et al. 2005; Lahoz et al. 2002] , causing shape changes upon
both heating and cooling.

Zhang et al. [2006] also reported that deep spherical indentation can induce
slip deformation that causes the indent to become shallow upon heating and deepen
upon cooling, as shown in Figure 1.27. When indents were removed by a
planarization step, thermally-reversible flat-to-bumpy surface transitions were
observed for both wrought NiTi shape memory alloys, and in NiTi thin films. We call
this transition Surface Form Memory (SFM).

Figure 1.27 WYKO optical profile measurement of the two-way indent depth change
of a deep spherical indent upon heating and cooling [Zhang et al. 2006].

38

Figure 1.28(a) shows SFM of a 3x3 array of spherical exdents made by
planarization of an array of spherical indents in a bulk martensite NiTi sample, after
heating to austenite, and cooling to martensite. This exdent-to-flat surface transition is
cyclical reversible indefinitely†. Figure 1.24(b) and (c) show the exdent-to-flat surface
transition of a scratch made by a conical indenter in a bulk martensitic NiTi sample
and a micro-indent made by a conical indenter in a thin film martensitic NiTi sample
[Zhang et al. 2006].

Figure 1.28 Three dimensional profiles of reversible surface exdents: (a) a 3x3 array
of spherical exdents on the surface of the austenite phase of NiTi at high temperature
which disappears when the sample is cooled to the martensite phase. (b) A protruding
line on the surface of the austenite phase that nearly disappears in the martensite
phase. (c) A nanoscale reversible protrusion on the surface a NiTi film. [Zhang et al.
2006]

†

Experiments have recorded tens of cycles with apparent convergence on a

steady state.

39

Figure 1.28 cont‟d.
1.4

Contact Mechanics
The concept of contact mechanics was first introduced by Hertz in 1888 [Hertz,

1896; Johnson, 1985]. However, the development of the theory stalled until the early
th

20 century, until it was revived, driven by engineering developments on the railways,
marine reduction gears, and in the rolling-contact bearing industries. The theory

40

generally applies to normal contact between two rigid smooth solids. Contact
problems are important in the context of wear, scratch testing, and indentation.

1.4.1

Elastic Contact
Elastic contact problems for spherical and sharp indenters into a half space

are important because they underlie the measurement of hardness, which is indicated
as the mean contact pressure Pm. Pm is defined as the normal load F on the indenter
divided by the contact area: Pm =F/a2. We can also refer to the mean contact
pressure as the “indentation stress” and 0.2a/r as the “indentation strain”. This then
connects the indentation stress-strain response with the commonly obtained
conventional uniaxial tension and compression stress-strain response.

Tabor proposed using the representative strain or the “indentation strain”, as a
function of the a/r ratio, to characterize spherical indentation into elasto-plastic solids
[Tabor, 1948, 1996]. The representative strain is r=0.2 a/r. It is deduced by plotting
the indentation and uniaxial deformation data of copper and steel (see Figure 1.29)
where the representative strain is equal to the true strain, and the hardness H is about
2.8 times of the true stress, Y. The empirical correlation between spherical indentation
and uniaxial tensile data shows that the representative strain can be used to measure
the true stress-strain relationship.

41

For elastic contact, several assumptions are generally made.
1.

The radii of curvature of the contact bodies are large compared with the in-plane

radius of the circle of contact. So we can treat each surface as an elastic half-space to
derive stresses and displacements information.

2.

The dimensions of each body are large compared with the radius of the circle of

contact. This allows indentation stresses and strains to be considered independently of
those arising from the geometry, method of attachment, and boundaries of each solid.

3.

The contact is frictionless. Only a normal pressure is transmitted between the

indenter and the specimen.

Spherical indentation contact problems are of special interest due to their
relationship to the present work. The most well-known scenario is the contact
between a rigid sphere and a flat surface, where Hertz found that the elastic contact
radius, a, can be calculated as: a  (

3FR 1/3
) , where F is the indenter load, R is the
4E *

radius of indenter, and E* is the reduced modulus, which is a combined function of
the modulus of the indenter and the substrate, as seen in Eq. (1.2):
2
1 1   2 1  ’


E*
E
E’

Eq. (1.2)

where E‟ and v‟, and E and v, are the elastic modulus and Poisson‟s ratio of the
indenter and the specimen respectively.

42

On the contact surface the contact pressure is not uniformly distributed. The
maximum contact pressure Po is 3/2 of Pm at the depth of contact radius a, as
expressed in Eq. (1.3) as

Po 

3F
2a

2

(

6 FE *2
3 2

 R

)1 / 3

Eq. (1.3)

Po reaches maximum when a=r.

The vertical displacement of the surface within the contact radius, measured
with respect to the original specimen free surface, is given by Eq. (1.4):

u Z (r )  

1   2 Po
(2a 2  r 2 ), r  a
E 4a

Eq. (1.4)

Cylindrical contact cannot be calculated solely by the Hertzian theory since it
is an elastic compression of a two-dimensional body, which needs adjustments of the
shape and size of the body and the way in which it is supported. Line contacts such as
gear teeth and roller bearings are difficult to calculate due to its support system while
the compression of circular cylinder in between two compressors can be analyzed
successfully.

For a cylindrical indenter, the compressive load per unit axial length, F, gives
rise to a Hertzian distribution of pressure P at the lower contact point

2F
x 2 1/3
4 FR
P
(1  2 ) , where a 2 
. The vertical displacement is
a
E *
a

43

1  2
4R
  2 P(
)[2 l n ( )  1]
E
a

1.4.2

Eq. (1.5)

Elastic-Plastic Contact
In metals, slip-plasticity develops when the elastic contact stress exceeds the

material‟s yield stress in compression, which means that Pm/Y is larger than 1. One
can visualize that as the indenter is pressed deep into the substrate, a plastic zone
appears below the indenter tip, surrounded by an elastic region. This is called the
“contained mode”, where Pm/Y is between 1 and 3. The plastic zone grows deep into
the substrate with indentation and also eventually breaks out to the free surface. From
then on, the displaced material is free to escape by plastic flow to the sides of the
indenter. This is the “uncontained mode” of indentation. This “uncontained mode”
begins when the mean contact pressure Pm reaches about triple the yield stress Y
[Johnson, 1986].

Both finite element analysis and semi-analytical models are commonly used in
analyzing elastic-plastic contact problems. A semi analytical spherical cavity model
was proposed by Johnson to explain the elastic-plastic indentation process [Johnson,
1986]. In this spherical cavity model, the contact surface of the indenter is encased in
a hemispherical “core” with a hydrostatic component, p. Outside of the core it is
assumed that the material will respond the same way as in an elastic half space
containing a spherical cavity under hydrostatic pressure, p. The elastic-plastic

44

boundary is somewhere outside the spherical cavity. The hydrostatic pressure, p,
below the indenter can be expressed in Eq. (1.6) as

E*t a n

 4(1  2 )
p 2
 {1  l n [ Y
]
Y 3
6(1   )

Eq. (1.6)

where Y is the yield stress of the indented material in compression.  is the angle of
the indenter at the edge of the contact (for example:  is 19.3o for Berkovich indenter
and =arcsin(a/r) for spherical indenter).

Figure 1.30 shows the finite element modeling of the expansion of plastic
zone during spherical indentation [Hardy et al. 1972]. Plastic flow leads to a flattening
of the pressure distribution as can be seen for the p/k value against r/aY. The
development of the plastic zone roughly follows the contours of J2, which is the
second invariant of the deviatoric part of the Cauchy stress, and is contained beneath
the contact area. In the fully plastic state for the spherical cavity model, P/PY equals
400, which refers to the elastic-plastic boundary. Sinclair et al. [1985] carried out a
detailed finite element analysis of the spherical indentation on elastic-plastic solids.
They confirmed Tabor's experimental correlations for work-hardening materials using
the representative strain, and found generally good agreement between theory and
other physical evidence. In addition, they also found there was high strain gradient
near the contact edge that was not predicted before.

45

Figure 1.29 Representative strain vs. mean stress in spherical indentation, and its
correlation with equivalent true stress-true strain behavior in uniaxial tension (Solid
line). Metal A is Steel and metal B is Copper. [Tabor 1948, 1996]

Figure 1.30 Finite element modeling of indentation of an elastic-plastic half space by
a rigid sphere: development of the plastic zone. Broken line: contour of J2. PY is the
normal load needed to first initiate plastic flow. [Hardy et al. 1971]

46

1.4.3

Indentation Related Contact Mechanics
The indentation process is a highly nonlinear problem where large strains and

deformations as well as material nonlinearities and contact are involved. Indentation
related contact problems need the understanding of elastic-plastic deformation regime
and material flow.

Figure 1.31 shows the plastic flow motion under Vickers indentation [Mata
et al. 2002]. The solid at the indenter‟s tip flows primarily toward the z direction
while the solid at the indenter‟s face flows toward the r direction because the
downward motion needs to be compensated by upward flow adjacent to it, based on
slip line field theory.

Figure 1.31 Flow patterns exhibited in a material that piles up [Mata et al. 2002].

47

During the process of indentation, the contact regime evolves from (i) an
elastic deformation, to (ii) an elastic-plastic transition, and to (iii) a fully plastic
regime. Sink-in and pile-up transitions can be observed for different materials.

Figure 1.32 compares the plastic zone size between spherical indent and
conical indent at different a/r ratios and illustrated the transition from sink-in to
pile-up for materials with different strain hardening exponent n [Mata et al. 2006].
The following observations are pertinent:
(i) For a material with large values of n and high yield strength (E = 200 GPa, s =
1000MPa, and n = 0.4, as shown Figure 1.32 (a)), it is noted that sinking-in of
material occurs around the contact boundary for both the conical and spherical indents.
Plastic deformation is contained by elastic deformation zone. It is also shown that the
size of the plastic zone for the conical indent with indenter half angle of 70.3o matches
up with that of the spherical indent when the a/r reaches 0.635, indicated by the
normalized plastic zone size, zys/hs.

(ii) As n and yield strength decrease (E =200 GPa, s = 400MPa, and n = 0.2), the
plastic zone breaks out to the free surface leading to an uncontained deformation
mode, as shown in Figure 1.32b. The plastic zone size of conical indent still matches
up with spherical indent at a/r of 0.635.

48

(iii) For even lower values of n and yield strength (E = 200 GPa, s = 50MPa, and n =
0.1), the material in the contact boundary flows primarily toward the free surface thus
leading to the development of pileup, as shown in Figure 1.32c. Still, the plastic zone
size of conical indent matches up with spherical indent at a/r of 0.635.

Figure 1.32 Comparison of the plastic zone size between conical and spherical
indentations at different values of a/R for (a) highly elasto-plastic solid (E=200 GPa,
s = 1000 MPa, and n = 0.4), (b) a solid with reduced elasto-plastic character (E =200
GPa, s = 400 MPa, and n = 0.2), and (c) a fully plastic solid (E = 200 GPa, s = 50
MPa, and n = 0.1). [Mata et al. 2006]

49

Figure 1.32 cont‟d.

50

Figure 1.32 cont‟d.

1.5

Finite Element Modeling of the Subsurface Deformation Zone

1.5.1 Introduction
In the present work it is hypothesized that the deformation zone under the
indent drives the indentation induced two-way response that underlies surface form
memory. Understanding the characteristics of the deformation zone is critical to
understanding the two-way response. Therefore, finite element modeling has been
used, in addition to experimental studies, to determine the size of the deformation
zone and the distribution of stress and strain under the indent.

51

Shape memory alloys are often modeled by either a phenomenological or a
micromechanical approach. The phenomenological models use measurable
parameters that can make the model computationally efficient and thus suitable for
engineering applications [Tanaka et al. 1982; Ivshin and Pence 1994; Brinson and
Huang 1996]. The micromechanical models follow closely with the crystallography
and use micromechanics to calculate the interaction energy during phase
transformation. Stresses and strains are represented in terms of volume averages and
martensite variants are not differentiated. Furthermore, the micromechanical models
are more computationally demanding since they keep track of many changing
variables. But based closely on physical metallurgy, the models can yield more
detailed information [Patoor et al. 1993; Huang et al. 1998; Goo et al. 1997].

1.5.2

The 3D Superelastic Model
Theoretical understanding of the indentation-induced phase transformations in

NiTi SMAs is challenging because of the coupling of martensitic phase
transformation and plastic slip deformation under complex contact loading conditions.
A number of three-dimensional constitutive models have been proposed based on
existing one-dimensional models. For phenomenological models, the expansion from
one to three dimensions is based on generalized plasticity theory, assuming that the
martensite-austenite phase transformation also holds for the assumptions of classical
plasticity theory. These assumptions are: 1) yield stress is insensitive to hydrostatic
stress; 2) inelastic deformation due to the shape memory effect, like plastic

52

deformation, is volume conserving; 3) the plastic strain increment in each direction is
proportional to the stress deviator component in that direction.

In this work, a three dimensional constitutive model is developed based on
Zhang‟s 3D superelastic model [Zhang et al. 2007] that develops from the
one-dimensional Ivshin-Pence constitutive model [Ivshin and Pence 1994], to
describe evolution of the subsurface deformation zone under contact loading
conditions. The Ivshin-Pence model carefully constructs transformation kinetics to
capture various aspects of the shape memory effect. Zhang‟s 3D superelastic model
simulates the superelasticity of NiTi SMAs in multiple loading directions.


The 3D superelastic model is presented in terms of total strain rate  ij . This
E
total strain rate is decomposed into elastic part  ij and a transformation strain
T
(inelastic) part  ij , as seen in Eq. (1.7)

 E T
 ij   ij   ij
E

－1

Eq. (1.7)
－1



The elastic strain rate is  ij  Eijkl kl . The stiffness tensor Eijkl is represented in
terms of Young‟s modulus E and Poisson‟s ratio .

Lubliner and Auricchio [1996] proposed a 3D framework for the shape
memory effect based on the generalized plasticity theory. The transformation strain
rate is determined by Eq. (1.8):

53

T 
 ij   (

Where

3
Sij)
2 J2

Eq. (1.8)

Sij is the deviatoric stress, J 2 is the second invariant of stress, and  is

the effective plastic strain, the rate of which is expressed in Eq. (1.9):



  

Eq. (1.9)

Here  is the austenite volume fraction, whereas δ is a material constant that gives
the maximum transformation strain that can be induced by the shape memory effect.
This maximum transformation strain δ is typically about 5%. The austenite volume
fraction  = (x, t) is a material dependant function of current von Mises stress and
the overall loading history. The  value is calculated using the Ivshin and Pence
constitutive model. Combining the elastic part and transformation part, the complete
strain rate was expressed as:
－1



 ij Eijkl kl   (

3
Sij)
2 J2

Eq. (1.10)

After elastic loading, the constitutive model switches algorithmically to the
slip-plasticity model. The plastic strain continues to increase according to Von Mises


plasticity so long as the change of the second invariant J 2  0 . Unloading is elastic
and linear, which ends when the stresses go to zero.

54

1.6

Summary
In this chapter, the literature on the discovery of shape memory alloys,

mechanisms of shape memory and the superelastic effect, contact mechanics, and
finite element modeling has been briefly reviewed. The thermoelastic nature of
martensitic phase transformations, twin boundary motion, and lattice correspondence
are three pillars of SME. Tension-compression asymmetry results from the different
Schmid factors between tension and compression directions due to low symmetry in
martensite variants.

Greater attention is given to the mechanisms of two-way shape memory effect,
indentation induced TWSME, and its finite element modeling, due to their importance
to the current work. Most of the literature is focused on training methods instead of
reaching deep into the mechanism of the two-way shape memory effect. The few
reports on TWSME that focused on mechanisms were reviewed. It was reported that
dislocations have been calculated to have lowest energy in the martensite variants
where they were originally formed. This helps to form thermo- dynamically
anisotropic martensite variants, which preferentially form during martensitic
transformation, thus remembering martensite cool shape.

Indentation in martensitic NiTi SMAs leads to a complete recovery for shallow
spherical indent when the representative strain is less than 5%. Deep spherical indents
show a two-way indent depth change upon heating and cooling. This two-way

55

response can be converted to a cyclic exdent-to-flat surface transition, giving rise to
Surface Form Memory (SFM) when the indent is removed by planarization.

56

CHAPTER 2:

2.1

EXPERIMENTAL METHODOLOGY

Introduction
Experiments on indent recovery and indentation/planarization SFM were

conducted in order to learn how to develop functional surfaces with various features
at multiple length scales, how to maximize the indentation-induced two-way response,
what drives the two-way shape memory effect, and how much work can be performed
via SFM.

Details of the experimental procedures are discussed in this chapter.
Specifically, methodologies for specimen planarization, such as hot vs cold
planarization, partial vs post-recovery M-planarization, and direct vs pre-recovery
M-planarization will be discussed in the detail. Procedures for the indent/exdent
measurements will be illustrated, followed by the methodologies for maximizing
TWSME and for measuring the indentation deformation zone. Details of constrained
recovery experiments that measure work output and energy density of SFM will be
discussed. A novel Laser Shock Peening process will also be introduced.

57

2.2

Materials

2.2.1

NiTi Shape Memory Alloys

2.2.1.1 Source and Thermal Response
A 3 mm thick 50.26 at % Ni wrought NiTi alloy sheet was acquired from
Special Metals Corporation. The transformation temperatures, Mf, Ms, As, and Af,
were determined by both Differential Scanning Calorimetry (DSC) and SQUID
magnetometry, as listed in table 2.1. The profiles of DSC and SQUID measurement
were shown in Figure 2.1. The enthalpy for transformation is 15.6 J/g.

Table 2.1 Transformation temperatures measured by DSC and SQUID
Transformation
Temperatures
Mf
Ms
As
Af

DSC, K

SQUID, K

Average, K

295
331
343
390

313
329
335
386

304 ±9
330 ±1
349 ±6
388 ±2

Specimens were first electro-discharge cut into 12.7 mm diameter discs,
ground, and polished, finishing with 0.05 µm colloidal silica. Before indentation
specimens were chilled to 243 K to ensure a fully martensitic condition at room
temperature.

58

Figure 2.1 DSC and SQUID measurements of the transformation temperatures of the
equi-atomic NiTi shape memory alloys.

2.2.1.2 Stress-Strain Behavior
Figure 2.2 shows the stress-strain curve of a NiTi specimen under
compression at a strain rate of 0.01/s. It is shown that the material exhibit two yield
stresses. The upper yield stress indicated as 1 is the yield stress for slip at a level of
315 MPa, while the lower yield stress indicated as 2 is the martensite detwinning
stress at a level of 93 MPa. The Young‟s Modulus was measured to be 5.6 GPa.
During compression, the material first detwins when the stress reaches 93 MPa at
around 1.5% compressive strain. During martensite detwinning, martensite variants
reorient to accommodate the deformation strain, which exhibit a plateau where stress
stays the same when the strain increases. Liu et al. [1998] conducted detailed study on

59

martensitic deformation using Ti-50.0 at% Ni plate and thought the plateau region
represents Luders deformation. Toward the end of the plateau, slip mechanism starts
to play a role. When the stress rises to 315 MPa, martensite detwinning mechanism
ends and slip deformation is the only mechanism that controls the stress-strain curve.

Figure 2.2 Compressive stress strain behavior for the martensitic NiTi used in the
study. 1 is the stress for slip deformation,  2 is the stress for martensite detwinning.

2.2.1.3 Indentation Response
Nanoindentation tests were conducted on the as-received NiTi SMA. Figure
2.3 shows the load-displacement curve during the indentation process. The NiTi
specimen was mechanically ground with 2000 Grit SiC paper and polished down to
0.05 m, using colloidal silica for 6 hours on the vibratory polisher. A Berkovich
diamond indenter was used to make an indent with a controlled depth of 1.5 m. The
actual indent depth, h, after unloading was 1.55 m. The peak load was 200 mN.

60

Therefore, the hardness of the NiTi specimen used can be calculated to be 3.4 GPa
2

based on H=P/A, where A=24.5h for a Berkovich indenter. The Young‟s Modulus
was calculated to be around 5.8 GPa using the stiffness measurement in the unloading
curve.

Figure 2.3 Nanoindentation in NiTi by MTS XP Nanoindenter with an indentation
depth of 1.55 m.

2.2.2

Stainless Steel, Aluminum, and Indenter Materials
Stainless steel grade 304 was used in constrained recovery experiments to

determine the work energy density associated with SFM. The Young‟s modulus and
Poisson‟s ratio of this material are 200 GPa and 0.3. The micro-hardness response
measured by CSM instrumented indenter is shown in Figure 2.4a. The hardness is
measured to be around 0.9 GPa.

61

6061-T6 aluminum has around 96% aluminum and 0.4% copper. The Young‟s
modulus and Poisson‟s ratio of this material are 70 GPa and 0.33. The hardness is
measured to be around 0.7 GPa, as shown in Figure 2.4b.

Figure 2.4 a) Load-Displacement curves of 304 Stainless Steel (Upper), b) 6061-T6
(Lower) used in the constrained recovery experiment.

62

2.2.3

Laser Shock Peening Materials
The materials used in the laser shock indentation experiments were 1/8” thick

1/2” in diameter NiTi discs as previously described, a 65 m thick aluminum foil tape
purchased from 3M Corporation, BK-7 type optical glass, and a Continuum Surelite III
laser for laser pulse generation. Details of the experiment are discussed further below.

2.3

Indentation Methods

2.3.1

Indenter Materials and Properties
The indenters that were used in the experiments are 1/32” and 1/16” diameter

tungsten carbide spherical balls with 6% cobalt (Young‟s modulus of 650 GPa and
Poisson‟s ratio of 0.26) purchased from McMaster-Carr Corporation; 1/100” and
1/16” diameter high purity tungsten rods (Young‟s modulus of 400 GPa and Poisson‟s
ratio of 0.28) from ESPI Metals Corporation; a Berkovich diamond nanoindenter with
Young‟s modulus of 1140 GPa and Poisson‟s ratio of 0.07; a Vickers diamond cone
indenter with a facial angle of 136 degrees; a 1.59 mm diameter tungsten rod used as
a punch indenter; and a 1cm x 1cm stainless steel screen mesh with thickness of 0.254
mm and mesh size of 0.5 mm.

2.3.2

Indentation Loading
All indentation specimens were cooled by liquid Freon to 243K before

indentation to ensure a fully martensitic state, and allowed to return to room
temperature before indentation. A screw driven Instron load frame was used to drive

63

the indenter into the specimen in the martensitic state. During indentation, the strain
rate was 0.01/s. The indentation tests were usually done at room temperature unless
otherwise specified, such as in the replica indentation test and the indentation
temperature variation test.

2.3.3

Post Indentation Thermal Cycling
Thermal cycling after indentation was used to stabilize the two-way indent

depth change prior to planarization. The indents were heated to 433 ±2 K (above Af)
using a thermo-electric cooler (TEC) and cooled to 243 ±2 K (below Mf) using liquid
Freon. The profiles of the indents during each heating and cooling cycle were
recorded optically.

2.3.4

Laser Shock Peening
Laser Shock Peening (LSP) was done through collaboration with Professor Gary

Cheng‟s group at Purdue University. LSP was used to improve indentation efficiency
and to observe how NiTi SMAs react to dynamic loading conditions at ultra high strain
rates. The NiTi specimens were polished down to 0.05 m colloidal silica in the
martensitic state. Before Laser Shock Peening, a 65 m thick aluminum foil tape
purchased from 3M Corporation was attached to the surface of NiTi disc. A confining
media of BK7 type optical glass was used to contain the plasma generated by laser
beam impingement on the aluminum foil. The contained pressure generated by the
plasma penetrated into the base material (NiTi) and made a dent with a large amount of

64

residual compressive stress. Figure 2.5 shows a schematic drawing of the LSP process
in NiTi shape memory alloys. The laser beam was generated by a Continuum Surelite
III laser with a wavelength of 1064 nm and pulse duration of 5 nanoseconds. The beam
2

sizes used were 0.5 and 1 mm. The laser intensities were varied from 1 GW/cm to 3.5
GW/cm2 to vary the indent depth. Figure 2.6 shows the optical image of the as-shock
peened specimens. The NiTi specimen on the left has indents made under a beam size
of 0.5 mm with indent spacing to be 2 mm. The NiTi specimen on the right has indents
made under a beam size of 1 mm with indent spacing to be 2 mm.

Figure 2.5 Schematic illustration of the laser shock peening process on NiTi shape
memory alloys (not to scale).

65

Figure 2.6 Optical image of the laser shock peened NiTi discs. The left sample has
indents made with beam size of 0.5 mm and the right with beam size of 1 mm.

2.4

Indent and Displacement Characterization
The WYKO optical profilometer is a white light interferometer that can

measure surface roughness and 3D image topography non-destructively, as well as
execute statistical computations on surface data. It has high lateral and vertical
resolution and can measure surface height up to 500 microns. It is used here to
measure the initial indent depth in the martensite, indent depth changes upon heating
and cooling, and exdent height and morphology on heating and cooling. Measurement
on heating was achieved by using a thermoelectric cooler (TEC) to 433K and on
cooling by liquid Freon down to 243K and then warming up to room temperature
(about 298K) in air. 2D line profiles and 3D topographical profiles are generated by
the WYKO software for further analysis, which are shown in the results and
discussion chapter.

66

2.5

Obtaining SFM by Indentation/Planarization

2.5.1

Thermo-mechanical Preparation
Various types of indentation were made in the martensitic NiTi to achieve the

indentation-induce TWSME. A heating and cooling cycle was conducted to obtain the
initial recovery of the indents. The indent profiles were recorded optically. Detailed
procedures are described in Section 2.4.1. After this, the indents were planarized to
obtain SFMs, the procedure of which is discussed in the following section. By
indenting with various forms, and by varying the planarization temperatures, various
classes of SFM were obtained (i.e: flat-to-exdented on heating vs flat-to-indented on
cooling.)

2.5.2

Grinding Technique

2.5.2.1 Cold Planarization
2.5.2.1.1

Post-recovery M-planarization

To explore the scope of surface transition responses, various geometries of
indenters were used for indentation, as listed in the indenter characteristics section.
After indentation, profiles of the indents were measured optically using temperature
profilometry procedures described above. After the first recovery (heating above Af
and cooling below Mf), the indents were planarized to restore to a flat surface. In the
planarization step, material was removed by mechanical grinding and polishing with

67

0.05 m colloidal silica to ensure a fine surface finish‡. The amount of material
removal below the base of the cool indent was carefully controlled via thickness
measurements with a resolution of 1-2 m. Since the indent was completely removed
by the planarization step, we call this “Post-recovery M-planarization”. After
planarization, the specimen surface profiles were recorded using the temperature
profilometry procedures discussed previously. Thermally-reversible surface
transitions, and SFM, were then observed after planarization.

2.5.2.1.2

Pre-recovery M-planarization

In the post-recovery M-planarization and partial planarization experiments,
the indents were planarized once they were thermally cycled to observe the initial
two-way indent depth change or “first recovery”. Pre-recovery M-planarization, on
the other hand, means planarization before indent recovery.

As shown in Figure 2.7, post-recovery generated an exdent with an amplitude
of 1 on heating, while pre-recovery generated an exdent with an amplitude of  2 on
heating.  2 is substantially larger than  1. This is because that the detwinned
martensite zone developed after initial indentation helps to push the material up due to

‡

True planarization cannot be accomplished using electrolytic methods because the
latter would simultaneously etch the base of the indent and degrade subsequent SFM
amplitude.

68

shape memory mechanisms. As the detwinned martensite zone had been recovered
during first recovery, the exdent formed on the post-recovery planarized specimen
was thus smaller. The larger exdent formed on the pre-recovery specimen has
potential application in one-way actuation, which was investigated in the constrained
recovery experiment where the exdent formed during austenitization can be used to
make replica indent into a base metal when they are coupled and compressed in the
Instron machine.

Figure 2.7 Comparison of post-recovery M-planarization after first recovery and
pre-recovery M-planarization without first recovery.

2.5.2.2

Post-recovery A-planarization
As illustrated above, deep spherical indents made in the martensitic state

exhibit two-way response either in the form of two-way indent depth change in the
as-indented specimens, or as an exdent-to-flat surface transition in as-planarized

69

specimens, both driven by thermal excursion between Mf and Af. The planarization
process just described was done in the martensitic state. It is also interesting to carry
out the planarization in the austenitic state. As will be shown, a flat-to-indent
transition (as opposed to an exdent-to-flat transition) on cooling may possibly be
achieved in the subsequent thermal cycling. That is, the indent may form upon
cooling and fully recover to a flat surface upon heating. This effect can also be cycled
for an indefinite time. Detailed experiment procedures are described below.

Spherical indents were made under loads of 200, 500, and 700 N with a 0.794
mm diameter spherical tungsten carbide ball. The surface profiles of the initial indents
were measured optically. The spherical indents were then heated to 473 K by a hot
plate to ensure a fully austenitic state. The specimens were then ground down to 2000
Grit sandpaper on the hot plate at 373 K in order to keep the specimens in the
austenitic state. The grinding procedure was carefully controlled so that the indent
was just ground away at 2000 Grit sandpaper without removing material beneath the
indent at the austenitic state. After planarization, the indents were fully martensitized
by cooling in liquid Freon to 243 K. The surface profiles of the indents after cooling
at room temperature and after subsequent heating to 433 K were recorded optically.
Thermal cycling was done 10 times at the temperature range between 243 K and 433
K to ensure a full reversibility in the change of surface features.

70

2.6

Maximizing Indentation-induced TWSME

2.6.1

Effect of a/r Ratio
For this series of experiments, spherical indents were made in martensitic

NiTi discs (12.7 mm in diameter) using a 0.794 mm diameter tungsten carbide ball
with loads of 200, 700, 1500, and 3000 N. Cylindrical indentations were made with a
0.254 mm diameter 6 mm long tungsten rod under loads of 500, 1500, 3500, and 5000
N, with the cylinder axis parallel to the indented surface. As described before, the a/r
ratio for spherical indent refers to the ratio of indent contact radius over indenter
radius. For cylindrical indents, the indenter radius is taken as the cylinder radius, and
the indent “radius” was taken as the indent half width, c.

Three dimensional profiles of the as-indented surfaces, and profiles for first
recovery of spherical and cylindrical indents were acquired optically using the
temperature profilometry procedures, as described in section 2.4.1.

2.6.2

Effect of Thermal Cycling
For the above spherical indents made under loads of 200 and 700 N, 25

heating and cooling cycles were applied to assess both the cyclic reversibility of the
shape change and the effect of cycling on the two-way response. Profiles of the
indents after each heating and cooling cycle were recorded. The indent depth change
suggests a thermal cycling effect which is further analyzed by measurement of
changes in transformation temperatures by a Superconducting Quantum Interference

71

Device (SQUID). Magnetic susceptibility is the ratio of an induced magnetization to
the applied magnetic field. The susceptibility of conduction electrons in a metal, i.e.
The Pauli susceptibility, depends on the density of states at the Fermi level. Different
crystallographic lattice structures, such as martensite and austenite, possess different
hole-carrier densities and thus display different susceptibilities. In the present study,
the SQUID was used to measure the transformation temperature change of the
as-indented specimen before and after thermal cycling. The specimen used in the
SQUID test was prepared by cutting a 3 gram 3 mm thick NiTi piece and making a
closed packed array of spherical indents on both sides of the specimen in order to
maximize both the dislocation density and the extent of the indentation deformation
zone. A control specimen with no indents was also used to serve as a baseline for
measuring transformation temperature changes.

2.6.3

Effect of Indentation Spacing
As a rule of thumb, indents need to be made away from each other and from

the edge of the sample in order to effectively measure the material‟s mechanical
properties such as hardness and stiffness (for instrumented indentation). Indents made
close to each other will be influenced by overlapping the indentation subsurface
deformation zones. As far as SFM is concerned, however, maximizing this interaction
may be a way to maximize SFM. Therefore, experiments with various indent spacings
were designed to evaluate how subsurface deformation zone overlap affects

72

indentation two-way response, and SFM, as the indent spacing is reduced from
infinity to zero (double indentation).

In this experiment, spherical indents were made with different spacing under
500 N using a 0.794 mm tungsten carbide ball and assessed the effect of indentation
spacing on the indentation-induced two-way response. The surface profiles of the
initial indents and the indents after heating to 433 ±2 K and cooling to 243 ±2 K
were measured optically. Two heating and cooling cycles for each indent were carried
out. Indentation at zero spacing is discussed below.

2.6.4

Effect of Multiple Indentations
A variation of maximizing the deformation zone by varying indent spacing is

to make indents at the same location for a number of times. This repeated indentation
method has been investigated by Zarudi et al [2003] to study the plastic deformation
behavior of monocrystalline silicon under cyclic microindentations with a spherical
indenter. In the present study, repeated deformation was used to increase plastic
deformation, thus the size of subsurface deformation zone, and finally the two-way
response. Specifically, spherical indents were made under 500N load using 0.794 mm
tungsten carbide ball in the same location 2, 5 and 10 times. The surface profiles of
the initial indents and the indents after heating to 433 ±2 K and cooling to 243 ±2 K
were measured optically. Two heating and cooling cycles for each indent were

73

observed. The effect of multiple indentations on the two-way response was then
extrapolated.

2.6.5

Effect of Indentation Temperature
Due to the transformational behaviors of NiTi SMAs, temperature was used

as a parameter to vary the mechanical and shape memory properties. Specifically,
spherical indents were made under 500 N using a 0.794 mm tungsten carbide ball at
temperatures of 333 K (16 K below As), 293 K (11 K below Mf), 273 K, and 213 K
(well below Mf). Temperatures were measured by the thermocouples spot welded on
both sides of the specimens. 333 K was reached by slowly heating the specimens
under convection using two heat guns. 273 K was reached by ice water and 213 K
was reached by using liquid nitrogen and an acetone slurry bath to cool the NiTi
specimen as it sat on an aluminum block inside the slurry bath. Three dimensional
profiles of the initial indents made under these temperatures, and the indents after
heating to 433 ±2 K and cooling to 243 ±2 K were measured optically.

2.6.6

Effect of Cold Rolling
Cold rolling is a metalworking process that increases the yield strength and

hardness of a metal by introducing defects into the metal‟s crystal structure. These
defects inhibit further slip. Interest in the study of the effect of cold rolling prior to
indentation was based on the fact that slip deformation is necessary for the
indentation-induced two-way effect. It is important to find out if cold rolling induced

74

deformation can sum with indentation-induced deformation to achieve a larger
two-way effect.

Spherical indents were made under 500 N load using an 0.794 mm tungsten
carbide ball in martensitic NiTi that had been pre-cold-rolled to 0%, 5%, 10%, and
30%. The surface profiles of the initial indents, and the indents after heating to
433 ±2 K and cooling to 243 ±2 K, were measured optically. Three heating and
cooling cycles for each indent were observed. The effect of cold rolling on the
two-way response will be further discussed in the Results and Discussion chapter.

2.7

Determining the Size of the Active Zone
The operating hypothesis for two-way indent depth change or SFM is that

indentation-induced two-way response is driven by the subsurface deformation zone.
The amplitude of two-way response, whether measured as an indent depth change, or
as an exdent-to-flat transition, is presumed to be directly related to the size of the
subsurface deformation zone, D*. Therefore, it is important to measure this size of
this zone, D*, which may provide direct information for understanding the
indentation-induced TWSME. This has been accomplished with a series of controlled
serial planarization experiments described below.

75

2.7.1

Front-Thinning Experiments
The front thinning experiments incrementally removed surface material below

the base of the as-recovered indent until the measurable exdent amplitude for each
increment decays to near zero during a subsequent thermal cycle. The depth of
removal by each planarization increment was determined by measuring the thickness
change at a chosen reference point with a high-precision micrometer having a
resolution of 1-2 m. The surface profiles of the exdent were independently measured
optically. The measured exdent amplitudes were recorded as a function of the depth of
material removal. Both cylindrical and spherical indents with 3 different a/r ratios
were used in the front-thinning experiments to determine the depth of the active
deformation zone.

2.7.2

Back-Thinning Experiments
Instead of removing material from the front surface, back thinning

experiments remove material from the back side, step by step, to an extent that the
two-way indent depth change or exdent amplitude (if the indent was planarized before
back thinning) just begins to decrease during a subsequent thermal cycle following the
most recent thinning step. This depth presumably corresponds to the deepest active
region in the deformation zone. The remaining depth of material up to the base of the
indent provides a second measurement of the depth of the active deformation zone.

76

A spherical indent was made under 500 N with a 0.794 mm tungsten carbide
ball in martensitic NiTi. The specimen was then heated and cooled to observe the
initial two-way response using the temperature profilometry procedure. Then the
specimen was thinned from the backside and the depth of removal was again
determined by measuring the thickness change at a chosen reference point with a
high-precision micrometer. The specimen was heated and cooled using the
temperature profilometry procedure after each back thinning experiment to observe
any change in two-way response.

2.7.3

Direct Observation of the Indent Deformation Zone
A spherical indent was made under a load of 7500 N using a 1.59 mm

stainless steel ball in a martensitic NiTi specimen. A cylindrical indent was made
under a load of 6000 N using a 6 mm long 1.59 mm in diameter tungsten rod with the
NiTi in the martensitic state. After indentation, the block containing the spherical
indent was cut into two halves along the indent centerline using a diamond saw. The
cross section face was then hot mounted and cooled to 243 K by liquid Freon before
being polished down to 0.05 m colloidal silica in the martensitic state at room
temperature (293 K). Due to hot mounting, the indent recovered which was not
intentionally designed. However, this does not greatly affect the study of the active
subsurface deformation zone.

77

The profiles of the initial indent, the residual half indent, and the
cross-sectioned surface after heating to 433 ±2 K and cooling to 243 ±2 K were
measured by optical profilometry.

2.8

Indent Replication and Energy Estimates

2.8.1

Introduction
NiTi SMAs are well known for their high work energy density and wide

application to actuation. SFM is different as compared to conventional actuators made
out of NiTi SMAs because it is a surface morphology change. However, SFM has
good potential in application into NEMS/MEMS devices, aero/hydrodynamic surfaces,
and robotic systems. Therefore, it is important to measure the energy density
associated with the formation of surface exdents in order to evaluate its application
potential. Constrained recovery experiments were designed to evaluate SFM in single
shot and reversible application conditions. When combined with measurements of the
deformation zone depth, the effective transformational energy density can be
estimated.

2.8.2

One Shot Replica Indentation
Spherical indents were made with a 0.794 mm diameter tungsten carbide ball

using loads of 200, 700, and 3000 N. Each sample was indented at the same location
twice since this had been found in the previous work to give a stronger training effect.
Surface profiles of the as-indented surfaces were acquired with the optical

78

profilometer. The loads used gave a/r ratios of 0.42, 0.68 and 1.0 respectively, as
summarized in Table 2.2. After indentation, but before any heating above Af, and
while still in the martensitic condition, the surfaces were ground and polished just
enough to remove visible traces of the indent. This method, called pre-recovery
M-planarization, was used in order to maximize the exdent amplitude during
exdentation in the constrained recovery experiment.

Table 2.2

Parameters of the Initial Spherical Indents in Martensitic NiTi
Load
N
200
700
3000

Depth
m
37
107
380

Contact radius a
m
167
271
397

a/r
ratio
0.42
0.68
1.00

Constrained recovery experiments were designed to obtain a rough estimate
of the work output capability of the exdents when they develop under strong reaction
forces.

After planarization, the optically-flat NiTi samples were placed in contact

with coupons of 3 mm thick 6061-T6 aluminum and positioned between compression
platens in a screw-driven load frame. The NiTi-aluminum sandwich was preloaded to
a compressive stress of 100 MPa. The assembly was then slowly heated by
convection to 423 K to austenitize the NiTi, thus allowing the NiTi exdent to indent
the aluminum coupon. (Thermal expansion in the assembly caused the stress to rise
to about three times the initial preload.) After reaching 423 K, the specimen was
unloaded immediately and cooled to room temperature (298 K) via air cooling.

79

Dimensional profiles of the indented aluminum surfaces were subsequently
acquired using the WYKO profilometer. Additionally, profiles were obtained for the
residual exdents in the NiTi samples, both directly following the indent replication
experiment (at room temperature), and also in the austenitic condition, with samples
heated to 423 K, in situ in the profilometer stage, using a thermoelectric module.

To provide a basis for estimation of the work output of the NiTi exdent,
conventional spherical indents were made in the same 6061-T6 aluminum plate using
a 6.3 mm diameter tungsten ball under various loads. The diameter of this ball was
chosen to match the radius of curvature of the replica indents.

Indents that were

found to have a/r ratios comparable to those of the replica indents were chosen to
provide force-displacement data to give a rough measure of the work of conventional
(Hertzian) indentation in 6061-T6 aluminum.

2.8.3 Reversible Replica Indentation by SFM
Reversible replica indentation by SFM is interesting in 2 aspects: 1) it
demonstrates the repeatability of SFM; 2) it shows the fatigue property of SFM. The
experimental procedures are described in the following.

Spherical indents were made with a 1.59mm diameter stainless steel ball
using a load of 3000 N. Surface profiles of the as-indented surfaces were acquired
with the optical profilometer. The initial indent depth was 240 microns and the a/r

80

ratio was calculated to be 0.71. 15 heating and cooling cycles were applied to assess
cyclic reversibility of the shape change. Temperatures were measured by
thermocouples spot-welded at points on either side of the specimen. After thermal
cycling, while still in the martensitic condition, the surfaces were ground and polished
just enough to remove visible traces of the indent.

A schematic illustration of the constrained recovery experiment is shown in
Figure 2.8. After planarization, the optically-flat NiTi samples were placed in contact
with coupons of standard 6061-T6 aluminum, 304 stainless steel, and 440 stainless
steel positioned between compression platens in a servo-hydraulic machine. A
servo-hydraulic apparatus was set up to maintain a constant load throughout the test,
since heating induced expansion will lead to an increase in applied load if placed in
Instron machine. The NiTi-replica material sandwich was preloaded to compressive
stresses of 50, 125, 200, 250, and 300 MPa using a servo hydraulic load frame to
maintain a constant load under conditions of changing temperature. The assembly was
then heated by four Cartridge Heaters placed next to the sandwich to 433 K to fully
austenitize the NiTi, thus allowing the NiTi exdent to indent the replica materials.
Two thermocouples were attached to the sandwich during heating and cooling to have
an accurate measurement of temperature. After heating to 433 K, the sandwich was
immediately unloaded to zero Newton, then cooled down to 293 K in air. Profiles of
the indented replica material surfaces were subsequently acquired optically.

81

Figure 2.8 Schematic illustration of the setup of the constrained recovery experiment.
304 stainless steel (SS) is used as an example of the replica material.

The effect of cooling under load was also evaluated by unloading the
sandwich after it was cooled down to 293 K. Experimental results show that cooling
under load leads to a larger response. The detailed results will be reported in Section
3.5.3.4. Dimensional profiles of the indented replica material surfaces were also
acquired optically.

To provide a basis for the estimation of the work output of the NiTi exdent,
spherical calibration indentation was conducted to measure the hardness of the
material being preloaded to 250 MPa and heated to 433 K using 1.59 mm diameter
tungsten ball under load of 30, 50, and 100 N. (250 MPa was chosen because this load
generated the maximum replica indent depth for these three materials.) The hardness,
measured by the load divided by the projected area, was used to calculate the
equivalent load needed to make the replica indent and for the estimation of the work
output of the NiTi exdent.

82

To estimate the volumetric energy density, these measurements were combined
with experimental estimations of D*, taken as the depth of a normally hemispherical
active deformation zone. This measurement effectively ignores any contribution from
elastic strain energy release in regions below D*.

2.9

Finite Element Modeling

2.9.1

The 3D Constitutive Model
A three dimensional constitutive model is developed based on Zhang‟s 3D

superelastic model [Zhang et al. 2007] that develops from the one-dimensional
Ivshin-Pence constitutive model [Ivshin and Pence 1994], to describe evolution of the
subsurface deformation zone under contact loading conditions. The Ivshin-Pence
model carefully constructs transformation kinetics to capture various aspects of the
shape memory effect. Zhang‟s 3D superelastic model simulates the superelasticity of
NiTi SMAs in multiple loading directions. The 3D constitutive model, however,
simplifies the superelastic model and simulates the indentation process into
elasto-plastic materials. The constitutive model is temperature independent since the
indentation process is modeled at 25oC and below Mf. Slip-plasticity is included in
this model to capture the mechanical response of NiTi SMA at large deformation
strains. NiTi material parameters were extracted from the tensile test, as shown in the
experimental section (Figure 2.2). Implementation of the model used a commercial
finite element program (ABAQUS) and a user material subroutine (UMAT).

83

The difference between the constitutive model and the superelastic model is
that the constitutive model does not account for the austenite volume fraction since
the material stays in the martensitic condition. Therefore, the effective plastic strain
cannot be expressed as a function of the austenite volume fraction. Instead, it is solely
a function of the current von Mises stress, as shown below:




1
3

Sij ij
h 2 J2

Eq. (2.1)

Where h is the plastic hardening slope with units of stress.

Thus the total strain can be calculated to show Eq. (2.2):
1 


 ij  Eijkl kl   (

2.9.2

3
Sij )
2 J2

Eq. (2.2)

Finite Element Program Implementation
The finite element implementation is accomplished using the User Material

Interface provided by ABAQUS. The strain increment is passed to the user material
subroutine and the corresponding stress increment is calculated from the constitutive
relations using a return mapping algorithm. The state variables, including effective
plastic strain, are updated. The strain increment can be expressed as Eq. (2. 3):
1
d ij  Eijkl d kl  1 3 Sij S pq d pq
h 4J 2

Eq.(2.3)

Multiplying both sides of the equation by the elastic modulus gives the trial stress
increment, shown in Eq. (2.4).

84

d t

E
ijkl
ij

d kl  d ij 

3
1
S S d
Eijkl
4J 2 kl pq pq
h

Eq.(2.4)

The elastic trial stress increment can also be expressed as Eq. (2.5):

d t

ij



( ip  jq



3
1
S S )d pq  Cijpq d pq
Eijkl
4J 2 kl pq
h

Eq.(2.5)

After Cijpq is calculated, the real stress increment d pq can be obtained by the
t

correction of the elastic trial stress increment d ij .

The mesh for the indentation model comprises 15,000 axisymmetric 4-node
bilinear elements. The indenter was modeled as a rigid body sphere for spherical
indentation and the friction coefficient between the indenter and the substrate is
assumed to be zero. For cylindrical indentation, the 3D model was reduced to 2D due
to the plain strain nature of cylindrical indentation. The indenter was simplified to a
quarter of a circle and the friction coefficient between the indenter and the substrate is
assumed to be zero.

85

CHAPTER 3:

3.1

RESULTS AND DISCUSSION

Introduction
Various indentation methods have been used in shape memory alloys (SMAs)

to study shape memory and superelastic characteristics. Ni and Zhang in Grummon's
group reported that shallow spherical indents made by a CSM instrumented indenter
can completely recover after heating above Af due to the shape memory mechanism,
while deep spherical indents became shallower but remained after heating due to
irreversible slip deformation. Following partial thermal recovery, cooling caused
these deep indents to deepen, reflecting a two-way depth change. When a
planarization step was applied to the indented martensite after the indent's initial
recovery, a thermally-reversible flat-to-bumpy transition was observed. We now call
this surface transition “Surface Form Memory” (SFM). These transitions are fully
reversible and cyclically stable.

In this chapter, experimental results will be presented on developing SFM
using different indenter geometries and different planarization steps (full vs partial,
hot vs cold, and direct vs indirect). Ways to maximize surface form memory will be
discussed considering the effects of a/r ratio, thermal cycling, indentation spacing,
multiple indentations, indentation temperature, and pre-cold-rolling. Experimental
results on probing the size of the active zone to understand surface form memory will

86

be compared to finite element modeling. The potential application of surface form
memory for actuation will be examined in terms of its work output and energy density
via one-shot and reversible constrained recovery experiments. Finally, an advanced
laser-based technique for making the initial indents will be demonstrated, allowing an
interesting comparison of indentation response between quasi-static (ball) and
dynamic (strain rate >106) conditions.

3.2

Developing and Maximizing the Indentation-Induced TWSME

3.2.1

Indent Recovery by the Indentation-Induced TWSME
Figure 3.1 schematically shows results for a typical indent made, in this case,

with a 794 micron diameter sphere at a load of 700 N. The indent had an initial depth
of 96.8 µm, giving an a/r ratio of 0.65, and thus a representative strain of 0.13. Upon
heating to above Af, the indent depth decreased from 96.8 µm to 56.9 µ by shape
m
memory mechanisms. This initial shape-change represents a depth recovery that is
41.2% of the initial indent depth. When cooling the specimen back below Mf the
indent depth increased to 65.9 µm by two-way shape-memory mechanisms. We call
this indent depth change upon heating and cooling right after indentation “indent first
recovery”. This two-way indent depth change induced by a single indentation method
represents a non-trivial shape memory phenomenon, as compared to various
thermo-mechanical training methods usually used in achieving TWSME. The
magnitude of the indent depth change will be compared with traditional TWSME and
its mechanism will be extrapolated based on experimental results.

87

Figure 3.1 Schematic illustration of initial indent depth recovery of the spherical
indent made with a 0.79 mm diameter tungsten carbide ball under 700N in the
martensitic state.

3.2.2

Maximizing Indent Recovery

3.2.2.1 Effect of a/r Ratio
3.2.2.1.1 Spherical Indents
Spherical indents are not geometrically the same at different indentation
depths. They are generally characterized by a/r, the ratio of the in-plane indent
contact radius a and the indenter radius r. This ratio determines the representative
strain (r=0.2 a/r), which characterizes the deformation strain in spherical indentation.

Spherical indents were made under various loads that resulted in a/r ratios
ranging from 0.4 to 0.95. (Note that for a/r =1.0 the sphere would be pushed down to

88

its equator.) Figure 3.2 plots the normalized indent depth change  over indenter
radius, r, as a function of the a/r ratio for the indentation experiments in this series.
The normalized cyclic indent depth change is seen to increase linearly with the a/r
ratio, reaching a maximum for a/r = 1.0 at which point the depth change is 4.3% of
the indenter radius.

Furthermore, the depth change extrapolates to zero for an a/r

ratio of 0.26, reflecting a representative strain of roughly 5%, consistent with our
previous observation that spherical indents producing less than 5% representative
strain have one-way (not two-way) shape memory effect [Ni et al. 2003]. The curve
fitting equation suggests an empirical way to predict the two-way amplitude based on
an a/r ratio, as shown in the figure.

100/r = 5.97±
0.15 a/r - 1.55

Eq. (3.1)

The higher the a/r ratio, the larger the two-way response is.

Eq. (3.1) can be modified to express the relation between two-way indent
depth and the initial indent depth, ho, as shown in Eq. (3.2)
100 /ho = 5.97±
0.15 sqrt(2 r/ho -1) -1.55 r/ho

Eq. (3.2)

Figure 3.3 shows the relationship between two-way response and a/r ratio
based on the experimental results and calculations based on Eq. (3.2). As shown in the
figure, the maximum two-way response is about 11.5% when normalized to the initial
indent depth. Based on equation 3.2, we can calculate that when a/r of 0.5 or

89

r/ho=7.95, the two-way response reaches a maximum of 10.8%. Based on the close
match between calculation and experimental results, it is reasonable to use Eq. (3.2) to
predict the amplitude of the two-way response based on the initial indent depth and
the indenter radius. This allows us to better control how deep the indent should be to
reach a certain two-way response on a given indenter. To predict based on Eq. (3.2)
what load to use when we change the size of the indenter, it is necessary to modify Eq.
(3.2) to include the indentation load, F, into the equation, as shown in Eq. (3.3):

100 = 5.97±
0.15 sqrt(F/*Hm) -1.55r

Eq.(3.3)

where the hardness of martensite is Hm=F/*a2.

Based on Eq. (3.3), we can predict the amplitude of the two-way response

Indent Depth Change/r, %

based on an indenter with any size and an indentation load at whichever magnitude.
5

y = 5.97±0.15x - 1.55
R2 = 0.9845

4

Spherical Indents
r=397 m

3
2

1
0
0

0.2

0.4

0.6

0.8

1

a/r ratio
Figure 3.2 Plot of the indent depth change (normalized to the indenter radius) as a
function of the a/r ratio for spherical indents.

90

Indent Depth Change Ratio,
/ho, %

Experimental
Results

Spherical Indents
r=397 m

15

Calculation
based on Eq.
(3.2)

12
9

6
3
0

0

0.2

0.4

0.6

a/r ratio

0.8

1

Figure 3.3 Plot of the two-way response, normalized to ho as a function of a/r ratio
based on the experimental results and calculations based on Eq. (3.2).

3.2.2.1.2 Cylindrical Indents
Similar to spherical indents, cylindrical indents are not geometrically the
same for different indentation depths. The c/r ratio, instead of a/r ratio, taken as the
cylindrical indent trench half-width, c, divided by cylinder radius, r, can be used to
represent the cylindrical indentation strain. Different from spherical indents, where
the representative strain is 0.2 a/r, the literature does not give a value for the
representative strain of a cylindrical compression. However, we can make a simple
model to estimate that constant, as shown below.

91

Assuming the threshold of representative strain to induced TWSME in NiTi
SMAs is 5%, for both spherical and cylindrical cases§, the empirical equation of
representative strain for the spherical case is r,(sph,crit)=sph a/r, where sph is 0.2.
As shown in Figure 3.4, the two-way response drops to zero at a c/r of 0.2, thereby
giving cyl=r,(cyl,crit)/0.2= 0.25 . This coefficient cyl of 0.25 can thus be used to
calculate the representative strain for the cylindrical compression.

Cylindrical indents were made with a tungsten rod lying parallel to the
surface and placed under various loads to give c/r ratios between 0.4 to 0.95. The
length of the cylindrical rod was about 6 mm, sufficiently greater than the indenter
radius, to eliminate end-effects.

Figure 3.4 plots the initial indent recovery

(normalized to the indenter radius, r, of 127 µm) as a function of the c/r ratio.
Unlike the spherical case, the cylindrical indent recovery amplitude increases
non-linearly with the c/r ratio, again reaching a maximum for a c/r = 1.0, with an
amplitude of 10.3% of the indenter radius (more than twice the corresponding
amplitude found for the spherical case).

We assume the threshold strain for indentation is also 5% even though NiTi
has a high tension and compression asymmetry.
§

92

Figure 3.4 Two-way indent depth changes for cylindrical indents, normalized to the
indenter radius, plotted as a function of the c/r ratio.

Curve fitting the dataset is used to get a sense of the trend of the relation
between indent depth change ratio and the c/r ratio, as shown in Eq. (3.4).
2

100 /r = 11.6(c/r) - 2.2 c/r

Eq. (3.4)

The initial indent depth can be included in the equation to show the relation
between indent depth change and the initial indent depth, as shown in Eq. (3.5).
100 / ho = 11.6(2-ho /2r) - 2.2 sqrt(2r/ho -1)

Eq. (3.5)

This equation can be further modified to show the relation between indent
depth change and the indentation load, as shown in Eq. (3.6).

100 = 11.6 F/2rL*Hm - 2.2 sqrt(F/2L*Hm)

93

Eq. (3.6)

Where F is the indentation load, L is the indenter length, Hm is the hardness of
2

martensite, and c = F/2L*Hm.

Table 3.1 Curve-fitting model for two-way indent depth change  for spherical and
cylindrical indents
Indentation Parameter
Affecting c (two-way

Spherical Indentation

Cylindrical Indentation

100 /r =5.97±
0.15 a/r
-1.55

100 /r = 11.6(c/r) - 2.2

100 /ho =5.97±
0.15

100 /ho = 11.6(2-ho /2r)

sqrt(2 r/ho -1) -1.55 r/ho

- 2.2 sqrt(2r/ho -1)

depth change)
Indenter Radius r

2

c/r

Indent Depth ho

3.2.2.2 Effect of Thermal Cycling
Thermal cycling is usually used as a method to stabilize the two-way response
and assess cyclic stability of the two-way response. Table 3.2 summarizes the thermal
cycling parameters and indent depth changes at a/r ratios of 0.4 and 0.65 for spherical
indents. As the table shows, thermal cycling increases indent depth change by about
40% at a/r of 0.4 and 0.65. Specifically, thermal cycling increases indent depth
change rapidly during the first few cycles, then the effect levels off, as shown in
Figure 3.5. This „negative-fatigue‟ behavior is yet to be understood, but one

94

possibility is that a cyclic softening effect involving dislocations in the subsurface
zone leads to a decrease of Af after thermal cycling.

More details of this cyclic evolution can be seen in Figure 3.6, which shows
the variation of the indent depth in the fully austenitic and the fully martensitic state
on thermal cycling, for spherical indents, at a/r of 0.4 and 0.65. It is apparent in
Figure 3.6 that the indent depth (plotted in negative value) in austenite increases with
thermal cycling, whereas the higher the a/r ratio, the larger the increase is. Note that it
is the austenite depth that changes, while upon cooling, the indent depth remains
unchanged along with thermal cycling. It may be that more and more retained
martensite transforms to austenite with thermal cycling, leading to an increase of
indent depth (negative value in Figure 3.6) in the austenitic state. Martensite may be
retained by the residual stress exerted by the defects in the deformation zone after
indentation. If the softening of the defects upon thermal cycling relaxes the residual
stress, it may allow more and more retained martensite to transform.

Table 3.2 Thermal cycling parameters and indent depth changes for spherical indents
with a/r ratios of 0.4 and 0.65
st

1 indent
a/r
0.4
0.65

recovery
(µm)
3.8
9

th

25 indent
Change%

95

Cooling
Temp

42.1%
38.9%

recovery
(µm)
5.4
12.5

Heating
Temp
433 K
433 K

243 K
243 K

Figure 3.5 Change of the indent depth upon thermal cycling for spherical indents with
a/r ratios of 0.4 and 0.65.

Figure 3.6 Change of the indent depth during heating and cooling for spherical indents
with a/r ratios of 0.4 and 0.65.

96

Another possible outcome of dislocation relaxation is the decrease of
transformation temperature Af, as is shown in Figure 3.7, which plots the change of
magnetic susceptibility due to phase transformation against temperature change. The
experiment was conducted in a Superconducting Quantum Interface Device (SQUID),
which can investigate deformation related properties and localize plastic deformation
during crack growth [Bonavolonta et al. 2007]. The samples used in the present study
were a three gram un-indented NiTi specimen and a four gram indented NiTi
specimen. Spherical indentations was made under a load of 500 N with a 0.79 mm
spherical ball. We know the measured magnetization or susceptibility is related to the
density of dislocations inside the specimen [Loloee et al. 1995]. The as-received
specimen shows the least amount of susceptibility, as shown in Figure 3.7, while the
indented specimen has larger susceptibility due to the indentation induced dislocations.
After thermal cycling of the indent, SQUID measurements show the transformation
temperatures do in fact shift towards lower temperatures. Af decreases from 400 K to
364 K and As decreases from 358 K to 330 K. Ms and Mf remain unchanged after
thermal cycling. This is a direct evidence for the measured thermal cycling effect and
the change of indent depth in the austenitic state with repeated heating. While Af
keeps decreasing, more and more retained martensite transforms into austenite at the
fixed thermal cycling temperature range used in the current experiment. Note too that
this suggests that „negative fatigue‟ result would vanish if a suitably high upper
temperature were used during thermal cycling.

97

3.2.2.3

Effect of Indentation Spacing
Assessing the effect of indent spacing is essentially asking about the effect of

overlapping the deformation zones beneath the indents. Figure 3.8 shows the effect of
the indentation spacing on the indent depth change for a spherical indent made under
500 N. The indentation spacing does not have a significant effect on the two-way
response until the indentation spacing is decreased to 2.9 a. This is a significant data
point because the indent made under 500 N with a r=400 m spherical ball in the
martensitic state has an a/r ratio around 0.6. Based on the front thinning experiments
which will be discussed later, the D* can be estimated to be around 1.4 a for a
spherical indent with a/r of 0.6. The overlapping effect is not supposed to play a role
until the spacing gets below 2.8 a, as shown in 3.8.

Figure 3.7 SQUID measurements of the as-indented specimen before and after
thermal cycling and the as-received NiTi specimen.

98

At a spacing of 2.9 a, where 10% of the indentation induced active zones
between two indents overlap, the indent depth change increases by a small amount
(about 5%) and the overlapping effect begins. When the two deformation zones
completely overlap (“double indentation”), a maximum increase in two-way response
of around 30% (from 6.8 to 9 µm) is observed. The two-way response further
increases to 10.7 µm (from 9 µm), showing another 20% increase. This shows double
indentation and thermal cycling can be coupled to maximize the two-way response.
Double indentation is thought to increase the size of the deformation zone that drives
the two-way response. Continuing the effect of double indentation, further tests on
multiple indentations are presented below.

Figure 3.8 Plot of the indent depth change against the normalized indentation spacing
d/a for 500 N spherical indents.

99

3.2.2.4 Effect of Multiple Indentations
Double indentation increases the two-way response as shown above in Figure
3.8. Multiple indentations to 5 and 10 repetitions of spherical indentation were used to
see if the two-way response can be further increased, with results shown in Figure 3.9.
As can be seen, multiple indentations have a larger two-way response than the double
indentation. This might be due to the fact that multiple indentations increases the
indent depth, as shown in Figure 3.10, where the initial indent depth is plotted as a
function of the number of repeated indentation. From the single indentation to 10
times of indentation, the initial indent depth increases from 70.7 m to 89.6 m. The
a/r ratio increases from 0.57 to 0.63. Based on Eq. (3.1), as shown in Figure 3.2, we
can calculate that the two-way response should change from 7.3 m to 8.9 m if the
a/r ratios changes from 0.57 to 0.63 based on the effect of a/r ratio. Since the results
show that the two-way response changes from 7 to 9.5 m, we can infer that it is the
increase of indent depth, thus a/r ratio, that accounts for the increase in two-way
response observed in the multiple indentation experiments.

Furthermore, as shown in Figure 3.10, the second thermal cycle increases the
two-way response by about 20% on average(from 9 to 11 m), which is almost three
times the increase of the single indentation (7%, from 7 to 7.5 m). Based on study of
the thermal cycling effect described in section 3.3.2, we know that the second thermal
cycle of a spherical indent made by single indentation has an a/r of 0.65, with an
increase in indent depth change by 10% (from 9 to 10 m). Therefore, multiple

100

indentations have a larger thermal cycling effect than single indentation. It also
suggests that multiple indentations not only increase a/r ratio, but also helps to
decrease Af faster to achieve full transformation between martensite and austenite
after indentation. However, this hypothesis is not verified and can be supported by the
thermal cycling study.

Figure 3.9 Change of indent depth as a function of repeated indentation times for 500
N spherical indents.

101

Figure 3.10 Plot of the initial indent depth as a function of repeated indentation times
for 500 N spherical indents.

3.2.2.5

Effect of Indentation Temperature
Mechanical properties such as hardness, yield stress, and young‟s modulus

are all different in martensite and austenite. Therefore, temperature can be effectively
used to change the indentation response in NiTi SMAs. Ms, Mf, As, Af are 4 major
characteristic transformation temperatures characterizing the phase transformation.

Spherical indents were made in both martensite and austenite for studying the
two-way response. Specifically, spherical indents were made at temperatures of 213,
293, 328, 333 and 383 K under a load of 500 N with a 794 m diameter spherical ball.
Since the specimens were first cooled in liquid Freon to 243K and Mf is 295 K, As is

102

343 K, the specimens were ensured to be in the martensitic state at a temperature
below 343 K while in the austenitic state at 383 K.

Figure 3.11 plots the initial indent depth as a function of the indentation
temperature. It is shown that the lower the temperature is, the shallower the indent
depth is for a given load for indentation of the martensite below As. This is due to the
fact that the material hardness increases with decreasing temperature in the
martensitic state. The flow stress for deformation increases accordingly, leading to a
decrease of the indent depth. It is shown that the maximum initial indent depth in the
martensitic state is 104 m (a/r is 0.67), which can be achieved at a temperature just
below As. The smallest initial indent depth in this experiment occurs at 213 K and is
69 m deep (a/r is 0.56), which is 50% less than the indent made at 333 K. Heating
above As transforms the specimen into the austenite phase. The indent depth in
austenite decreases because austenite is generally harder than martensite.

Figure 3.12 is a plot of the indent depth change as a function of indentation
temperature. We can see that the lower the indentation temperature is, the smaller the
indent depth change is. This is probably because the harder the material is at low
temperature, the less slip deformation the indentation load can induce, thus generating
less two-way response.

103

The maximum indent depth change occurs in a specimen that has the largest
initial indent depth. It is more than twice as large as for the indent with the lowest
initial indent depth. It is interesting to see that two-way response also occurs for the
indent made in the austenite phase. It is possible that slip deformation occurred during
deformation in austenite and remained in the specimen after unloading. This suggests
that the indentation induced two-way response is directly related to slip deformation.

Figure 3.13 plots the indent depth change ratio (normalized over the initial
indent depth) as a function of the indentation temperature. The maximum indent depth
change ratio is 13.4%, which is about 50% larger than the ratio of the indent made at
the lowest temperature.

Figure 3.11 Plot of initial indent depth as a function of indentation temperature for
500 N spherical indents.

104

Figure 3.12 Plot of indent depth change as a function of indentation temperature for
500 N spherical indents.

Figure 3.13 Plot of the indent depth change ratio (normalized to initial indent depth)
as a function of indentation temperature for 500 N spherical indents.

105

3.2.2.6

Effect of Prior Cold Rolling
It is hypothesized that the slip deformation zone under the indent helps induce

the two-way shape memory effect underlying SFM phenomena. Therefore, the cold
rolling experiment was designed to plastically deform the specimen before indentation
to see if this would maximize the slip deformation zone and to increase the two-way
response.

Spherical indents were made under 500 N in the pre-cold rolled specimens
with 5%, 10%, and 30% compressive pre-strains produced by cold rolling in the
M-state. The 500 N spherical indent made in the un-deformed specimen was used as a
control experiment. Three thermal cycles were applied to the specimens to obtain the
two-way indent depth change. Figure 3.14 plots the change of the initial indent depth
against the percentage of cold rolling. It can be seen that cold rolling work hardens
NiTi specimen and leads to a decrease of indent depth up to 50% for the 30% cold
rolled specimen, as compared to the un-deformed specimen. Figure 3.15 plots the
two-way indent depth change as a function of the cold rolling percentage. Prior cold
rolling greatly decreases the two-way indent depth change at the first thermal cycle by
up to 90% for the 30% cold rolling specimen, as compared to the control specimen.
nd

2

rd

and 3 thermal cycles increase the indent depth change for the indents in the

control specimen and the 5% cold-rolled specimen. The 10% and 30% cold rolled
specimens may be too heavily deformed to show any change in two-way response
upon thermal cycling.

106

Admittedly, the conclusion on the effect of cold rolling could be better
supported if spherical indents with same a/r ratios were made in the cold rolled
specimens. However, making indents with same a/r ratios in specimens cold rolled to
various percentages requires a lot of trial and error and is not practical in engineering
applications, thus is not the concern of this study.

Figure 3.14 Plot of the initial indent depth as a function of the percentage of cold
rolling for 500 N spherical indents.

107

Indent Depth Change, m

10
1st thermal cycle
2nd thermal cycle
3rd thermal cycle

8
6
4

2
0

0

10

20
30
Cold Rolling, %

40

Figure 3.15 Plot of indent depth change as a function of the percentage of cold rolling
for 500 N spherical indents.

3.2.2.7 Maximum Two-way Response
Table 3.3 summarizes the effects of all the parameters used to improve
indentation induced two-way response. It is shown that an indentation temperature of
333 K, double indentation, and thermal cycling can be effectively used to increase the
two-way response. Therefore, experiments were designed to combine all these effects.
A spherical indent was double indented under load of 500 N at temperature of 333 K
with a r=400 m spherical ball. After indentation, the indent was thermal cycled 15
st

th

th

th

times and measurement of indent depth change was made at the 1 , 5 , 10 , and 15
thermal cycle.

108

Table 3.3 The parameters and their effects on maximizing the two-way responses of
the spherical indents.
Parameters

Initial response

Max two-way
response, %

Change %

a/r

11.8 % at a/r of 0.4

11.8 % at a/r of 0.4

Baseline

Thermal
Cycling*
Indentation
Spacing
Multiple
Indentation
Indentation
Temp
Cold Rolling

11.8 % at a/r of 0.4
9.9% at a/r of 0.6
9.9% at a/r of 0.6
9.9% at a/r of 0.56

16.7 % after thermal
cycling
10.9 % with 0
indent spacing
10.9 % with double
indentation
13.4% with 60oC
indentation

9.9% at a/r of 0.56

9.9 % at a/r of 0.56

42%
10%
10%
34%
0

*Indent two-way response is 9.9-10.2% of the initial indent depth at a/r of 0.56.

The initial indent was measured to be 90.3 m, with a/r being calculated to be
0.64. The indent depth change on the first heating and cooling cycle was measured to
be 12 m. This is 13.3% of the initial indent depth, larger than the maximum effect
(11.8%) observed in the a/r ratio experiments. Figure 3.16 plots the thermal cycling
effect on the double indented indent made at 333 K, compared to the single indented
indent made under 700 N at 293 K at an a/r of 0.65. These two experimental results
are comparable due to similar a/r ratio. As shown in the figure, double indentation at
higher temperature indicates a larger initial recovery and a drastic increase at the
initial 5 thermal cycles. The maximum increase from 12 to 17 m for double
indentation by thermal cycling again demonstrates a 40% increase, the same as that of
a single indentation (from 9 to 12.5 m, a 39% increase). Interestingly, the two-way

109

response of the double indented spherical indent decreases thereafter and stabilizes at
13 m, matching the maximum response (12.5 m) of the single indent with similar
a/r ratio. This result again adds support to the hypothesis that thermal cycling
decreases Af and completes the phase transformation between martensite and
austenite. The result also suggests that double indentation at temperature close to As
does not necessarily make the active zone larger, but facilitates phase transformation,
thus having a larger initial recovery and reaches a saturation point faster.

To summarize, various methods such as a/r ratio, thermal cycling, indentation
spacing, multiple indentation, indentation temperature, and cold rolling have been
used to evaluate their effects on the two-way response of the spherical indents. Only
cold rolling has demonstrated a negative effect on maximizing two-way response, as
shown above. Table 3.3 summarizes the parameters used and their effects on
maximizing the two-way responses. It is shown that a combination of a/r ratio,
indentation temperature, and thermal cycling can be used to most efficiently increase
the two-way response. It is estimated that the increase can be up to 90% for an indent
with a/r of 0.56 and the maximum two-way response can be 19.1% --[10.2%*(1+34%)*(1+40%)]. The experimental results support the hypothesis by
showing that the maximum response 17 m is 19% of the initial indent depth (90.3
m) or as a 90% increase from the two-way response of a single indent (9 m).

110

Indent Depth Change, m

20

Spherical Indents
a/r=0.65
a/r=0.64

15

10

5
0

10
20
# of Thermal Cycle

30

Figure 3.16 Comparison of the two-way response due to thermal cycling for spherical
indents made by double indentation at 333 K and single indentation at 293 K with
similar a/r ratios.

3.3

Planarization for Surface Form Memory

3.3.1 Introduction to the Indentation-Planarization Method
The indentation-planarization method used to develop two-way surface form
memory is schematically shown in Figure 3.17. A spherical indent is made in the
martensitic NiTi using a spherical ball with a radius of r at a load P to a depth h. The
indentation develops three subsurface deformation zones: a represents the slip zone; b
represents martensite detwinned zone; c represents elastic deformation zone, as
indicated in Figure 3.17b. Thermal cycling between Af and Mf leads to a two-way
indent depth change with amplitude 1, as shown in Figure 3.17c. Thermal cycling
after planarization (in Figure 3.17d) results in a reversible exdent-to-flat transition

111

with amplitude 2. Both indent depth change and exdent-to-flat transitions have been
demonstrated to occur for up to 25 thermal cycles.

3.3.2

Post-recovery M-planarization
Surface profiles of the exdent-to-flat transition for a specimen made by

spherical indentation-planarization are shown in Figure 3.18, as measured optically.
The spherical indent was made under 700 N with a 0.794 mm tungsten carbide ball.
The a/r ratio is 0.65. The exdent height is around 9 m. The size of the exdent is
around 500 m. This makes the contact angle of around 2o. Here, the contact angle is
used to compare exdent responses at different a/r ratios and to characterize various
exdent shapes made from different indenters. For spherical exdents, the maximum
contact angle is calculated to be 2.4o, where the exdent amplitude is around 16 m,
the size of the exdent is around 800 m, and the a/r ratio of the indent is 0.95.

Figure 3.19 shows (a) a reversible cylindrical exdent-to-flat transition, (b) a
punch exdent-to-flat transition, (c) a mesh exdent-to-flat transition, (d) a pyramidal
exdent-to-flat transition and (e) a comparison between a pyramidal exdent and a
spherical exdent. In figure 3.19 (e), the vertical scale is adjusted for better
visualization.

112

Figure 3.17 Indent training (a-c) and planarization (d) to obtain Surface Form
Memory (SFM) „exdents‟ in NiTi. Thermally cycling between (b)  (c) and (d) 
(e) is indefinitely repeatable. Indentation results in three deformation zones under the
indent: a: Plastic Slip Zone, b: Detwinned Martensite Zone, c: Elastic Deformation
Zone (Semi infinite volume).

A cylindrical indent was made by indenting a 0.254 mm diameter tungsten
rod with cylinder axis parallel to the indented surface under 1500 N. The cylindrical
indent can be thought as an extension of spherical indents with a 2 dimensional
deformation zone. The maximum contact angle is calculated to be around 5.7o when
the exdent amplitude is 12 m, the size of exdent is 240 m, and the a/r ratio is 0.95.

113

Flat punch indents were made by indenting a 0.254 mm in diameter and 2 mm
long tungsten rod with rod axis perpendicular to the sample surface. The exdent
replicates the punch indent shape, as shown in Figure 3.19 (b). Owing to the flat
punch geometry, a maximum contact angle (90o) can be achieved. The flat punch
indents can be placed close to each other to achieve maximum exdent density. These
closely packed punch exdents can be used to generate larger actuation response.

Mesh indents were made by indenting a stainless steel screen mesh with a
mesh size of 0.5 mm. The junction points in the mesh indent can be thought as stress
concentration points which induce larger indentation deformation zone underneath it
and generate higher exdent response, as shown in Figure 3.19 (c).

A Vickers indenter having a half angle of 68o was used to develop pyramidal
exdents. As shown in Figure 3.19 (d), a pyramidal exdent-to-flat transition occurs
repeatedly upon heating and cooling after planarizing a Vickers indent. The exdent
shares the same contact angle as is the half angle of the Vicker‟s indenter (70.3o).

Figure 3.19 (e) compares the shape of a spherical exdent with a pyramidal
exdent. Spherical exdent has a smooth transition in curvature. Pyramidal exdent
replicates the shape of the Vickers indenter with a sharp peak in the center. The
plastic strain at the peak is assumed to reach singularity due to stress concentration.
The vertical scale is exaggerated for better visualization.

114

Figure 3.18 Cyclic exdent-to-flat surface transitions after planarization of a spherical
indent.

Figure 3.19 Profiles of various exdent shapes measured after indentation-planarization
in a series of hot-cold-hot: a) cylindrical bumpy-to-flat transition; b) punch
bumpy-to-flat transition; c) mesh bumpy-to-flat transition; d) pyramidal bumpy-to-flat
transition; e) spherical exdent is compared to pyramidal exdent. (The vertical scale is
adjusted for better visualization).

115

Figure 3.19 cont‟d.

116

Figure 3.19 cont‟d.

3.3.3

Pre-recovery M-planarization
To further the understanding of first recovery, a planarization step was

conducted on indents before recovery in the martensitic state. A spherical indent was
made under a load of 500N with a r=0.794 mm spherical ball in the martensitic state.
The initial indent depth was optically measured to be 78.6 m. The a/r ratio was
calculated to be 0.6. After indentation, the indent was planarized in the martensitic
state to restore a flat surface without first recovery. We call this procedure
“Pre-recovery M-planarization”. The specimen was then heated to 433 ±2 K (above

117

Af) and cooled to 283 ±2 K (below Mf) using a thermo-electric cooler (TEC). The
surface profiles of the specimen upon heating and cooling were obtained optically, as
shown in Figure 3.20. Heating after pre-recovery M-planarization generated an exdent
with amplitude of 38 m, which is around 50% of the initial indent depth. Cooling,
however, could not lead to a transition to complete flatness, but to an exdent with
amplitude of around 28.8 m. This confirms that the detwinned martensite zone
recovered after heating cannot fully recover upon cooling. Interestingly, the decrease
in amplitude of around 9 m upon cooling was found to be the same as the first
recovery of a spherical indent with a similar a/r ratio. This suggests that planarization
does not change the slip deformation zone. The material with a larger volume
underneath the indent plays an important role in the indentation-induced TWSME.
Apparently the material in the radial zone around the indent contributes little to the
effect.

The large exdent formation after pre-recovery M-planarization generates
appreciable energy and can be applied to high energy actuation devices. To this end,
replica indentation experiments were carried out to study the work output and energy
density of NiTi SFM, the detail of which will be illustrated in section 3.5.2.

118

Figure 3.20 Surface profiles of the initial 500 N spherical indent, exdent upon heating
and cooling after pre-recovery m-planarization.

3.3.4

Post-recovery A-planarization
In the current research, NiTi specimens at a given temperature have one of

three possible surface forms: exdented, flat, and indented. There are consequently
totally 32=9 possible classes of surface form memory. Exdent-exdent, flat-flat, and
indent-indent transitions are trivial cases, since there is no change in form. Therefore,
6 variations are left to be developed, as shown in Figure 3.21, with variation 1 being
the class demonstrated up to this point. In order to develop the remaining five
variations, indentation and exdentation methods in the martensitic state were

119

developed and planarizations at hot, median, and cold temperatures were used.
Median temperature refers to a temperature at which martensite and austenite coexist
(optimal condition is 50% martensite and 50% austenite). Indent-to-exdent or
exdent-to-indent transition (variation 5 and 6) can possibly be developed using
median temperature planarization method.

Figure 3.21 Six possible variations of Surface Form Memory in NiTi SMAs.

Thus far it has been demonstrated that indentation and planarization in the
martensitic condition turns the indent depth change into reversible exdent-to-flat
surface transition (variation 1). It is possible to achieve a flat-to-indent transition
(variation 4) by planarizing the indent in the austenitic state. This planarization
method is called “Post-recovery A-planarization”. To this end, a spherical indent was
made under 200 N with a r=0.794 mm spherical ball in the martensitic state. Figure
3.22 shows the surface profiles of the 200 N spherical indent, specimen under cooling

120

after post-recovery A-planarization, and under subsequent heating. The initial indent
depth was measured to be 38.2 m and the a/r ratio was calculated to be 0.42. After
post-recovery A-planarization, the specimen was cooled to 243 K and a spherical
indent was observed with a measured depth of 5.6 m at 293 K. This indent depth
after first recovery is 14.7% of the initial indent depth. This effect is larger than what
we have observed for the initial indent depth change ratio (the maximum is 11.4%
[Fei et al. 2009]). The surface roughness Ra for the specimen is optically measured to
be 500 nm. The specimen was then heated to 433K and the surface profile measured
optically shows a flat surface with roughness Ra to be 100 nm and Peak-to-Valley
difference Rt on the order of 1 micron. Apparently, heating has smoothend the surface
due to shape memory recovery.

Subsequent cooling drove the formation of the indent again, while heating
afterward led to a flat surface with a small exdent having amplitude of around 0.6 m.
This exdent occurred consistently on the following thermal cycles, as shown in Figure
th

3.23. At the 10 thermal cycle, the indent depth in cooling was measured to be 6.4
m, while the exdent height in heating was measured to be 1.1 m. This difference of
7.5 m is about 34% increase from the initial indent recovery (5.6 m).This result
matches with what we observed for the spherical indent that has been thermal cycled
25 times and shown a 40% increase in amplitude. The small exdent amplitude on
heating is unexpected but can be removed by a subsequent fine polish of the surface

121

in austenite. The details of the thermal cycling effect have been discussed in Section
3.2.2.2.

To conclude, the surface exdents are thought to be related to the indenter
geometries and deformation zones under the indents. The maximum contact angle of
the surface exdents is 90o for flat punch exdent. Spherical exdents have the lowest
contact angle due to the smooth transition of the curvature of the spherical exdent.
The mechanism of forming surface exdent is further studied in Section 3.4. Methods
such as indentation, exdentation, and hot, median, cold planarizations can generate six
variations of surface form memory, which can be possibly applied into many areas.

3.4

Characterization of the Active Deformation Zone

3.4.1

Front Thinning Experiments

3.4.1.1 Spherical Indents
It was previously shown that the material above the base of the indent does not
affect the two-way response based on the comparison between post-recovery
M-planarization after first recovery and pre-recovery M-planarization without first
recovery. Therefore, it becomes important to study the deformation zone underneath
the indent in order to understand the indentation-induced TWSME and SFM. To this
end, step by step front-thinning experiments were conducted in order to locate a
critical depth of material removal (dpl) that would cause the exdent effect to vanish, as

122

shown in Figure 3.24. This critical depth is represented by D* in the following
discussion and refers to an active zone that drives two-way shape recovery.

Figure 3.25 shows the relationship between exdent height (he) and the amount
of material removed by polishing (dpl) for the case of spherical indents with a/r of 0.4,
0.65, and 0.95, respectively. Not surprisingly, as successive layers of material are
removed the exdent amplitude decreases. The decrease is nonlinear but can be fitted
to a parabolic curve. Extrapolation of these curves to zero exdent amplitude is a
gauge of the depth of the subsurface deformation zone responsible for the exdent
effect (referred to the initially planarized flat surface). This depth D* has values of
179, 378, and 718 m for the spherical indents with a/r ratios of 0.4, 0.65 and 0.95,
respectively. Normalized to the initial indent contact radius, a, these depths are
1.13a, 1.47a, and 1.90a, respectively.

123

Figure 3.22 Surface Profiles of 200N spherical indent, indent upon cooling after
post-recovery A-planarization, and subsequent heating.

124

Figure 3.23 Subsequent thermal cycling of 200 N spherical indentation after
post-recovery A-planarization to ensure the reversibility of surface feature transition.

Figure 3.24 Schematic illustration of the front thinning experiment.

125

Figure 3.25 Results of the front-thinning experiments. The stable cyclic exdent height
is plotted as a function of surface removal depth for spherical indents with a/r ratios
between 0.4 and 0.95.

3.4.1.2 Cylindrical Indents
The cylindrical indentation deformation zone was also studied using
front-thinning experiments.

Figure 3.26 shows exdent height vs. material removal

for cylindrical indents with a/r ratios of 0.42, 0.64, and 0.95. This data can be
linearly curve-fitted and could again be extrapolated to determine the removal depth,
D*, expected to cause zero exdent amplitude. D* values for the cylindrical indents
were 95.2, 186 and 502 m, or 1.78a , 2.28a , and 4.15a, for a/r ratios of 0.42, 0.64,
and 0.95 respectively, all substantially larger than found for spherical indents having

126

comparable a/r ratios. Table 3.4 summarizes the indent responses (hmax, D*) for both
spherical and cylindrical indents at various a/r ratios.

Stable Exdent Height, he, micron

12

Cylindrical Indents:
Rind = 127 microns

8

4
a/r=0.95

a/r=0.4

0

0

100

a/r=0.64

200

300

400

500

Planar Depth Removed, dpl, micron
Figure 3.26 Results of successive planarization experiments. The stable cyclic
exdent height is plotted as a function of surface removal depth for cylindrical indents
with a/r ratios between 0.4 and 0.95.

Table 3.4 Responses of spherical and cylindrical indents at different a/r ratios
Indent Type
Spherical

Cylindrical

a/r

hmax(m)

D*(m)

D*/a

0.4
0.65
0.95
0.4
0.64
0.95

4.1
9.5
16.1
2.1
4.3
12.5

179
378
718
94
185
500

1.13
1.47
1.9
1.78
2.28
4.15

127

3.4.2

Back Thinning Experiments
Front thinning experiment registers the size of deformation zone, D*,

successfully. However, it cannot provide support whether it is the plastic deformation
zone itself that induces two-way response in NiTi SMAs. It may well be that the
elastic deformation zone under the indent serves as a “bias spring” that pushes the
indent up upon heating and pulls the indent down upon cooling. To address this
problem, back thinning experiments were designed to: 1) verify the size of the active
zone, D*; 2) show whether the elastic deformation zone is responsible for indentation
induced two-way response. (Note that the zone laterally outside the D* hemisphere
may contribute.)

A spherical indent with initial indent depth of 91.3 m, thus a/r of 0.64, was
used in back-thinning experiments in which material from the backside of the
specimen was removed by successive grinding up to a depth close to the base of the
indent, as shown in Figure 3.27.

With back thinning, the difference between the indent depth in heating and
cooling did not decrease, but instead increased by about 12% due to the thermal
cycling effect, as shown in Figure 3.28, which plots the indent depth change upon
heating and cooling as a function of the material remained, divided by over contact

128

Figure 3.27 Schematic illustration of the back thinning experiment.  decreases after a
critical depth of material was removed from the backside of the specimen.

radius a. It was until the deformation zone was just beginning to be polished away
where the indent depth change rapidly decreased. In Figure 3.28, zero refers to the
bottom of the indent after first recovery, thus matching closely with the results of the
incremental front thinning experiment. The result shows the size of the deformation
zone measured by back-thinning experiment was 1.7a for a spherical indent with an
a/r of 0.64. Figure 3.29 plots the indent depth in martensite and austenite as a function
of material remained, normalized to the contact radius. It is shown that the indent
depth in martensite decreases quickly after the active zone starts being polished away.

Figure 3.30 shows the results of front and back thinning experiments, and the
spherical indentation deformation zone modeled by ABAQUS [Fei et al, 2010]. The
spherical indentation deformation zone was modeled by indenting an elasto-plastic
material under general elasticity and Von Mises plasticity mechanisms. The material
properties used in the model were measured by tensile testing of the martensitic NiTi

129

dog-bone shaped specimen. In Figure 3.30, the experimentally measured D* from
back thinning matches with the front thinning experiment relatively well and
corresponds to a deformation zone that has deformation strain of around 7% which is
beyond the strain that can be accommodated by martensite deformation twinning
mechanism. Therefore, it is concluded that slip deformation occurs in the active zone
and dislocations are responsible for the indentation-induced TWSME. Elastic
deformation zone is ruled out in the back thinning experiment as being responsible for
two-way response.

Figure 3.28 Plot of indent depth change as a function of the remained material,
normalized over the contact radius a.

130

Figure 3.29 Plot of the indent in martensite and austenite, normalized over the contact
radius a, as a function of the remained material, normalized over the contact radius a.

Figure 3.30 Plot of D* based on front and back thinning experiments with spherical
indentation deformation zone modeled by ABAQUS.

131

3.4.3

Two-way Strain in the Active Subsurface Zone
Deep spherical indentation in the martensitic NiTi (Figure 3.17a) creates a

slip zone, a detwinned martensite zone, and an elastic deformation zone, as shown in
Fig. 3.17b. Both the amplitude of reversible indent depth changes (  1 in Figure 3.17),
and the height  2 of the surface protrusions (or exdents) were found to be related to
the depth of a subsurface zone in which indentation has induced strains beyond those
that can be mediated by deformation twinning reactions in the martensite [Fei et al.
2009], as discussed in detail below.

Successive material removal experiments for larger a/r ratios (larger than 0.25)
have identified a critical depth below the planarized indent, D*, that defines the depth
of material driving the observed two-way cyclic exdent phenomenon. Comparing
the maximum exdent amplitude with the value of D* determined by successive
grinding reveals a simple relationship between these two quantities. Figure 3.31,
which plots he as a function of D* for both spherical and cylindrical indenters, shows
that the data collapses onto a single line. A linear fit (forced through 0,0) gives a line
with a slope of 0.024. It may thus be concluded that for both indenter types, and over
a range of a/r ratios, that the cyclic exdent amplitude is consistently 2.4% of the depth
of the „active zone‟ below the base of the indent. This value apparently represents the
average shape-memory recovery strain occurring within the active zone.

132

It has also been found that successive planarization results for spherical
indents at various a/r ratios, which showed a non-linear dependence on material
removal depth, can be collapsed onto a single curve using a fitting relation with a
single adjustable parameter.

Table 3.5 shows the a/r ratios and D* of various spherical indents measured
in the front and back thinning experiments, which are used in the plot of stable cyclic
exdent height as a function of the depth of the subsurface active zone, D*, as shown in
Figure 3.32. The slopes of the linear curve fit in the front and back thinning
experiments give an average strain in the volume defined by D* of ~2.4% and 1.9%
respectively.

Table 3.5 Summarize the a/r ratios and D* measured in the front and back thinning
experiments
Pln
Methods
a/r
D*

Front Thinning
0.4
1.13a

0.65
1.47a

Back Thinning
0.95
1.9a

133

0.37
1.35a

0.64
1.7a

0.87
2a

Exdent Height, he, micron

Figure 3.31 Exdent heights for both spherical and cylindrical indents with various a/r
ratios between 0.4 and 0.95 plotted as a function of the size of the subsurface active
zone, D*. The slope of the linear curve fit gives the average strain in the volume
defined by D*.

20

Front Thinning
Back Thinning

15

Curve Fitting
through 0,0

0.024

10

0.019

5

0

0

200

400

600

800

D*, micron
Figure 3.32 Stable cyclic exdent height for spherical indents as a function of the depth
of the subsurface active zone, D*, in the front and back thinning experiments.

134

Figure 3.33 show cyclic exdent amplitude data for three a/r ratios fitted by the
relationship:

   d 2   d 2 
he

pl
pl
  
 
 C 31  
 D *    1  D *  
D*
  
 
  
 


Eq. (3.7)

Here, dpl is the material depth removed by grinding (referenced to the flat, planarized
surface), he is the cyclic exdent amplitude for dpl = 0, and C is a constant found to be
0.006. This relationship is similar to that found by Zhang (Zhang et al. 2008) except
that the contact radius, a, used in the latter, has been replaced by the active zone depth,
D*, which allows collapse of curves for various a/r ratios on to a single plot.

Figure 3.33 Data for spherical indents for all a/r ratios studied normalized to the
experimentally determined depth of the subsurface active zone, D*. The data may
be adequately fitted to eq. (3.7) which has a single adjustable parameter.

135

3.4.3.1 The Spherical Cap Model
Based on the spherical cavity model [Johnson, 1986] in contact mechanics, as
discussed in section 1.4, we speculate that the deformation zone is hemispherical
under the indent. To understand the relationship between the exdent height and
residual deformation zone after planarization, we consider a mathematical model
analyzing the residual volume of hemisphere underneath the indent after planarization
in the following discussion.

The volume of residual spherical hemisphere can be calculated based on
Figure 3.34 which shows a 2D cross-section of spherical hemisphere after a depth of
material being removed, dpl. This residual volume of the spherical hemisphere, Vres, is
given by:

1
Vres   ( D * d pl )[3( D *2 d pl 2 )  ( D * d pl ) 2 ]
6

Eq. (3.8)

Where D* is the size of the spherical deformation zone.

Figure 3.34 2D illustration of residual spherical cap after a depth of dpl of material
being removed.

136

Vtot is the hemisphere with radius of D* and equals to

2
D *3 . Thus
3

residual volume ratio can be calculated as shown in Eq. (3.9),

Vres

Vtot

2
2
 d pl     d pl    d pl  
 
 
1 
 3 1 
  1
 

D *    D *   
D* 
 
 

 
 
 

Eq. (3.9)

4

Zhang [et al. 2008] found that the exdent height is proportional to the residual
volume of the spherical hemisphere divided by the residual depth and the contact
radius for an a/r ratio of 0.57. The normalized exdent height versus normalized depth
removal can be closely fitted by Eq. (3.10) shown as below.

d pl 2
d pl 2
he
 C{3[1  (
) ]  (1 
) }
a
a
a

Eq. (3.10)

where C is a constant (independent of indentation depth and indenter radius), a which
is the contact radius is exactly the radius of spherical hemisphere under the indent in
the reference.

Here, we use D* as a generalized term for the radius of the spherical
hemisphere. Equation 3.7 fits the dataset for spherical front thinning experiment well
and closely matches with Eq. 3.10.

The relationship between the normalized exdent and the normalized residual
volume can be derived as following based on equation 3.7 and 3.9,

137

d pl 1 Vr e s
he
 4C (1 
) (
)
D*
D*
Vt o t

Eq. (3.11)

Equation 3.11also suggests that the as-planarized exdent height is 2.4% of the
size of deformation zone, D*, for spherical indents regardless of the a/r ratios. The
exdent amplitude can be related to the residual spherical volume after planarization
with a coefficient (D*- dpl) or residual depth.

3.4.3.2 The Cylindrical Cap Model
As seen in Figure 3.35, results for the cylindrical indents can fit into a single
curve by normalizing the data to the respective values of D* for various a/r ratios.
The linear fitting curve fits the data points very well, showing the same result as for
the spherical indents. That is, the exdent height is 2.4% of D* regardless of the a/r
ratios.

Assuming that the deformation zone under the cylindrical indent can be
modeled as a cylindrical cap, we can construct a 2D mathematical model to describe
the relation between the exdent and the residual volume per unit length (or cross
section area), using the schematic drawing shown in Figure 3.36.

The residual cylindrical cross section area is calculated in Eq. (3.12),

Ares  D *2 c o 1(
s

d pl
D*

138

)  d pl D *2 d pl 2

Eq. (3.12)

where Afan is the area of the fan with angle theta, Atri is the area of a triangle inside
the fan. Ares is double the rest area of the fan. Atot is the area of the half circle with
radius of D*.

The ratio between residual cross section and total cross section area is then
given by Eq. (3.13):

Ares

Atot

1
2( c os (

d pl
D*

)

d pl
D*

1 (



d pl 2
) )
D*

Eq. (3.13)

Similar to the spherical indentation case, we assume that the exdent height is
related to the residual cylindrical cap volume ratio. We then obtain Eq. (3.14):

d pl
d pl 2
he
1 d pl
 Ccylinder ( c os (
)
1 (
) )
D*
D* D*
D*

Eq. (3.14)

Figure 3.35 shows that Eq. (3.14) fits very well the experimental data with two
fitting parameters, Ccylinder = 1.5 x 10-2 and D* = 1.85a, 2.35a and 4.35a.
Based on equation 3.13 and 3.14, the relation between exdent height and
residual cross section area ratio is shown in equation 3.15.

he 
A
 C( r e s
)
D* 2
At o t

Eq. (3.15)

Equation 3.15 also suggests that the as-planarized exdent heights for cylindrical
indents are 2.4% of D* regardless of the a/r ratios. The exdent amplitude is linearly
correlated with residual area after planarization with a coefficient of 2.4%.

139

Figure 3.35 Data for cylindrical indents for all a/r ratios studied normalized to the
experimentally determined depth of the subsurface active zone, D*.

Figure 3.36

Schematic illustration of residual area in the cylindrical cap after a

depth of dpl of material away. Afan is the area of the fan with angle ,

Atri is the

area of a triangle inside the fan. Ares is double the rest area of the fan. Atot is the area
of the hemisphere. dpl is the depth of material being removed and D* is the size of the
deformation zone.

140

3.4.4

Direct Observation of the Indentation Deformation Zone

3.4.4.1 Spherical Indent and Deformation Zone
We demonstrated that the behavior of the deformation zone under the
spherical indent is consistent with a hemispherical shape. Here, we directly observe
the shape of the spherical indentation deformation zone in NiTi shape memory alloys
due to its response to thermal stimulus. A spherical indent was made under 7.5 KN
with a r=800 m spherical ball. The initial indent was too deep to be measured
optically (the limit is 400 m). After cutting along the center line of indent, the
cross-section area was hot mounted and polished and measured to have an indent
depth of 315 m. (The indent recovered due to hot mounting.) Figure 3.37
schematically illustrates the cross-sectional view of the spherical indent. Figure 3.38
shows the surface profiles of the cross-sectional view of a spherical indent at room
temperature, after heating to 433 K, and after cooling to 243 K, measured optically.
The indentation deformation zone is clearly shown in Figure 3.38 after the specimen
was heated to 433 K. The shape of the exdent in the cross-section of the spherical
indent after heating represents the shape of the subsurface deformation zone, which is
hemispherical under the base of the indent. This agrees with our speculation about the
shape of the deformation zone and supports our method to use D* as an important
parameter to represent the subsurface deformation zone size.

The residual indent depth was measured to be 315 m for a r=800 m
indenter. The indent depth changes from 315 m to 290 m on heating, representing a

141

25 m two-way amplitude. It has been reported that the two-way response is 2.4% of
D*. Therefore, the D* can be calculated to be around 1036 m, as labeled in the
middle graph of Figure 3.38. As can be seen in the figure, the exdent showing the
deformation zone matches with D* closely.

3.4.4.2 Cylindrical Indent and Deformation Zone
A cylindrical indent was made with 1.59 mm diameter 6 mm tungsten rod
under 6 KN. The initial cylindrical indent was measured to have a depth of 94 m and
the a/r ratio of 0.47. The first recovery of this indent was measured to be 16 m. After
this, the indent was cut along the centerline by diamond saw in the martensitic state.

Figure 3.37 Schematic drawing of the cross-sectional view of the spherical indent
after cutting.

142

Figure 3.38 A cross-sectional optical image of a spherical indent at room temperature
(293 K) after polishing, after heating to 433 K, and after cooling to 243 K and
measured optically at 293 K.
Figure 3.39 schematically illustrates the cross-sectional view of the after-cut
cylindrical indent. Figure 3.40 shows the surface profiles of the cross-sectional view
of the after-cut cylindrical indent at room temperature, after heating to 433 K, and
after cooling to 243 K, measured optically. The residual indent depth could not be
easily determined optically because of the resolution of the color gradient and the fact
that the indent depth is quite small compared to the size of scanning window.
However, the indentation deformation zone can still be clearly observed optically.

143

The shape of the exdent in the cross-section view of the cylindrical indent is linear in
the horizontal direction which represents the geometrical property of a cylinder. Since
the indent was cut along the centerline, the half indent can still represent the
deformation zone of a cylindrical indent. Based on the result that the two-way indent
depth change is 2.4% of D* , D* is calculated to be around 660 m, as in Figure 3.40,
which correlates with the size of the cylindrical deformation zone.

Figure 3.39 Schematic drawing of the cylindrical indent after cutting.

144

Figure 3.40 A cross-sectional optical image of a cylindrical indent at room
temperature (293 K) after polishing, after heating to 433 K, and after cooling to 243 K
and measured optically at 293 K.

3.5

Indent Replication

3.5.1 Introduction
NiTi shape memory actuators can produce stresses excessing 400 MPa
--1000 times larger than the human muscle [Hunter et al. 1992]. Their energy density,

145

on the order of 10 MJ/m3, is 10 times larger than that of piezoelectrics [Park et al.
1997]. These high stress and energy density characteristics allow SMA actuators to be
used in various applications such as medical devices [Maeda et al. 1996], robots [Lee
et al. 2004], and smart structures [Michaud et al. 2002]. It is important to determine
the apparent work energy density associated with SFM events.

3.5.2

One Shot Spherical Replica Indentation
Due to the interest in the application of surface exdent into actuators, replica

indentation experiments were made in aluminum using spherical exdents during a
constrained austenitization process. Combined with previously described
measurements of D*, this allows estimation of work-energy density in SFM.
Spherical indents were made using double indentation under loads of 200, 700, and
3000 N with a 0.794 mm diameter tungsten carbide ball in the martensitic state and
directly planarized in order to maximize the exdent amplitude. Planarized NiTi
specimen was placed against 6061-T6 aluminum and inserted in between two Instron
platens. After heating to 150 oC, an exdent formed and indented aluminum under a
preload of 100 MPa. Surface profiles of the replica indents are shown in Figure 3.41,
together with profiles of the corresponding NiTi exdents taken at room temperature
after the replica indents were made. Measurements derived from these profiles are
summarized in Table 3.4, which additionally includes the measurements of the
exdents after the replication experiments as re-heated to 423 K in the profilometer.

146

The latter measurement may be construed as the free-recovery amplitudes of the
exdents.

Figure 3.41 Residual exdents in NiTi (left) and Replica indents in aluminum (right) at
293 K made directly following the replication experiments. Images (a,b), (c,d) and
(e,f) correspond to initial indent a/r ratios of 0.42, 0.68 and 1.00 respectively. In
each image the field size is 1.9 mm x 2.5 mm. Vertical scales vary, as shown.
Images on the left have been inverted to correspond to the relative orientations of the
features during their formation.

It is of interest to compare the replica indent depth to the height of the
corresponding NiTi exdent in the austenitic condition at 423 K (see right most column

147

in Table 3.6). This is shown in Figure 3.42 which plots these measurements.

For

the present case of indent replication in 6061-T6 aluminum, the replica indent depth is
approximately half of the austenitic exdent height.

Table 3.6
Initial
indent
a/r ratio
0.42
0.68
1.00

Replica Indent and Residual Exdent Dimensions

Replica
indent
depth, drepl
m
7.1
11.4
25.8

Replica
radius of
curvature,
m
3808
8812
15443

Replica
indent
a/r ratio

NiTi Exdent
height, m
(293 K)

NiTi Exdent
height, m
(423 K)

0.061
0.051
0.058

10.7
20
42

16.4
27.6
51.2

Figure 3.42 Depths of replica indents plotted as a function of the post-replication
height of the corresponding NiTi exdents when re-warmed to 423 K (the austenitic
condition).

148

3.5.3 Reversible Spherical Replica Indentation
3.5.3.1 Spherical Exdent and Replica Indent
A spherical indent was made under 3000 N with a 1.59 mm in diameter
stainless steel ball in the martensitic state. After indentation, the indent was thermal
cycled 15 times to maximize and stabilize the two-way response and then planarized
to be used in a constrained recovery experiment. We know exdents will form upon
heating and are cyclically reversible. Besides, the exdent height is thought to be the
same as the stabilized two-way indent depth change, which was 30 m.

Replica indentations were made in 6061-T6 Aluminum, 304, and 440
stainless steels using these reversible exdents. After exdentation, the samples were
cooled either under load or without load, (the difference of which will be discussed in
the following section). Figure 3.43 shows the optical image of the NiTi exdent (left)
and replica indent in 304 SS (right) after replica indentation with a preload of 250
MPa. This exdent was acquired by heating the NiTi specimen after replica indentation.
The exdent has a diameter around 2 mm and height about 24.5 m as shown in Figure
3.44. The indent in 304 SS qualitatively demonstrates how large is the force that the
exdent exerted on the 304 SS to make such a replica indent.

149

Figure 3.43 The optical image of the NiTi exdent and replica indent in 304 stainless
steel after replica indentation with a preload of 250 MPa.

Figure 3.44 shows the WYKO surface profile of the NiTi exdent taken at 433
K after the replica indentation, together with the replica indent in 304 SS made at a
preload of 250 MPa. The line profiles of the exdent and the indent match up well. The
indent depth (17 m) is around 56.7% of the original exdent height (30 m). The
exdent height after replica indentation decreased by 18.3% to 24.5 m. This is
because that cooling under load plastically deformed the martensite at a stress of 250
MPa. These plastically deformed martensite variants are not only un-transformative
upon heating to austenite, but also restrain the matensite underneath the plastic
deformation zone from transforming, thus substantially decreasing the exdent height.

150

Figure 3.44 The surface profiles of NiTi exdent heated to 433 K (the upper left figure)
and a replica indent in 304 stainless steel at 293 K (the lower left figure) after
replication under preload of 250 MPa. The dimensions of the exdent and replica
indent are plotted in the right figure.

The relationship between the replica indent depth and the preload for these
three replica materials was investigated and shown in Figure 3.45. The replica indent
depth increases with the increase of preload stress up to 250 MPa which is close to the
yield stress of the austenite phase (250-300 MPa). Further increasing the preload
stress leads to a drop of the replica indent depth because plastic deformation of NiTi
decreases the NiTi exdent height. 6061 T6 Al has the deepest indents under replica
indentation due to its lowest hardness value among three replica materials. Fei et al.
[2009] shows that after planarization, the exdent height is about the same as the
two-way indent depth change. Thus we use the indent depth change at the 15th cycle
as the stabilized exdent amplitude. When the replica indent depth is normalized to the
stabilized exdent height, aluminum has the largest indent/ exdent ratio of 0.7. This

151

result is 40% higher than what we achieved (0.5) using exdents made without prior

Replica Indent Depth, micron

thermal cycling in [Fei et al. 2009].

304 SS
6061 T6 Al
440 SS

30

20

10

0
0

100

200
Preload, MPa

300

400

Figure 3.45 The relation between replica indent depth and preload stress for stainless
steel and aluminum. No replica indent is assumed at 0 preload stress.

3.5.3.2 Effect of Preload Stress
The constrained recovery experiments demonstrate that NiTi exdents can be
used to make replica indents in a variety of materials at various preload stresses.
Material hardness determines how deep the indent can be for a fixed preload stress.
Figure 3.46 plots the replica indent depth against the material hardness for a preload
stress of 250 MPa. The fitting curve as shown in the figure is a linear line, which
achieved an R2 value of 0.9968. The data point at the hardness of zero means that the
exdent can make a replica indent with a depth as large as the exdent amplitude. It is
shown in the curve that NiTi exdent cannot make a replica indent at a hardness of

152

above 2.6 GPa. We call this the hardness limit (H*). This value is close to the
hardness of austenitic NiTi, showing that the limit of NiTi exdent is controlled by its
bulk material property.

Replica Indent Depth,
micron

40
y = -11.605x + 30.21
R2 = 0.9968

30
20
10
0
0

1
2
Hardness, GPa

3

Figure 3.46 The relation between the replica indent depth and material hardness at
preload stress of 250 MPa.

Figure 3.47 plots the replica indent depth with material hardness at preload
stresses of 125, 200, 250, and 300 MPa. Curve-fitting lines suggest that H* for these
preload stresses are 1.87 GPa, 2.29 GPa, 2.6 GPa, and 2.26 GPa.

Figure 3.48 plots H* as a function of the preload stress. Parabolic curve fitting
of the data points leads to a peak around 250 MPa, which is the optimum preload to
generate replica indent. As also shown by the curve fitting is a preload stress limit of
around 480 MPa. This means at 480 MPa, no replica indent can be made because
there is no exdent formed when heated under the preload stress of 480 MPa. Figure

153

3.49 plots H*/preload as a function of preload. Linear curve fitting also suggests a
preload stress limit of around 480 MPa.

The preload level of 480 MPa can increase the transformation temperature by
80K, making As 429 K and Af 468 K according to the Clausius-Clapeyron equation

d H

 6MPa / K . So heating to 433 K (As+4 K) cannot transform the matensite
dT T 0
to austenite, thus causing no exdent formation.

Figure 3.47 Plot of replica indent depth against the material hardness at preload
stresses of 125, 200, 250, and 300 MPa.

154

Figure 3.48 Plot of H* against the preload stress. Parabolic curve fitting suggests 480
MPa is the largest preload stress that can be used to make a replica indent.

Figure 3.49 Plot of the preload stress against H*/ preload. Linear curve fitting
suggests 482 MPa is the largest preload stress that can be used to make a replica
indent.

3.5.3.3 Low Cycle Fatigue Test
In order to understand the fatigue response of the NiTi exdent, we studied the
low cycle fatigue effect of NiTi exdent under preloads of 100 and 250 MPa for 440

155

stainless steel. 440 stainless steel was chosen simply because it is the hardest material
used in these experiments and consequently it should generate the largest fatigue
response.

Figure 3.50 plots the replica indent depth with the number of replica
indentation cycles at preload stresses of 100 and 250 MPa. The higher preload stress
generates a larger fatigue response out of the NiTi exdent due to the accumulation of
plastic deformation at 250 MPa. Interestingly, the fatigue response stabilizes and
converges after 6 cycles of replica indentation, showing the limit of the replica indent
depth that each preload can generate after multiple indentations. This can be
explained by the effect of NiTi work hardening playing against plastic deformation.
Due to the work hardening effect, the NiTi cannot be further plastically deformed
after a number of replica indentations. Thus, the exdent height becomes stable in the
pre-stress and temperature regime. The higher the pre-stress, the larger number of
replica indentation cycles it takes to work harden and to stabilize the exdent height.
Since the replica indent depth is positively related to the exdent height, the higher the
pre-stress, the slower the indent depth drops down and stabilizes, which is shown in
Figure 3.50, demonstrating that the replica indent depth stabilizes slower at 250 MPa
preload than at 100 MPa preload.

156

Replica Indent Depth, micron

15
250 MPa
100 MPa
10

5

0
0

2

4

6

8

Number of Replica Indentation
Figure 3.50 Low cycle fatigue of NiTi exdent in the constrained recovery
experiments.

3.5.3.4 Factors Affecting the Constrained Recovery
3.5.3.4.1 Effect of Temperature
Stressing can affect the phase transformation temperatures of NiTi SMAs,
such that Af increases after loading. Therefore, it is important to know how much the
transformation temperature changes in order to determine the thermal cycling
temperature range to be used in the experimental setting to achieve a full
transformation between martensite and austenite.

The Clausius-Clapeyron equation is often used to estimate the change in
transformation temperature due to stressing. The enthalpy of the SMAs measured by
DSC is 15.6 J/g. This leads to a maximum increase in transformation temperature

157

about 50 K at a preload of 300 MPa since the Clausius-Claperyon equation indicates

d H
d

 6MPa / K , where
is first derivative of stress over temperature,
dT T 0
dT
H is enthalpy which is 15.6 J/g, T is temperature which is the average of Ms and As,

and o is strain limit before plastic deformation which is 0.05.

The specimens were heated up to 433 K in the constrained recovery
experiment. Such heating temperature almost accommodates the increase of Af (438 K)
by 50 K. Therefore, in the present experiments there is confidence that martensite was
close to fully transformed to austenite during heating, and the maximum response was
achieved.

3.5.3.4.2 Path Dependency of Replica Indentation
We observed the path dependency effect of replica indentation based on the
different indent depth measured on cooling without load (path A) and cooling under
load (path B). Figure 3.51 schematically draws these two paths and explains the
difference. The replica indent depth is smaller in path A than in path B because, upon
cooling, the load exerts a large deformation into martensite, pushing the residual
exdent further into the replica material (see 304 SS in the graph). This further
indentation in the replica material causes the replica indent depth to increase. The
harder the replica material, the larger the deformation into martensite, and the larger is
the replica indent depth or the path dependency effect.

158

Figure 3.51 Path dependency of replica indentation: A) cooling without load; B)
cooling under load.

Table 3.7 summarizes the effect of path dependency on the replica indent
depth at the preload of 250 MPa. For aluminum samples, the replica indent depth
under path A is about 30% lower than that under path B. For 440 SS, the replica
indent depth under path A is about 37.5% lower than that under path b. Since 440 SS
is harder than aluminum, we may conclude that the path dependency effect is related
to the hardness of the replica material. The harder the material, the larger the effect is.

159

Table 3.7 Path dependency effect on the replica indent depth at the preload of 250
MPa

Preload, MPa

Path
A: cooling
without load
B: cooling under
load

Replica Indent
Depth, m (Al)

Replica Indent
Depth, m
(440 SS)

250

14.3

8

250

20.1

12.8

3.5.3.5 Work Output and Energy Density Calculation
3.5.3.5.1 Spherical Calibration Indentation at 433K
In order to estimate the energy density associated with exdent formation
during the constrained recovery experiment, we made conventional spherical indents
in the replica materials at 433 K using a ball with a radius of 1.59 mm and loads of 35,
50, and 100 N. Results for these calibration indentations are given in Table 3.8. The
average hardness of each material obtained by spherical calibration indentation is
used in the calculation of equivalent load that NiTi exdents generate to make the
replica indents.

Table 3.8 Calibration Indentation Data for 440 SS, 304 SS, and Al 6061-T6 at 433K
440C SS
Indent
depth,
m
5.0
7.7
12.5

Contact

Load,
N

Indenter
radius, m

a/r
ratio

Contact
radius, m

area, m

2

Hardness,
H440, Pa

35
55
100

8E-04
8E-04
8E-04

0.112
0.138
0.176

0.89E-04
1.11E-04
1.41E-04

2.51E-08
3.85E-08
6.25E-08

1.39E+09
1.42E+09
1.6E+09

160

Table 3.8 cont‟d
Indent
depth,
m
7.9
11.3
19.0

Load,
N

Indenter
radius, m

304 SS
a/r
ratio

38
56
105

8E-04
8E-04
8E-04

0.140
0.167
0.217

Contact
radius, m
1.12E-04
1.34E-04
1.73E-04

Contact
2

Hardness,
H304 SS, Pa

3.94E-08
5.64E-08
9.40E-08

0.96E+09
0.99E+09
1.06E+09

area, m

6061 T6 Al
Indent
depth,
m
7.5
11.8
23.3

Contact

Load,
N

Indenter
radius, m

a/r
ratio

Contact
radius, m

area, m

2

Hardness,
HAL, Pa

35
55
103

8E-04
8E-04
8E-04

0.137
0.171
0.240

1.09E-04
1.37E-04
1.92E-04

3.73E-08
5.89E-08
11.6E-08

0.94E+09
0.93E+09
0.89E+09

3.5.3.5.2 Energy Density Calculation
The projected contact area during replica indentation is calculated as
2
Arepl  arepl allowing us to estimate the peak replication force (assumed to be an

increment above the pre-load force) as Prepl  Arepl H Al . The indentation energy
then becomes Ui  1 / 2Prepl drepl . This is a rough estimation of the energy of
indentation. It is correct only when there is no work hardening effect in the material
during indentation. The Load-Displacement curves (see Figure 2.4) measured by the
CSM instrumented indentation machine in General Motors Technical Center show no
work hardening effect for the aluminum and stainless steel specimens that we used in
the experiments.

To obtain the effective energy density it is necessary to make an estimate of
the volume of material in which shape-memory recovery occurs, i.e., the active

161

subsurface volume contributing to the formation of the exdent.

Previous

experimentally determined active volume that drives exdent formation in [Fei et al.
2009] gives a D*=1.56a when a/r ratio is 0.71. Back planarization may possibly give
a D*=1.74a based on Table 3.1. Therefore, the D* on average is 1.65a when a/r ratio
is 0.71.

With a simplifying assumption that D* represents the radius of a hemisphere
centered on the indent, we estimate the volume of the active zone as

V *  (1 / 2)[4 3D *3 ] . Dividing the energy of indentation, Ui, by this quantity
then provides an estimate of the energy density associated with exdent formation in
NiTi during an SFM event.

Results of this calculation are summarized in Table 3.9.

The calculation is based on the replica indent depth under path B. Energy densities
3

estimated in this way have an average about 12.1 MJ/m for 3 materials at 250 MPa
preload.

Since path a leads to an average 35% reduction of the replica indent depth, the
3

average energy density can be recalculated to be 7.84 MJ/m under path A. This may
be compared to the energy density exerted by a NiTi actuator that undergoes a shape
memory strain of 4% under constraint conditions that cause a reaction stress to rise
linearly from zero to 400 MPa during the actuation stroke. This energy is
3

U 1/2 , or about 8 MJ/m .



162

Table 3.9 Calculation of Energy Density for Replica Indent made under 250 MPa
Repli
Ind contac
dept, radius
m
arepl,
m
12.8 868.7
E-06 E-06
18.6 837.7
E-06 E-06
20.1 832.7
E-06 E-06

Mater
440
SS
304
SS
6061
Al

3.6

Repli
a/r
ratio
0.029
0.044
0.048

Replica
contact
area,
Arepl,
m^2
2.37
E-06
2.20
E-06
2.18
E-06

Equiv
replica
load,
Prepl, N

Indent
energy,
U,
Joules

Estim
D*, m

3.49
E+03
2.20
E+03
2.00
E+03

2.23
E-02
2.05
E-02
2.01
E-02

0.94
E-03
0.94
E-03
0.94
E-03

Est.
Ene
den.
MJ/
m^3
12.8
11.8
11.6

SFM from Dynamic Indentation (Laser Shock Peening)

3.6.1 Indent Recovery and SFM
Laser shock peening produces very high shock pressures, ultra high strain
6

rates (>10 /s) [Peyre et al. 1998], and allows precise control for localized treatment.
It can introduce a strong work-hardened layer and compressive residual stresses that
have been often used to improve fatigue performance, corrosion resistance, wear
resistance and service life in various materials [Hu et al. 2006].

There are few reports on the effect of ultra-high strain rates on the deformation
of NiTi-based SMAs, especially for the case of martensite behavior. Millett et al,
[2002] reported that the strain rate sensitivity of austenitic NiTi increases the yield
strength of NiTi from 500 to 794 MPa when deformation is caused by a shock wave
instead of quasistatically.

In the present case, martensite deformation is of interest,

163

which can proceed either by deformation twinning reactions, or by dislocation
creation and slip mechanism. Both are of crucial significance in the case of two-way
memory effects, and rather little is known about either of these mechanisms at
ultra-high strain rates.

Therefore, dynamic indentation was conducted via the Laser Shock Peening
(LSP) process in the martensitic NiTi specimen. Details of experimental procedures
were given in Chapter 2. Two-way indent depth change and exdent-to-flat transition
were evaluated and compared to the quasi-static spherical indent response, as discussed
below.

Figure 3.52a shows the schematic drawing of the indent made under laser
2

intensity of 2.5 GW/cm and beam size of 0.5 mm, after heating, and after cooling. The
initial indent depth is 5.7 m. Upon heating to above Af, the indent became shallow and
reaches 0.6 m by the shape memory mechanism. This depth change represents a
recovery of 89.5% of the initial indent depth. Upon cooling to below Mf, the indent
deepens to 2 m. The two-way recovery upon heating and cooling is 1.4 m and
accounts for 24.6% of the initial indent depth, a ratio more than twice the maximum
recovery ratio of the spherical indent (11.8%) made by quasi-static indentation [Fei et
al. 2009].

164

The trace of the indent was removed by mechanical grinding, as in previous
examples of planarization. After planarization, specimens were heated and cooled
again over the same temperature range. Thermally-reversible surface transitions were
then observed after planarization.

Figure 3.52b shows the schematic drawing of the thermally-reversible
exdent-to-flat transition. An exdent with amplitude of 1.2 m was observed and was
comparable to the two-way indent depth change.

Figure 3.53 shows a typical 3D profile for the thermally reversible
exdent-to-flat transition illustrating the high degree of reversibility for this transition.
The surface becomes bumpy when the specimen is warmed and returns to flatness when
it is cooled. Due to the small scale of the exdent, the surface roughness change and
thermal expansion and contraction during heating and cooling affect the surface
flatness in a large degree. That is why the surface is not totally flat when the specimen is
cooled to the martensitic state.

165

Figure 3.52 Schematic illustrations of (a) initial indent depth-recovery, and (b)
exdent-to-flat surface transition during heating and cooling after planarization of the
LSP indent.

166

Figure 3.53 Cyclic exdent-to-flat surface transition after planarization of the dynamic
indent.

Having shown surface form memory developed by LSP with ultra high strain
rate, it is interesting to compare high strain rate dynamic surface form memory with
quasi-static indentation induced surface form memory with regard to both indent/
exdent responses and deformation zones. Figure 3.54 shows the comparison of initial
indent depth between the spherical indent made under 700 N with a r =400 m
spherical ball and the dynamic indent under laser intensity of 2.5 GW/cm2 and beam
size of 0.5 mm. The spherical indent has an initial indent depth of 96.8 m, thus a/r of
0.65 and a=260 m. The indent diameter is thus around 0.5mm, which is close to that of
the dynamic indent. For the two indents with same indent diameter, the dynamic indent
is significantly shallower than the quasi-static indent. Assuming the dynamic indent is
spherical in shape, the equivalent indenter radius r to make such indent can be

167

2

2

calculated to be 5485 m by r = (a +h )/2h, where a is the contact radius which is 250
m in this case, h is the indent depth which is 5.7 m. The a/r ratio for the dynamic
indent is then calculated to be 0.05.

Ni et al. [2003] shows spherical indents with a/r less than 0.25 can fully recover
after heating due to the absence of dislocations when the deformation strain is less than
5 %. As the result, no indent remains when the initial indent having an a/r of 0.05 is
heated above Af. Furthermore, for a/r <0.5, no TWSME is observed. However, the
dynamic indent shows a two-way indent depth change of 1.4 m, as shown in Figure
3.55, which shows the comparison of two-way recovery between the quasi-static
spherical indent and the dynamic indent. It is clearly shown that the normalized
two-way recovery ratio of the dynamic indent is larger than that of the spherical indent,
i.e, when the dynamic indent is magnified to have the same vertical length as the
spherical indent, shown in the right side of the figure. The dynamic indent has a
two-way recovery ratio of 24.6% while the spherical indent under mechanical loading
has a two-way recovery ratio of 9.3% in this case. This larger recovery ratio of the
dynamic indent may be due to the strain rate sensitivity effect of NiTi. Table 3.10
summarizes the differences between the quasi-static spherical indent and the dynamic
indent. It can be concluded that the indent response of the dynamic indent is 2.1 times
larger than that of the quasi-static indent, when the indent diameter is the same.

168

Another way to compare these two types of indents is shown in the following.
In the previous work [Fei et al, 2009], we found that the reversible indent depth
change, , for quasi-static indents depends on both the indenter radius rqs, and the
indent contact radius a, following such relation = 0.06a - 0.0155rqs. Therefore, the
radius of a quasistatic indenter expected to give a 1.4 micron depth change when the
in-plane indent radius is 250 microns is rqs = (.06a - )/.0155 = 877 microns. This
corresponds to an a/rqs ratio of 0.285 for the quasistatic case. Figure 3.56 shows the
differences between the quasi-static spherical indent and the dynamic indent when the
two-way indent depth is kept the same. Indent and recovery parameters for both
indent types are summarized in Table 3.11 below. It can be concluded that reversible
indent or exdent displacement can be generated from a much smaller indent
dynamically than quasi-statically, by a factor of approximately 6.5. This is significant
because it reduces the SMA substrate thickness to express a similar SFM effect in
application and also decreases the amount of material removal to achieve SFM.

169

Figure 3.54 Comparison of the initial indent depth between the spherical indent made
under 700 N with r =400 m spherical ball and the dynamic indent under laser intensity
of 2.5 GW/cm2 and beam size of 0.5 mm.

Figure 3.55 Comparisons of the initial indent depth, two-way recovery between the
quasi-static spherical indent and the dynamic indent. The dynamic indent is magnified
to have the same vertical length as the spherical indent.

170

Table 3.10

Comparison of Quasistatic and LSP Indent with same contact radius
1

2

Quasistatic

LSP

0.65

LSP%/QS%

0.05

a/r Ratio
m

%Change

m

%Change

Indenter Radius

397

5500

Contact Radius

260

250

Initial Indent Depth

96.8

5.7

Initial Recovery

40.8

42.1

5.1

89.5

2.1 x

Reversible Depth Change

9

9.3

1.4

24.6

2.6 x

1

Quasistatic parameters are interpolations based on previous experiments.

2

LSP a/r ratio estimated assuming a spherical indent profile.

Figure 3.56 Comparisons of initial indent depth and two-way recovery between the
spherical indent and the LSP indent. Vertical scales are identical for each impression
but the base of the quasistatic indent is off the scale.

171

Table 3.11

Comparison of Quasistatic and LSP Indent Parameters with same
two-way indent depth change
1

2

Quasistatic

LSP

0.285

LSP%/QS%

0.05

a/r Ratio
m

%Change

m

%Change

Indenter Radius

877

5500

Initial Indent Depth

36.5

5.7

Initial Recovery

31.9

87.5

89.5

Reversible Depth Change

1.4

3.8

24.6

6.5 x

Reversible Exdent Height

1.4

3.8

17.5

4.6 x

(6.5 x)

-1

1

Quasistatic parameters are interpolations based on previous experiments.

2

LSP a/r ratio estimated assuming a spherical indent profile.

3.6.2 Mechanism of SFM in Dynamic Indents
Laser shock peening produces a high energy shock wave by laser pulse which
penetrates into material substrate. The shock wave propagates faster than the speed of
sound and at a strain rate larger than 106/s. The shock wave propagation speed consists
of two parts: The elastic wave propagation speed CEL, and the plastic wave
propagation speed CPL, both of which are defined in the following equations [Ding et
al. 2006]:

CEL 

  2


CPL 

  2 / 3


172

Eq. (3.17)

Eq. (3.18)

Here,  is Lame‟s constant which is 17.31 GPa in this case,  is shear
modulus which is 11.54 GPa,  is density of the material (NiTi) which is 6.45 g/cm3.
So Cpl is calculated to around 2,000 m/s, which is beyond the speed of twin boundary
motion of martensite detwinning mechanism (the speed of which equals to 1/3 square
root of young‟s modulus over density, which is around 700 m/s [Otsuka et al. 2005]).
Therefore, plastic deformation occurs when the shock wave propagates into NiTi,
generating a subsurface dislocation deformation zone. The dislocation deformation
zone accounts for the observed TWSME even when the a/r ratio is 0.05.

Millett et al. [2002] reported that the strain rate sensitivity of austenitic NiTi
increases the yield strength of NiTi from 500 MPa to 794 MPa by shock wave. As the
shock wave propagates into the metallic target, plastic deformation occurs to a depth at
which the peak pressure no longer exceeds the metal‟s Hugoniot elastic limit (HEL). A
metal‟s HEL is related to the dynamic yield strength according to Johnson and Rhode
[1971]:

H E L

1 v d y n
y
1  2v

Eq. (3.19)

dyn
where v is Poisson‟s ratio and  y is the dynamic yield strength at high strain rates.

The literature contains one experimental estimate of the HEL for austenitic
NiTi (3.5 GPa [Millett et al, 2002]), but there is considerable uncertainty regarding
this rather high value, which in any case was determined in experiments on austenitic

173

(rather than martensitic) NiTi. The measured quasi-static compressive stress strain
behavior of the NiTi material used in the present study is shown in Figure 2.2 for the
-2

data collected at a strain rate of 1x10 /s.

Two yield features can be discerned. The

first is unambiguously associated with the onset of plastic strain by martensite variant
detwinning and occurs at 93 MPa, whereas the second, which can be associated with
the onset of dislocation activity after dewtinning mechanisms have been exhausted, is
found at 315 MPa. For two-way shape memory to occur, both detwinning and slip
must occur, and must do so in that temporal order.

This is because memory of cool

shape in the martensite requires that dislocations be created in the martensite after it
has been largely detwinned. Only in this way can such dislocations stabilize the
variant distribution of the deformed martensite.

We therefore proceed on the

assumption that the shock interacts with the martensitic structure on the basis of one
HEL value for martensite detwinning, and another (higher) HEL applicable to slip
deformation.

An estimate for the value of the dynamic (high strain rate) yield stress for the
martensitic material in the present study can be obtained based on results presented in
[Adharapurapu et al, 2006] and [Liu et al, 1999], both of which measured yield
stresses for the martensite phase at different strain rates, in both the quasi-static and
dynamic regimes. We first develop a logarithmic curve fit to low and high strain
rate data for martensite deformation in these two studies, and use this regression to
6

extrapolate the results to a strain rate of 10 /s (assumed to apply to deformation under

174

-2

LSP conditions), and additionally to interpolate them for a strain rate of 10 /s -- the
rate used to obtain the data shown in Figure 2.2.

Table 3.12 summarizes the data

values used for this estimate, where regression parameters m and b are used to
6

-2

extrapolate dyn at d/dt = 10 /s and 10 /s, for each of the two studies, according to

 (d/dt) = [m ln(d/dt) + b]

Eq.(3.20)

From this we extract the ratio, R =  10^6/ 10^-2 , for each study, take the
average of the two values, and finally estimate 10^6 for the present material as equal
to R times the measured quasistatic slip-yield strength of 315 MPa. This procedure
predicts that the dynamic yield stress for slip is 427 MPa. Assuming a Poisson ratio
of 0.3, this value yields an HEL of 747 MPa for slip deformation.

It is of interest then to estimate the peak pressures, to determine the plastified
depth likely produced by the LSP approach, and to compare these estimates to our
previous experimental observations of the plastic zone produced by quasistatic
spherical indents. The peak pressure of the shock produced using laser ablation is
given by
P = 0.01[Z Io /(2+3)]

1/2

Eq. (3.21)

-2

where Io is the incident laser power density (GWcm ), P is the pressure (GPa), Z is the
reduced acoustic impedance between the target and the confining medium.  is the
efficiency of the plasma-material interaction. (  = 0.1-0.2) [Fabbro et al. 1998]. In the

175

6

2

glass confinement mode, the acoustic impedance of glass is 1.3 x10 g/cm s, while the
6

2

acoustic impedance of NiTi is about 1.34 x10 g/cm s. As the laser intensity used for
2

LSP was 2.5 GW/cm , the estimated peak pressure ranges from 3.2 GPa, assuming  =
0.1. The shock pressure of 3.2 GPa is therefore about four times the HEL and will
clearly cause extensive plastic deformation.

Applying the same procedure to the

yield stress for detwinning gives an HEL estimate for the dynamic case of 133 MPa.

The plastified depth depends on the HEL, the pressure, P, and the pressure
duration, p, and may be estimated according to [Peyre et al, 1996] as
Dp = CEL CPLp [(P-HEL)/2 HEL)]/(CEL – CPL)

Eq. (3.22)

The latter two parameters are the elastic and plastic wave propagation speeds:
CEL = [(+2)/]

1/2

and CPL = [( +2 /3)/ ]

1/2

respectively [Johnson et al, 1971],

where  is Lame‟s constant (16.2 GPa in this case), µ is the shear modulus of the
3

martensite, taken to be 10.77 GPa, and  is density of NiTi which is 6.45 g/cm .
CEL and CPL are therefore about 2420 and 1900 m/s, respectively. The pressure
pulse duration, p, is longer than the laser pulse duration by a factor of 1.8 [Fabbro et
al, 1990] and is taken to be 9 ns in the present case.

These parameters give an

estimate for the depth, DS,P, to be 131 microns, at which slip deformation begins.
The strain in this zone saturates at P = 2.5 HEL at a magnitude

P = -2 HEL (P/HEL - 1)/(3+2)
176

Eq. (3.23)

giving S = 3.2% in compression (for P = 2.5 HEL) for the present experiment.
saturation depth, DS,2.5 , by eq. 4 is 60.2 microns.

The

Similar calculations for the

detwinning HEL of 133 Mpa give a saturation strain, T, of 1% at the same saturation
depth (DT,2.5 = 60.2 microns) However, for the case of the detwinning yield, the
depth at the HEL, DT,P =512 microns, is much greater than that for slip deformation.

Table 3.12 Data and regression coefficients used in estimating the dynamic yield
stress for slip deformation in the martensite for the present work
Strain Rate
d/dt

Ref.

[16]

[17]

Presen
work,
slip
Presen
Work,
Detwi
nning

1 x 10E-3
1.2 x 10E2
1 x 10E-2,
interp.
1 x 10E6,
extrap.
3 x 10E-4
3 x 10E3
1 x 10E-2,
interp.
1 x 10E6,
extrap.
1 x 10E-2,
meas.
1 x 10E6, est.
1 x 10E-2
meas.
1 x 10E6, est.

Compressive
Yield Stress
MPa
1290
1720

Yield Stress
Ratio R, d/dt
@ 10E6/10E2

Regress
Parameter
m

Regress
Parameter
b

30.719

1502.2

21.094

1421.1

1360
1.416
1927
1250
1590
1324
1.293
1713
315
427
1.355

93
133

It is useful to assess the above estimates in terms of the observed initial LSP
indent depth of 5.7 microns and the initial recovered depth of 0.6 microns.

177

The

former should be consistent with the sum of strain contributions from detwinning and
slip. Here it is assumed that the strain decays from the saturation value at the
saturation depth (DS,2.5 and DT,2.5) to zero at the HEL. The total indent depth
should therefore be

DTOT = 1/2S (DS,P+ DS,2.5)+ 1/2T (DT,P+ DT,2.5)

Eq. (3.24)

which is 6 microns if the above estimates are used, in reasonable agreement with the
observed depth of 5.7 microns. The residual indent depth after heating should reflect
only the slip component of the deformation, giving DResid. = 1/2S (DS,P+ DS,2.5)
which is 3.1 microns – substantially greater than the 0.6 micron residual depth
observed. This suggests that the above analysis has overestimated the plastic strain
due to slip, or that the effect of surface relief waves has reversed some of the slip
deformation. It has been reported [Peyre et al, 1996] that when the peak load is above
2.5 HEL, surface release waves focus and amplify from the edge of the impact.

The

peak load 3.2 GPa is considerably higher than 2.5 HEL for slip (1.87 GPa), which
may additionally explain the observed 0.5 micron pile up effect observed in Figure
3.55.

Finally, for quasistatic spherical indents, extensive experimental measurements
of the depth of the active zone, beneath the indent, show that this depth, D*, to be
approximately 40 times the height of the stable cyclic exdent amplitude. Conversely,
the stable exdent amplitude is consistently found to be about 2.4% of the active zone

178

depth.

If LSP indents respond in the same way, and estimate of the active zone

depth in the present experiment would 1.2/.024 = 58 microns, which is in reasonable
agreement with the 60 micron depth estimated above.

To conclude, LSP was demonstrated to be able to create two-way indent
recovery or reversible exdent-to-flat transition by coupling a planarization step in NiTi
SMAs. This is due to the subsurface slip deformation zone induced by large peak load,
high intensity, and ultra high strain rate of LSP. The two-way effect of the indent made
by LSP is more than twice the maximum effect made by quasi-static indent in low
strain rate condition, which maybe due to the strain rate sensitivity effect induced by
ultra high strain rate LSP. Further experiments can be designed to ascertain
deformation strain, residual stress and the size of the subsurface deformation zone,
either by experimentation or by finite element modeling.

Thermally-reversible surface topographical transitions have potential
application to micro-electrical-mechanical systems (MEMS), variable friction surfaces,
information storage media, controllable optical devices, tribological surfaces, and
mechanical joining.

179

3.7

Modeling Indent Recovery and SFM

3.7.1

Finite Element Modeling of the Deformation Zone
Finite element analysis was carried out using a commercial program

ABAQUS (SIMULIA, Providence, RI). The mesh used has 15,000 nodes with mesh
refinement in the contact deformation region as shown in Figure 3.57. The mechanical
properties of NiTi used in the model were measured in a tensile test. The Young‟s
modulus was measured to be 28GPa, the yield stress was 180 MPa. The work
hardening rate was 6.5 GPa and the poisson‟s ratio was 0.3.

Figure 3.57 Finite element mesh with 15,000 nodes and a refined area as the contact
region and 4 mm spherical indenter.

3.7.1.1 Spherical Indentation and the Deformation Zone
Spherical indentation was modeled by indenting a half-space shape memory
alloy using a 4 mm radius indenter. The mechanical properties of the shape memory
alloy were defined above. The indentation depth was varied in order to achieve a

180

range of a/r ratios. Figure 3.58 shows the development of plastic strains inside the
shape memory alloy upon indentation. As we can see, the maximum plastic strain
almost stays at a constant distance below the indent, while the plastic strains progress
downwards as the indenter goes into the material progressively.

For the spherical indent with an a/r of 0.4, the maximum plastic strain is
about 16.6%. The deformation zone at a depth of D* which was measured to be about
1.1-1.35a has a minimum plastic strain of about 8-9%, as shown in Figure 3.59(a).
For the spherical indent with an a/r ratio of 0.65, the plastic strain can be as high as
about 30%, as shown in Figure 3.59(b). The deformation zone at a depth of D*,
measured by the front and back thinning experiments to be about 1.5-1.7a for such
indent, has a minimum plastic strain of about 7-9%. For the spherical indent with an
a/r of 0.95 has a maximum plastic strain about 80%. The deformation zone at a depth
of D* which was measured to be about 1.9-2.0a has a minimum plastic strain of about
9-10%, as shown in Figure 3.59(c).

Therefore, the finite element model consistently shows a deformation zone at
a depth of D* has a plastic strain larger than 7%, which is beyond the strain that can
be accommodated by the martensite deformation twinning. Thus slip plastic
deformation takes control and dislocations generated in the detwinned martensite
variants achieve the lowest free energy and favor the formation of certain martensite
variant in the subsequent cooling, enabling a two-way response.

181

Figure 3.58 Finite element model of the spherical indentation process: a) 10% of the
indentation steps; b) 60% of the indentation steps; c) 100% of the indentation steps
(Indentation step finished).

Figure 3.59 Effective plastic strain developed after spherical indentation with an a/r
of a) 0.4, b) 0.65, and c) 0.95 simulated by ABAQUS finite element model.

182

Figure 3.59 cont‟d.

3.7.1.2 Cylindrical Indentation and the Deformation Zone
The difference between spherical and cylindrical indentation lies in the
indenter geometry. As spherical indentation involves an axisymmetric problem,
cylindrical indentation relates to a plane strain problem. Thus, the cylindrical
indentation experiment was modeled by indenting a half-space shape memory alloy
using a 4 mm radius cylindrical rod.

Figure 3.60(a) shows that for the cylindrical indent with an a/r ratio of 0.41,
the plastic strain can be as high as about 10%. The deformation zone at a depth of D*,
measured by the front thinning experiment to be about 1.8a for such cylindrical indent,
has a minimum plastic strain of about 7%.

183

Figure 3.60(b) shows that for the cylindrical indent with an a/r ratio of 0.65,
the plastic strain can be as high as about 17%. The deformation zone at a depth of D*,
measured by the front thinning experiment to be about 2.3a, has a minimum plastic
strain of about 8%.

Figure 3.60(c) shows that for the cylindrical indent with an a/r ratio of 0.95,
the plastic strain can be as high as about 31%. The deformation zone at a depth of D*,
measured by the front thinning experiment to be about 4.15a, has a minimum plastic
strain of about 7.5%.

Figure 3.60 Effective plastic strain developed after cylindrical indentation with an a/r
of a) 0.41, b) 0.65, and c) 0.95 simulated by ABAQUS finite element model.

184

Figure 3.60 cont‟d.
As we can see, the sizes of the spherical deformation zones are around 50%
smaller than those of the cylindrical deformation zones. The difference in the size of
the deformation zone accounts for the difference in the observed two-way response
(cylindrical indents have about twice as large two-way response as spherical indents).

185

However, both spherical and cylindrical deformation zones correspond to strains
around 7-9%, which can induce slip deformation. This again supports the conclusion
that TWSME induced by indentation is due to the dislocations in the subsurface
deformation zone. Dislocations in this zone are retained in the austenite after warming
and, on cooling, encourage the formation of the same variants in which dislocations
were originally formed and presumed minimal free energy, thus stabilizing the
original martensite variants, and resulting in a degree of „memory‟ of both the warm
and cool shapes.

186

CHAPTER 4 CONCLUSIONS AND FUTURE WORK

4.1

Conclusions

1. An indentation-planarization method has been demonstrated not only to induce
two-way cyclic depth change but also to create thermally-driven reversible surface
form transitions in NiTi alloys. These surface features can be easily modified by
various indenter geometries, such as spheres, cylinders, punches, pyramids and many
others.

2. Indentation-induced two-way response increases with increasing a/r ratio for both
spherical and cylindrical indents. Thermal cycling after indentation can increase
two-way response by about 40% due to the completion of phase transformation.

3. Reducing indent spacing can also increase two-way response due to the interplay of
the deformation zone of each indent. Zero indent spacing or double indentation
maximizes the two-way response. Further increase of indentation repetitions does not
have much effect on increasing two-way response. Indentation at temperature below
As could decrease the two-way response because the material flow stress increases
below As, causing the material work harden faster and reducing plastic deformation.

4. This indentation-induced two-way shape memory effect is thought to be enabled by
a deformation zone under the indent in which the martensite has been plastically

187

strained beyond the detwinning plateau strain of ~5%, causing production of defects
associated with slip deformation.

The depth of this active zone, D*, has been

experimentally determined by successive grinding experiments and shown that, for
both spherical and cylindrical indents and over a range of a/r ratios between 0.4 and
0.95, the maximum cyclic exdent height amplitude is approximately 2.4% of the
magnitude of D*.

5. Cylindrical indentation is shown to produce substantially larger cyclic exdent
response than spherical indentations with similar a/r ratios. This increased response
can be directly related to the depth of the active zone for these indents.

D* for

cylindrical indent is greater than that for spherical indent. This is due to different local
constraint conditions, with cylindrical indentation leading to plane-strain deformation
and larger hydrostatic stress components in the deformation zone.

6. Finite element modeling shows the deformation zone at a depth of D* has a
minimum plastic strain of 7% for spherical and cylindrical indents at various a/r ratios.
This confirms the conclusion that slip deformation is a necessary condition for
indentation-induced TWSME. It is due to the preferential formation of certain
martensite variants, where the dislocations were developed by slip deformation, that
lead to a selective shape change in martensitic transformation. The driving force for
the selection of martensite variants is the system‟s tendency to achieve lowest free

188

energy in martensitic transformation. Only in the martensite variants in which the
dislocations initially generated can they find the lowest free energy.

7. Surface Form Memory has been demonstrated to exert large specific work output.
The energy density of the spherical exdent in constrained recovery can be close to 107
J/m3, the magnitude which is close to the maximum of energy density of conventional
NiTi actuators.

8. Dynamically made indents by the Laser Shock Peening process were found to have
larger two-way response at smaller indentation depth or smaller a/r ratio level than
the quasi-static indent made by conventional indenters. This was due to the larger
plastic depth, thus larger deformation zone under the LSP indent. High efficiency,
large two-way response, and high energy density of SFM exdents make the material
an ideal candidate for miniature actuators for MEMS.

4.2

Future Development

1. Mechanisms of the indentation-induced shape memory, super-elasticity and
two-way shape memory effect. New experimental tools such as synchrotron X-ray
diffraction, Neutron Diffraction, Electron Back Scattering Diffraction and in-situ
TEM indentation may be used to assess the critical information like the one-one
correspondence between austenite and martensite variants in the indentation
deformation zone in NiTi alloys.

189

2. Tribological applications of the surface form memory. Surfaces with exdent-to-flat
surface transition or flat surface-to-indent transition can be made to vary friction
coefficients with differences on the order of a magnitude. Special interest can be
placed in the wheel-roll contact in the high speed railway system.

3) Developing finite element modeling of shape memory effect is the next step after
current modeling of the indentation deformation zone. FEA of shape memory effect
can provide stress-strain distributions beneath indents in NiTi before and after phase
transformations, which can be used to explain indentation-induced shape memory and
two-way shape memory effect.

Coupling the thermal field with the current model is

essential in developing shape memory model.

190

APPENDIX

Figure a. Schematic illustration of the indentation-planarization process.

1

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