Mechanics of Elastomeric Copolymers
by
Hansohl Cho
Bachelor of Science, Seoul National University
Master of Science, Massachusetts Institute of Technology
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
ARCHIE
Doctor of Philosophy in Mechanical Engineering
OF TECHNOLOGY
at the
MAY 0 8 2014
Massachusetts Institute of Technology
LIBRARIES
February 2014
© Massachusetts Institute of Technology 2014. All rights reserved
A uthor................................................................a....
.......................................
Department of Mechanical Engineering
December 19, 2013
C ertified by..........................................
/
Mary C. Boyce
Ford Professor of Engineering
Thesis Supervisor
I
/
Accepted by...........................................
David E. Hardt
Ralph E. and Eloise F. Cross Professor of Mechanical Engineering
Department Graduate Program Officer
2
Mechanics of Elastomeric Copolymers
by
Hansohl Cho
Submitted to the Department of Mechanical Engineering
on January 31, 2014, in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
Elastomeric copolymers have been versatile materials for a broad variety of engineering
applications of critical importance ranging from ballistic protective coatings to self-healing
microstructures, possessing a backbone structure composed of alternate hard and soft segments,
where the hard/soft nomenclature corresponds to the thermodynamic glassy/rubbery state at
ambient temperature. The thermodynamic incompatibility of microstructures often lead to a
phase-separated morphology of the hard and soft domains which can be tailored depending on
the chemical composition, molecular dispersion, processing and synthesis to give a variety of
physical properties. The mechanical behavior of elastomeric copolymers is hence governed by
the chemical composition as well as the morphology providing a hybrid performance by virtue of
simultaneous contributions from constituent homopolymers often offering new and unique
properties. In this research, the mechanics and physics of large deformation behavior of
elastomeric copolymers are addressed in terms of their resilience and dissipation involving
elastomeric "segmented" copolymers and elastomeric "ionic" copolymers.
The presence of hard and soft domains yields to multiple molecular relaxations and hence
multiple viscous dissipation mechanisms in elastomeric copolymers. In addition to the viscous
dissipation, stretch-induced softening due to microstructural evolution revealed via x-ray
scattering observation during deformation provides another major dissipation pathway.
Furthermore the segmented copolymers exhibit a substantial shape recovery upon unloading in
tandem with a remarkable amount of hysteresis. A microstructurally-informed constitutive model
is proposed to address the main features of mechanical behavior of exemplar copolymers under a
variety of loading conditions, employing multiple micro-rheological mechanisms representing
hard and soft domains. The proposed model was found to be capable of capturing the salient
features of resilient yet dissipative stress-strain behavior of materials at a wide range of strains
and strain rates. The model was then furthered to examine the effect of weight fraction,
morphology and segmental dynamics of hard and soft microstructures.
Next, the resilience and dissipation in elastomeric segmented copolymers are examined in
their connections to shape recovery under microindentation testing in experiments and numerical
simulations. Numerical simulations imparting the proposed constitutive model were found to be
3
capable of capturing the microindentation behavior of materials including force-displacement
responses under complicated loading scenarios. Additionally, the microindentation behavior
revealed a substantial shape recovery of indented surfaces which was due to inelastic flow
beyond elastic resilience. The elastically- and inelastically-driven shape recovery provides
critical insight into a better understanding of shape memory, recovery and self-healing
mechanisms in this class of segmented elastomers.
The extreme nature of elastomeric copolymers under harsh mechanical environments is
then addressed via Taylor impact testing, where an ultrafast deformation event is incurred.
Numerical simulations of Taylor impact behavior of elastomeric copolymers are compared to
experimental results in terms of overall and localized deformation profiles, revealing a threedimensional capability of our framework under dynamic, inhomogeneous deformation field.
Furthermore, energy dissipation under such an extreme event is examined by comparison to that
found in "model" glassy and rubbery polymers revealing that copolymeric materials enable a
highly recoverable, protective composite architecture for shock and ballistic mitigation by taking
advantages of hybrid performance of glassy and rubbery polymers.
Lastly, the mechanics of elastomeric "ionic" copolymers is addressed for a broad
understanding of their resilience, dissipation and shape recovery under a wide range of
mechanical loading conditions. Our viscoelastic-viscoplastic constitutive framework is further
developed to address the large deformation behavior of ionic elastomers including ethylene
methacrylic acid (EMAA) copolymer and its chemically-modified counterparts which are
recently finding new avenues towards multi-functional soft materials involving their self-healing
ability under severe deformation events.
This study provides a simple yet intuitive framework to rationalize physically-sound
deformation mechanisms of diverse elastomeric copolymers by employing a combination of
novel modeling, experimentation and computation. Finally, potential topics for further research,
to which the present work can directly contribute, are discussed in a wide variety of engineering
contexts.
Thesis Committee:
Mary C. Boyce, Ford Professor of Engineering (Chair),
David M. Parks, Professor of Mechanical Engineering,
Robert E. Cohen, St. Laurent Professor of Chemical Engineering,
Raul A. Radovitzky, Professor of Aeronautics and Astronautics
4
Acknowledgements
I wish to express my sincere gratitude towards my advisor, Professor Mary Cunningham Boyce
for her support and guidance throughout my graduate education and research at MIT. Every
single phase throughout my research, Professor Boyce gave me critical insight and motivation
with highly supportive and positive ways. I'd also like to express my appreciation to my thesis
committee members for their advice and support: Professor David M. Parks (my academic
grandfather), Professor Robert E. Cohen and Professor Raul Radovitzky.
Many thanks to Juliette Pickering, Leslie Regan and Una Sheehan for their support
throughout my academic life; and Professor Lallit Anand, Professor Simona Socrate, Professor
Gregory C. Rutledge, Aidan P. Thompson, Alan Schwartzman, Willis Mock Jr., Susan Bartyczak,
Alex Hsieh, Aurelie Jean, Martin Hautefeuille and former Boyce group members for their
support for my research; I'd also like to thank wonderful friends in Boyce group and MechE
department. In particular, I've got great support from friends in KGSAME throughout my life at
MIT.
I wish to express my sincere gratitude towards my family: Minjoe, Kyungsook, Yonghwan,
Sunhee, Insook, Ah-Ram and
10IEF. Lastly, I could not have finished this work without
love and support from my Tough Girls: Hyunsook and Dawn Hayoung. . .-
l
-L'Le }I
JA~t LI r-F
This work was generously supported by the Office of Naval Research and the Kwan-Jeong
Fellowship.
5
6
Table of Contents
Chapter 1 Introduction and Research Objective..................................................................
23
1.1.
General Introduction ..................................................................................................
23
1.2.
Research Objectives ....................................................................................................
24
1.3.
Structure of the Thesis................................................................................................
25
1.4.
Research Highlight....................................................................................................
27
Chapter 2 Structure, Property and Chemistry of Elastomeric Copolymers......................30
2.1.
Introduction and Background.....................................................................................
30
2.2.
Segmented Copolymer Polyurea and Polyurethane ....................................................
32
2.3.
Ionic Copolymer Ethylene Methacrylic Acid (EMAA) and Ethylene Methacrylic Acid
Butyl Acrylate (EMAABA)..................................................................................................
39
2.4 .
43
Reference........................................................................................................................
Chapter 3 Constitutive Modeling of Resilient yet Dissipative Large Deformation of
Elastomeric Segmented Copolymers.....................................................................................
3.1.
B ackground ....................................................................................................................
47
47
3.1.1.
Inelasticity of amorphous polymers ........................................................................
47
3.1.2.
Elasticity of rubbers................................................................................................
49
3.1.3.
Stretch-induced softening: Mullins effect .............................................................
49
3.1.4.
Viscoelasticity of elastomeric materials..................................................................
50
3.1.5.
Recent advances in continuum mechanical studies of polymeric materials............ 52
3.2.
Mechanical Behavior of Exemplar Elastomeric Segmented Copolymer Polyurea........ 52
3.3.
Model Development....................................................................................................
56
3.4.
Result: Experiment vs Model....................................................................................
63
3.4.1.
Low Strain Rate Behavior ......................................................................................
66
3.4.2.
High Strain Rate Behavior.......................................................................................
71
7
3.4.3.
Constrained Behavior under Biaxial Tensile Testing..............................................
73
3.5.
Concluding Remark and Future Work ......................................................................
3.6.
Procedure for Determination of Material Parameters in the Constitutive Model..... 80
3.7.
R eference........................................................................................................................
78
85
Chapter 4 Computational Procedures for Simulations of Large Deformation Elastic-Plastic
Behavior of Elastomeric Copolymers.....................................................................................
90
4 .1.
Introduction ....................................................................................................................
90
4.2.
Formulation of Initial and Boundary Value Problems ................................................
91
4.3.
Finite Element Procedures for Dynamic and Inhomogeneous Large Deformation....... 94
4.4.
Numerical Updates of Large Deformation Elastic-Plastic Kinematics and Relevant
Internal V ariables ......................................................................................................................
96
4.5.
Aspects of Numerical Procedures of Thermomechanically-coupled Deformation ....... 99
4.6.
Future Work for Computational Implementation.........................................................
4.7.
Reference......................................................................................................................102
100
Chapter 5 Resilient yet Dissipative Large Deformation of Elastomeric Segmented
Copolymers: Effect of Weight Fraction and Segmental Dynamics of Microstructure ...... 104
5.1.
Introduction ..................................................................................................................
5.2.
Segmental Microstructure and its Connection to Stress-Strain Behavior of PU 1000 and
PU6 5 0 ......................................................................................................................................
5.3.
104
10 5
Constitutive Models of PU650: Effect of Weight Fraction and Segmental Structures of
Hard, Soft Phases and Their Mixtures ....................................................................................
110
5.4.
Micromechanical Modeling of Co-continuous Morphology .......................................
114
5.5.
Conclusion and Future Work .......................................................................................
120
5.6.
Material Parameters for PU650 Mixed Phase..............................................................
120
5.7 .
R eference......................................................................................................................
124
Chapter 6 Extreme Behavior of Elastomeric Copolymers under Harsh Environments ... 127
8
6.1.
High Strain Rate Behavior of Elastomeric Copolymers ..............................................
127
6.2.
Experimental and Computational Methods for Taylor Impact Testing .......................
130
6.3.
Shape Evolution in Polyurea Rods during Taylor Impact Testing...............................
132
6.4.
Taylor Impact Behavior of "Model" Rubbery and Glassy Polymers...........................
141
6.5.
Effect of Adiabatic Heating and Temperature Rise due to Inelastic Deformation ...... 146
6.6.
Taylor Impact Behavior of "Model" Linear Viscoelastic Polyurea.............................
149
6.7.
Discussion and Future Work........................................................................................
152
6.8.
R eference......................................................................................................................155
Chapter 7 Resilience,
Dissipation and Shape Recovery
of Elastomeric Segmented
Copolymers under Microindentation......................................................................................158
7.1.
Introduction ..................................................................................................................
158
7.2.
Shape Recovery Mechanism in Elastomeric Materials................................................
160
7.3.
Microindentation Behavior of Elastomeric Copolymer Polyureas ..............................
162
7.3.1.
In situ Micro-Indentation Test..................................................................................
163
7.3.2.
Load-Displacement Behavior under Microindentation ............................................
164
7.3.3.
Shape Recovery under Microindentation .................................................................
170
7.4.
C onclusion ....................................................................................................................
177
7.5.
Reference......................................................................................................................
179
Chapter 8 Mechanics of Elastomeric Ionic Copolymers.......................................................
183
183
8.1.
Introduction ..................................................................................................................
8.2.
Mechanical Behavior of Ethylene-based Ionic Copolymers: Constitutive Modeling
Framew ork ..............................................................................................................................
184
192
8.3.
Microindentation Behavior of EMAA .........................................................................
8.4.
Resilience, Dissipation and Shape Recovery of EMAABA and EMAABA-Na ......... 197
8.5.
C onclu sion ....................................................................................................................
203
8.6.
Material Parameters for EMAA and EMAABA ..........................................................
204
9
8.7 .
R eference......................................................................................................................
Chapter 9 Concluding Remark and Future Work ................................................................
208
211
9.1.
Summary and General Conclusion...............................................................................
211
9.2.
Future Work ..........................................................................................................
212
9.2.1.
Micromechanics of various co-continuous or occluded morphologies for energy
dissipation and shape recovery................................................................................................
9.2.2.
Multi-scale mechanics of elastomeric copolymers: atomistic, coarse-grained and
continuum models and their coupling .....................................................................................
9.2.3.
212
213
Mechanics of other elastomeric copolymers including thermoplastic polyurethane and
polyurethane-urea....................................................................................................................
2 14
9.2.4.
Crazing, cavitation and failure in elastomeric copolymers under high strain rates.. 215
9.2.5.
Modeling of healing behavior of elastomeric copolymers .......................................
9 .3.
Reference......................................................................................................................
216
2 17
10
11
Table of Figures
Figure 1-1 Research summary and highlight in this thesis ......................................................
28
Figure 2-1 Schematic of chemical structures of polyurea (polyurethane): (a) Reactants:
functionalized amine (soft) and diisocyanate (hard); (b) Exemplar chemical formula for polyurea
studied in this research (PU650 and PU 1000)...........................................................................
34
Figure 2-2 Microstructure and dynamic properties of polyurea: (a) Schematic for phase-separated
morphology of hard and soft segment; (b) Dynamic mechanical properties: storage modulus and
loss factor ......................................................................................................................................
35
Figure 2-3 Micrographs of polyurethane and polyurea: (a) Transmission electron microscope
(TEM) image of polyurethane (57% soft phase, 43% hard phase, from Qi et al. 4); (b) Tapping
mode atomic force microscope (AFM) phase image of polyurea 1000 (64% soft phase, 36% hard
phase, from C astagna et al.13 ...............................................................................................
. 36
Figure 2-4 In situ X-ray scattering data of polyurea under cyclic tension: (a) WAXS and SAXS
evolution during cyclic loading-unloading-reloading; (b) Stress-strain data under cyclic tensile
test (from R inaldi et al. 5)...............................................................................................................
38
Figure 2-5 Schematic of chemical structures of EMAA and its neutralization: EMAA formed
from ethylene and m ethacrylic acid...........................................................................................
40
Figure 2-6 Dynamic mechanical property of ethylene methacrylic acid copolymers; storage
modulus, loss modulus and loss tangent curves as a function of temperature: (a) EMAA; (b)
EMAABA; (c) EMAABA-Na (from Deschanel et al.4).........................................................
40
Figure 3-1 Uniaxial compression and tension data for polyurea (a) dynamic mechanical analysis
explaining the co-continuous microstructures, from Rinaldi et al. 41; (b) rate-dependent stressstrain data under low strain rate compression with an inset of flow stress as a function of strain
rate, from Yi et al. 40; (c) rate-dependent stress-strain data under low to high strain rate
compression, from Sarva et al. 66 (d) flow stress as a function of strain rate under low to high
strain rate compression (e) asymmetric stress-strain data under compression and tension at low
strain rates of Il = 0.001s-1 ,0.01s-1 and 0.ls1 with an inset (from Choi et al.
65)
of stress-strain
data up to locking phenomena under tension; and (f) stress-strain data under cyclic tension at a
strain rate of
= 0.003 s-,
from Rinaldi et al. ......................................................................
54
12
Figure 3-2 Schematic representations for the constitutive model (a) a polyurea microstructure
representing a segmented copolymer with interpenetrating hard and soft domains; (b) a onedimensional micro-rheological interpretation for the hard and soft domains in the proposed
constitutive m odel.........................................................................................................................57
Figure 3-3 Schematic representations for the constitutive model and the large deformation
kinematics (a) the intermolecular and network element in the hard domain for low strain rate
behavior; (b) a kinematics map in the large deformation elastic-plasticity ...............................
59
Figure 3-4 Monotonic compression and tension data, comparing model results (lines) with
experimental data (open symbols) at low strain rates of 0.001, 0.01 and 0.1 s-1 (a) compression
stress-strain curves (b) flow stress at strains of 0.4 and 0.9 as a function of strain rate (c) tension
stress-strain curves (d) asymmetry in tension and compression stress-strain behavior at a strain
rate of 0.0 1 s- ...............................................................................................................................
67
Figure 3-5 Cyclic compression and tension data, comparing model results with experimental data
(a) stress-strain curves under cyclic compression at a strain rate of 0.1 s-1 (b) stress-strain curves
under cyclic tension at a strain rate of 0.015 s-1.......................................
........... ...... ............... . .
68
Figure 3-6 Multiple cyclic tension data, comparing model results (lines) with experimental data
(open symbols) (a) stress-strain curves with an increasing strain under multiple cyclic tension at
a strain rate of 0.005 s- (the inset shows the stretch-induced elastic softening in the hard domain
network) (b) stress-strain curves under cyclic tension up to a maximum strain of 1.25 at a strain
rate of 0 .0 5 s ' ...............................................................................................................................
70
Figure 3-7 Energy dissipation during cyclic tension tests (a) a schematic of the data reduction for
a dissipated work density in consecutive cycles with an increasing strain (b) dissipated work
density in consecutive cycles with an increasing strain in model and experiment (the two tests
presented in Figure 6(a) and (b) are evaluated) ........................................................................
71
Figure 3-8 High strain rate behavior under compression (a) stress-strain curves in experiment
(open symbol) and model at strain rates ranging from 0.01 to 6500 s1 (b) flow stress at strains of
0.4 and 0.9 as a function of strain rate ......................................................................................
72
Figure 3-9 Decomposition of the simulated low to high strain rate behavior into hard and soft
contribution (a) stress-strain curves in hard (black) and soft (gray) component (b) schematic for
the micro-rheological hard and soft components in the simulation...........................................
73
13
Figure 3-10 Schematic of biaxial tensile testing at two different biaxial ratios (B=
A.
-4=0
and 1.0)
.......................................................................................................................................................
74
Figure 3-11 Engineering stress-engineering strain behavior under biaxial tensile testing in
experim ents and simulations....................................................................................................
75
Figure 3-12 Contours of stress field under biaxial tensile testing at a strain of 0.125 in the central
biaxial region in the cruciform specimen: (a) stress in x direction at biaxiality of 1.0; (b) stress in
y direction at biaxiality of 1.0; (c) stress in x direction at biaxiality of 0.0; (a) stress in y direction
76
at biaxiality of 1.0 .........................................................................................................................
Figure 3-13 Contours of strain field under biaxial tensile testing at biaxiality of 1.0 at a strain of
0.125 in the central biaxial region in the cruciform specimen: (a) axial strain field in experiment;
(b) axial strain field in simulation; (c) shear strain field in experiment; (d) shear strain field in
77
simulation ......................................................................................................................................
Figure 3-14 Contours of strain field under biaxial tensile testing at biaxiality of 0.0 at a strain of
0.125 in the central biaxial region in the cruciform specimen: (a) axial strain field in experiment;
(b) axial strain field in simulation; (c) shear strain field in experiment; (d) shear strain field in
78
simu lation ......................................................................................................................................
Figure 3-15 Data used to determine material parameters in hard domain intermolecular and
network component (a) stress relaxation data at a strain rate of 0.1 s-1 under compression with a
model prediction from the hard network component (b) shear yield stress
data as a function of shear strain rate (
~N
uniaxial )
(
((1H [3)
OIH uniaxial
under compression at low strain rates (c)
cyclic tension data at a strain rate of 0.005 s-1 for stretch-induced softening model parameters the
in hard network component (NH) (d) stress-strain curve at a moderate strain rate of 0.2 s- under
tension, revealing the dramatic stress hardening due to the finite extensibility (the saturated value
of chain limiting extensibility is estimated via
NI ~
r
+
) from Choi et al. ........ 82
Figure 3-16 Data used to determine material parameters in soft intermolecular and network
component (a) shear yield stress data as a function of shear strain rate under compression at low
to high strain rate (b) shear viscosity data as a function of strain rate......................................
84
Figure 4-1 Solution procedure in finite element solver with constitutive model subroutines...... 91
14
Figure 4-2 Schematic of boundary conditions for equations of motion ....................................
94
Figure 5-1 Mechanical behavior of PU650: (a) stress-strain data of PUOGO and PU650 under
monotonic compression at low strain rate; (b) stress-strain data of PU1000 and PU650 under
cyclic tension at a strain rate of 0.005 /s (PUlOOG data reused from Chapter 3); (c) stress-strain
data of PU650 at low to high strain rate; (d) flow stress of PU650 as a function of strain rate at
strains of 0.4 and 0.9 ...................................................................................................................
107
Figure 5-2 Model predictions of PU650 stress-strain behavior via weight fractional scaling.... 109
Figure 5-3 Dynamic mechanical analysis (DMA) data of PU1000 and PU650: (a) storage
m odulus; (b) loss m odulus; (c) loss factor..................................................................................
110
Figure 5-4 Stress-strain behavior of PU650 under compression in experiment and model: (a)
Stress-strain curves at low to high strain rate; (b) Decomposition of simulated stress response
into hard, mixed and soft contribution at strain rates of 0.01 and 2000 /s..................................
112
Figure 5-5 Flow stress as a function of strain rate at increasing strains of 0.4 and 0.9 in PU1000
an d PU 650 ...................................................................................................................................
113
Figure 5-6 Resilient yet dissipative mechanical behavior of PU650 and PU10: Stress-strain
curves under multiple consecutive cyclic tensile tests at a strain rate of -0.005 /s (a) experiment;
(b) model; (c) dissipated work density as a function of strain during 1st and 2 "d cycle in PUOOG
(d) dissipated work density as a function of strain during 1st and 2 "d cycle in PU650 (See Chapter
3 for detailed PU 1000 data in model and experiment) ...............................................................
113
Figure 5-7 Schematic of representative volume elements of exemplar co-continuous network 115
Figure 5-8 Contours of axial strain field in RVEs (Di,/Dout=1.0) of co-continuous morphology
subjected to tensile loading up to a true strain of 1.0 (a) 46% hard phase (b) 34% hard phase. 116
Figure 5-9 Contours of axial strain field in RVEs (Di,/DOut=0.85) of co-continuous morphology
subjected to tensile loading up to a true strain of 1.0 (a) 46% hard phase (b) 34% hard phase. 116
Figure 5-10 Stress-strain behavior of RVEs: (a) Di,/DOut=1.0; (b) Di,/DOut=0.85 (here, the RVE
stress was normalized by a stress magnitude of 46% RVE, Di,/DOut=1.0 at a strain of 0.6) ...... 117
Figure 5-11 Ratios of RVE stress in 34% to 48% hard phase for Di,/Dout=0.85 (red) and
D in/D out= 1.0 (black).....................................................................................................................
118
Figure 5-12 Exemplar microstructures found in the elastomeric copolymers: (a) unit-cells for bicontinuous and occluded heterostructures (volume fraction: 50%/50%); (b) stress field in a bicontinuous structure; (c) stress field in occluded hard spheres in soft matrix (occluded four
15
spheres sitting on a primitive cubic (cP) lattice); (d) stress field in a occluded hard sphere in soft
matrix (occluded one sphere sitting on a primitive cubic (cP) lattice) .......................................
119
Figure 6-1 Flow stress as a function of strain rates at increasing strains: (a) PUOOO and PU650
(See Chapter 5 for details); (b) ethylene methacrylic acid (EMAA) copolymer (See Chapter 8 for
details).........................................................................................................................................
12 8
Figure 6-2 Experimental setup for Taylor impact testing (a) schematic of target region prior to
projectile impact (b) undeformed polyurea rod (L/D
-
2.0) (c) deformed polyurea rod (L/D ~ 2.0)
for im pact velocity of - 245 m/s.................................................................................................
131
Figure 6-3 Schematic for axisymmetric finite element simulations with an exemplar three-node
triangular m esh ............................................................................................................................
Figure 6-4 Deformation profiles during a Taylor impact test with L/D
-
132
4/3 and V ~ 245 m/s:
high-speed photographs (red dots: digitized deformation profiles) and deformation profile
prediction made using an axisymmetric finite element simulation (a) loading (b) unloading ... 133
Figure 6-5 Evolution of selected geometric dimensions and kinetic energy in the Taylor impact
test with L/D
-
4/3 and V ~ 245 m/s (a) evolution of normalized length and diameter at the
impact surface in experiment and simulation (b) evolution of normalized kinetic energy with
selected deformed profiles under loading and unloading ...........................................................
134
Figure 6-6 Contours of axial-stress at various stages in the Taylor impact test with l/D ~ 4/3 and
V - 245 m/s.................................................................................................................................
135
Figure 6-7 Contours of inelastic strain rate in hard intermolecular component at various stages in
the Taylor impact test with LID
-
4/3 and V
-
245 m/s .............................................................
136
Figure 6-8 Contours of elastic shear modulus in hard network component at various stages in the
Taylor impact test with L/D ~ 4/3 and V
-
245 m/s ...................................................................
Figure 6-9 Deformation profiles during a Taylor impact test with LID
-
2 and V
-
137
245 m/s: high-
speed photographs (red dots: digitized deformation profiles) and deformation profile prediction
made using an axisymmetric finite element simulation (a) loading (b) unloading.....................
138
Figure 6-10 Evolution of selected geometric dimensions and kinetic energy in the Taylor impact
test with L/D
-
2 and V
-
245 m/s (a) evolution of normalized length and diameter at the impact
surface in experiment and simulation (b) evolution of normalized kinetic energy with selected
deformed profiles under loading and unloading .........................................................................
139
16
Figure 6-11 Comparison of simulated 270' axisymmetric sweep rods and contours of elastic
shear modulus at t = 0.75 ms with recovered rods for tests in (a) L/D
L/D
-
4 (more softening achieved in the rod of L/D
-
-
4/3 (b) L/D ~ 2 and (c)
4/3)........................................................
140
Figure 6-12 Evolution of normalized kinetic energy in the Taylor impact tests with rods of L/D =
4/3and 4 at a velocity of
-
245 m/s.............................................................................................
140
Figure 6-13 Schematic for constitutive behavior in polymeric materials (a) stress-strain in
dissipative yet resilient copolymers (b) stress-strain in "elastic-plastic" glassy polymers and
"hyperelastic" rubbery polym ers ................................................................................................
142
Figure 6-14 Stress-strain behavior under uniaxial compression at a strain rate of 1000 /s (a)
glassy constitutive model with hard/soft intermolecular components (b) original copolymeric
polyurea constitutive model (c) rubbery constitutive model with hyperelastic components...... 142
Figure 6-15 Evolution of selected geometric dimensions in Taylor impact tests with L/D ~ 2 and
V
-
245 m/s (a) evolution of normalized length and diameter in copolymeric, glassy, and rubbery
constitutive model (b) simulated deformation profiles at maximum spreading in copolymeric,
glassy, and rubbery m odel..........................................................................................................
143
Figure 6-16 Evolution of simulated deformation profiles in Taylor impact tests with L/D ~ 2 and
V
-
245 m/s (a) copolymeric constitutive model (repeated), (b) viscoplastic glassy constitutive
model, and (c) hyperelastic rubbery constitutive model.............................................................
145
Figure 6-17 Evolution of kinetic energy and deformation profile in Taylor impact tests for a
variety of "model" polymers (a) evolution of normalized kinetic energy in copolymeric, glassy
and rubbery constitutive model (L/D
-
2, V
245 m/s) (b) deformation profiles of the "model"
-
hyperelastic rubbery polymer at local minima and maxima of kinetic energy...........................
145
Figure 6-18 Evolution of kinetic energy and deformation profile in Taylor impact tests for the
"model" hyperelastic rubbery polymer (L/D
-
4) (a) evolution of normalized kinetic energy at
velocities of 245 m/s and 330 m/s (b) deformation profiles at 245 m/s (c) deformation profiles at
330 m/s........................................................................................................................................
14 6
Figure 6-19 Contours of temperature evolution at various stages in the Taylor impact test with
L/D - 4/3 and V - 245 m/s .............................................................................................
149
Figure 6-20 Stress-strain behavior in viscoelastic constitutive model 18at low to high strain rate
.....................................................................................................................................................
150
17
Figure 6-21 Deformation profiles during a Taylor impact test with JD ~ 4/3 and V ~ 245 m/s
using the viscoelastic constitutive model; all of the numerical details are the same as those used
in Figure 4 including finite element mesh, time-step and boundary conditions.........................
151
Figure 6-22 Evolution of selected geometric dimensions in the Taylor impact test with JD - 4/3
and V
-
245 m/ in viscoelastic model, PU model (this study) and experiment..........................
152
Figure 6-23 Cavitation and radial crack pattern in polyurea rod under an impact velocity of 450
2415
m/s. ............................................................................................................................................
154
Figure 7-1 Shape recovery mechanism in elastomeric materials: (a) schematics of shape recovery
under cyclic compression; (b) residual strain-time curve during recovery; (c) total stress-strain
curve; (d) intermolecular stress component during loading-unloading; (e) network stress
component during loading-unloading; (f) individual stress component-time curves during
reco v ery .......................................................................................................................................
16 1
Figure 7-2 Shape recovery at increasing imposed strains of 0.1 and 0.5: (a) Strain vs time at an
imposed strain of 0.1 (inset: total stress-strain behavior); (b) Individual stress component vs time
in intermolecular and network mechanism at an imposed strain of 0.1; (c) Strain vs time at an
imposed strain of 0.5 (inset: total stress-strain behavior); (d) Intermolecular/network stress
component vs time at an imposed strain of 0.5; (black lines: loading-unloading, red lines:
recovery after unloading)............................................................................................................
162
Figure 7-3 Schematics of microindentation testing: (a) load-displacement curve with an inset of
load function composed of loading, creep, unloading and zero-force creep (a zero force
maintained to monitor shape recovery after unloading); (b) displacement-time curve .............. 164
Figure 7-4 Microindentation behavior of PU650 in experiment and numerical simulation: (a-c)
Force-displacement curves at increasing peak forces of 1.5, 2.5 and 3.5 mN (inset: load functions
of loading, creep and unloading); (d) Displace-time curves (dashed line: experiment; solid line:
simu lation) ..................................................................................................................................
165
Figure 7-5 Contours of axial strain field of PU650 under microindentation at a peak force of 1.5
mN : loading, creep and unloading..............................................................................................
166
Figure 7-6 Contours of axial strain field of PU650 under microindentation at a peak force of 2.5
mN : loading, creep and unloading ..............................................................................................
166
Figure 7-7 Contours of axial strain field of PU650 under microindentation at a peak force of 3.5
mN : loading, creep and unloading ..............................................................................................
167
18
Figure 7-8 Contours of inelastic strain rate of PU650 under microindentation at the end of
unloading (t=30s) in order of increasing peak forces: (a) 1.5 mN; (b) 2.5 mN; (c) 3.5 mN ...... 167
Figure 7-9 Microindentation behavior of PUOOG in experiment and numerical simulation: (a-c)
Force-displacement curves at increasing peak forces of 1.5, 2.0 and 2.5 mN (inset: load functions
of loading, creep and unloading); (d) Displace-time curves (dashed line: experiment; solid line:
168
simu lation ) ..................................................................................................................................
Figure 7-10 Contours of axial strain field of PU1000 under microindentation at a peak force of
1.5 mN : loading, creep and unloading ........................................................................................
169
Figure 7-11 Contours of axial strain field of PUOOG under microindentation at a peak force of
2.0 mN : loading, creep and unloading ........................................................................................
169
Figure 7-12 Contours of axial strain field of PUOOG under microindentation at a peak force of
2.5 m N : loading, creep and unloading ........................................................................................
169
Figure 7-13 Contours of inelastic strain rate of PU1000 under microindentation at the end of
unloading (t=30s) in order of increasing peak forces: (a) 1.5 mN; (b) 2.0 mN; (c) 2.5 mN ...... 170
Figure 7-14 Dissipation in microindentation testing of PU650 and PUOGO
at increasing
indentation displacement (See Figure 7-4 and 7-9 for load-displacement curves).....................
170
Figure 7-15 Shape recovery of PU650 surface at a peak force of 1.5 mN: (a) load-displacement
curve; (b) load function composed of loading-creep-unloading and "recovery"; (c) timedisplacement curve during loading and unloading; (d) time-displacement curve during recovery
(here, time was zeroed from the end of unloading)....................................................................
172
Figure 7-16 Contours of displacement field of PU650 under microindentation at a peak force of
1.5 mN (inset: load-displacement curve); (a) displacement field at loading-creep-unloading; (b)
displacement field during recovery (after unloading).................................................................
173
Figure 7-17 Shape recovery of PU650 surface at a peak force of 2.5 mN: (a) load-displacement
curve; (b) load function composed of loading-creep-unloading and "recovery"; (c) timedisplacement curve during loading and unloading; (d) time-displacement curve during recovery
(here, time was zeroed from the end of unloading)....................................................................
174
Figure 7-18 Contours of displacement field of PU650 under microindentation at a peak force of
2.5 mN (a) displacement field at loading-creep-unloading (inset: load-displacement curve); (b)
displacement field during recovery (after unloading).................................................................
175
19
Figure 7-19 Shape recovery of PU650 surface at a peak force of 3.5 mN: (a) load-displacement
curve; (b) load function composed of loading-creep-unloading and "recovery"; (c) timedisplacement curve during loading and unloading; (d) time-displacement curve during recovery
(here, time was zeroed from the end of unloading)....................................................................
176
Figure 7-20 Contours of displacement field of PU650 under microindentation at a peak force of
3.5 mN (a) displacement field at loading-creep-unloading (inset: load-displacement curve); (b)
displacement field during recovery (after unloading).................................................................
177
Figure 8-1 Mechanical behavior of EMAA: (a) dynamic mechanical analysis data: storage
modulus and loss factor as a function of increasing temperature; (b) stress-strain behavior at
strain rates of 0.0016, 0.016, 0.16, 16, 115, 1300 and 6000 /s (black: low rate, gray: intermediate
rate, light gray: high rate); (c) flow stress as a function of strain rate at increasing strains of 0.3
and 0.8 (from D eschanel et al. 1)................................................................................................
185
Figure 8-2 Schematic of constitutive models of EMAA: (a) multiple micro-rheological
mechanisms (b) kinematics of elastic-plastic deformation.........................................................
187
Figure 8-3 Stress-strain behavior of EMAA in experiments and simulations: (a) stress-strain
curves under compression at strain rates of 0.0016, 0.016, 0.16, 16, 115, 1300 and 6000 /s (solid
line: simulation, symbol: experiment) (b) flow stress as a function of strain rate at strains of 0.4
and 0.8 (experimental data reproduced from Deschanel et al.")................................................
191
Figure 8-4 Force-displacement behavior of EMAA under microindentation in experiments and
numerical simulations: force-displacement behavior (inset: load functions) at increasing peak
forces of (a) 1.5; (b) 2.0; (c) 2.5; and 3.0 mN (open symbol: experiment, solid line: simulation)
.....................................................................................................................................................
194
Figure 8-5 Displacement-time curves of EMAA under microindentation in experiments and
numerical simulations (dashed line: experiment, solid line: simulation) ...................................
195
Figure 8-6 Contours of inelastic strain rates of EMAA under microindentation at t = 10 sec (a)
Fmax=1.5 mN; (b) Fmax=2.0 mN; (c) Fmax=2.5 mN; (b) Fmax=3.0 mN .........................................
195
Figure 8-7 Contours of axial strain field of EMAA under microindentation at Fmax=1.5 mN in
indentation direction (a) loading; (b) creep; (c) unloading .........................................................
196
Figure 8-8 Contours of axial strain field of EMAA under microindentation at Fmax=2.0 mN in
indentation direction (a) loading; (b) creep; (c) unloading .........................................................
196
20
Figure 8-9 Contours of axial strain field of EMAA under microindentation at Fmax=2.5 mN in
indentation direction (a) loading; (b) creep; (c) unloading .........................................................
196
Figure 8-10 Contours of axial strain field of EMAA under microindentation at Fmax=3.0 mN in
indentation direction (a) loading; (b) creep; (c) unloading .........................................................
197
Figure 8-11 Stress-strain data of EMAABA and EMAABA-Na under cyclic compression at a
strain rate of 0.0016 /s (data reproduced from Deschanel et al.13) .............................................
198
Figure 8-12 Stress-strain behavior of EMAABA and EMAABA-Na under compression at low to
high strain rates (a) EMAABA at strain rates of 0.001, 0.01, 0.1, and 3000 /s (solid lines:
simulations, symbols: experiments); (b) EMAABA-Na at strain rates of 0.001, 0.01, 0.1, 68, 600,
and 6500 /s (line: simulations, symbol: experiments); experimental data reproduced from
G reviskes et al .13 . . .
.......................................................................................................
200
Figure 8-13 Flow stress as a function of strain rate at strains of 0.35 and 0.7 in EMAABA and
E M A A BA -N a .............................................................................................................................
20 1
Figure 8-14 Stress-strain behavior of EMAABA under cyclic compression at a strain rate of 0.01
/s (a) maximum imposed strain of 0.7 (b) maximum imposed strain of 1.0 (experimental data
reproduced from G reviskes et al. 13)............................................................................................
201
Figure 8-15 Stress-strain behavior of EMAABA-Na under cyclic compression at a strain rate of
0.01 /s (a) maximum imposed strain of 0.7 (b) maximum imposed strain of 1.0 (experimental
data reproduced from Greviskes et al. 13 ) ....................................................................................
202
Figure 8-16 Dissipated work during cyclic deformation at an imposed strain of 1.0 in (a)
EM A A BA ; (b) EM A A BA -Na....................................................................................................
202
21
22
Chapter 1
Introduction and Research Objective
1.1.
General Introduction
Elastomeric copolymers are soft block copolymeric materials able to take advantages of
hybrid mechanical, chemical and thermal properties by virtue of simultaneous contributions from
constituent homopolymers, which often exhibit a phase-separated morphology due to their
thermodynamic incompatibility. They have been hence a focal point of research interest in a
broad range of engineering and scientific contexts via their unique properties, which can be
tailored at microscopic- and macroscopic-levels. The presence of multiple constituents in varying
morphologies of blocks or segments has been found to lead to complicated mechanical and
thermal behavior travelling between those in the constituent homopolymers. Tailoring the
copolymeric molecular structures was therefore found to be essential to achieve a variety of
tunable physical properties for resulting macroscopic functionalities on demand. In this research,
we aim at a broad understanding of the critical aspects of mechanics and physics of elastomeric
copolymers to provide physically-sound underlying mechanisms of deformation.
Over the past several decades, the mechanical behavior of elastomeric copolymers has
been widely studied to address the physical basis of deformation mechanisms involved to the
elastic and inelastic properties of materials under a broad variety of mechanical environments.
Furthermore, physically-based constitutive theories have been proposed to model the large
deformation behavior of materials. In particular, numerous viscoelasticity models have been
widely accepted to capture the rate- and temperature-dependent stress-strain behavior, employing
classical linear theories of viscous flows and relaxations in the materials. The linear
viscoelasticity models have been found to provide a simple framework valid at a relatively
narrow range of deformation and deformation rates, relying on complicated parametric studies to
23
fit the models to experimental data without rigorous physical foundations of deformation
mechanisms. Understanding the physically-sound mechanisms responsible for the complicated
mechanical and thermal behavior of elastomeric copolymers is essential to enable the "tunable"
material architectures. However, the constitutive framework inferred from the microstructural
features has not been established well, posing challenges towards the predictive design principles
of materials for a wide range of mechanical functionalities.
In this research, we explore the mechanics of elastomeric copolymers in a continuum
mechanical framework employing a combination of modeling, experimentation and computation.
In particular, we are focused on the development of constitutive models of elastomeric
copolymers, which involves the multi-phase components, through this research providing a
simple yet systematic framework that can be directly applied to other soft materials whose
mechanical behavior is similar to that in the elastomeric materials covered in this thesis. We
begin with the development of constitutive models informed from stress-strain data of an
exemplar copolymeric material under simple loading histories and extend the modeling
framework to understand complicated deformation phenomena involving time- and constraintdependent loading histories. We aim at a broad understanding of deformation mechanisms
involved in mechanical behavior of elastomeric copolymers throughout this research. In
particular, our efforts have been focused on elastomeric segmented copolymer (polyurea and
polyurethane) and elastomeric ionic copolymers, which have received great attention for a
diversity of engineering applications for the past several decades.
1.2.
Research Objectives
The mechanical behavior of elastomeric copolymers has been widely studied by a number
of research groups. The prior experimental data revealed main features of deformation as follows;
highly nonlinear elasticity with substantial stretch over a strain of 1.0; highly rate-dependent
yield-like behavior followed by stress rollover; highly nonlinear hardening beyond the stress
rollover; highly nonlinear unloading accompanied by a substantial amount of hysteresis;
substantial deformation recovery; strain- and strain-rate dependent elastic-inelastic softening
24
known as Mullins effect upon cyclic loading; dramatically enhanced stress levels at high strain
rate; and multiple relaxation processes in dynamic mechanical tests. Throughout this research we
are focused on the development of an unified constitutive framework to address the "resilient"
yet "dissipative" large deformation behavior of elastomeric copolymers including segmented
copolymers (polyurea copolymers with varying segmental structures, morphologies and weight
fractions of hard/soft contents), ionic copolymers (ethylene methacrylic acid copolymers,
ethylene
methacrylic
acid butyl acrylate
copolymers),
and their chemically-modified
counterparts. The main objectives of this research are summarized as follows:
(a) The physically-sound deformation mechanisms are rationalized by combinations of novel
modeling, experimentation and numerical simulation. In particular, the constitutive
models of materials are developed in systematic ways, inspired by microstructural
information under a variety of loading conditions.
(b) The physically-informed constitutive models are validated for complicated deformation
processes in conjunction with relevant experimental data and numerical simulations
incorporating the proposed constitutive models.
(c) The constitutive modeling framework of elastomeric "segmented" copolymers is
furthered to capture the main features of stress-strain behavior of elastomeric "ionic"
copolymers and their chemically-modified counterparts.
(d) The nature of resilience and dissipation of materials is elucidated for extreme
deformation events; here, we address the Taylor impact behavior of elastomeric
copolymers and "model" glassy and rubbery polymers.
(e) The resilient yet dissipative deformation at small scales is investigated in terms of
viscoelastic-viscoplastic
behavior
and
inelastically-driven
shape
recovery
via
microindentation testing in experiments and numerical simulations
(f) Finally, we propose further research topics with simple yet intuitive examples to which
the present thesis work may contribute directly.
1.3.
Structure of the Thesis
25
This dissertation is composed of two main parts: (1) mechanics of elastomeric segmented
copolymers
(polyurea 1000 and polyurea 650) and (2) mechanics of elastomeric ionic
copolymers (ethylene methacrylic acid and ethylene methacrylic acid butyl acrylate).
First, the mechanical behavior of elastomeric copolymers and its connections to the
microstructural features of constituents are outlined involving the chemistry, synthesis and
processing of materials in Chapter 2. In particular, the multi-phase morphology of elastomeric
copolymers is extensively reviewed in the context of their hybrid mechanical behavior of
constituents, which provides insight into the microstructurally-informed constitutive modeling
framework of the materials through this thesis.
Second, a predictive constitutive modeling framework to capture the large deformation
behavior of elastomeric copolymers is proposed for segmented copolymers and ethylene-based
ionic copolymers in Chapter 3 (polyurea 1000), Chapter 5 (polyurea 650) and Chapter 8
(EMAA, EMAABA and EMAABA neutralized with metallic salt cations, especially, sodiumneutralized counterpart). In addition, a constrained mechanical behavior of polyurea thin films is
addressed via biaxial tensile tests in experiments and numerical simulations validating a multidimensional predictive capability of the proposed constitutive model in Chapter 3.
Third, the mechanics of resilience and dissipation of elastomeric copolymers under
ultrafast deformation is discussed via Taylor impact testing in experiments and numerical
simulations in Chapter 6. Also, computational procedures for simulations of dynamic,
inhomogeneous viscoelastic-viscoplastic large deformation of elastomeric copolymers are
detailed by nonlinear finite element formulations of initial and boundary value problems and
elastic-plastic kinematics update of constitutive models in Chapter 4.
Fourth, the constitutive modeling framework is further discussed to understand the
underlying mechanisms of shape recovery of elastomeric copolymers via in-situ microindentation testing in experiments and numerical simulations in Chapter 7 (polyurea 1000 and
polyurea 650) and Chapter 8 (EMAA) Additionally, the mechanical principle responsible for
substantial shape recovery is briefly discussed in terms of its implications for self-healing in the
elastomeric copolymers in Chapter 8.
26
Finally, potential topics for further research, to which the present work can directly
contribute, are proposed ranging from micro- and molecular mechanics of phase-separated
microstructures to coupled modeling frameworks of thermal, mechanical and healing behavior of
ionic copolymers in Chapter 9.
Keyword: elastomeric segmented copolymer (polyurea, polyurethane and polyurethane-urea),
elastomeric ionic copolymer (ethylene methacrylic acid copolymer and ethylene methacrylic acid
butyl acrylate copolymer neutralized with salt cations), large deformation, viscoelasticviscoplastic constitutive model, nonlinear finite element simulation, resilience, dissipation, shape
recovery and memory, extreme rate behavior
1.4.
Research Highlight
Figure 1-1 shows a graphical highlight of research roadmaps in this thesis. By beginning
with the development of preliminary constitutive models for simple, homogeneous deformations,
we extend our scope to complicated deformation processes summarized above.
27
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u
OI"a4PsdWoA(
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'F
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tIi~i~
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I
ii
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I
I
2.
0 p
I
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woof
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Figure 1-1 Research summary and highlight in this thesis
28
29
Chapter 2
Structure, Property and Chemistry of Elastomeric
Copolymers
2.1.
Introduction and Background
In this chapter, the chemistry, synthesis, structure and processing of elastomeric
copolymers are briefly reviewed for polyurea/urethane copolymers and ionic copolymers.
In
particular, the chemical and microstructural features of elastomeric copolymers are extensively
discussed in terms of their multi-component nature and its connections to the structure-property
relationship. The multiple relaxation processes found in the materials are then detailed via
dynamic mechanical properties. Furthermore, the morphological features of materials are
detailed for background of the underlying mechanisms of deformation-induced microstructural
change, which is further investigated in the following chapters on the constitutive modeling of
materials.
Elastomeric copolymers have been versatile material systems since the relevant chemical
processes were developed in the early 1900's. In particular, the segmented copolymeric
elastomers have been employed for a variety of structural and functional materials as a
replacement for natural rubbers from the early stage of polyurethane- and polyurea-chemistry
first introduced by Bayer.I The development of polyurethane- and polyurea-based elastomers
began as an active research field with global commercialization and those elastomers exhibited
great advantages over natural rubbers and other synthetic rubbers, which had been previously
developed, in terms of elastic properties and flexibility in a variety of coatings and structural
applications. 2 After the first commercialization of the segmented elastomers by Bayer and
Goodrich in the 1950's, the field of polyurea- and polyurethane-chemistry has rapidly grown up
30
by the major chemical companies including Bayer, Goodrich, Dow Chemicals and BASF to date.
In particular, tailoring molecular structures of hard and soft segments were found to be relatively
flexible for the purpose of manipulating the mechanical, thermal and chemical properties
travelling between hard and soft phases and their mixtures.
1,2
The mechanical and thermal behavior of elastomeric copolymers was found to exhibit
hybrid features of glassy and rubbery polymers, which result from the two-phase molecular
structures of hard and soft phases. The dynamic mechanical properties revealed the soft phase is
above a glassy transition at room temperature while the hard phase is in the glassy state, which
results from the distinct relaxation processes of pre-polymers: isocyanate and diamine for hard
and soft phase, respectively for an exemplar polyurea 1000. The mechanical behavior of
elastomeric copolymers is hence found to travel between those in the glassy and the rubbery
polymers. The multiple processes of relaxation was also found to lead to a natural transition in
the rate-sensitivity in stress-strain curves, which exhibited (1) highly nonlinear elastic-plasticity;
(2) isotropic hardening due to effects of alignment and locking of molecular chain networks in
post-yield regime; (3) dramatic stress-upturn and asymmetry between compression and tension
due to the network elasticity; (4) highly nonlinear elastic-plastic softening and stretch-induced
softening known as Mullins effect during cyclic loading-unloading; (5) a large amount of
hysteresis; and (6) substantially nonlinear elastic recovery with a residual deformation upon
unloading. 3-1
A better understanding of the underlying mechanical principles for the
complicated constitutive responses has been a focal point of research interest to provide the
design principles of material architectures of elastomeric copolymers at a wide range of lengthscales in tandem with the chemical principles of molecular microstructures of hard and soft
phases and their mixtures.6,
12, 13
In particular, the highly "resilient" yet "dissipative" large
deformation of the materials has received a significant attention in a variety of research efforts
via experimentation, constitutive modeling to numerical simulations in diverse contexts of
polyurea- and polyurethane-based systems.3-7, 14-16 Recently, the features of "resilience" and
"dissipation" are finding new avenues towards new material architectures facilitating substantial
energy dissipation as well as shape recovery in a broad variety of engineering and military
applications ranging from impact and shock protective coatings 14' 17' 18 to self-healing
microstructures. 19-21
31
Elastomeric copolymers were found to significantly enhance the protective performance of
composites against impact and ballistic penetration. Recent studies on blast-elastomer
interactions also revealed that the highly resilient yet dissipative features of the materials under
extreme deformation played a key role for shock mitigation of combat helmets against extreme
blast loading.22,23 The polyurea- and polyurethane-based elastomers have been hence "potential"
materials for protective systems such extreme strain and strain rate environments as impact,
ballistic and blast loading. The complex stress-strain responses of the materials were hence found
to be of central importance, posing significant challenges in developing appropriate theoretical
and experimental frameworks spanning a wide range of mechanical loading histories. In
particular, there has been an increasing interest for physically- and microstructurally-informed
constitutive theories and their computational implementation, which can be validated through
experiments. In this research, we are focused on the development of generalized constitutive
modeling framework for elastomeric copolymers including segmented elastomers and ionic
elastomers
and their chemically-modified
counterparts
imparting
segmental dynamics,
morphology and weight fractions of hard and soft phases and their mixtures which can be
tailored by a simple chemical processing.
2.2.
Segmented Copolymer Polyurea and Polyurethane
Segmented copolymers possess a backbone structure comprised of alternate "hard" and
"soft" segments, where the hard/soft nomenclature corresponds to the thermodynamic
glassy/rubbery state at ambient temperature.
'''
The thermodynamic incompatibility leads to
a phase-separated morphology of the hard and soft domains which can be tailored depending on
the chemical composition, molecular dispersion, processing, and synthesis to give occluded soft
domains in a hard matrix, occluded hard domains in a soft matrix or a co-continuous network.1 5 '
26-28 The mechanical performance of the segmented copolymer is governed
by the chemical
composition as well as the morphology providing a hybrid performance of its constituents often
offering new and unique properties. Herein, the mechanics of the large strain behavior of
elastomeric segmented copolymers is examined by taking an exemplar polyurea. Polyurea
exhibits a highly resilient yet dissipative behavior, wherein its "rubbery" soft domain provides
32
enhanced resilience while its "glassy" hard domain provides enhanced stiffness as well as
substantial dissipation mechanisms. The presence of hard and soft domains yields to multiple
molecular relaxations and hence multiple viscous dissipation mechanisms.'
6 25
In addition to the
viscous dissipation, in-situ X-ray scattering data revealed microstructural evolution governs
stretch-induced softening during deformation and provides another major dissipation pathway in
5,6,10
polyurea.,5,29,30 Furthermore, polyurea exhibits a substantial shape recovery upon unloading.'
'
Through the highly dissipative yet resilient features, polyurea has become a versatile material for
a myriad of applications ranging from the ballistic protective composites'3 1 3 2 to the self-healing
microstructures 9-21 and the multi-functional active coatings.33,
34
Furthermore, elastomeric
segmented copolymers are finding new opportunities to design tunable adhesive and frictional
surfaces for biologically-inspired robotics.:3-37
The elastomeric segmented copolymer polyurea, polyurethane and polyurethane-urea
share the very similar chemical features comprising almost identical hard and soft segments with
urea-, urethane- and urethane-urea-links, respectively. Such macromolecular structures of hard
phases connected to soft phases via specific links can be achieved since they are derived from the
same prepolymer (diisocyanate) for hard segments and chain extenders (amine and alcohol for
urea and urethane respectively) for soft segments. The large chain molecules comprising of hard
and soft segments are mainly connected via hydrogen bonds. Figure 2-1 illustrates schematics of
chemical structures of polyurea which is polymerized from diisocyanate and amine. Methylene
diphenyl diisocyanates (MDI) react with functionalized amines to form urea (N2H2 CO) links in
polyurea copolymers. If the amine is replaced by a polyol (alcohol) such as an ethylene glycol,
polyurethane copolymers are formed with urethane (NHCO 2) links. To date, a number of
polymerization pathways have been developed based on the exemplar urea and urethane
chemistry including step polymerizations (poly-addition polymerization) and free radical
polymerization.
33
0
(a)
NH 2
C
0
-
0 -OCH2CH 2CH 2CH 2);nO-0
NH 2
Polytetramethyleneoxide-dI-p-aminobenzoate (Versalink P Series, Air Products)
OCN-R-N-C=N-R-NCO
IOCN
- R- NCO
- R- NCO
O=C-N
Uritoneimine modified diphenylmethane dilsocyanate (Isonate 143L, Dow)
R:=
0
Hard
0
C- (0CH2CH 2CH2CH 2)-
-
N-C-
R-N
-
Soft
0
"IT---"1;
C= N-R-C-
-
N-R
C -(OCH
0
-(OCH 2CH 2CH2CH);;-0-
LH
C -
N-C-
2CH
2CH 2CH2)-0-C
-- C>N
Figure 2-1 Schematic of chemical structures of polyurea (polyurethane): (a) Reactants:
functionalized amine (soft) and diisocyanate (hard); (b) Exemplar chemical formula for polyurea
studied in this research (PU650 and PU 1000)
The chain extenders in soft phase prepolymers can be simply modified to provide tunable
properties
of
polyurea
and
polyurethane.
In
particular,
the
molecular
weight
of
polytetramethyleneoxide (PTMO) chains in diamine prepolymers in Figure 2-1 was found to
significantly affect the overall segmental microstructures and the weight fractions of hard and
soft phases specifically in polyurea chemistry. We will discuss the effect of molecular weight
control of diamine prepolymers to macroscopic constitutive responses of two different polyureas
in terms of resilience and dissipation in Chapter 5. Additionally, the constitutive modeling
framework proposed in the following sections will be furthered to capture the effects of
variations of segmental microstructures and weight fractions of hard and soft phases and their
mixtures. Figure 2-2a presents a schematic for the segmental microstructures of polyurea and
polyurethane
comprising
the
multiple
components
and
the hydrogen
bonds
between
macromolecular networks. Here, a variety of morphologies of hard and soft segmented can be
made via the bulk polymerizations including bi-continuous, interpenetrating networks and soft
34
or hard phase aggregates occluded in hard or soft matrices. Figure 2-2b shows a dynamic
mechanical property of a polyurea studied in this research (PU1000), revealing the presence of
two phase nature. The two peaks in the loss factor represent major relaxation processes of hard
and soft phases revealing the hard segments are in glassy phase while the soft segments are
above glass transition at room temperature. The multiple relaxation processes with the phaseseparated morphology of hard and soft segments provide a basic idea for the constitutive models
comprising multiple micro-rheological mechanisms in the following chapters.
(b)
Isolated Hard Segment (HS)
1
Bonded Hard
\egment
-.
Z:
-X-
.
0.10
Occluded
Soft
Hard
0.15,
CL
io
Hydrogen
soft
0.20
L
2 10,:
-0.05.3
Domain
0
macromolecule
100
150
0O
S0
100
150.0
Temperature, T ['CI
Figure 2-2 Microstructure and dynamic properties of polyurea: (a) Schematic for phaseseparated morphology of hard and soft segment; (b) Dynamic mechanical properties: storage
modulus and loss factor
35
(a)
(b)
Figure 2-3 Micrographs of polyurethane and polyurea: (a) Transmission electron microscope
4
(TEM) image of polyurethane (57% soft phase, 43% hard phase, from Qi et al. ); (b) Tapping
mode atomic force microscope (AFM) phase image of polyurea 1000 (64% soft phase, 36% hard
phase, from Castagna et al.13 )
As revealed in Figure 2-3 on the micrographs of exemplar polyurea and polyurethane,
there are two-distinct phases of hard (bright) and soft (dark) constituents which often exhibit a
variety of morphologies. The microstructures and morphologies of polyurea and polyurethane
have been studied in many literatures using X-ray scattering. A variety of synthesis and
processing pathways were proposed to control the morphological features of constituents which
6
lead to significant changes in mechanical and thermal properties on demand. '
15, 27-29
In this
research, we will introduce a simple micromechanical modeling of an exemplar co-continuous
morphology to address the morphological effects on the macroscopic mechanical response of
materials in conjunction with the weight fraction effects of hard and soft phases, which provides
a simple yet critical insight into a morphologically-tunable mechanical property in this class of
elastomeric materials.
The morphological and microstructural changes were also found to critically affect the
mechanical performance of the materials upon mechanical deformation. In particular, the stretchinduced softening was extensively reported in polyurea and polyurethane elastomers as a
precursor for the microstructural change due to mechanical deformation. The stretch-induced
softening, first reported by Mullins and coworkers was found to provide a major source of
36
hysteresis in diverse natural and synthetic elastomers. Recent studies,
, " on in situ x-ray
scattering observation during tensile deformation of polyurea revealed that the irreversible
anisotropic evolution with the microstructural breakdown in hard phases observed in the small
angle x-ray data is responsible for the stretch-induced softening. In Rinaldi et al., the detailed
evolution of small (SAXS) and wide angle x-ray data (WAXS) provided quantitative information
on microstructural change in inter- and intra-domains of segments. Figure 2-4 shows the
evolution of SAXS and WAXS during a cyclic tensile test. Here, the SAXS data represent interdomain distance (long distance for aggregate structures) while the WAXS data show intradomain spacing (short distance for internal structures). As shown in Figure 2-4a, there was no
significant evolution of WAXS data during cyclic loading and unloading in Figure 2-4b. On the
other hand, the irreversible anisotropic evolution of SAXS data revealed that the breakdown in
hard domain aggregates is a governing mechanism for softening and hysteresis providing new
microstructures upon loading; i.e. the internal properties related to elastic and inelastic behavior
significantly change upon loading; but there was no significant evolution of SAXS during
unloading and reloading up to a prior maximum imposed strain. These observations on the
microstructural evolution of hard phases provide significant insight into microstructurallyinformed constitutive modeling framework discussed in the next chapters.
37
(a)
loading
Z,
k~~ 1(b)
2
~dreloading
1
.
S12
20
0.0
unloading
0.2
0.4
0.6
0.8
1.0
True Strain
Figure 2-4 In situ X-ray scattering data of polyurea under cyclic tension: (a) WAXS and SAXS
evolution during cyclic loading-unloading-reloading; (b) Stress-strain data under cyclic tensile
test (from Rinaldi et al.5 )
The stretch-induced softening provides a large amount of energy dissipation in tandem
with other inelastic mechanisms such as viscoelasticity and viscoplasticity which arise in the
materials via molecular relaxations (Rouse or Zimm mode 38) and reptation of chain networks. 39
The presence of viscous inelasticity and stretch-induced softening often leads to complicated
mechanical behavior exhibiting a "weaving" stress-strain path (See Chapter 3) which was never
achieved in purely elastomeric materials, posing a challenge in predictive models capable of
capturing the salient features of deformation. In addition, the viscous, inelastic mechanism was
found to play a key role for substantial shape recovery of localized residual deformation beyond
purely elastic recovery via network elastic resistance. The shape recovery mechanisms will be
further discussed in the following chapters and will provide critical insight into the shape
memory and the self-healing processes which have been widely reported in this class of
elastomeric copolymers.
In this research, we investigate polyurea 1000 and polyurea 650 as exemplar segmented
elastomers. The samples have been synthesized by mixing procedures developed by Naval
Surface Warfare Center, Carderock and Dahlgren division.
38
2.3.
Ionic Copolymer Ethylene Methacrylic Acid (EMAA) and Ethylene Methacrylic
Acid Butyl Acrylate (EMAABA)
Ionomers are polymers possessing ionic functional groups partially through the backbone
structures.4
42
In this research, we are focused on the mechanical behavior of ethylene-based
ionic elastomers developed by Dupont. Ethylene methacrylic acid (EMAA) copolymers were
first developed about four decades ago.43 EMAA copolymers are chemically homogeneous and
known as ionomers when not neutralized. They are formed in the high-pressure, free-radical
polymerization processes which are essentially the same as those in synthesizing low-density
polyethylene. Therefore, when there is no methacrylic acid content, their chemical, thermal and
mechanical behavior approaches those in low-density polyethylene. It was hence found that
varying the methacrylic acids (MAA) in EMAA substantially affect the macroscopic response of
materials. 42'
44
Figure 2-5 shows an exemplar chemical structure of EMAA derived from
ethylene and methacrylic acids illustrating the methacrylic groups and the methacrylic acid
groups which are connected via the branched or non-branched ethylene chains. Here, the
methacrylic acid groups can be neutralized with salt metallic cations (M') to provide the salt
counterparts of EMAA. The ionic functional groups can be processed to be pendant to the
polymer backbone structures leading to a variety of morphologies of constituents. The
mechanical, thermal and chemical properties of ionic copolymers exhibit an outstanding range
depending on the microstructures of materials, which often comprise soft amorphous domains,
ionic clusters, and hard crystalline domains. 42 45, 46 More specifically, the multiple domains of
phase-separated morphology include non-branched linear ethylene chains, amorphous, branched
ethylene chains, and ionic aggregates. At ion-poor regions, the ionic aggregates known as
multiplets act like physical crosslinks while they act as both physical crosslinks and crystallites
at ion-rich regions providing enhanced stiffness to the materials.
A chemically-modified counterpart of EMAA has been introduced by adding butyl
acrylate contents to the base material, where the branched ethylene butyl acrylate regions are
formed in microstructures leading to ethylene methacrylic acid butyl acrylate (EMAABA)
terpolymers. In addition, the methacrylic acid groups in EMAA and EMAABA can be partially
and fully neutralized with metallic salt cations such as sodium (Na'), zinc (Zn'), magnesium
39
(Mg'), lithium (Li') and copper (Cu') providing neutralized counterparts of methacrylic acids in
the base materials.
CH 3
I
CH 3
+
CH 2 = CH 2
CH 2 = C
-m~-
-(CH2)n
-
CH 3
I
C -(CH2)n
0
I
C=O
I
H
H
H
C=o
-
C -
(CH2) n-
I
C=O
I
O-
Figure 2-5 Schematic of chemical structures of EMAA and its neutralization: EMAA formed
from ethylene and methacrylic acid
(a)
(b)
So
300
200
102
100
-100
.50
0
00
Tempoinature (*C)
10
L
Tangent
i
I.I.
0.3
400
io'
if
0.10
0.25
Storage Mod\
0.2
I
3003
J102
200
I
0
0
Los Mouu
100
0.1
10,~
-150
Soo
Storage
Modulus
.0.
0.5
400
400
Storage Modulu
LoeModulus
Loss Tangent
(c)
50
... ..
...
0.25
0
--1p-100
)
Temperature (IC)
0
0.2
Loss %
Modulus\
0.2
200
io,t
0.15*
CA
100
01
-
Te10a-100
r0
0
0.1
0
so
Temperature (*C)
Figure 2-6 Dynamic mechanical property of ethylene methacrylic acid copolymers; storage
modulus, loss modulus and loss tangent curves as a function of temperature: (a) EMAA; (b)
EMAABA; (c) EMAABA-Na (from Deschanel et al.16)
The structures and properties of ionomers strongly depend on the distribution of ionic
groups along the polymer backbone chains. The multi-phase morphological nature of ionic
copolymers has been extensively reported by X-ray scattering 44 and various physical
measurements.47 ' 48 In particular, the dynamic mechanical properties reported in McKnight et
al.42 and Deschanel et al.4 6 revealed the presence of multiple relaxation processes of EMAA and
sodium-neutralized EMAA copolymers. Figure 2-6 shows dynamic mechanical analysis (DMA)
40
'
data of EMAA, EMAABA and EMAABA-Na revealing the presence of r , 8, 8' and C
relaxations at increasing temperature. In order of increasing temperature, the y peak occurs at -120 C which is essentially the same as that found in a linear polyethylene homopolymer. Then,
a 8 peak is present at ~- 10 'C and a higher temperature peak occurs at
-
50 C labeled as "C'.
The 8 peak corresponds to the relaxation of ion-poor within the amorphous phases; and the ionrich regions partially crystallized are responsible for the C peak. However, due to the
devitrification of ion-rich regions and the simultaneous melting of secondary crystals, the
interpretation and observation of C relaxation are very complicated. Additionally, another
relaxation (/') can be observed between the two peaks for 8 and C in EMAA.
The multiple transitions in the ionic copolymers often lead to a hybrid mechanical
behavior of constituent phases over a wide range of strain rates. In the previous studies of
elastomeric ionic copolymers (EMAA, EMAABA and EMAABA neutralized with cations 45,46'
49-51),
the mechanical properties were found to be very similar to those in elastomeric segmented
copolymers such that (1) highly rate-dependent yield-like behavior and its transition near to a
moderate strain rate; (2) nonlinear, isotropic hardening after a rollover; (3) highly nonlinear
unloading behavior accompanied by remarkable hysteresis and shape recovery; and (4)
substantial elastic-plastic softening under cyclic loading histories. Specifically, the highly
resilient yet dissipative mechanical behavior of elastomeric ionic copolymers was examined in
terms of energy storage and dissipation in Greviskes et at.45 In their research, the energy storage
and dissipation at an increasing imposed strain were extensively quantified for the ethylene
methacrylic acid butyl acrylate terpolymers which were non-neutralized and neutralized with
metallic cations. The chemically-modified counterparts of ionomers were found to provide an
outstanding range of "tunable" resilience and dissipation functionality in tandem with varying
the weight fractions of constituents.
In addition to a remarkable range of tunable mechanical behavior, the elastomeric ionic
copolymers have received great attention by virtue of their "self-healing" behavior by which
fractured interfaces often exhibit an immediate closing via a rapid transport and rearrangement of
chain molecules across the interfaces that underwent an opening.20,
21
The self-healing
phenomena in ethylene methacrylic acid copolymers and their chemically-modified counterparts
52,5
have been extensively studied using puncture and impact tests.:'53 At present, the underlying
41
mechanisms responsible for healing behavior have not been addressed well. Though healing has
been considered a "thermally-activated"
process dependent upon deformation rate and
temperature, there is no direct observation of interfacial diffusion of chain molecules which is
strongly dependent upon local concentration and molecular weight at the surfaces. In addition, a
driving force for initiation of healing remains unclear to date. The interfacial mobility of chain
molecules has been found to strongly depend on viscoelastic-viscoplastic
deformation,
temperature and pressure at the polymeric interfaces. Furthermore the shape recovery was found
to be essential to initiate healing since it is clear that healing should be accompanied by a
physical contact of fractured interfaces. In the following chapters, we will investigate the
remarkable shape recovery of elastomeric ionic copolymers under large deformation to provide
physical insight into the mechanically-driven shape memory mechanisms in tandem with those
found in elastomeric segmented copolymers.
In this research, we investigate emaa, emaaba and emaaba neutralized with sodium cations
as exemplar ionic elastomers. All of the samples have been provided by DuPont.
42
2.4.
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45
46
Chapter 3
Constitutive Modeling of Resilient yet Dissipative
Large Deformation of Elastomeric Segmented
Copolymers
Portions of this chapter were published in a journal paper, H. Cho, R. G. Rinaldi and M. C.
Boyce, "Constitutive Modeling of the Resilient and Dissipative Large Deformation Behavior of
Elastomeric Copolymer Polyurea", Soft Matter, 2012
3.1.
Background
We herein review the previous studies that provided basic ideas for the development of
constitutive models of elastomeric copolymers covered in this research including (1) inelasticity
of glassy polymers; (2) rubber elasticity; (3) viscoelastic-viscoplastic constitutive models of
elastomeric materials; (4) Mullins effect of stretch-induced softening; and (5) recent advances in
continuum mechanical framework of large deformation behavior of amorphous materials. In
particular, the overall theoretical framework built by Argon, Boyce, Parks and their coworkers'
" has broadly contributed to constitutive theories of glassy/rubbery polymers and copolymers
travelling between the two physical phases, providing the physically-sound deformation
mechanisms of various polymeric materials at microscopic- and macroscopic scales; and
constituting the basis for many ideas in the constitutive modeling framework proposed in this
research.
3.1.1. Inelasticity of amorphous polymers
47
Viscous flows of inelastic materials were found to mainly rely upon the activation
processes of molecular transport and slip under mechanical loading. Eyring and Ree pioneered
the laws of kinetics in non-Newtonian flows of various plastic solids employing the principles of
absolute reaction rate under shear force.' 2 ' 4 The non-Newtonian viscous flow model of Eyring
and Ree theories was found to provide a useful framework accepted for viscoplasticity in a
moderate range of temperature and strain rates. Argon furthered the thermally-activated inelastic
flow model originally proposed by Robertson 15 to address yielding of glassy solids at low
temperature including amorphous polymers' and metallic glasses.16 In his work, yielding
relevant to inelastic flows of glassy solids was explained by local inelastic straining produced by
pairs of molecular kinks under shear stress. Furthermore, the free energy for formation of a pair
of molecular kinks was derived leading to a simple phenomenological formulation of inelastic
strain rate in terms of applied shear stress and temperature.
The Argon model was then further generalized by Boyce et al.7 involving temperature-,
rate- and pressure-dependent inelastic large deformation of glassy polymers over a wide range of
strains and strain rates. In addition, the multiple micro-rheological models comprising
intermolecular and network resistance mechanisms were then experimentally validated by the
differential scanning calorimetric (DSC) observation of the two distinct mechanisms in glassy
polymers by Hasan and Boyce.10'
17,
18
The constitutive modeling approaches employing the
multiple micro-rheological elements have been hence widely accepted and successful for
modeling various polymeric materials. Furthermore, their early studies were recently modified to
model multiple relaxation processes observed in glassy polymers such as polycarbonate (PC) and
polymethylmethacrylate (PMMA) incorporating a dramatic transition in the rate-sensitivity of
yielding."
Boyce et al.8 also examined the multiplicatively decomposing elastic-plastic kinematics of
amorphous polymers and single crystals that undergo finite strain plastic deformation. Their
analysis showed that the kinematic decomposition may be chosen to best analyze the specific
material model of interest since the choice of elastically-relaxed configuration is not essential to
describe finite strain elastic-plastic deformation. A generalized kinematic framework of isotropic,
viscoplastic solids which allow for finite deformation was recently provided with critical reviews
48
on the elastic-plastic decomposition, material symmetry and frame-indifference of the
constitutive theories by Gurtin and Anand 9 ' 2 0 and Gurtin et al.
3.1.2. Elasticity of rubbers
The force-extension models of natural and synthetic rubbers comprising a number of chain
segments (or Kuhn effective segments) were originally proposed by Wang and Guth 22, Flory and
Rehner2 3 , and Treloar 24 -2 6 based on the Gaussian statistics of chain molecules, by which the
entropic force arises without any intermolecular interactions. The entropic free energy without
internal energy due to the intermolecular interactions was found to naturally result in a purely
configurational force in large molecular networks. Rubber (or entropic) elasticity was also
further developed within the continuum mechanical frameworks of invariant-based models in
many literatures including Mooney27 , Rivlin 2 8 , 29, Ogden30,
31
and Anand.32 The non-Gaussian
statistics of chain molecules was also investigated to allow for larger stretches which cannot be
afforded by the Gaussian statistics incorporating networks of three, four or an infinite number of
chains. Following the use of Langevin statistics of chain molecules by Kuhn and Grun33, Arruda
and Boyce3,
6, 34
provided continuum- and statistical-mechanical treatment of the evolution of
orientation and locking of chain molecules during deformation such that elastic stress
dramatically increases near to the limiting extensibility of chains employing an eight-chain
spatial configuration of networks. The Arruda-Boyce eight-chain model has been highly
successful and accepted in modeling a broad variety of soft materials which exhibit the locking
behavior with the orientation effect ranging from synthetic rubbers to biological materials.
3.1.3. Stretch-induced softening: Mullins effect
Stretch-induced softening has been extensively observed in cross-linked rubbers beginning
with the early studies of Mullins and coworkers 35 -38 as well as in segmented copolymers
including polyurethane 3 9, 40 and polyurea. 40' 41 At present, a number of theories to capture the
49
softening of the rubbery response have been proposed based on damage or other forms of
microstructural evolution and breakdown. 42 -47 The model for stretch-induced softening in elasticplastic responses has built based on the microstructural evolution models for intermolecular and
network resistance. Specifically, extended from micromechanical modeling 48' 49 of rigid particle
filled elastomers and thermoplastic vulcanizates with occluded volume effect of filler and matrix,
a constitutive model for the evolution model in the hard and soft domain microstructure has been
proposed to account for the softening, where the effective volume fraction of constituent
domains evolves due to relative motions and deformation of the hard domain. Based on the
evolution model of "effective" volume fractions of constituents combined with thermallyactivated viscoplasticity, the mechanical behavior of thermoplastic copolymer polyurethane
including cyclic softening events has been successfully captured by Qi and Boyce.:'
47
Although
Qi and Boyce model is successful in cyclic compressive tests in low strain rate, it lacks capturing
the rate dependence of network elasticity assuming a time-independent behavior of the network
resistance and is not capable of describing high strain rate behavior.
3.1.4. Viscoelasticity of elastomeric materials
Large deformation viscoelastic behavior of elastomers has been extensively investigated
over the last decade. Bergstrom and Boyce '
proposed a two-phase constitutive model capable
of predicting the rate-dependent nonlinear stress-strain behavior of carbon-black filled rubbers
exhibiting a substantial shape recovery that accompanies a significant hysteresis. In their work, a
nonlinear viscoelasticity model was introduced following the use of the Doi and Edward theory
5 of reptation for chain transport and slip within a topological restriction. The reptation-based
nonlinear viscoelasticity was also found to be capable of capturing the orientation-dependent
molecular relaxations of chain networks in polyethylene terephthalate (PET) and polyethylene
terephthalate glycol modified (PETG) in and above their glassy transition temperature in Dupaix
and Boyce.9,
52
Constitutive models of elastomers have been recently furthered by Qi and
Boyce 39' 47 to capture the stress-strain behavior of thermoplastic polyurethane. In the Qi and
Boyce model, the Mullins effect was successfully captured by a phenomenological model that an
50
occluded effective volume of soft phases evolves upon deformation leading to substantial
stretch-induced softening.
Over the past decade, constitutive models to capture the stress-strain behavior of
elastomeric segmented copolymer polyurea and polyurethane have been proposed by a number
of research groups. Most of the constitutive models have been constructed based on the theory of
linear viscoelasticity including the William-Landel-Ferry (WLF) models for time-temperature
superposition. In addition, many prior models have employed a Prony series for multiple
relaxation processes with a number of time constants and weighting factors which were not
experimentally verified. Linear viscoelasticity models with multiple time constants to capture a
time-temperature equivalence of polyurea have been recently proposed by Amirkhizi et al." and
Chevellard et al. 54 Furthermore, a rate-dependent constitutive model was proposed to capture
uniaxial compression behavior of polyurea under low to moderate strain rate55 based upon prior
modeling of the large strain behavior of polymeric materials. 4' 3 9,49 Additionally, a nonlinear
viscoelasticity model was recently reported to model monotonic stress-strain behavior of
polyurea not including unloading behavior.56 Thus these models had predictive capabilities in a
relatively narrow range of strains and strain rates including a moderate level of resilience and
hysteresis. In particular, their models were not able to capture the mechanical behavior at large
strain regimes. By a restricted magnitude of inelastic flows, their models usually overestimated
stress levels at large strains. Therefore, the highly nonlinear elastic-plastic behavior including
nonlinear unloading and stretch-induced softening due to structural changes was never captured
in their models. Also, in terms of timescales or strain rates, their models have been restricted at
relatively short timescales or very high strain rate regimes. Though their models have been
successful for modeling the high strain rate behavior and confined deformation by which an
imposed strain level is relatively small, they still lack predictive capabilities for the main features
of this class of materials under a variety of loading histories up to a true strain of 1.0. In addition,
at unconfined extreme strain rate conditions, their models exhibited numerous limitations in
capturing the resilient yet dissipative features of materials. We will briefly discuss the limitations
of prior constitutive models using linear viscoelasticity by comparison to our framework in
Chapter 6 on the Taylor impact behavior of elastomeric materials. The prior work on the
constitutive models of this class of copolymeric elastomers has provided a strong motivation for
51
a new constitutive modeling framework that has better predictive capability over a broad range
of strains and strain rates under a variety of loading conditions.
3.1.5. Recent advances in continuum mechanical studies of polymeric materials
A continuum mechanical framework, which is thermodynamically consistent, was recently
proposed for elastic-plastic amorphous solids by Anand and Gurtin.19 The thermodynamically
consistent theory was then furthered for amorphous polymers at a wide range of strain rates and
58
temperatures spanning the glass transition in Anand and Ames557 , Anand et al.8
and Ames et al. 59
In their research, the constitutive theory was thermomechanically-coupled to model the large
deformation of materials accompanied by transient thermal transport. A very rigorous treatment
of evolution models for internal variables involving plastic flows was performed for numerical
simulations of complex deformation processes in glassy polymers6 0 and other functional soft
materials such as shape memory polymers 6 1 and hydrogels. 62 Anand and coworkers rationalized
theoretical foundations that rigorously satisfy the thermodynamic principles in classical
continuum theories dealing with the frame indifference of constitutive theories and the principle
of virtual power to derive micro- and macroscopic force balances. Additionally, they provided
appropriate computational procedures for finite element implementation of their constitutive
theories, which can be widely accepted for numerical modeling of coupled phenomena involving
mechanical deformation and transient heat and mass transport in amorphous polymers.
3.2.
Mechanical Behavior of Exemplar Elastomeric Segmented Copolymer Polyurea
Here we consider an exemplar polyurea (PUl000) with 34% hard and 66% soft content.
The hard and soft domain structure of polyurea materials form a phase-separated morphology at
the length-scale of tens of nanometers as found via microscopy and small angle X-ray scattering
(SAXS).41'61-6' Figure 3-iFigure 3-1a plots the storage modulus (E') and the loss factor ( tan 5 )
as a function of temperature measured via dynamic mechanical analysis (DMA) at a frequency of
52
1 Hz. The two peaks in the loss factor at -40 OC and 135 'C are associated with the main
molecular relaxations of the soft and hard domains, respectively. Thus DMA supports the phaseseparated morphology of polyurea where the soft aliphaticdiamine is in a rubbery state while the
hard diisocyanate is in a glassy state at room temperature (25 C). The storage modulus of 100
MPa at room temperature implies that the hard domain structure is continuous in the initial
undeformed state.
53
o a-0.4
(a)
0.20
e-El
VI
15
P 0.15
..- tan
10
"
(b)
-
12
ta0.25"-
train RooV/s)
9
0.01 8
6
2
10
0.0
S0
10 1
3
Te 0.00
0.2
0.0
Temperature [*C]
(d)
50
0.8
0.6
0.4
True Strain
1.0
50
6sos
30
()
0
40
40
0
0
30
1.
20
0
20
0 0
.
10
0.901 a
r"
0.2
0.0
0.01. 8".1a
0.8
0.6
0.4
-0
10
10-3
1.0
True Strain
4W0
(f)
0.20
300
30
N
'U
2810'.5
1.0
1.5
103 10'
20
tension
20
0 10 10
10-2 1
Strain Rate [s]
a.
2.0
-N2
15
10
-
10
0
.
0
1001$ -
'I)
'U
2
I-
compression
. .20.0015.
.0
0.2
0.6
0.4
True Strain
5
0.08a' 0.l3
"I
0.8
1A(0
$.0
0.2
0.6
0.4
True Strain
0.8
1.0
Figure 3-1 Uniaxial compression and tension data for polyurea (a) dynamic mechanical analysis
explaining the co-continuous microstructures, from Rinaldi et al. 41 ; (b) rate-dependent stressstrain data under low strain rate compression with an inset of flow stress as a function of strain
rate, from Yi et al.
40;
(c) rate-dependent stress-strain data under low to high strain rate
compression, from Sarva et al. 66 (d) flow stress as a function of strain rate under low to high
strain rate compression (e) asymmetric stress-strain data under compression and tension at low
strain rates of Iel= 0.001s-1 ,0.01s-' and 0.1s- with an inset (from Choi et al.
65)
of stress-strain
54
data up to locking phenomena under tension; and (f) stress-strain data under cyclic tension at a
strain rate of t
=
0.003 s-1, from Rinaldi et al.41
Uniaxial tension and compression behavior were extensively studied over a broad range of
strain rates in Rinaldi et al. 41 , Yi et al. 40 , Sarva et al. 66 , and Roland et al. 67 The large deformation
rate-dependent stress-strain behavior features a relatively stiff initial response with a ratedependent roll-over or yield to a more compliant response (Figure 3-1 (b)). As shown in the
inset of Figure 3-1 (b) there is a single main relaxation mechanism at low strain rate, where flow
stress is logarithmically proportional to strain rate up to 1 s-1. An increase in rate-sensitivity is
observed after a critical strain rate of 1 s- (Figure 3-1 (c) and (d)), after which the soft domains
are not fully relaxed and provide an additional resistance to deformation at high strain rate.
Figure 3-1 (e) shows the asymmetry in the tension and compression stress-strain behavior as the
applied strain increases. As shown in the inset of Figure 3-1 (e), the dramatic stress hardening is
observed around a strain of 2.0 under tension, which arises from the orientation-based chain
limiting extensibility of network structures in hard domains. Unloading is observed to be highly
nonlinear and reveals substantial hysteresis and resilience (Figure 3-1(b) and (f)). A stretchinduced elastic-plastic softening is observed upon reloading (Figure 3-1(f)).
The stretch-induced softening that results from the microstructural breakdown is a
significant mechanism of energy dissipation in segmented copolymers. Extensive cyclic tension
tests accompanied by in-situ small (SAXS) and wide (WAXS) angle X-ray scattering
measurements were reported by Rinaldi et al.
to assess the microstructural evolution during
deformation. The SAXS measurements revealed the irreversible rearrangement and breakdown
of aggregate networks in the hard domains during loading, which lead to a new structure with a
substantially
softened behavior
observed
during reloading.
The strain-dependence
of
microstructural breakdown observed in the SAXS data and the cyclic tension data provides
insight into capturing the microstructural evolution as the mechanism for stretch-induced
softening in the constitutive model.
In summary, a constitutive model should be able to capture the main features of the
mechanical behavior of polyurea as follows:
(1) Large strain nonlinear elasticity
55
(2) Asymmetric stress-strain behavior in compression and tension
(3) Rate-dependent viscoelastic-viscoplasticity including a transition in rate-sensitivity
(4) Nonlinear unloading accompanied by substantial hysteresis and shape recovery
(5) Stretch-induced elastic-plastic softening
3.3.
Model Development
In summary, the overall framework for the constitutive model will build upon prior
modeling of the three-dimensional rate-dependent large deformation of polymeric materials,
including models of nonlinear viscoelasticity of elastomeric materials 4'
68,
models of elastic-
viscoplastic behavior of glassy polymers'',7'11'59 and models which travel between rubbery and
glassy states. 9' 49 The macromolecular network of a polymer responds to mechanical deformation
through its intermolecular and network resistances, where the networked structures are formed
by chemical crosslinks, physical entanglements, or a combination of the two components. When
above the glass transition, the polymer chains undergo stretch and rotation during deformation
and the equilibrium behavior is well captured by statistical mechanics models of rubber
elasticity3, 6,69, 70 and/or continuum mechanics based hyperelasticity models. 29-31 Furthermore, the
chains and network junctions can slip via reptation giving viscoelastic behavior.451 ' 6 8 7 1' 72 When
in the glassy state, the intermolecular interactions provide a significant resistance, giving a
dramatic increase in the elastic stiffness.
The thermally-activated nature of the rate- and
temperature-dependent yield has been well-captured by an Eyring type viscous flow model1.
When the range of temperature and strain rate spans multiple molecular relaxation mechanisms,
a Ree-Eyring type viscous flow model' 3 has successfully captured the distinct relaxation
processes that govern viscoelastic or viscoplastic flow." The network elastic resistance is
described to act in parallel to the intermolecular resistance and, once the intermolecular
resistance yields, the large strain enables a substantial amount of network evolution with
deformation, giving the post-yield stiffness, "strain hardening" behavior, and nonlinear recovery.
56
The constitutive model considers the hard and soft domains to be co-continuous. The
overall resistance of each domain is considered to consist of an intermolecular resistance and a
network resistance as represented by the micro-rheological elements in Figure 3-2. The material
properties of the two elements in each domain are determined to capture their individual
contribution to overall responses as described in detail later.
Isolated Hard Segment (HS)
(a)
(a)
(b)
Hydrogen
Bonded Hard
soft
Domain
Hard
J
segmnent
.
Occluded
E
00
E
soft
L
Dmin
Doma n
Hard Domain
Soft Domain
macromolecule
Figure 3-2 Schematic representations for the constitutive model (a) a polyurea microstructure
representing a segmented copolymer with interpenetrating hard and soft domains; (b) a onedimensional micro-rheological interpretation for the hard and soft domains in the proposed
constitutive model
The large deformation kinematics of the present constitutive model follows the framework
presented in Bergstrim and Boyce, Boyce et al.7 '
8,
, Qi and Boyce39 , and Mulliken and
Boyce." Terms relating to the viscoelastic-viscoplastic intermolecular component and the elastic
network component in the hard domains shall be given by a subscript a of "IH" or "NH",
respectively. Similarly, soft domain components will be given by "IS" and "NS".
The total deformation gradient F Vx=
ax
that maps a material point in an undeformed
referential configuration x to a deformed spatial configuration
x
acts on each of the four
components of the model. The total deformation gradient is multiplicatively decomposed into
elastic (e) and inelastic (p or v) parts following a Kroner decomposition 73, which is schematically
illustrated in Figure 3-3,
57
F= FP
1H=
F=FNH
P
N =FsFs
isisNs =F
-S(1)
The elastic and plastic deformation gradients are decomposed into the right stretch (u ) and
rotation (R)
or the left stretch (v ) and rotation (R ) terms following a polar decomposition:
F =R Ue =Ve R e;
FP= RPU
=VPRP.
velocity gradient L,= gradv=
The deformation rate is examined through the spatial
. The velocity gradient is decomposed into elastic and plastic
contributions,
La = #aF- 1 = L
ia =a
+ FeLP
F -I =
P F
+ F pF~
1
F7~1 = Le +
Lp
(2)
(3)
+WC,
where ii; and W represent the rate of plastic stretching and spin, which are the symmetric and
skew part of LP, respectively. With no loss on generality , the viscoplastic flow in the current
configuration is taken to be irrotational, giving
Nj = F.~ 1 DPFeFP = F~ 1 DPFa
The rate of plastic deformation gradientit
(4)
is then numerically integrated to obtain the plastic
deformation gradient FP and, consequently, the elastic deformation gradient is obtained via
FP-i
a
. The rate of plastic stretching frj under a given stress state is constitutively
a
prescribed:
D
=
(5)
j NP,
where the plastic flow is taken to be co-axial to the deviatoric stress tensor, and hence NP is the
normalized deviatoric stress tensor,
(6)
N,a _ T,
'
" Ta
1
3
where T" =T, -- tr(T,)I.
58
(b)
(a)
F = F FP
..........................
Soft Domain
Hard Domain
Figure 3-3 Schematic representations for the constitutive model and the large deformation
kinematics (a) the intermolecular and network element in the hard domain for low strain rate
behavior; (b) a kinematics map in the large deformation elastic-plasticity
The rate-dependent yielding event is considered to be a thermally-activated process,
whereby the energy barrier to intermolecular interactions must be overcome.",
12, 13
The
thermally-activated inelastic process can be expressed in terms of the thermodynamic parameters
such as the activation free energy, the activation volume and the intrinsic shear resistance against
inelastic flow. The magnitude of inelastic flow g is constitutivelyprescribed by,
-
faf =-I T
+ K'a exp
AGa
k 0 sinh
kB
k7
AGa
-(7
"~-"
,
(7)
kB9.a)
where Yo,,a is the reference viscoplasticrate for the magnitude of flow, A G a is the activation free
energy of inelastic flow, k, is Boltzmann's constant, o is the absolute temperature, 3 is the
athermal shear strength, ); is the initial shear viscositythat captures the initial linear nature of the
viscoelastic behavior,
and
a
is a magnitude of the deviatoric stress tensor. The
activation free energy as the intermolecular energy barrier to inelastic flow can be related to the
activation volume through the internal shear strength via A Ga = a a. The shear resistance
against yielding was found to substantiallyevolve during deformation, revealing the "softened"
intermolecular structures with inelastic straining. Additionally, the shear resistance may be
59
further modified to account for the pressure sensitivity in the viscoplastic flow as in Boyce et al.7
using
+ a - p , where the mean pressure p =
s,=,
--
1
trTa and is a the pressure sensitivity
3
coefficient. The pressure sensitivity coefficient a may be taken to be 0.075 ~ 0.15. In particular,
the pressure-sensitive shear strength becomes important at high strain rate behavior, where a
significant mean pressure develops. Furthermore, under confined deformation conditions
including the indentation testing at Chapter 7 and 8, the pressure sensitivity coefficient was
taken to be 0.125 and 0.135 for PU1000 and PU650 (See Chapter 5 for PU650 constitutive
behavior).
The intermolecular stress term is taken to follow a compressible Neo-Hookean31
representation with the Cauchy stress, given by
T, =
a
/,\2BIA
)2
(1)
(Ba
-(A'
)
I +
j-J-7)I,
where the subscript a stands for "IH" or "IS", J =
volume change, and B
FFT
=
(8)
det F, = det F' det FP = det Fe
is the elastic
is the left Cauchy-Green tensor, p, is the elastic shear modulus,
B is the initial bulk modulus of the material, A =
is an "average" stretch, I;e
tr(Be)
is the
first invariant of B e, and i is the second order identity tensor. A Lennard-Jones type volumetric
free energy is employed in order to capture the nonlinear volumetric stress contribution under
extreme pressure, which has been validated up to a few GPa in the pressure-shear impact tests on
polyurea by Jiao et al.74 The bulk resistance to volumetric strain may be lumped into the
intermolecular elasticity in the hard domain.
The network stretching and rotation are captured by the Arruda-Boyce eight-chain
elasticity model.3 '6 ' 3 4 The network stress is taken to be deviatoric and the Cauchy stress is given
by
Ta
-
J
3JA,,
r
"
IN)
B
-(A,
)2 1)
(9)
60
where the subscript a stands for "NH" or "NS",
L-'
VIT. denotes the inverse Langevin
function which is asymptotically estimated by a Pade's approximate 75
/
3-
2
c"
-(10)
where, jj
Ack .a
is the chain limiting extensibility,
e =
etr(B;)
is
an "average" chain
stretch in the eight chain network, and p, is the initial elastic shear modulus of the network. The
eight-chain model captures the strain-dependent stiffening and the asymmetric nature of
stiffening as evidenced in the dramatic difference in hardening behavior in tension as compared
to compression, indicating the hardening in these materials is dominated by an orientation
mechanism. The dramatic stress hardening due to the orientation-based finite extensibility in
these materials has been extensively reported in Choi et al.6 5 and Roland et al.67 Additionally, in
Rinaldi et al.4 , the evolution of anisotropy and orientation due to stretch in the hard domain
structures was monitored in the WAXS measurements, supporting the stress hardening behavior
due to the finite extensibility.
A phenomenological evolution which develops with inelastic straining is employed to
capture the elastic-plastic softening in the intermolecular component. Following Boyce et al.,
the athermal shear strength evolves to a preferred state with plastic straining,
a =ha 1-
-i
Sssa
Ia,
(11)
0.077,u,
where ha is the softening slope, s,,
(12)
is the steady state shear strength, va is the initial Poisson's
ratio, and ma controls the rate of evolution to the steady state. The intermolecular elastic shear
modulus pa also evolves with plastic straining,
61
ssa
(13)
,u
where ,.a is the steady state elastic shear modulus.
As evident in the SAXS measurements of Rinaldi et al.4 , the microstructural breakdown
that occurs during straining provides a substantially softened hard domain structure and is
responsible for the large hysteresis and dissipation. The irreversible breakdown of hard domain
aggregates monitored in the SAXS measurements during loading leads to a new structure with a
substantially softened behavior observed during reloading. Motivated from observations on the
microstructural breakdown of the hard domain network, a phenomenological evolution model for
the chain limiting extensibility
2lOCkNH
(t) = VNNH (t) in the hard domains is proposed in the
present constitutive model to capture the microstructural breakdown as a softening of the
networked structures during deformation. As the network breakdown proceeds, it may be
interpreted that the chain limiting extensibility increases during loading leading to an increase in
NH (t) and hence a decrease inUNH W -This relation may be given by,
SNo
=pm
(t)N t).
(14)
The chain limiting extensibility is taken to follow a single evolution rule,
AockNH
Slock,NH
O c NH
SONH
ck.NH /Assock,NH
(2ock,NH
Krran
2
A, H=r
k.o
where
AsslockNH
chain stretch,
tn
/
r2H
AkC
sinh
oNH
ySlockNH);
(15)
(16)
ockNH (
(17)
I1
9) *IOck.NH
,
(0)
is a maximum achievable chain limiting extensibility dependent upon the rate of
Slock,NH
is the internal resistance to the breakdown of chain networks which also
evolves during deformation, and 5NH = .TNH:
TNH
is a magnitude of the current stress tensor in
62
the hard network. The evolution of the chain limiting extensibility also captures the ratedependence of the network structural breakdown.
Nonlinear viscoelasticity is employed to capture the time dependent relaxation of the
network response of the soft domains which is observed to be strongly dependent on strain rate.
Following Dupaix and Boyce9 and the Doi-Edwards reptation theory 4'
51,
nonlinear
viscoelasticity in the soft network component is captured as follows:
&v
= f()
(D( )
(18)
N)l7Ns,
-
(19)
where 1/ D(9) is the reference shear viscosity for the thermally-activated viscoelastic flow,
T
NS TS
Ns TNs
2NSN
is a magnitude of the stress tensor, and # is an orientation parameter, which
provides a quantitative assessment of molecular chain alignment. The orientation parameter# is
found to be:
#=--cos
2
where
{141 is
{
-
,
(20)
tr(BNS)
a set of principal stretches in the soft network. The nonlinear viscoelastic flow
stopped when the orientation parameter approaches the cutoff value (,).
3.4.
Result: Experiment vs Model
The material parameters for the constitutive model in this work were systematically
identified using a procedure provided in 3.6 and are listed in Table 1. In brief, the low strain rate
data are taken to be governed by the hard domain contribution since the soft domain is fully
relaxed and hence the stress-strain data at low strain rates from 10- 3 to 10-1 s-1 is used to obtain
the hard domain material parameters. The hard domain model prediction is then extrapolated to
63
high strain rates. The soft domain material parameters can be identified using the high strain rate
data and subtracting the predicted hard domain contribution at these rates.
Table 1 Material parameter for intermolecular component
Intermolecular
Hard
Soft
Intermolecular Stress
+
J
-
ujs =7.0[MPa]
=14.9[MPa]
_IH
3
B=1.95[GPa]
Viscoelastic-Viscoplastic Flow
AGIH = 6.1[10
1
=
+
qa
exp
KAG,a
sinh
G.
kB9s
kBO
0 J]
AGIs =5.5 [10-2 1 J]
= 15 .6 [s-
=3.H
[ sIr
IH
is
q=1.2-1.5[GPa -s]
.
Elastic-Plastic Softening
2
IH
hH = 2.0[MPa]
=IH
Is
ssIH
SHs/IH
IH
*
hIH
flH
sO,IH
s{s,IH /4,H
0H
0.5
=0.25
The shear resistance sIH may include the pressure sensitivity via
IH
SIH +ap,
where the
pressure sensitivity coefficient a may be taken to be 0.05 - 0.125 and p -- 1trT is the mean
3
normal pressure.
64
Table 2 Material parameter for network component
Network
Hard
Soft
Network Stress
T
"
iiJp.K
J.
0
3JKA)
pNH =10.4[MPa]
e
I
a aN
-j
NNH
4. 5
p NS =5.8[MPa]
N
NNS =14.0
Viscoelastic Flow
(D
ris =
1/ D=1.7 [10 5Pa - so.33
1NS) I.
.A
c _ min{A1,A2 ,A}
0=--cos 1
=
0.33
N
tr(BNS
2
n
0
=0.1
Stretch-induced Softening
MNH(0NN
N
( )NH
()
ock,NH
ockNH
ss,NH
2Lkj
.
IO.7/
0.,s~0..8s
.
k
C
I
Wt
N= (tH
r
)5.0\)S~
1
'7NH
so=
0.5 5 [MPa]
SNH
r = 2.2
SNH = SO.NH
(ZlockNH
/
2
lock,NH
(o
r2 =
sslkNH =
rthrAo
r, rt
-
NH
V;,okH=
c
ckH
XokNH
(0)
0.5
AA~ =20[s-]
65
3.4.1. Low Strain Rate Behavior
The stress-strain behavior of polyurea at low strain rates ranging from 10-3 to 10-1 S-1 is
simulated to verify the ability of the model to capture the low strain rate compression behavior
and to predict the tension-compression asymmetry.
Figure 3-4(a) shows the comparison
between the model and experimental data under monotonic compression at strain rates of 10- 3,
10-2
and 10'
s-1. The simulation results agree well with the experimental data and clearly
capture the rate-dependence of the stress-strain behavior. In Figure 3-4 (b), flow stress levels at
strains of 0.4 and 0.9 are displayed as a function of strain rate. The model is shown to naturally
capture the single main relaxation mechanism over low strain rate, which results in the constant
rate-sensitivity over this range in strain rate. The stress contributions of the intermolecular and
network component of the soft domains were found to be negligible due to the complete
relaxation of the soft domain at low strain rate.
Figure 3-4 (c) shows the model to provide excellent predictions of the uniaxial tension
stress-strain curves at strain rates of 10-, 10-2 and 10-' s-'. The stress-strain behavior was found
to be asymmetric between uniaxial tension and uniaxial compression due to the well-known
difference in the evolution of orientation of network structures as captured by model (Figure 3-4
(d)). The stress responses at large strains in uniaxial tension are greater than those in uniaxial
compression whereas the initial elastic stiffness and yield stress are almost identical.
66
(a)
(b)
-
35
S.g.A,
a. 151
.g
:0 I
10
o0-0
0
20
0
2
wde
compression
0a=.9~,mowde:
0.9O, exp
25
-
exp
0 sA,modeli tension
0 e.a modeJ
15
*j
10
0
0
0.2
0.4
0.6
0.8
1
10
True Strain
(c)
10
(d)
5
O_
10
Strain Rate [/s]
-
2
2
0
0
I0
0
0
eso
U- 15
d1
0D
0
20.
0 0
E
0O
0)
a
10-
5
b
o
1
compression
0.2
0.4
0.6
True Strain
0.8
0.2
0.4
0.6
0.8
True Strain
Figure 3-4 Monotonic compression and tension data, comparing model results (lines) with
experimental data (open symbols) at low strain rates of 0.001, 0.01 and 0.1 s-1 (a) compression
stress-strain curves (b) flow stress at strains of 0.4 and 0.9 as a function of strain rate (c) tension
stress-strain curves (d) asymmetry in tension and compression stress-strain behavior at a strain
rate of 0.01 s-1
The constitutive model is tested under cyclic compression and tension to assess the
model's ability to capture the stretch-induced softening. Figure 3-5 (a) and (b) show stress-strain
curves under cyclic compression and tension, respectively. The model results agree well with the
experimental data exhibiting the features of substantial resilience and dissipation including
stretch-induced softening during cyclic compression and tension. First, a highly nonlinear
unloading behavior is observed and captured by the model. Second, the stress-strain curve in the
subsequent cycle is more compliant than the initial stress-strain response observed in the first
67
cycle. Third, as the reloading strain approaches the maximum strain in the prior cycle, the stress
level tends to reach that in the first cycle. Fourth, in subsequent reloading, further softening tends
to be delayed until reaching the maximum strain obtained in the prior cycle, where a larger strain
produces greater softening. Additionally, the permanent set is reduced in the subsequent cycle.
The stretch-induced softening arises from the evolution in the chain limiting extensibility, which
directly leads to a substantial decrease in the elastic modulus in the network component. Also,
the elastic-plastic softening in the intermolecular component leads to a dramatic decrease in the
initial elastic stiffness and yield stress in the subsequent reloading. In the cyclic compression
testing (Figure 3-5a), a substantial shape recovery was achieved within tens of minutes between
the two cycles in experiment and simulation. More detailed analysis on the elastically- and
inelastically-driven shape recovery in this class of materials will be discussed in Chapter 7 and
Chapter 8 on polyurea elastomers and ethylene methacrylic acid elastomers, respectively, in
conjunction with resilience and dissipation; and their roles in shape recovery and memory of the
materials.
(a)
2
2G(b)
-15
.
25
()20
--- N=2. sim
,
N=2, sim
.
N=1,
15.
a N=2, exp
0.2
20.
'
to
recovery
sim
-N=1,
N=1. sim
0.4
0.6
0.8
1
True Strain
0
-
exp
N=2, exp
0.2
0
0.4
0.6
0.8
1
True Strain
Figure 3-5 Cyclic compression and tension data, comparing mnodel results with experimental
data (a) stress-strain curves under cyclic compression at a strain rate of 0.1 s-1 (b) stress-strain
curves under cyclic tension at a strain rate of 0.015 s-I
A multiple cyclic tensile test with a strain increasing with each cycle is simulated and
68
compared to experimental data in Figure 3-6(a). The simulation result clearly indicates that the
constitutive model is capable of capturing the main features such as a progressive decrease in the
elastic stiffness and yield stress; stretch-induced softening, which increases with the increasing
magnitude of the imposed strain; and the stress-strain curve which tends to merge into the
original stress-strain path near the strain where it was unloaded. Additionally, Figure 3-6(b)
shows stress-strain curves under cyclic tension up to a large true strain of 1.25, indicating a
significant stress upturn approaching the finite extensibility as well as a large amount of energy
dissipation with substantial softening during a loading-unloading-reloading cycle. The multiple
cyclic tension results from experiment and model in Figure 3-6 are further reduced to quantify
the dissipated work density, which results from the multiple dissipation mechanisms, as a
function of the maximum strain of each cycle. Following the procedure presented by Rinaldi et
al. 41 and Greviskes et al. 76
during loading
Tnl: d
,
the dissipated work density is estimated by taking the total work
Tmax: dc and subtracting the recovered work density during unloading
in the stress-strain curves, which is schematically illustrated in Figure 3-7(a). The
dissipated work density in model and experiment is presented in Figure 3-7 (b) as a function of
the maximum strain for two consecutive cycles. Here, the two tests presented in Figure 3-6(a)
and (b) are evaluated. As expected, the dissipated work density increases as the maximum
imposed strain increases and there is excellent agreement between the model and the experiment.
Furthermore, we observed the dramatic reduction in second cycle dissipation in comparison to
first cycle dissipation by examining the results for loading (first cycle) and reloading (second
cycle). The evolution in the elastic shear modulus (the inset of Figure 3-6 (a)), which indicates
the level of stretch-induced softening in the network component, exhibits a substantial decrease
with a imposed strain during first cycle loading but no substantial evolution during unloading
and reloading until reaching the prior maximum strain in the second cycle, which is consistent
with the microstructural evolution seen in the SAXS data under cyclic tension.41 Thus, it is
interpreted that most dissipation during second cycle is governed by the viscoelastic-viscoplastic
relaxation in the intermolecular component while dissipation during first cycle is governed by
the stretch-induced softening in the network component. The constitutive model is found to be
capable of capturing the main contribution and features of the energy dissipation in the material
69
under cyclic loading based on the quantitative comparison of the dissipated work density in the
simulated stress-strain curve to the experimental data.
-
(a)
12
(b)
-10
30
25
a-(U
cc
W1-
40
-N=1,
simn
--- N=2, simn
0 N=1, exp
r-4 30
20
1
%0.5
True Strainaa
CL
SN=2, exp
20
15
0
U,
10
a
a1
a
I
a
5
C)
10
a3
-0n
a
000
000
an
I0 ---- --
I
0.2
0.4
0.6
True Strain
0.8
1
0
0.2
0.4
0.6
0.8
1
1.2
True Strain
Figure 3-6 Multiple cyclic tension data, comparing model results (lines) with experimental data
(open symbols) (a) stress-strain curves with an increasing strain under multiple cyclic tension at
a strain rate of 0.005 s-1 (the inset shows the stretch-induced elastic softening in the hard domain
network) (b) stress-strain curves under cyclic tension up to a maximum strain of 1.25 at a strain
rate of 0.05 s-1
70
(a)
(b)
14
Dissipated work density
in 111load
12 -0
1
1
10
o
Cr
8
6-
Dissipated work density
Loading exp
Reloading, OXP
Loading, aim
M Reloading, aim
E Loading, exp
E Reloading, exp
- Loading, aim
Reloading, sim
-
Fig.6(a)
Fig. 6( b)
In V reload
1
-
*Z 20
0.0
0.3
0.6
0.9
1.2
1.5
True Strain
Figure 3-7 Energy dissipation during cyclic tension tests (a) a schematic of the data reduction
for a dissipated work density in consecutive cycles with an increasing strain (b) dissipated work
density in consecutive cycles with an increasing strain in model and experiment (the two tests
presented in Figure 6(a) and (b) are evaluated)
3.4.2.
High Strain Rate Behavior
High strain rate behavior of polyurea has been experimentally studied by Sarva et al.66 and
Yi et al.40 by conducting the split Hopkinson pressure bar tests (SHPB). As shown in Figure
3-1(c) and (d), the presence of the distinct relaxation processes in the soft and hard domain leads
to a transition in rate-sensitivity at a moderate strain rate, where there is a dramatic increase in
the initial elastic modulus and the flow stress at increasing strains.
71
(a)
(b)
70
G0
,-, 50
o
-
0.01s650 /s, exp
0.01 -6soo/s,sim
6500
/s
60-
1200/s
IL
S 40
1/ "14/s
S30
+- 2
20
0
I
50
0
40-
Il. 30U)
0
ML
10
-o
0.2
.2/s
0.01 /s
0.6
0.8
0.4
True Strain
1
20
10-
o
z=0.4, exp
=.,U
0 s=0.4, sim
S8=0.9, Sim
200
!
|
0'* 4
2
10~- 10-2
-
0
1--
-1 100 10
2
3-
102 103 104
Strain Rate [is]
Figure 3-8 High strain rate behavior under compression (a) stress-strain curves in experiment
(open symbol) and model at strain rates ranging from 0.01 to 6500 s- (b) flow stress at strains of
0.4 and 0.9 as a function of strain rate
The constitutive model predictions of the stress-strain behavior under high strain rate
compression are presented in Figure 3-8. As shown in Figure 3-8(a) and (b), the proposed
constitutive model clearly captures the main features of high strain rate behavior including a
transition in rate-sensitivity in the vicinity of a strain rate of 1 s-1. The overall stress response is
decomposed into contributions of the hard and soft components in Figure 3-9. As expected, the
soft domain component substantially contributes to the overall stress response at high strain rate
whereas it is negligible at low strain rate. This additional contribution from the soft domains at
high strain rates gives the transition in rate sensitivity.
72
(a)
(b)
...............
V-------------------
40
3530
Ern~-
75
25
T~
a:
0
206-
U)
210-
Hard component: IH + NH
-----------------
0.2
0.4
0.6
0.8
Soft component:
IS+ NS
1
True Strain
Figure 3-9 Decomposition of the simulated low to high strain rate behavior into hard and soft
contribution (a) stress-strain curves in hard (black) and soft (gray) component (b) schematic for
the micro-rheological hard and soft components in the simulation
3.4.3. Constrained Behavior under Biaxial Tensile Testing
In this section, the multi-axial mechanical behavior of the exemplar elastomeric copolymer
(PU1000) is addressed under biaxial tensile testing in experiments and numerical simulations.
The constrained behavior of thin elastomeric films has been found to be interesting in terms of
their durability under in-plane multi-axial deformation.
As schematically illustrated in Figure 3-10, we performed biaxial tensile testing on the
polyurea thin films under different biaxial loading ratios (- -). Here, we used a biaxial tensile
tester manufactured by Deben Inc.; the polyurea thin films have been derived via bulk
polymerization of Isonate 143L (Dow Chemical) and Versalink P-1000 (Air Products) following
the mixing and curing procedures of PU1000 developed by Naval Surface Warfare Center,
Carderock Division. In particular, we employed a film applicator (Sheen Instruments) to make
the polyurea films with thickness of 100 ~ 150 micrometers since the maximum load in the
biaxial tester was limited to 50 N, i.e. in order to meet yield behavior, we had to make very thin
films under such a small load capacity in the machine. The polyurea thin films were then
73
fabricated to cruciform specimens (30 mm by 30mm; cruciform width: 10 mm) using a laser
cutter (Epilog Laser Inc.).
Biaxiality
-....
x
Undeformed
Deformed
Figure 3-10 Schematic of biaxial tensile testing at two different biaxial ratios (B=
-yy
=0 and 1.0)
Figure 3-11 shows the engineering stress-engineering strain behavior of polyurea 1000
films under two different biaxial ratios of B=0.0 (no stretching in y direction) and 1.0 (equibiaxial stretching in x and y directions) in experiments and simulations. In call cases, the stressstrain behavior exhibits an elastic-plastic response with the features similar to the uniaxial
behavior. The initial elastic stiffness of polyurea under biaxial deformation with B=0.0 and 1.0
was found to be greater than that found in uniaxial behavior. However, the initial elastic stiffness
in y direction under biaxial deformation with B=0.0 (no stretching in y direction and
displacement in y direction constrained) was found to be much smaller than uniaxial stiffness.
This observation can be further confirmed by recalling the biaxial moduli in terms of the uniaxial
modulus E as follows,
EB=
E,=OV
1-v 2
1.E.1
En <
Eun, < ExxB=O
1-v
2
E
<E
B
=
1
V21£
E
1-v
(1
The yield stress level required for biaxial yielding is also greater than that observed in the
uniaxial yielding; the biaxial stress components at yield were expected to be equal to the uniaxial
yield stress based on a Mises yield criterion. This unexpected discrepancy between the biaxial
and the uniaxial behavior in yield stress level may result from the inhomogeneous stress field in
74
the central region of cruciform specimen which was actually under inhomogeneous biaxial
deformation as shown in Figure 3-12. Furthermore, the stress-rollover was initiated more rapidly
in the biaxial deformation. This may lead to the greater yield stress in the biaxial deformation in
conjunction with the effect of inhomogeneous deformation field and specimen geometry. The
overall biaxial deformation behavior was found to be predicted well in numerical simulations for
two different biaxial ratios of 1.0 and 0.0
including the constrained stress evolution in y
direction under a biaxial ratio of B=0.0.
The biaxial behavior was furthered by quantification of inhomogeneous strain fields in
experiments and numerical simulations. Deformation images at each time frame were processed
using a digital image correlation (DIC) in order to produce strain fields in experiments by
tracking the Lagrangian material points on the thin samples made by a spray. Figure 3-13 and
Figure 3-14 show contours of axial and shear strain field at a strain of 0.125 in the central region
of cruciform specimen under a biaxial ratio of B=1.0 and B=0.0. Numerical simulations were
found to well predict the inhomogeneous deformation field in the central biaxial region and the
uniaxial regions in the cruciform specimen.
12
M uniaxial
C..
0 B=1, exp
0 .O=0, exp
CO
9
A
sW exp%=0,
EngB=1,
sim
---rB=, sim
6-
- - -- B=0O,s
siC
CO.
LU
0
0.000 0.025 0.050 0.075 0.100 0.125
Engineering Strain, EXX
Figure 3-11 Engineering stress-engineering strain behavior under biaxial tensile testing in
experiments and simulations
75
(a)
(C)
(b)
10 MPa
10 MPa
0
0
10 MPa
0
(d)
4 MPa
0
Figure 3-12 Contours of stress field under biaxial tensile testing at a strain of 0.125 in the central
biaxial region in the cruciform specimen: (a) stress in x direction at biaxiality of 1.0; (b) stress in
y direction at biaxiality of 1.0; (c) stress in x direction at biaxiality of 0.0; (a) stress in y direction
at biaxiality of 1.0
The biaxial behavior of thin polyurea samples has been addressed in experiments and
finite element simulations, where the proposed constitutive model was numerically implemented.
The numerical simulations predicted well the main features of biaxial behavior in terms of
elastic-plastic properties supporting a multi-dimensional predictive capability of the constitutive
model under inhomogeneous deformation.
76
(a)
(b)
-0.1
0.4
-0.1
0.4
0.0
-0.18
-0.23
x
Figure 3-13 Contours of strain field under biaxial tensile testing at biaxiality of 1.0 at a strain of
0.125 in the central biaxial region in the cruciform specimen: (a) axial strain field in experiment;
(b) axial strain field in simulation; (c) shear strain field in experiment; (d) shear strain field in
simulation
77
(a)
46X X
(c)
(b)
-0 .0 1
-0.01
(d)
0.0
I-0.12
0.4
0.0
Y
-0.17
Figure 3-14 Contours of strain field under biaxial tensile testing at biaxiality of 0.0 at a strain of
0.125 in the central biaxial region in the cruciform specimen: (a) axial strain field in experiment;
(b) axial strain field in simulation; (c) shear strain field in experiment; (d) shear strain field in
simulation
3.5.
Concluding Remark and Future Work
A viscoelastic-viscoplastic constitutive model is proposed to capture the large strain
behavior of a segmented elastomeric copolymer polyurea under a variety of uniaxial and biaxial
loading scenarios over a broad range of strain rates. The elastic-viscoplastic constitutive model is
developed based on the experimentally observed features in the mechanical behavior of polyurea.
The macroscopic material response is decomposed into contributions from the hard and soft
components which results from the phase-separated morphology. To capture the multiple
relaxation processes over a wide range of strain rates, the viscoelastic-viscoplastic intermolecular
resistance with the distinct time constant is employed in each domain. The Arruda-Boyce eight78
chain hyperelasticity is used to prescribe the stress arising from the network resistance. A
phenomenological evolution model of the chain limiting extensibility successfully captures the
stretch-induced softening.
The model was able to capture all of the main features of the large strain behavior of
polyurea using the relatively small number of material parameters. The proposed constitutive
model can be reduced to only two hard domain components without the soft components in order
to capture the mechanical behavior of the material at low strain rate. The model is found to be
capable of capturing the multiple dissipation pathways such as viscoelasticity, viscoplasticity and
the stretch-induced softening, quantifying the dissipated and stored energy during deformation.
The success in capturing the energy storage behavior is further evidenced by the ability of the
model to capture the material resilience as well as its softened behavior upon reloading. In
addition to the multiple dissipation mechanisms captured by the constitutive model, the multiple
and distinct molecular relaxation processes in the phase-separated morphology were successfully
captured in the constitutive model, which leads to a substantial change in the rate-sensitivity of
the stress-strain response from low to high strain rate.
In future work, the constitutive model will be furthered to capture the effect of hard/soft
domain volume fractions as well as their morphological evolution to the overall mechanical
behavior in Chapter 5. Additionally, it may be thermomechanically-coupled to capture the
mechanical behavior of the material under a wide range of temperature. The proposed
microstructurally-based constitutive model is also capable of providing a general framework to
model the large strain behavior of other phase-separated copolymers such as a transparent
polyurethane-urea (TPUU) 77, an ethylene methacrylic acid (EMAA) 78, an ethylene methacrylic
acid butyl acrylate (EMAABA) 76' 78, and their chemically-modified counterparts in which the
features of the stress-strain response are analogous to those in polyurea in conjunction with the
multiple dissipation mechanisms, transition in strain rate sensitivity, and substantial resilience as
detailed in the following chapters.
79
3.6.
Procedure for Determination of Material Parameters in the Constitutive Model
In this section, a simple procedure is provided to identify the material parameters in the
proposed constitutive model. In this procedure, the material parameters are clearly identified
based on physical mechanisms and features of the stress-strain behavior and hence avoid
unphysical empiricisms.
A. Hard component
The material parameters for the intermolecular and network component in the hard
domains are determined using experimental data at low strain rate. The initial small strain shear
modulus at low strain rate is found to be p = 25.2 MPa using low strain rate stress-strain curves.
This value corresponds to the initial elastic shear modulus in the hard domain. The
intermolecular and network elastic contribution can be identified using the stress-relaxation data
in Figure 3-15a. The relaxed elastic shear modulus is found to be 10.4 MPa giving the
corresponding network contribution (pNH = 10.4 MPa) and the intermolecular contribution
(pIH
=14.9 MPa).
The parameters associated with the intermolecular viscoplastic flow are determined using
the rate-dependent shear yield stress data. The magnitude of the rate of viscoplastic flow is
prescribed by,
)"P = Yf, exp
_
" exp
"
.a
(Al)
Equation (Al) was derived by taking a forward process from the original flow model; and it can
be rewritten for the effective shear stress r,, as a function of the shear strain rate ?p
Ta
= A -in
A=
B=
a
s,
?' + B,
(A2)
-('kBA3
AG)
- A -Infla,
(A4)
80
wherei~s taken to be an initial value of
of shear yield stress (- ~
uniaxia,
0.077p
" . As shown in Figure 3-15b, a least-square fit
1-va
) data as a function of shear strain rate (
,,na
-
) under
compression provides A and B in Equation (A2). Using an initial elastic shear modulus ratio of
the intermolecular component to the network component, shear yield stress ( r, ) in the
intermolecular component in Equation (A2) is taken to be 65% of shear yield stress in
compression data (r1H ~0.65 -overai ). Consequently, a set of material parameters {nt, AGa } for
the viscoplastic flow in the intermolecular component is determined by Equation (A3) and (A4).
The evolution of intermolecular shear strength (sH) and elastic shear modulus (a,, ) governs the
elastic-plastic softening which is observed in the cyclic loading data. Elastic shear modulus and
yield stress during reloading in Figure 3-5 give the saturated values of shear strength
elastic shear modulus (pssIH ). In reloading stress-strain data,
0.5
-.
and 0.25
1H
0.,H
s,
(
) and
and p,, IH are found to be
, respectively. The saturated values of shear strength and elastic shear
modulus may be taken to be rate-dependent. The elastic-plastic softening slope h,, and the
nonlinearity mIH in Equation (11) and (13) are determined based on the shape of evolution of sIH
and p, .
The material parameters in stretch-induced softening model in the hard network are
identified using cyclic tension data. As shown in Figure 3-15c, the unloading curves in the
multiple cyclic tension data give quantitative measurement on the network elastic shear modulus
and the corresponding chain limiting extensibility ( A,,
pNH (t)
(t) =
saturated value of chain limiting extensibility was found to be 2 ,,,lok= 2.
r
rates in Figure 3-15d. Additionally
AssNH
lock.NH (0)
=
-
tanh
,
2
NNH
2
(t) ), where the
,, at low strain
o,
is found to be substantially rate-dependent via
, which leads to restricted evolution at high strain
Ac
rates. The initial internal resistance (soNH) to the breakdown of chain networks is estimated from
the rubbery modulus and the network stress level at a strain where the stretch-induce softening is
initiated.
soNH
was found to be approximated via 0.55 MPa. The intermediate values of the chain
81
limiting extensibility and the shear modulus give an insight into estimating the reference
softening coefficient (c)
and the nonlinearities in Equation (15) which control the shape of
evolution in stretch-induced softening.
(a)
10
(b)
8
ILo
6
Contbution frm hardoewm
r1
0
2.0
k
0000
4
1..
,*.
0
3.0
2.
2.5-
0
0**
o
22
0.00
hadnwo
0.1 Is, up
=
1.0
A -In
,,+
B
.A
.0.5
0.05
0.10 0.15
True Strain
0.20
0.0
0.25
104
10
10*1
100
Strain Rate f/si
(d) 400
(c)
pl.,(t)
2520
*
1
*
pMa (t)
15
10
200
~5.0 MPa
#w(0)
~-93 MP
MPaa
0
0.0
300
2.OMPa
0.2
2 100
0.4
.
0.6
0.8
1.0
.0
*
0.5
True Strain
1.0
1.5
2.0
2.5
True Strain
Figure 3-15 Data used to determine material parameters in hard domain intermolecular and
network component (a) stress relaxation data at a strain rate of 0.1 s1 under compression with a
model prediction from the hard network component (b) shear yield stress (-r,"
data as a function of shear strain rate (
(I
f[)
Ir,,niaia,
)
~ i tuniaxial) under compression at low strain rates (c)
cyclic tension data at a strain rate of 0.005 s- for stretch-induced softening model parameters the
in hard network component (NH) (d) stress-strain curve at a moderate strain rate of 0.2 s4 under
tension, revealing the dramatic stress hardening due to the finite extensibility (the saturated value
of chain limiting extensibility is estimated via
I~
2m
+
J)
from Choi et al. 65
B. Soft Components (intermolecular/network)
82
The material parameters in soft components can be identified using the high strain rate
data and the model predictions from hard components. The stress-strain curve at a high strain
rate gives the elastic shear modulus (uSf =12.9 MPa) in soft domains via p,,
=
pta
-
Phard
The
elastic shear modulus in the intermolecular and network component is found to be p,, = 7.0 MPa
and
aNS
=5.8 MPa, respectively based on the high strain rate data. The viscoplastic flow
parameters in the intermolecular component are identified using shear yield stress data at low to
high strain rate. The magnitude of the rate of viscoplastic flow is prescribed by Equation (Al).
As shown in Figure 3-16a, a least-square fit of shear yield stress data as a function of shear
strain rate provides the coefficients, A and B for the total intermolecular response at high strain
rate. The parameters {K,
G
} for the
soft intermolecular component are then determined by
subtracting the hard intermolecular model predictions from the total intermolecular component
fit. This process enables the model to capture a transition in rate-sensitivity of the initial yield
behavior by naturally taking distinct relaxations of the hard and soft domains. Since the
intermolecular elastic-plastic softening in soft domain is not experimentally evident, no softening
model is employed for the shear strength and the elastic shear modulus in the soft components.
The material parameters for the soft network component are finally identified using the
high strain rate data and the model predictions from the hard/soft intermolecular components and
the hard network component at high strain rate. The nonlinear viscoelasticity is employed to
capture the rate-dependent viscoelastic slip in the network component in soft domains. This also
enables the dramatic transition in the rate-sensitivity in flow stress at large strains in conjunction
with the two distinct viscoplastic flows in the intermolecular components. Figure 3-16b shows
shear viscosity as a function of strain rate for the soft network.
83
(b)
(a)
0. 6
__
_
_
_
_
_
_
_
_
'S
s. =A,$1
4-
1n
P+
10
1
7
1
1
1
1
102
10 3
104
106!
3
-c~.*u0.46
w 0.37:
*vue-
20
j
-
0
3
103 102 10- 100 101 102 10 104
5
e~-*.21
1021
100
Strain Rate i/si
101
Strain Rate i/si
Figure 3-16 Data used to determine material parameters in soft intermolecular and network
component (a) shear yield stress data as a function of shear strain rate under compression at low
to high strain rate (b) shear viscosity data as a function of strain rate
The viscosity is simply estimated by dividing each stress value by the strain rate. Since the
viscosity versus strain rate curve give a straight line on a log-log plot, it follows that a power-law
should capture the nonlinear viscoelastic flow. A curve fit for shear viscosity as a function of
shear strain rate gives the reference viscosity q7=
1
D (0)
and the power-law exponent
n.
Here,
the effective shear stress at each orientation 0 is calculated by subtracting the stress predictions of
the hard/soft intermolecular components and the hard network component from the experimental
data. In the proposed model, D (9) is assumed to be a constant at an ambient temperature. In
future work, it will be furthered to capture the temperature-dependent
D (9) = D0
exp
I2i
viscosity using
as in Bergstr~rm and Boyce 4, and Dupaix and Boyce. 9
The material parameter for the network elasticity is identified using the stress-strain data at
a strain rate of 1200 s 1 since the viscoelastic flow is negligible at high strain rates through the
nonlinear viscoelasticity. The chain limiting extensibility (AOCkNS ) in the soft network is found to
be V-JKI =
i4 -40.Also,
no softening for the chain limiting extensibility is introduced in the soft
network since there is no experimental evidence for the stretch-induced softening in soft domains.
84
3.7.
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89
Chapter 4
Computational Procedures for Simulations of Large
Deformation Elastic-Plastic Behavior of Elastomeric
Copolymers
4.1.
Introduction
In this chapter, computational procedures for simulations of the large deformation elasticplastic behavior of copolymeric materials are discussed including; finite element formulations
and solution procedures of initial and boundary value problems of inhomogeneous and dynamic
large deformation fields; numerical procedures for the updates of elastic-plastic kinematics and
relevant internal field variables such as shear strength and elastic moduli that evolve
simultaneously with deformation; numerical validation of proposed constitutive models which
are used in the finite element simulations.
Throughout this thesis, a commercially-available finite element solver, ABAQUS has been
extensively employed for the simulations of elastic-plastic large deformation fields of segmented
copolymers (polyurea 650 and polyurea 1000) and ethylene-based ionic copolymers (EMAA,
EMAABA and EMAABA-Na+) In addition, the proposed constitutive model framework was
implemented for a general use in a large-scale finite element solver, SUMMIT, developed by
Raul Radovitzky Group at the Institute of Soldier Nanotechnology, MIT. Flows and structures
of internal data for general finite element procedures used in SUMMIT are very similar to those
in ABAQUS. In particular, numerical algorithms employed in ABAQUS are essentially the same
as those used in SUMMIT. The structures and flows of internal data and numerical procedures in
ABAQUS and SUMMIT are schematically shown in Figure 4-1. Fortran subroutines for the
90
constitutive models as user-defined material models in ABAQUS were therefore employed in
SUMMIT without any further major modifications.
Finite Element Solver
Vo P+b = p.
J
(P: V0 4 + pO -q)dV
( -,)dS
0 =0
-
Mi +fn = fe
Fr+d
Fr
Constitutive Model
T
F = Fe FP
_J C
T =Z Ta
Hard Domain
I
Ai
Soft Domain
{s, (t)} Internal variables
Figure 4-1 Solution procedure in finite element solver with constitutive model subroutines
4.2.
Formulation of Initial and Boundary Value Problems
From the notion of conservation of mass, linear and angular momentum and energy in a
Lagrangian framework (referential formulation), we have governing equations for deformation
field of continuous media as follows[1-3],
A local balance form of the conservation of mass is expressed,
91
Avo
(1)
Sv;
Here, the subscript ( ). stands for the undeformed reference configuration; p0 and p represent
density at the reference and deformed configuration, respectively; (5v0 and
6v
represent local
volume at the reference and deformed configuration, respectively.
Equations of motion are then formulated from the conservation of linear momentum in the
undeformed reference configuration for finite deformation,
VO *P+bo = pi ,
(2)
where, F is the Piola stress tensor (engineering stress); b, is the body force; and
<p =
x (x;t) is the
motion which maps material points to spatial points. The Piola stress is defined as follows,
P= JTF-,
(3)
where J=
9vO
is the volumetric change, which is determined via J = det (F), T is the Cauchy
stress tensor (true stress), and F
=
Vp is the deformation gradient.
The angular momentum balance provides a symmetry relationship for the Piola stress via
the deformation gradient as follows,
PFT = FPT .
(4)
The given partial differential equations in a strong form in Equation (2) - (4) require
appropriate traction and displacement boundary conditions on the partial surfaces aBO of the
reference body BO as illustrated in Figure 4-2 which are prescribed by,
P -n0 =
f on aBo
(p =<~ on 3BO
=
= Stractin
,
Sdisplacement-
(5)
(6)
92
Here, n, denotes the unit normal vector to the reference body; i is the traction force acting on a
partial reference surface; and j is the prescribed motion of a partial reference surface. In
addition, the initial conditions are given by,
(P= X0 in 4,
(7)
00 =VO in B..
(8)
The equation of motion in Equation (2) is then solved numerically in the finite element
framework with the initial and boundary conditions in Equation (5) - (8). The Lagrangian
formulation and the finite element procedures have been notably useful for a broad variety of
initial and boundary value problems of large deformation of elastic-plastic solids. In addition, the
Lagrangian approach was recently found to be applicable to initial and boundary value problems
of fluids involving Newtonian viscous flows[4], dense granular flows[5], and fluid-solid
interactions [6], which have been solved in the Eulerian framework and/or the arbitrary
Lagrangian-Eulerian framework in a fixed space and/or a control volume. If the constitutive
models of solids, fluids and their interactions are well formulated in terms of the deformation
gradient and its derivatives, i.e., P=P(F,F) and the variational structures can be postulated
appropriately, the Lagrangian equations of motion can be solved with appropriate finite element
procedures as described in the following sections. In the following sections, a simple variational
approach is summarized for temporal and spatial finite element discretization of the initial and
boundary value problems. In addition, a solution procedure of the finite element equations is
briefly discussed in explicit and implicit methods.
93
P-no =f
Figure 4-2 Schematic of boundary conditions for equations of motion
4.3.
Finite Element Procedures for Dynamic and Inhomogeneous Large Deformation
In this section, a finite element procedure for time and space is briefly discussed for
exemplar simulations of Taylor impact testing, where the driving force for deformation was due
to an "inertial" effect; it will be further detailed in Chapter 6. A weak formulation for an
admissible variational field q (x;t), which satisfies the initial and boundary conditions, is derived
by the principle of virtual work, recasting the equations of motion in Equation (2) and neglecting
the non-inertial body force, as follows[l, 7],
o
B
dVo = 0 for Vi 1 .
-o)-
(9)
The initial and boundary conditions are also prescribed as follows,
P -n0
<p
=
=
f on aBO =
i on aBO
=
Sdisplacement-
p =X0 in B,
(0
= Vmpc,
Sractioncontact 9
in BO.
(10)
(11)
(12)
(13)
94
Here, a fixed displacement boundary condition is given at the symmetric axis (x, =0) and the
traction force vector at the impact surface is calculated through the normal (hard) and tangential
(frictional) contact conditions. Otherwise, a traction-free condition is applied for all of noncontact free surfaces. By integration by parts, the weak form leads to,
fBO(P:OVO+po.q)BdV -
Sas,
f -q)dso =o for vtj.
(14)
Equation (14) constitutes the basis for the initial and boundary value problem which is solved
with finite element discretization as follows.
The deformation field (, ) and the variational field (, ) are then discretized with finite
elements in the reference configuration sharing the same finite element interpolation functions
from the standard Galerkin's method[l] via,
l,
--
(ph =
ia Na ,
(15)
Na.
(16)
qa
a=1
Here, Na (X) is the "global" finite element interpolation function acting on each node a. The
finite element discretization on the spatial deformation field and the admissible variation is then
inserted into the weak form in Equation (14). Since the weak form holds for all admissible
variations, it leads to the finite element balance equations comprising systems of 2 nd order
ordinary differential equations as follows,
M
t
(17)
t
+fin = fex ,
where, M is the consistent mass tensor,
fin
is the internal force vector which is determined from
the stress tensor via the constitutive models, and f"' is the external force vector which is
determined from the traction boundary conditions. The elements of the mass tensor, the internal
force vector, and the external force vector can be expressed as follows,
Mi., =
o
0NaNaNbdV,
(18)
95
fi"'
=n
f"X
=
(19)
f jNdSO,
(20)
B
i,"axi0dV ,
where,
gk
= I
],, is the identity tensor.
The system of 2 nd order ordinary equations in Equation (17) discretized via the finite elements in
Equation (18) - (20) is then numerically integrated in time. The time integration can be
performed explicitly or implicitly with an appropriate time-stepping algorithm such a Newmark
method. [2, 7]
4.4.
Numerical Updates of Large Deformation Elastic-Plastic Kinematics and Relevant
Internal Variables
The deformation gradient decomposes into the elastic and inelastic parts in each
constitutive element through a geometric compatibility by an assumption of homogenized
motion of a body. [8]
F = FtF.,
(21)
where a stands for a micro-rheological mechanism in the constitutive model; F, is the elastic
part of the deformation gradient; and FP is the inelastic (viscoelastic or viscoplastic) part of the
deformation gradient. The elastic-inelastic decomposition was schematically illustrated in
Chapter 3. The intermediate configuration that undergoes inelastic deformation can be chosen
arbitrarily without any loss of generality if the appropriate stress measure is chosen. If the
kinematics of inelastic deformation is prescribed beginning at the intermediate configuration, the
corresponding stress measure should be Mandel stress which is converted from Cauchy stress via
the elastic deformation gradient. We explore the kinematics of inelastic deformation beginning at
the spatial configuration and the Cauchy stress measure is hence used in order to calculate the
magnitude of driving force for inelastic flows. More general reviews on the kinematics of elastic-
96
inelastic solids undergoing large deformation can be found in Boyce et al.[9] and Anand and
Gurtin[3, 10] including amorphous polymers and crystalline materials.
Revisiting the finite deformation kinematics of elastic-plastic materials, the deformation
rate is examined through the spatial velocity gradient L,=gradv=-.
The spatial velocity
gradient is decomposed into elastic and plastic contributions via,
L
= FaFFa = L
f/,
=6;
+ F LFe-1 = #
-+ FjFj-
1
F7- =L'+
t,
(22)
(23)
+ *j,
where fiP and Wj represent the rate of plastic stretching and spin, which are the symmetric and
skew part of Qr4, respectively. The viscoplastic flow in the current configuration is taken to be
irrotational, giving,
P = F"-'DF)F
=FP=
e-
(24)
Fa .
At this point, the rate of plastic stretching DYP under a given stress state is constitutively
prescribed via,
f
=
(25)
PNaP
where the plastic flow is taken to be co-axial to the deviatoric stress tensor, and hence NP is the
normalized deviatoric stress tensor. The magnitude of inelastic flow 7;
is constitutively
prescribed by an appropriate function of the stress tensor Ta, the internal variables s, and the
inelastic flow parameters q via ? =
f(T
,s,,)
as we described in Chapter 3. Once the
magnitude of inelastic flow is determined, the rate of plastic deformation gradient in Equation
(24) is then integrated explicitly or implicitly using appropriate numerical procedures to obtain
the plastic deformation gradient Fj' using the deformation gradients F, (t) and F, (t + t) provided
by the finite element solver; and consequently the elastic deformation gradient is finally obtained
via Fe(t+&)=Fa(t+&t)F-
1
(t+St) at t+&. The internal variables including the inelastic flow
resistance and the softening parameters which evolve with elastic and inelastic straining are also
implicitly or explicitly updated using the evolution models in Chapter 3. The elastic-plastic
97
constitutive model may be updated with fully implicit algorithms to guarantee the unconditional
stability of internal variables which are updated simultaneously with the stress and the elasticplastic kinematics.[8] Once the elastic-plastic constitutive model including the internal variables
is updated in each micro-rheological element, the total stress tensor at t+&r can be obtained via
T(t + St)=
T, (t + St) . Finally, the internal and external force vectors in Equation (19) and (20)
can be obtained from the updated stress tensor.
Additionally, an implicit finite element solution procedure usually requires the material
Jacobian at every time step since the implicit equations are solved using a Newton iteration
scheme. Thus an analytical or numerical material Jacobian should be provided through the usermaterial subroutines with the updates of elastic-plastic kinematics and relevant internal variables.
The material Jacobian may be numerically implemented in each micro-rheological element via,
[J] =
at
(6
(26)
.
Here t is a vector of the Cauchy stress tensor components, i.e. t={T,T, 2 ,T 3 3,T,2,T,TT
}T
and Ac
is a vector of perturbed strain tensor components, i.e. AF ={AE 1 ,AE ,AE ,2AE ,2AE,2AE2} .
Consequently, the material Jacobian is the stress perturbation in all directions of deformation at
the current time step.
The constitutive model of the large deformation behavior of an exemplar copolymer
polyurea is validated through a one-element simulation of "homogeneous" uniaxial compression
(or tension) in the finite element solvers. The homogeneous deformation simulations with one
element in the finite element solvers should throw the same stress-strain data simulated in
MATLAB implementations of the constitutive models. The constitutive models of all
elastomeric copolymers (PU650, PU1000, EMAA, EMAABA, and EMAABA-Na) covered in
this research have been verified with ABAQUS for 2- and 3-dimensional elements including
three-node linear triangular elements, four-node bilinear elements, and eight-node solid elements.
In addition, the finite element simulations performed in ABAQUS involving the
constitutive models are validated through simulations in SUMMIT that shared the same usermaterial subroutines as ABAQUS as described in the previous sections. In particular, high strain
98
rate performance of the constitutive model was validated through 2- and 3-dimensional
simulations of shock-tube testing and Taylor impact testing of polyurea samples.
4.5.
Aspects of Numerical Procedures of Thermomechanically-coupled Deformation
The equation of heat transport in a rigid conductor is formulated from the thermodynamics
first law, the conservation of energy as follows,
c D=-VqR
at
q.
(27)
Here, e is the temperature field in the reference configuration; c is the heat capacity per a unit
reference volume; qR is the heat flux; and qR is the heat generation.
The diffusive heat flux is expressed by a simple Fourier's law,
(28)
qR =-K -V.
Thermal conduction is here assumed to be isotropic, by which the thermal conductivity tensor is
reduced into a scalar thermal conductivity k. The equation of heat transport is hence expressed
by,
c-=
at
(29)
kVE+qR'
When the heat transport is coupled to the mechanical deformation, the heat supply due to
inelastic dissipation should be considered. In addition, a thermo-elastic coupling may be added to
Equation (29) in terms of derivatives of elastic free energies. Here, neglecting the thermo-elastic
coupling effect, the balance equation of scalar temperature field is reformulated involving the
heat generation due to the inelastic power density as follows,
c
ata = kV2e+
T' : fD .
(30)
99
Here, T' is the deviatoric stress of each micro-rheological element; and fr is the rate of
inelastic stretching of each micro-rheological element. In addition, all of the inelastic work is
assumed to convert into heat generation.
Finite element simulations of transient, thermomechanically-coupled deformation can be
performed by temporal and spatial finite element discretization of the temperature field equation
in Equation (30). In particular, the thermal diffusion term in Equation (30) should be discretized
spatially with appropriate finite elements when the local heat flux is non-negligible. However, in
most of amorphous polymers which are thermally-insulated, the diffusion term can be neglected
at macroscopic length-scales since the thermal diffusion length is often much less than the
macroscopic characteristic length of interest. Neglecting the spatially diffusion term in Equation
(30), the thermomechanically-coupled
simulations can be performed via a simple finite
difference approximation for the time derivative at each material point. This approximation
makes the thermomechanically-coupled problems much simpler without any loss of physical
justifications. In summary, the field equations in Equation (2) are solved via a finite element
discretization in Equation (17) - (20) simultaneously with the updates of elastic-plastic
kinematics and constitutive variables as well as of temperature field at each material point.
Furthermore, the constitutive parameters should be updated at each material point at every time
step based on the resulting temperature field and the thermal expansion should be considered via
in the constitutive models of stress tensors. In the simulations of Taylor impact testing of
polyurea rods described in Chapter 6, we have applied the present scheme for temperature
evolution, where a simple scaling analysis was addressed for the thermal transport and relevant
length-scales.
4.6.
Future Work for Computational Implementation
In this chapter, computational methods were briefly introduced for the initial and boundary
value problems of elastomeric materials that undergo elastic-plastic finite deformation. Finite
element procedures were detailed with numerical updates for constitutive models and relevant
kinematics. In particular, the constitutive modeling framework developed through this research
100
was numerically implemented for a general use in finite element solvers including ABAQUS and
SUMMIT and tested and validated for simulations of high strain rate behavior of elastomers. The
finite element implementation of constitutive models was found to be numerically accurate and
stable in both the commercially-available code (ABAQUS) and the Linux-based large-scale code
(SUMMIT) for simulations of extreme mechanical deformation.
There has been a significant attention on design principles for elastomeric-coated
composites for shock-mitigation. In particular, the elastomeric coatings have been utilized for
protective armor systems which are exposed to severe impact and ballistic threatening. Recently,
they are finding new avenues towards light-weight composite design e.g. military helmet against
traumatic-brain-injuries (TBI), which results from extreme air-blast wave loading into human
head. The highly dissipative features of elastomers have been found to substantially increase the
shock-protective performance of primary structures such as steels and glassy polymers. In future
work, we will investigate the salient features of blast-elastomer interactions via numerical
simulations in terms of shock-mitigation and energy absorption and dissipation of the polyurea
copolymers (PU1000 and PU650) for Kevlar plates. The computational results of blast-elastomer
interactions will provide significant insight into better design principles of energy dissipatingcomposite structures on demand. In order to realize such in silico designs of high performance
elastomeric composites, computational issues should be well addressed involving coupled
theories of mechanical deformation, heat transfer and mass transport; high strain rate failure
behavior of elastomeric materials and relevant cohesive zone models, and solid-fluid interactions
which arises in blast-induced shock loading into composites.
101
4.7.
Reference
1.
Hughes, T.J., The finite element method: linearstatic and dynamic finite element analysis.
2012: DoverPublications. com.
2.
Belytschko, T. and T.J. Hughes, Computational methods for transient analysis.
Computational Methods in Mechanics, 2013. 1.
3.
Gurtin, M.E., E. Fried, and L. Anand, The mechanics and thermodynamics of continua.
2010: Cambridge University Press.
4.
Radovitzky, R. and M. Ortiz, Lagrangianfinite element analysis of Newtonianfluidflows.
International Journal for Numerical Methods in Engineering, 1998. 43(4): p. 607-619.
5.
Kamrin, K., Nonlinearelasto-plastic modelfor dense granularflow. International Journal
of Plasticity, 2010. 26(2): p. 167-188.
6.
Kambouchev, N., L. Noels, and R. Radovitzky, Nonlinearcompressibility effects influidstructure interaction and their implications on the air-blastloading of structures. Journal
of Applied Physics, 2006. 100(6): p. 063519-063519-11.
7.
Belytschko, T., B. Moran, and W.K. Liu, Nonlinearfinite element analysis for continua
and structures.Vol. 1. 1999: Wiley.
8.
de Souza Neto, E.A., D. Peric, and D.R.J. Owen, Computational methods for plasticity:
theory and applications. 2011: John Wiley & Sons.
9.
Boyce, M.C., G. Weber, and D.M. Parks, On the kinematics of finite strain plasticity.
Journal of the Mechanics and Physics of Solids, 1989. 37(5): p. 647-665.
10.
Gurtin, M.E. and L. Anand, The decomposition F= F< sup> e</sup> F< sup> p</sup>,
materialsymmetry, and plastic irrotationalityfor solids that are isotropic-viscoplasticor
amorphous. International Journal of Plasticity, 2005. 21(9): p. 1686-1719.
102
103
Chapter 5
Resilient yet Dissipative Large Deformation of
Elastomeric Segmented Copolymers: Effect of Weight
Fraction and Segmental Dynamics of Microstructure
Portions of this chapter will be submitted to a journal paper, H. Cho and M. C. Boyce,
"Resilience, Dissipation and Shape Recovery of Elastomeric Segmented Copolymers under
Microindentation", in preparation,2013
5.1.
Introduction
The thermodynamic immiscibility of hard and soft constituents often leads to a phaseseparated morphology of bi-continuous, interpenetrating networks of segmental microstructures
in elastomeric segmented copolymers. The phase-separated segmental structures were found to
provide "tunable" resilience and dissipation, travelling between glassy and rubbery polymers and
their combinations as we discussed in the previous chapters. The underlying mechanical
principles responsible for the multiple energy storage and dissipation pathways of polyurea have
been widely investigated for an exemplar copolymer polyurea with a specific weight fraction
(-34% and -66%) of hard and soft contents at low to high strain rate conditions. A physicallybased constitutive model of the resilient yet dissipative large deformation was proposed to
address the resilient yet dissipation large deformation behavior at a broad range of strains and
strain rates in Chapter 3. We have limited our attention to a polyurea copolymer (PU1000)
possessing a distinct separation of hard and soft phases in a co-continuous morphology. Recent
studies on various polyureas revealed that the molecular weight of polytetramethylene oxide
chains (PTMO) in soft diamine prepolymers significantly affects the degree of phase-separation;
It was also found that a decreased molecular weight for PTMO chains in diamines (Versalink P104
1000, 650 and 250) leads to less weight fraction of soft phases and to dramatic decrease in the
degree of phase-separation of hard and soft segments in bulk-polymerized PTMO-based
polyureas and polyurethanes.1~3 Macroscopic mechanical behavior of polyureas was found to
changes
substantially due to the varied weight fractions, morphology and segmental
microstructures. Furthermore, the main features of dissipation and resilience are affected by
varied weight fractions and segmental microstructures of hard and soft contents, hence posing
challenges for the predictive design of elastomeric materials with "tunable" functionality.
In this chapter, the constitutive model of elastomeric segmented copolymers is furthered to
capture the mechanical behavior of polyureas possessing a different weight fraction and
morphology. The nature of the different phases and their segmental dynamics was found to
significantly affect the stress-strain behavior in tandem with the weight fractions, providing
critical insight into a multi-component (more than two phases) constitutive modeling framework.
Additionally, a simple micromechanical modeling is employed to further understand the
morphological effect of a co-continuous network of hard and soft phases to the macroscopic
mechanical behavior. The micromechanical study outlined here provides a simple yet critical
insight into further studies on the multi-scale modeling of phase-separated copolymers which
connects the molecular-scale information with the macroscopic continuum mechanical
framework.
Keyword: Polyurea 650, polytetramethylene oxide (PTMO), phase-separation, morphology,
hard/mixed/soft segments, micromechanics
5.2.
Segmental Microstructure and its Connection to Stress-Strain Behavior of PU 1000
and PU650
Polyurea 1000 and 650 have been derived via a bulk polymerization of prepolymers of
diisocyanate (Isonate 143L, Dow Chemicals
P650, Air Products
5) based
4)
and functionalized diamine (Versalink P1000 and
on the synthesis procedures developed by Naval Surface Warfare
105
Center, Carderock Division.6 The molecular weight of polytetramethyleneoxide chains was
estimated in -1000 g/mole or -650 g/mole for amine prepolymers. The relative weight fraction
of hard and soft segments in the cured PUOOO and PU650 polymers is controlled via the varied
numbers of PTMO repeating units in the soft diamine. Furthermore, a recent study revealed the
relative weight fraction can be tailored by modification of the molecular weight of hard phase
prepolymer, isocyanate.' Here, we are focused on the weight fraction controls by changing the
soft prepolymers. The relative weight fraction of hard and soft phases is summarized for PUOOG
and PU650 in Table 3.
As shown in Table 3, the weight fraction of hard phases increases as the molecular weight
of PTMO decreases in the soft phase prepolymers. Consequently, the mechanical behavior of
copolymers was found to exhibit greater "glassy" features involving much greater stiffness and
yield stress (Figure 5-1a) with a greater residual strain and hysteresis upon unloading (Figure
5-1b). However, the overall features of stress-strain behavior of PU650 are very similar to those
found in PU 1000 including the highly rate-dependent features (Figure 5-1c) and the transition in
the rate-sensitivity (Figure 5-1d).
Hard Segment [%]
Soft Segment [%]
PU1000
34
66
PU650
46
54 (Soft + Mixed)
Table 3 Weight Fraction of Hard and Soft Phase in PU1000 and PU6501' 6
A simple scaling rule for PU650 is first examined to account for the effect of an increased
weight fraction of hard phases. Here, we assume a perfect phase separation in both PU1000 and
PU650. Thus we can simply estimate the Cauchy stress of PU650 using the constitutive
components of PU1000 without any change in the material parameters via,
T
= O ar
puThar
loo1000 + 0,f, ToPI
u 1000 1
(1)
106
where
P,,,,,
and o,
are the relative weight fractions of hard and soft domains of PU650 and
PU1000. The "effective" scaling factors (r,
9,,,,) for hard and soft phases are estimated as
follows; when scaling the stress components of hard and soft phases for PU650, we have relative
scaling factors of 46%/34%
1.35 (hard) and 54%/66% ~ 0.84 (soft) based on the weight
-
fractions given in Table 3; however, the sum of relative scaling factors is 2.11 in PU650; the
"effective"
scaling factors for hard and soft phases should be rescaled by (1+1
in
PUl000)/(1.35+0.82 in PU650) ~ 0.92 since we began with 1:1 contribution from hard and soft
phases in the constitutive modeling of PUlOOG with a perfect phase separation "not" including
the weight fraction of hard and soft phase; i.e. Tlooo
=1-
T,,,,1.100 +1. Tou1
effective scaling factors for hard and soft phases are found to be
h,,
; therefore, the
-1.2 and A
0 ,ft
0.8. Clearly,
this scaling will not be sufficient since we see the flow stress in PU650 is a factor of -2 greater
than the PU1000 flow stress, not a factor of 1.2. However, we will follow this scaling rule to
demonstrate the stress-strain behavior predicted in this approach at this point.
(a)
30--___(b)_
12.s00
- -- PUIOOO, 0.011
---,20PU660
30
PU6500.0011:
0 -PU50
0.OlIs
0
320-.,
0
10
.. .
0.2
W.
10
......
0.6
0.4
True Strain
0.8
1.0
,
0.2
WO
0.8
0.6
0.4
True Strain
(d)
(c)
100
us
s80
---puSM, o.0oi
.0 Is
a.60-
PUSm
-
, w-0.35
0 -oPU650, a-0.75
0 -0
j 0*
2o-
0)
.
0.2
.
0.4
0
0
0I
W.
C.
*
--. PUOB, 0.01 Is
"
1W 75....PU660,0.1
75
2 Is Is16
a.
PU6850 2200 Is
2
0-
.--..-..-
0.6
True Strain
0.8
1.0
1
10
10, io, 101 102 1
Strain Rate [is]
Figure 5-1 Mechanical behavior of PU650: (a) stress-strain data of PU1000 and PU650 under
monotonic compression at low strain rate; (b) stress-strain data of PU1000 and PU650 under
cyclic tension at a strain rate of 0.005 /s (PU1000 data reused from Chapter 3); (c) stress-strain
107
data of PU650 at low to high strain rate; (d) flow stress of PU650 as a function of strain rate at
strains of 0.4 and 0.9
Figure 5-2 shows the stress-strain behavior of PU1000 and PU650 in experiments and
simulations. As expected, the overall stress of PU650 was amplified by the factor of 0,.d by
comparison to PU1000 low strain rate data. However, the simple scaling based on the relative
weight fractions of hard phases in PU650 and PUl000 was not able to capture the dramatic
elevation in stress observed in PU650, which is at least a factor of 2 greater than the stress
observed in PU1000. As we discussed previously, the dramatic increase in stiffness and flow
stress in PU650 was due to the effect of weight fraction, morphology and segmental structures of
hard, soft and mixed domains. In particular, the presence of mixed phases in PU650 due to the
lower degree of phase-separation via the smaller chain length of PTMO in the amine
prepolymers provides a strong motivation for another constitutive component for PU650 at low
to high strain rates. A modified constitutive modeling framework is developed based on the
three-phase nature of PU650 employing mixed phase components.
The phase-separation of hard and soft segments has been widely investigated for a broad
variety of polyurethane and polyurethane-urea in terms of relative weight fractions of constituent
homopolymers.
8,9
The degree of phase separation of hard and soft segments was also found to
be significantly affected by molecular weight of PTMO in polyurea in a recent study on the
microstructures of PUOOG and PU650."
2 In
their studies, it was found that PU650 possessed
"three" phases of hard, soft and mixed segments while PU1000 exhibited a perfect phase
separation of hard and soft segments. The degree of miscibility in PUOGO, 650 and 250 was
estimated based on tapping model AFM phase images and small angle X-ray (SAXS) data by
which the electron density variance for the segment demixing was determined from the total
scattering intensity using the background and absolute intensity corrected SAXS intensities. The
electron density variance and the corrected SAXS intensity were found to significantly decrease
in PU650 and PU250 revealing that decreasing soft segment molecular weight had a marked
effect on the degree of unlike segment demixing. The presence of "three" phases in PU650 was
also confirmed by dynamic mechanical analysis data. Figure 5-3 shows dynamic mechanical
analysis of PUOGO and PU650. A greater storage modulus (Figure 5-3a) was observed over a
108
wide range of temperature around room temperature, where a very gradual reduction in E" with
0
increasing temperature as seen in the supposed "glass transition" range of -25 C and 25" C. The
loss factor of PU650 in Figure 5-3c strongly supports the presence of a mixed phase or interphase regions of hard and soft segments, seen by another peak in loss factor located at - 25"C
0
between the two peaks of -50 0 C and 135 C corresponding to relaxations of hard and soft phase.
Therefore, in order to capture the constitutive responses of PU650 with the base material
properties of PUOOG, the effects of phase-mixed segmental microstructures should be involved
in conjunction with the effect of weight fractions of constituents.
30
0.001 /s
0.01 /s
-
CL
20
-- --.
PU650, 0.001 Is,exp
PU660, 0.01 /s, exp
- PU660, 0.001 Is, model
PU60, 0.01 Is model
.PU1000,
.e*-.PU1OOG,
U)
I-
ni
0.0
*
dp
101
0.2
0.4
0.6
0.8
.P.
PU650 model: Weight
fractional scaling from
PU1000 model
1.0
True Strain
Figure 5-2 Model predictions of PU650 stress-strain behavior via weight fractional scaling
109
(a)
(b)
10000
-o-PU1000
-a- PU650
0
(c)
1
AM,
1000
-0-
100
--o-PUSS0
CL 300
vi.
-,.-Pu1000
-o-PU5O
PU1000
T
Ur
200-
0
40
0
0
2 100
a
0.1
-j
0
-J
-15
-100 -50
0
50 100 150
Temperature [C]
-150 -100
-50
0
50 100
Temperature [0C]
15 0
-150 -100
0
50 100
-50
Temperature [C]
150
Figure 5-3 Dynamic mechanical analysis (DMA) data of PU1000 and PU650: (a) storage
modulus; (b) loss modulus; (c) loss factor
5.3.
Constitutive Models of PU650: Effect of Weight Fraction and Segmental Structures
of Hard, Soft Phases and Their Mixtures
Detailed properties of the mixed phase in PU650 have not been characterized
experimentally yet in terms of the morphology and the weight fraction. However the constitutive
response of mixed phases can be estimated using the low strain rate data of PU650 since
contribution from soft phases is still negligible at low strain rates. Mathematical formulations of
intermolecular and network component in the mixed phase follow those in the hard phases.
However, to capture the rate-dependent network resistance, we introduced an orientationdependent power-law viscoelasticity in the network component, by which the mixed phase
contribution dramatically increases at high strain rate via,
7NM
(NMDNM
(2)
NMF,
where pNM is the orientation angle;
I
is the nominal shear viscosity; and rM is a magnitude of
DNM
deviatoric stress in the mixed network (NM) component.
An extrapolation of scaled intermolecular and network components at low strain rates
gives a first estimation of the mixed phase contribution; the stress level of mixed phases is
estimated by subtracting the scaled stress of hard phases via Equation (4) - (8) (in 5.6) from the
110
low strain rate data of PU650. Furthermore, the stress-strain data under multiple consecutive
cyclic tests with an increasing, imposed strain provide detailed information about the evolution
of softening parameters in mixed components. The high strain rate data also gives the "effective"
scaling factor (p,, ~ 0.6) for soft components with the presence of mixed phases. Using Equation
(4) - (8), the material parameters of soft components were then scaled for PU650. Finally, the
Cauchy stress of PU650 is calculated by the sum of all of contributions from mixed and scaled
hard/soft component via,
TP165
=' Ohard
(TIH
+ TNH
)PU1
(TIM + TNM
)±soft (TIs +
TNS
PU(3)
Detailed information about the material parameters for mixed components and the scale
parameters for hard and soft components can be found in 5.6.
Figure 5-4a shows the stress-strain behavior of PU650 in experiments and simulations at
low to high strain rate. The modified constitutive model with "mixed" phase components was
found to capture well the dramatically increased stress in PU650 at a broad range of strain rates.
The stress-strain curves of PU650 at two different strain rates (0.01 /s and 2200 /s) were then
decomposed into individual contributions from hard, soft and mixed components in Figure 5-4b.
The constitutive response of mixed components was found to be naturally located between hard
and soft responses. Furthermore the rate-dependent behavior of mixed phases was well captured
at low to high strain rates. The transition in the rate-sensitivity was also well captured in PU650
models as shown in Figure 5-5 on flow stress as a function of strain rate at increasing strains of
0.4 and 0.9. The transition behavior was naturally captured in the modified constitutive model
since the stress contribution from the soft components was still negligible due to complete
relaxation at low strain rates.
The modified constitutive model was examined for the resilient yet dissipative behavior of
PU650 under cyclic tensile testing. Figure 5-6a and b shows the stress-strain curves of PU650
and PU1000 under a multiple consecutive cyclic tension test subjected to an increasing
maximum strain at a strain rate of
-
0.005 /s in experiments and simulations. The overall
behavior of cyclic tensile test was similar to that in PUlOOG as discussed in Chapter 3 including
the dramatic stretch-induced softening that provided a major dissipation source in the first cycle.
The simulated stress-strain curves agreed well with the experimental data. In addition, there was
111
a clear asymmetry between compression (Figure 5-4) and tension (Figure 5-6), where a greater
hardening was observed in the post-yield regime due to a greater orientation in the network
elasticity.
To quantify the deformation-dependent dissipation of PU650 (Figure 5-6d), the
stress-strain data were reduced to dissipated work density as a function of increasing strains in
the first and second cycle. As expected, most of dissipation was achieved in the first cycle due to
the stretch-induced softening via microstructural breakdown in hard and mixed phases. Though
dissipation dramatically decreased in the second cycle, it still exhibited "finite" level mainly due
to viscous flows in each phase. By comparison to PU1000 (Figure 5-6c; and see Chapter 3), the
dissipated work density dramatically increased with greater residual strains under the same
imposed strains revealing the greater "glassy" features of PU650 under deformation; i.e. tailoring
the molecular structures and the weight fractions of hard/soft phases resulted in substantial
change in dissipation and resilience in the elastomeric segmented copolymer PU 1000 and PU650.
(a) 100
1
(b) 40
model, 0.001 /s
model, 0.01 /s
-- model, 0.1 /s
--- model, 2200 /s
* OXP, 0.001 /s
---
800..
60 -
Hard, 2200/s
A exp, 0.1 /
40
2
20
0.
Vexp, 0.01 /s
@ exp,2200Is
2
W 30 -
L
*0
20-
Mixed, 2200/5
,A
0
.*
Had,0.-/
o,*
I
0S.22%
e,0.01/s
0.2
0.4
0.6
True Strain
0.8
1
0.2
0.4
0.6
True Strain
Soft, 0.01/s
0.8
1
Figure 5-4 Stress-strain behavior of PU650 under compression in experiment and model: (a)
Stress-strain curves at low to high strain rate; (b) Decomposition of simulated stress response
into hard, mixed and soft contribution at strain rates of 0.01 and 2200 /s
112
...
100
PU1000, e=0.35, exp
PUIOO , e=0.75, exp
o PUIOOO, e=0.35, model
*
75 l-
E
-
50
2
-
PUIOOO, e=0.75, model
PU650, e=0.35, exp
PU6B0, E=0. 75 , exp
o PU650, a=0.35, model
0 PU6S0, e=0.75, model
*
U
A
3
~ 0.
rd~h.r
+
+Tid
0,oftTsof,
E3.
Co
25
o
TPUIOO
U.
0
~"
hrd +Tsft
10"3 10-2 10 10 10 102 103 10
Strain Rate [/s]
Figure 5-5 Flow stress as a function of strain rate at increasing strains of 0.4 and 0.9 in PU1000
and PU650
(b)
(a)
j.PU650I
30
CL
20
S3
2C
;q
0
C%,
10
I
0.0
(c)
0.2
0.6
0.4
True Strain
b
0.8
6
5
CL
4'
10
o Loading, exp
o Reloading, exp
e Loading, sim
m Reloading, sim
CL
ia
3.
0.
2
-I
1
'3
b.0
0. 8
0.6
True Strain
7
Lw
0.4
0.2
3:
8
6
o Loading, exp
c3 Reloading, exp
* Loading, model
m Reloading, model
0
0
4
0.
V
2
U
0.2
0.4
True Strain
0g
0.6
0.8
0.0
S
0.2
M
0.4
0.6
0.8
True Strain
Figure 5-6 Resilient yet dissipative mechanical behavior of PU650 and PUlO: Stress-strain
curves under multiple consecutive cyclic tensile tests at a strain rate of -0.005 /s (a) experiment;
113
(b) model; (c) dissipated work density as a function of strain during 1st and
(d) dissipated work density as a function of strain during
1 st
and
2 nd
2 nd
cycle in PUl000
cycle in PU650 (See
Chapter 3 for detailed PU 1000 data in model and experiment)
5.4.
Micromechanical Modeling of Co-continuous Morphology
The micromechanical modeling framework has been extensively employed to provide
critical insight into underlying mechanisms of elastic-plastic deformation of various polymeric
materials.1' " In particular, micromechanical analysis of multi-phase polymeric materials
-
revealed that the morphological features of thermodynamically-incompatible phases strongly
affect the macroscopic response of materials. Inspired from the co-continuous morphology often
found in a broad variety of copolymers, Wang et al. demonstrated multi-material composites to
tailor the mechanical properties. 12 In their research, co-continuous composites were fabricated
via 3D printing, comprising hard and soft materials, which has been widely used to demonstrate
a broad variety of "digital" composites with complex geometries. In particular, various cocontinuous morphologies were suggested to account for the tunable elastic and plastic properties
based on the close-packed lattice structures often found in crystalline metals. Furthermore, a
computational analysis proposed design principles for micro-frames to provide tunable plastic
deformation and energy dissipation15' 16 revealing that the co-continuous composites may be
highly promising in terms of tunable mechanical performances.
In this section, we address a simple micromechanical modeling of co-continuous networks
of hard and soft phases in elastomeric segmented copolymers to account for the morphological
effect of networks to the macroscopic mechanical responses. In particular, we are focused on an
eight-chain type network of hard segments as one of the likely bi-continuous morphologies
observed in diverse segmented copolymeric materials possessing thermodynamically-separated
constituent homopolymers.
17-21
114
Dout
D.~
Din
Figure 5-7 Schematic of representative volume elements of exemplar co-continuous network
A representative volume element (RVE) comprising a network of eight hard ligaments was
constructed as illustrated in Figure 5-7 neglecting individual force contribution from soft phases
as described previously, where the soft phase is simply voided to enforce the nearly
incompressible nature of deformation. This allows for a simple micromechanical study of this
particular co-continuous geometry without any loss of physical intuition. We constructed a set of
RVEs composed of 34% and 46% fractions of hard phases. Additionally, we examined RVEs
with two different ratios of internal to external diameter of chains to quantify the geometric
effect at network joints, where a very large stress is localized. The RVEs for four co-continuous
to uniaxial
networks (Dif/D
0out=1.0 and -0.86; 34% and 46% hard phases) are then subjected
tension up to a "global" true strain of 1.0 to account for the morphological effects to the
macroscopic mechanical response; here, a kinematic constraint in lateral directions was
employed such that a total volume for RVE was preserved during deformation with an
assumption of incompressible deformation often found in elastomeric materials. In addition, a
simple hyperelasticity model was used for the constitutive response of "model" materials.
115
(a)
(b)
Figure 5-8 Contours of axial strain field in RVEs (Din/D 0,t=1.0) of co-continuous morphology
subjected to tensile loading up to a true strain of 1.0 (a) 46% hard phase (b) 34% hard phase
(a)
(b)
*
Figure 5-9 Contours of axial strain field in RVEs (Di,/D 0 ut=0.85) of co-continuous morphology
subjected to tensile loading up to a true strain of 1.0 (a) 46% hard phase (b) 34% hard phase
Figure 5-8 and Figure 5-9 show the undeformed and deformed configurations of RVEs
with contours of axial strain field. Very large deformation developed in each chain network at a
true strain of 1.0. As expected, the maximum strains were found to be localized at joints
connecting the eight ligaments due to large stress concentration. During deformation, the
reaction force was monitored in each RVE to quantify the effect of eight chain networks to the
macroscopic response.
116
(a)
(b)
1.0
.
0.8 -
46%
34%
.
£0.6-
.
0.8-
.
*
46%
34%
0.6-
0
Ew
U
>0.4
0.4-
0.2
0.0
0.0
1.0
0.2
0.1
0.2
0.3
Axial Strain
0.4
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Axial Strain
Figure 5-10 Stress-strain behavior of RVEs: (a) Di,/D 0o.t=1.0; (b) Dfn/Dout=0.85 (here, the RVE
stress was normalized by a stress magnitude of 46% RVE, Di/Dout=1.0 at a strain of 0.6)
Figure 5-10 shows the stress-strain behavior in each RVE under tensile loading up to an
axial RVE true strain of 0.5. Here, RVE stress was normalized by the true stress of 46% RVE
(Din/D 0out=l.0) at an axial true strain of 0.6 for both joint geometries of Din/Dout=1.0 and 0.86. As
expected, much greater stress was developed in the RVEs of 46% hard phase in both cases.
Stress amplification in the RVEs was quantified in Figure 5-11 on the relative level of RVE
stress during deformation. As shown in the RVE stress ratios, there was a dramatic stiffening
effect in the RVEs of 48% hard phase in both cases of Din/Dout=0.86 and Din/D0out=1.0. The level
of stiffening was found to be much greater than that found in the simple volumetric scaling (1.25). The significant increase in stiffening is due to combined effects of stretching, bending, and
rotation of the hard domains to the RVE stress and it indicates a simple scaling is not a
reasonable approach even if there is distinct phase separation. In this particular co-continuous
morphology, there was a significant effect from the rotation of microstructures that resulted in a
remarkable increase in stress in addition to the volume fractional effect.
Furthermore, the rotational effect was found to be strongly dependent upon the shape of
joints that connect the microstructural eight chains of hard phases; much greater stiffening was
observed in the case of Din/Dout=1.0 due to greater bending stiffness at joints; i.e. the bending at
the joint is driving the higher stiffening effect since the resistance to the rotation of eight
117
ligaments dramatically increases in the case of Di/D,t=1.0. The stress concentration occurs at
the joint nodes since they are acting as a fixed end in a simple beam deflection; there is likely a
bending moment effect with greater enhancement to the resistance since a moment of inertia in
the case of Din/Do
0 t=1.0 is much greater than that in Din/Dot=0.86. This is also supported by that
the ratio of RVE stress gradually decreases as an imposed axial RVE strain increases in the both
cases; i.e. the effect of bending and rotation of eight ligaments gradually decreases at increasing
strains, where the stiffening in RVEs is governed by stretching rather than bending and rotation.
The ratio (~ 1.85) of RVE stress with 46% and 34% hard phases at the case of Din/D 0out=l.0 was
found to be very close to that (- 1.95) found in the stress-strain curves of PU650 and PU1000
(Figure 5-4).
The simple micromechanical analysis supports the co-continuous morphological effect can
also provide for a greater stiffening and enhanced flow stress level in copolymers in addition to
the influence of phase mixing. Additionally, the local geometric features in co-continuous
morphology may significantly affect the macroscopic mechanical responses, which can be
rationalized via a simple structural mechanical perspective.
3.0
-
2.5-
0
cc
11*I'
2.0U
E1.
*
*
*
*
Volume fraction scale
* Di /Dout=1.0
0.5
* Di/Do =0.86
0.0.0.0
-
I
0.1
0.2
-
0.3
0.4
0.5
Axial Strain
Figure 5-11 Ratios of RVE stress in 34% to 48% hard phase for Din/Dout=0.85 (red) and
Din/D out= 1.0 (black)
118
In future, further studies should be conducted to better understand how the geometric
features of microstructures affect the macroscopic mechanical behavior including a broad variety
of morphologies available in this class of segmented elastomeric copolymers. Though we
explored a bi-continuous morphology composed of (ribbon-like) eight-chain ligaments for an
exemplar analysis, a variety of morphologies of hard and soft phases have been found in this
class of elastomeric materials involving non-continuous morphologies. In addition to the cocontinuous structures, hard (or soft) phases can be isolated and occluded in soft (hard) matrices
dependent upon volume fractions and chemical modification of the constituents. Figure 5-12
shows co-continuous and non-continuous microstructures available in the materials including the
bi-continuous heterostructures and the occluded spherical inclusions in the surrounding matrices.
Further micromechanical analysis on these heterostructures will provide an insight into microand macroscopic design principles of the elastomeric copolymers for geometrically-tunable
mechanical performances.
(a)
(b)
Figure 5-12 Exemplar microstructures found in the elastomeric copolymers: (a) unit-cells for bicontinuous and occluded heterostructures (volume fraction: 50%/50%); (b) stress field in a bicontinuous structure; (c) stress field in occluded hard spheres in soft matrix (occluded spheres
sitting on primitive cubic (cP) lattices)
119
5.5.
Conclusion and Future Work
In this chapter, we investigated mechanical behavior of PU650 possessing a greater weight
fraction of hard phases considering the effects of varied weight fractions, hard/soft morphology
and segmental microstructures. Though still lacking detailed information about the mixed phases,
the proposed constitutive model was found to nicely capture the stress-strain behavior of PU650
at low to high strain rates and the relevant resilient yet dissipative features. In addition, a
micromechanical modeling of co-continuous microstructures provided a simple yet intuitive
outlook into further studies towards the morphologically-tunable mechanical properties of phaseseparated copolymers. The co-continuous morphology comprising a network of eight ligaments
was found to be able to provide significant stiffening and enhanced flow stress in copolymeric
materials in conjunction with the effect of weight fraction and phase mixing.
In future research, a thermodynamic analysis of miscibility of phases in the polyurea
copolymers should be rationalized in terms of relative fractions of hard and soft constituents and
their connections to the phase separation. Constitutive theories of multiple-microstructures in the
materials may be furthered based on the quantitative information about the phase-mixing and the
relevant features of mixed domains. Furthermore, a broad variety of morphologies of hard,
mixed and soft domains which can be formed in the materials should be explored via x-ray
scattering experiments on the microstructures. In addition to experimental approaches, a
multiscale theoretical study via a combination of fully- or coarse-grained atomistic and
micromechanical
models may provide further insight into the morphologically-tunable
mechanical performance, which can be facilitated in a predictive design of composites derived
from this class of materials at macroscopic levels.
5.6.
Material Parameters for PU650 Mixed Phase
For PU650 constitutive models, we have scaled the material parameters of hard and soft
components used in PU1000 models as given in Chapter 3.
120
Using the scaling factors (1.2 and 0.7) determined for the hard and soft phase with the
presence of mixed phase, we can estimate the elastic and inelastic material parameters in hard
and soft components for PU650. Revisiting the yield stress-strain rate relationship used to
determine the inelastic parameters in the thermally-activated flow model, we had,
r= Ain k
+B ,
(4)
where, A and B are linearly dependent on the rate-dependence and the magnitude of yield,
respectively, which can be expressed by functions of major inelastic parameters including the
activation free energy, shear strength, and reference strain rate. (See Chapter 3) The parameter
set of A and B for PU650 may be further estimated via Ap1650 = Aulwo and BP 6 5 ,
= OBPUioM
since we
have no experimental evidence for the change of rate-dependence of hard phase in PU650.
However, the yield stress was found to increase as the weight fraction of hard phases increased.
These relations lead to scaling rules as follows,
J-PU650 =PPUIOOO,
(5)
SPU650
(6)
PUIOW
AGpU650 = #AGu10w ,
(7)
n'IP
U650
(8)
=
( OIUO) 0
where p is the elastic shear modulus;
s
is the shear strength; AG is the activation free energy
against viscoplastic flow; and gp is the reference inelastic strain rate. The "scaled" material
parameters for hard and soft components for PU650 are hence estimated using the relations in (4
- (8) and the original material parameters for hard and soft components for PUOOG in Chapter
3.
Here, the constitutive equations and the material parameters for the intermolecular and
network mechanism in mixed phases are provided in Table 2 and 3.
121
Table 2 Material parameter for intermolecular component in mixed phase
Intermolecular
Mixed
Intermolecular Stress
16.8 [MPa
a= " Bap
J
Viscoelastic-Viscoplastic Flow
AGIM = 2.1[10- 2 0
=OP
S(
j]
sinh
kBG)
kBG'Sa'IM
=1.74[s-1]
Elastic-Plastic Softening
1
him = 2.0[MPa]
sa
assa
2
p=h
L-&a J;
'Pss,a)
yss,IM
UssIM
/O,IM
=0.5
lu0,
==0.5
122
Table 3 Material parameter for network component in mixed phase
Network
Mixed
Network Stress
T
"Ba
-
_=_"
3J
=
/NM
~ae
N
8.1[MPa]
= 6.5
NM
Viscoelastic Flow
YNM
,
C S-1 min{
=
1/D=3.3[10 6Pa
= (DorNM 1 /n
2
0.167]
,In}
tr(BNM
Stretch-induced Softening
,uNM
(0)NNM (0)=NM (t)NNM
(t)
c0
(t)
jNm
sinh
)sNM
ock,NM
I ,eAM
AockNM=
(t);2NM
'NM
S 0
=0.1
= 0.5 [MPa]
SNM
SNM
= SO.NM
(lockNM
2
lock ,NM (0))
2.5
=0.3
11r
Ass.lock,NM
r
tanh
AA
HAockNM
i=1.4
(0)
) jAC=20s]
123
5.7.
Reference
1.
A. M. Castagna, A. Pangon, T. Choi, G. P. Dillon and J. Runt, Macromolecules 45 (20),
8438-8444 (2012).
2.
A. M. Castagna, A. Pangon, G. P. Dillon and J. Runt, Macromolecules 46 (16), 65206527 (2013).
3.
D. Fragiadakis and J. Runt, Macromolecules 46 (10), 4184-4190 (2013).
4.
Dow-Chemicals.
5.
Air-Products.
6.
J. J. Fedderly, (NSWC, Carderock Division, 2012).
7.
K. Holzworth, Z. Jia, A. V. Amirkhizi, J. Qiao and S. Nemat-Nasser, Polymer 54 (12),
3079-3085 (2013).
8.
R. Hernandez, J. Weksler, A. Padsalgikar, T. Choi, E. Angelo, J. S. Lin, L.-C. Xu, C. A.
Siedlecki and J. Runt, Macromolecules 41 (24), 9767-9776 (2008).
9.
L. M. Leung and J. T. Koberstein, Journal of Polymer Science: Polymer Physics Edition
23 (9), 1883-1913 (1985).
10.
J. A. W. van Dommelen, D. M. Parks, M. C. Boyce, W. A. M. Brekelmans and F. P. T.
Baaijens, Journal of the Mechanics and Physics of Solids 51 (3), 519-541 (2003).
11.
M. Danielsson, D. M. Parks and M. C. Boyce, Journal of the Mechanics and Physics of
Solids 50 (2), 351-379 (2002).
12.
L. Wang, J. Lau, E. L. Thomas and M. C. Boyce, Advanced Materials 23 (13), 15241529 (2011).
13.
P. A. Tzika, M. C. Boyce and D. M. Parks, Journal of the Mechanics and Physics of
Solids 48 (9), 1893-1929 (2000).
14.
N. Sheng, M. C. Boyce, D. M. Parks, G. C. Rutledge, J. I. Abes and R. E. Cohen,
Polymer 45 (2), 487-506 (2004).
15.
L. Wang, M. C. Boyce, C.-Y. Wen and E. L. Thomas, Advanced Functional Materials 19
(9), 1343-1350 (2009).
16.
L. Wang and M. C. Boyce, Advanced Functional Materials 20 (18), 3025-3030 (2010).
17.
G. H. Fredrickson and F. S. Bates, Annual Review of Materials Science 26, 501-550
(1996).
124
18.
F. S. Bates, Science 251 (4996), 898-905 (1991).
19.
P. R. Laity, J. E. Taylor, S. S. Wong, P. Khunkamchoo, M. Cable, G. T. Andrews, A. F.
Johnson and R. E. Cameron, Macromolecular Materials and Engineering 291 (4), 301324 (2006).
20.
W. Li, A. J. Ryan and I. K. Meier, Macromolecules 35 (13), 5034-5042 (2002).
21.
R. G. Rinaldi, M. C. Boyce, S. J. Weigand, D. J. Londono and M. W. Guise, Journal of
Polymer Science Part B: Polymer Physics 49 (23), 1660-1671 (2011).
125
126
Chapter 6
Extreme Behavior of Elastomeric Copolymers under
Harsh Environments
Portions of this chapter were published in a journal paper, H. Cho, M. C. Boyce et al.,
"Resilience and Dissipationof Elastomeric Copolymers under Extreme Strain Rate", Polymer,
2013
6.1.
High Strain Rate Behavior of Elastomeric Copolymers
High strain rate behavior has been attractive to research communities of polymeric
composites for numerous applications. The remarkable resistance of polymeric materials against
impact and shock loading has been hence a focal point of research interest to design light-weight
material architectures for impact and shock mitigation. Over the past several decades, the high
strain
rate
behavior
of
glassy
polymers
including
polycarbonate
(PC)
and
polymethylmethacrylate (PMMA) has been extensively studied revealing the critical importance
of multiple distinct processes including primary and secondary relaxations for macroscopic
mechanical responses.
Elastomeric materials were also found to be highly rate-dependent over
a wide range of strain rates as we discussed in Chapter 3 and 5. In particular, the presence of
thermodynamically-separated morphology of constituent phases we discussed was found to be
responsible for a "transition" in the rate-dependent stress-strain behavior. In addition to the
change in the rate-sensitivity of yield (or yield-like, near a strain of 0.05-0.15) stress at which the
stress rolls over, there exists changes at increasing strains at post-yield regimes. In particular, the
slope of rate-sensitivity was found to increase at increasing strains e.g. strains of 0.4 and 0.9.
This revealed that the overall transition behavior of rate-sensitivity at different strain levels
should be captured in the constitutive models by taking the multiple thermally-activated
127
intermolecular elastic-plastic components as well as the rate-dependence of network elasticity
via the rate-dependent stretch-induced softening and the nonlinear viscoelastic flows in network
resistances. In Chapter 3 and 5, we reported that the proposed constitutive models of
elastomeric segmented copolymers (PU 1000 and PU650) captured well the transition behavior at
different levels of strains. Furthermore, the transition behavior of stress-strain responses has been
extensively observed in elastomeric ionic copolymers which will be covered in Chapter 8.
Figure 6-1 Error! Reference source not found. shows the flow stress as a function of strain rates
at two different strains in PUl000, PU650 and EMAA. We can clearly observe the transition
behavior in all the materials covered in this research.
(a)
(b)
100
75
(U
0
'N
* PU1000, a=0.35, exp
0 PUIOO, e=0.75, exp
o PUIOOO, P=0.35, model
o PUIOO, 0.75, model
* PU650, e=0.35, exp
* PUB5O, em0.75, exp
o
50
75, model
50- o
0U
a: 25
-2 10-
I
1
1
.....
1..
E=0.4,
model
3
6=0.8, model
0
2-
0
0
13
10-3
o
ca.
-.
25
0
....
* e=0.4, exp
m e=0.8, exp
CU
PU650, =0.35, model
0 PU-S,
.
0
10
10 1
Strain Rate [/s]
1
10
10- 10-2 10-
10
10
102
10
1
Strain Rate [/s]
Figure 6-1 Flow stress as a function of strain rates at increasing strains: (a) PU1000 and PU650
(See Chapter 5 for details); (b) ethylene methacrylic acid (EMAA) copolymer (See Chapter 8
for details)
For the high strain rate behavior of glassy, rubbery and copolymeric materials, we have
extensively used a Split Hopkinson (or Kolsky) Pressure Bar (SHPB facility, Institute of Soldier
Nanotechnology, MIT) following a detailed procedure by Mulliken. 4 However, the maximum
level of strain rates incurred without "inertial" effects was found to be near to 103 and 104 /s in
SHPB testing. In realistic impact or ballistic penetration events, the material can reach over strain
128
rates of 105 ~ 106 /s with remarkable inertial effects. The nature of resilience and dissipation of
materials under such "ultrafast" events should be addressed to provide a better design principle
of shock- or impact-protective polymeric structures. Experimental studies of this class of
extreme events have been found to be extremely challenging and expensive. Therefore, to date,
computational studies of polyurea and polyurethane copolymers under shock loading have been
widely performed employing atomistic and coarse-grained molecular dynamics simulations
revealing detailed information on explicit atomistic and molecular structures undergoing the
extreme deformation events. However, the deformation rates, which can be incurred in such
atomistic and molecular computations, are extremely high in comparison to the rates in actual
ballistic and impact loading events since the timescales available in such simulations are
extremely short at order of nano-seconds. Though computational ability and resource are
growing rapidly, time- and length-scales in atomistic and molecular simulations still lack reality
in comparison to experimental scales. In this chapter, the ultrafast deformation behavior of
polyurea rods (PUl000) is studied in Taylor impact tests in experiments and numerical
simulations at continuum scales revealing the extreme nature of resilient- and energy-dissipating
polyurea copolymers.
The Taylor impact test has been utilized to study the high strain rate flow stress and
constitutive behavior of metallic materials. 5-9 Although widely used on metallic and ceramic
materials, relatively few studies on the Taylor impact behavior of polymeric materials have been
reported. Briscoe and Hutchings employed Taylor impact tests to quantify the rate- and
temperature-dependent flow stress of high density polyethylene (HDPE).10 Also, the Taylor
impact behavior of polyetheretherketone (PEEK) was used to investigate dynamic failure
behavior" and high strain rate mechanical properties. 12 A comprehensive work on Taylor impact
testing of polycarbonate (PC) reported the mechanics of high strain rate behavior of glassy
polymers by Sarva et al.3 However, the study of extreme deformation behavior of elastomeric
copolymers which possess hybrid properties of "glassy" and "rubbery" polymers has been
largely unexplored at present. In this paper, the mechanics of extreme strain rate behavior of
polyurea is elucidated in experiments and computational modeling using Taylor impact tests,
where inhomogeneous dynamic deformation processes with ultrafast strain rates greater than 105
/s are achieved. The highly resilient yet dissipative features of polyurea rods are examined under
such extreme-rate events by quantifying evolution of characteristic geometries and localized
129
deformation profiles. Finally, the "glassy" and "rubbery" features of elastomeric copolymers are
discussed by comparing kinetic energy evolution and deformation profiles of the two-phase
copolymer during Taylor impact to those of a purely hyperelastic rubber and those of a
viscoplastic glassy polymer. This work provides a predictive framework to quantify the ultrafast
deformation which can be incurred in realistic ballistic or blast loading events on this important
class of copolymeric materials.
Keyword: high rate behavior, copolymer, glassy polymer, rubbery polymer, Taylor impact test,
finite element simulation
6.2.
Experimental and Computational Methods for Taylor Impact Testing
The exemplar copolymer polyurea (referred to as PU1000 in this work, a weight fraction
of 34%/66% for hard and soft contents) was derived from the hard diisocyanate (functionality >
2, Isonate 143L, Dow Chemical) and the soft aliphatic diamine (Versalink P-1000, Air Products
and Chemicals) with a weight fraction of 1:4 for the prepolymers. Test samples were molded in a
cylindrical shape of length, L and diameter, D. The Taylor impact tests on the exemplar polyurea
were performed using the Naval Surface Warfare Center Gas-Gun Facility. 13 A schematic for the
muzzle region is shown in Figure 6-2. The cylindrical polyurea rods are subjected to high-speed
impact loading by a rigid steel flyer launched by a high-pressure breech gas. Deformation
profiles of the polyurea rods are recorded by a high-speed camera (Hadland Ultra 68, frame rate:
62,500 - 125,000 /s) during impact loading and unloading up to separation of the impacted rods
from the rigid surface. Exemplar deformation profiles (undeformed and deformed) under Taylor
impact testing of a polyurea rod with L = 25.7 mm and D = 12.6 mm (L/D
-
2.0) for a flyer plate
velocity of ~ 245 m/s are presented in Figure 6-2b and c.
130
(a)
VACUUM
ROD
*ULE
CLOSURANG
FOU.
GE
GAS GUN
(b)
(c)
BUARRELAGE
ASEGAS
to
Figure 6-2 Experimental setup for Taylor impact testing (a) schematic of target region prior
projectile impact (b) undeformed polyurea rod (l/D ~2.0) (c) deformed polyurea rod (IJD ~2.0)
for impact velocity of ~ 245 m/s
Computational modeling of the Taylor impact test was performed using finite element
simulations where the large deformation constitutive model of Cho et al." was numerically
value
implemented. Numerical procedures for the finite element solutions of initial and boundary
are
problem, the finite deformation kinematics and the elastic-plastic constitutive update
in
provided in Chapter 4 involving material parameters for the constitutive model provided
~0.85
Chapter 3. (We have used the elastic shear modulus in the soft network component: 0.75
p1 NS given in Chapter 3 for the Taylor impact simulations; however there is no significant
change in the constitutive stress-strain responses at low to high strain rates discussed in Chapter
3 with this modification.) Taking advantage of symmetry, axisymmetric simulations were
employed, where the rod was discretized with three-node triangular elements and four-node
bilinear elements. The four-node bilinear elements were used at the rear end of rods to avoid
A
unexpected free surface oscillations which were incurred with three-node elements.
6-3.
representative finite element mesh for a 2-D axisymmetric simulation is shown in Figure
The mesh size was varied from coarse to fine to verify that the chosen mesh structures ensured
stability and convergence of numerical solutions. A hard contact constraint with a Coulomb
friction (p,. = 0.1~- 0.15 ) was employed for the interfacial behavior. Also, an isothermal condition
was assumed during impact loading and unloading events since temperature rise due to adiabatic
heating during inelastic straining was found to have only a negligible difference. The effect of
flow
temperature rise and adiabatic heating is discussed in Appendix B. Additionally, the viscous
in the soft network mechanism may be turned off for computational efficiency in the Taylor
131
impact simulations since it was found to be negligible at high strain rates over 10
-
100 /s as
detailed in Chapter 3.
V
impact
Rigid surface
R
Figure 6-3 Schematic for axisymmetric finite element simulations with an exemplar three-node
triangular mesh
6.3.
Shape Evolution in Polyurea Rods during Taylor Impact Testing
In this section, experimental and computational results are presented for Taylor impact
testing of polyurea rods of UD
-
and D = 12.6 mm) and L/D
4 (L = 25.7 mm and D = 6.4 mm) subject to impact by a flyer
-
4/3 (L = 25.7 mm and D = 18.9 mm), L/D ~ 2 (L = 25.7 mm
plate at a velocity of ~ 245 m/s. Figure 6-4 shows deformed images with digitized deformation
profiles at different stages of loading (Figure 4a) and unloading (Figure 4b) for an experiment
with L/D
-
4/3 and Vimpact
245 m/s. The deformation results in the formation of a mushroom
head, which enlarges rapidly as the event progresses. The maximum spreading occurs at the
impact surface. The rod displays a significant shape recovery during unloading and separation.
Figure 6-4 also shows the simulated predictions of deformation profiles at times corresponding
to the high-speed photographs. The finite element simulation well captures the localized
deformation profiles during loading and unloading as evidenced by comparison to the digitized
deformation profiles.
Figure 6-5 shows the overall evolution of selected geometric dimensions taking the
overall length and the diameter at the impact surface during deformation, comparing experiment
132
and simulation results. The values of current length and end diameter were taken at each frame
and normalized by the original values. The overall shape evolution in both L(t) and D(t) was
found to be asymmetric between loading and unloading with good agreement between model and
experiment. Figure 6-5b shows the kinetic energy evolution during deformation, where a
nonlinear damped oscillation is observed. As expected, a significant amount of energy
dissipation (~ 81 %) is observed. This energy dissipation was also accompanied by the dramatic
level of shape recovery observed in Figure 6-4. In addition, a rebound speed in the simulation
was found to be 72 m/s, which agrees well with the experimentally measured relative rebound
speed, 67 m/s and further confirms the prediction of energy dissipation vs. energy storage.
Experiment
LOADING
Simulation
0 s
8
s
24 ps
40 ps
56 ps
72 ps
88 ps
184 ps
25 6
Experimen
UNLOADING
Simulation
104 ps
120 ps
136 s
152 ps
168 ps
s
320 s
Figure 6-4 Deformation profiles during a Taylor impact test with LD - 4/3 and V - 245 m/s:
high-speed photographs (red dots: digitized deformation profiles) and deformation profile
prediction made using an axisymmetric finite element simulation (a) loading (b) unloading
133
(a)
(b)
e
exp, D(t)
E exp, L(t)
* model, D(t)
model, L(t)
2.5
2.0 90
41.5
01
0
50
E
046-
~0.4
0
Ii
0.8
42
9EDe
0
E6E
0.50.0
Dissipated energy
1.0
3.0
EP02
100
150
200
250
0.0
0
Time [ps]
100
200
300
400
Time [ps]
Figure 6-5 Evolution of selected geometric dimensions and kinetic energy in the Taylor impact
test with IJD - 4/3 and V - 245 m/s (a) evolution of normalized length and diameter at the
impact surface in experiment and simulation (b) evolution of normalized kinetic energy with
selected deformed profiles under loading and unloading
The evolutions in stress (Figure 6-6) and strain rate contours (Figure 6-7) are presented to
understand the propagation of deformation and shape change in the polyurea rod; the
deformation-induced material softening is examined by contours in the elastic shear modulus
(Figure 8) (which evolves with inelastic strain).
Figure 6-6 shows axial stress contours at various stages of impact. Extremely high
compressive stress (- 600 MPa) is induced immediately upon impact at the head. Within10ps,
the magnitude of the peak compressive stress rapidly decreases and deformation axially
propagates in the specimen. These observations indicate the extremely rapid transient nature of
the stress evolution induced during the first few microseconds after impact. The compressive
wave front rapidly propagates towards the rear end of the rod. The deformation profiles show the
initial elastic compressive front is followed by a slower propagating inelastic deformation front
(Figure 6-7). As the inelastic front proceeds, the initial elastic compressive front reflects back
from the rear end and interacts with the tensile front. Also, the compressive and tensile wave
fronts are fairly stabilized, travelling between the two rear ends of the rod beyond the maximum
spreading at the impact surface. These observations clearly indicate the extreme nature of stress
and deformation gradients induced in the rod during the Taylor impact test.
134
Axial Stress [MPa]
t
t =3.3ps
1.5ps
-640.0
670.0
t-67.8ps
50.0
-130.0
t
7.0
-50.0
320.0
490.0
t 122ps
,
85.pt=9.9ps
t
20.0
-
20.0
165ps
t
t =14.0g
-130.0
190ps
-10.0
t = 29.1 ps
-340.0
t
30030.0
2.
separation
220pa
30.0
5.0
10.0
Figure 6-6 Contours of axial-stress at various stages in the Taylor impact test with L/D
V
-
-
4/3 and
245 m/s
Figure 6-7 shows the evolution of inelastic strain rate in the rod during deformation.
During the initial 6,us , ultrafast inelastic strain rates as high as 3.0 x 105 / s are achieved. The
peak inelastic strain rate drops to ~ 1.0 x10 / s before the maximum spreading. Between 70pfs
and 120pfs, the peak plastic strain rate decreases substantially due to change in loading direction
(loading to unloading). The peak plastic strain rate was observed not to diminish significantly
even during unloading. Dramatic plastic strain rates are still sustained at 2.0 x104 / s
-
8.Ox104 / s ,
resulting in significant plastic back-flows during unloading, which lead to a substantial shape
recovery in conjunction with the elastic resilience in the material.
135
Plastic Strain Rate I"4]
1.0 x 10,
t
I .5 ps
5.X0z
102
0.0
.Ox
.0X
0
10, t
3.3
1W
*
.
t
t =67.8ps
6.0 X 104
5.0 x102
5. x 1W0
0.0
3.0 X 105 t
=6.6 ps
1.5 X10W t = 9.9 ps
S0 X1
0.0
OX10
0'0.0
122ps
1.0 X IGS
0.0
ps
t =165ps
.0
0.0
10
4.0
X102
2
4
X 10
5.0 X 10
0.0
1.5 x 105 t
=14.lip
5.0 x102
10.0
t
2.0 X 2W,
0.0
X
2.0 x 1W
OX
0.0
1O9.
=29.1
102
t=220p
19ops
t
separation
2.OXW
0o.0
Figure 6-7 Contours of inelastic strain rate in hard intermolecular component at various stages in
the Taylor impact test with [UD ~ 4/3 and V
-
245 m/s
Figure 6-8 shows the evolution of elastic shear modulus in the rod during deformation. In
the prior study on the constitutive modeling of the stress-strain behavior of polyurea, stretchinduced softening resulting from the microstructural breakdown in hard-phase networks was
found to provide a significant source of energy dissipation in addition to viscous dissipation
during deformation. A substantial stretch-induced softening during loading leads to a large
amount of hysteresis, which was found to be due to irreversible change in microstructures via insitu X-ray scattering observation. 15 The microstructural rearrangement may lead to a significant
change in internal energy storage mechanisms during deformation, where the stretch-induced
softening results in decrease in the internal energy storage capability of the material upon the
high strain rate loading. Additionally, as shown in Figure 6-8, the elastic shear modulus drops
from 10.5 MPa to a steady state value of 3.4 MPa during impact loading while there is negligible
stretch-induced softening during unloading.
136
Shear Modulus [MPaJ
t-1.5ps
10510.5
1.
=6.6ps
t -3.3ps
t0.
F7.8
t=67ips
t=122ps
=9.9
5.3
.5
7.06A
10.
105A
0.
10.
t
-14.
5.0
t=19Ops
t=165pa
t
t0. 29.1ps
s
.
t=220ps
separation
7.87.
Figure 6-8 Contours of elastic shear modulus in hard network component at various stages in the
Taylor impact test with LD
-
4/3 and V - 245 m/s
Figure 6-9 shows results for the rod of UD = 2 at a velocity of ~ 245 m/s. As is observed
in the experimental images, a mushroom head first forms; the maximum spreading occurs at the
impact surface; and the rod displays a significant shape recovery during unloading and
separation. In comparison to the test on the rod of UD = 4/3 at the same velocity, the event
duration has shortened. The successive photographic deformation profiles are again compared
with numerical simulations. The finite element simulation was found to well capture the
localized deformation profiles during loading and unloading as evidenced by comparison to the
digitized deformation profiles, supporting the predictive capability of the copolymeric
constitutive model.
137
Experiment
LOADING
Simulation
0 ps
16 gs
32 ps
48 ps
64 ps
80 ps
Experiment
UNLOADING
Simulation
96 s
112 ps
128 s 144 ps
160 ps 176 ps
Figure 6-9 Deformation profiles during a Taylor impact test with LJD
200 Ps 232 ps
-
2 and V - 245 m/s:
high-speed photographs (red dots: digitized deformation profiles) and deformation profile
prediction made using an axisymmetric finite element simulation (a) loading (b) unloading
Figure 6-10 shows overall evolution of selected geometric dimensions during deformation
in experiment and simulation. Under the same impact velocity (-245 m/s) as the previous test,
the lateral maximum spreading at the impact surface was found to increase (-7.5 %) for the
greater aspect ratio (L/D = 2) while the time at maximum spreading and the separation time
decrease. The dissipated energy is quantified by kinetic energy evolution in Figure 6-10b.
Though the nominal strains for the greater aspect ratio (L/D = 2) increased with the shortened
event duration and the decreased transient time to the nonlinear damped oscillation, the level of
energy dissipation (-75 %) was found to decrease by comparison to the energy dissipation
(~
81 %) for the lower aspect ratio (L/D = 4/3).
138
(b)
(a)
E
e
2.62.0-
ae
Dissipated energy
1.0
3.0
0)
exp, D(t)
exp, L(t)
* model, D(t)
0 model, L(t)
~
0.8
.
LU
I
0.6
1.5
0.4
1.0
EB
8B
0.5
0.0
03
*
-
0
40
-n
A
DMB
E
120
160
0.2
BE
M
80
200
240
Time [Ips]
0.0
0
100
200
300
400
Time [ps]
Figure 6-10 Evolution of selected geometric dimensions and kinetic energy in the Taylor impact
test with LD ~ 2 and V - 245 m/s (a) evolution of normalized length and diameter at the impact
surface in experiment and simulation (b) evolution of normalized kinetic energy with selected
deformed profiles under loading and unloading
Figure 6-11 shows that the model predictions of fully recovered shapes match well the
experiments, exhibiting a significant resilience in the rods of L/D = 4/3, 2, and 4 under a velocity
245 m/s. As shown in the recovered rods in experiments and simulations, a perfect shape
recovery upon impact loading-unloading was achieved within a very short timescale, which
of
-
resulted from the elastic resilience as well as the mutual balancing between intermolecular and
network resistances. Additionally, the contours of elastic shear modulus at 0.75 ms are presented
to quantify the terminal level of stretch-induced softening in the rods. As shown in the contours,
the softened region was found to be greatest for the rod of ID = 4/3, due to the greater extent of
deformation. Figure 6-12 shows the evolution of normalized kinetic energy in the rods of IAD =
4/3 and 4. As is observed, the level of energy dissipation was found to decrease as the aspect
ratio increases under the same impact velocity. This is consistent with the decrease in stretchinduced softening for the greater aspect ratio presented in Figure 6-11, which corresponds to the
decreased hysteresis. Additionally, the total time for event duration of impact loading and
unloading was found to decrease in the polyurea rod of the greater aspect ratio as evidenced in
the phase shift to the left in time.
139
(a)
(b)
(W
Shear Modulus [MPs]
9.8
7.6
3A
3.4
3.4
Figure 6-11 Comparison of simulated 270' axisymmetric sweep rods and contours of elastic
shear modulus at t = 0.75 ms with recovered rods for tests in (a) UD
L/D - 4 (more softening achieved in the rod of L/D
-
-
4/3 (b) IAD
-
2 and (c)
4/3)
1.0
UD = 4/3
0.8
0)
0h..
w
-UD=4-
0.6
U
0
0.4
0.2
0.0
-
0
-
100
200
300
400
Time [ps]
Figure 6-12 Evolution of normalized kinetic energy in the Taylor impact tests with rods of L/D
= 4/3and 4 at a velocity of - 245 m/s
140
6.4.
Taylor Impact Behavior of "Model" Rubbery and Glassy Polymers
The segmented copolymer polyurea exhibits a mechanical response with both the "glassylike" and "rubbery-like" polymeric aspects due to the two-phase morphology. The constitutive
behavior of polyurea possesses hybrid properties travelling between glassy and rubbery
responses. A schematic of the stress-strain behavior of polyurea under deformation is presented
in Figure 6-13, which was built based on a combination of "dissipative" elastic-plastic glassy
behavior and "resilient" elastic rubbery behavior. The role of energy dissipation and shape
recovery mechanisms in the segmented copolymers under extreme deformation is now discussed
by comparison to the behavior of purely hyperelastic (rubbery) and purely viscoplastic (glassy)
materials. The purely hyperelastic rubbery behavior is modeled by taking only the purely
hyperelastic components of the constitutive model while the viscoplastic glassy behavior is
modeled by taking only the elastic-plastic intermolecular components as shown in Figure 6-14.
Figure 6-14 shows the uniaxial stress-strain behavior under high strain rate compression (1000 /s)
for the three material cases: the glassy constitutive model (highly dissipative, Figure 6-14a), the
copolymeric polyurea constitutive model (both dissipative and resilient, Figure 6-14b), and the
rubbery constitutive model (highly resilient, Figure 6-14c). The "model" glassy behavior is
similar to that in other glassy polymers such as PMMA and PET/PETG under large deformation
up to true strains of 0.8 - 1.2.2,16 The Taylor impact testing of a polyurea rod of I/D = 2 with a
flyer plate velocity of 245 m/s was simulated for the copolymer, the glassy polymer and the
rubbery polymer. Figure 6-15 shows overall shape evolution of the three simulated cases in
comparison to the experimental data of the polyurea rod. As expected, the viscoplastic glassy
constitutive behavior leads to a substantial residual deformation with a very modest level of
shape recovery while the hyperelastic rubbery constitutive behavior exhibits a substantial shape
recovery without an asymmetric evolution of length and diameter between loading-unloading,
which was clearly observed in the copolymeric behavior.
141
(a)
(b)
A
IA
U)
U)
copolymeric
a,
I.
4'
I
rubbery
U)
A;-
+
glassy
strain
strain
Figure 6-13 Schematic for constitutive behavior in polymeric materials (a) stress-strain in
dissipative yet resilient copolymers (b) stress-strain in "elastic-plastic" glassy polymers and
"hyperelastic" rubbery polymers
Resilient
(a)
10o
(b)
. .
- - - Glassy py
4.
80.
40,
model
.
..... 0.
!20-
0 21
10-
0.25
0.75
0.50
True Strain
1.00
[;;.Rubbey
30
30
*
e4'
5.00
Copolymerio PU mod
80
Io rmodel
100.0.0
0. 60.
9)30-
0.25
0.50
True Strain
0.75
1.00
.00
0.25
0.50
0.75
1.00
True Strain
Dissipative
Figure 6-14 Stress-strain behavior under uniaxial compression at a strain rate of 1000 /s (a)
glassy constitutive model with hard/soft intermolecular components (b) original copolymeric
polyurea constitutive model (c) rubbery constitutive model with hyperelastic components
Simulated results of deformation profiles for the polyurea copolymer, the glassy polymer
and the rubbery polymer are shown in Figure 6-16. The glassy polymer (Figure 6-16b) displays
a dramatically increased lateral spreading in comparison to that in the polyurea copolymer
142
(Figure 6-16a), which results from a significant inelastic flow under impact. On the other hand,
the rubbery polymer (Figure 6-16c) displays a decreased lateral spreading due to extremely
rapid wave propagation from the impact surface, followed by a perfect elastic shape recovery.
The overall time to separation was found to significantly decrease in the hyperelastic behavior
while the glassy behavior exhibits a retarded separation with a substantial residual deformation.
The level of energy dissipation is quantified in the evolution of kinetic energy during
deformation for the three constitutive laws in Figure 6-17a. The kinetic energy in the purely
hyperelastic rubbery behavior is substantially recovered whereas it is entirely dissipated in the
viscoplastic glassy behavior. A nonlinear free oscillation with large amplitudes is observed due
to the hyperelasticity and the ongoing wave reflection back and forth between the front and rear
end. In contrast, a nonlinear damped oscillation after reaching zero kinetic energy is observed in
the glassy behavior, where the damped amplitudes are much smaller than those in the
copolymeric constitutive behavior. In the copolymeric constitutive law, not only can a large
amount of energy dissipation be achieved from a variety of mechanisms comprising viscoelasticviscoplastic flows and stretch-induced softening, but also multiple elastic energy storage
mechanisms in the intermolecular and network resistance in the hard and soft domains enable a
substantial shape recovery taking advantages from both "glassy" and "rubbery" constituents.
(b)
(a)
4
0
3-
0 000000
D(t), copolymer
D(t),
D(t), glaSSY
gs
2- 0
1
17
D(t), rubbery 0
0
rubbry 000L(t),
-
0
01
0
60
"
9
M
t)
1 5
150
100
g
L(t), exp
1B SMga L(t), c polymer
200
260
Time [ps]
Figure 6-15 Evolution of selected geometric dimensions in Taylor impact tests with LID - 2 and
V ~ 245 m/s (a) evolution of normalized length and diameter in copolymeric, glassy, and rubbery
143
constitutive model (b) simulated deformation profiles at maximum spreading in copolymeric,
glassy, and rubbery model
The nonlinear hyperelastic oscillation is further examined for the geometric features of
purely hyperelastic rubbery polymers. Figure 6-18a shows the evolution of kinetic energy and
deformation profile in the rod of L/D = 4 (with the same diameter of 12.7 mm as L/D = 2 case in
Figure 6-17) at impact velocities of 245 m/s and 330 m/s. By comparison to the hyperelastic rod
with L/D = 2 (Figure 6-16c and Figure 6-17b), the event duration was found to dramatically
increase with the larger characteristic time to the steady state oscillation while the amplitude of
kinetic energy oscillation substantially decreases in the hyperelastic rod of L/D = 4. Though
there is no significant difference in the level of kinetic energy recovery for the rods of L/D = 2
and 4, the higher oscillation modes were observed in the rod of L/D = 4, due to the more active
interactions between hyperelastic waves in the body. However, for the impact velocity of 330
m/s, the level of kinetic energy recovery was found to remarkably decrease, exhibiting the larger
elastic energy stored during deformation due to the higher modes of oscillations (b-b' and d-d' in
Figure 6-18a). These higher modes of oscillations sustained between the local minima (a and c
in Figure 6-18a) in kinetic energy evolution, which results in the larger elastic energy storage
through the deformed body. These observations suggest that the "model" hyperelastic rubbery
polymer is capable of storing the elastic energy via a wide variety of deformation modes under
high speed impact.
144
(a)
Polyurea
Copolymer
s
32 ps
64 ps
96
ps
128 ps
0 ps
45 ps
90 Is
135 ps
180 ps
0 ps
20
0
200 s
160 ps
23
0 ps
(b)
Glassy
Polymer
225 ps
270 ps
315 s
120 s
140 s
(c)
Hyperelastic
Rubber
40 s
s
6
0 ps
80 ps
100Lps
Figure 6-16 Evolution of simulated deformation profiles in Taylor impact tests with IJD - 2 and
V - 245 m/s (a) copolymeric constitutive model (repeated), (b) viscoplastic glassy constitutive
model, and (c) hyperelastic rubbery constitutive model
(b)
(a)
1.0
b
0.8
0M
wi
f
f\.
I
........
A
0.6L
I
-
-~
0.4
.Rubbery
'I
0.2
0.0
0J
a
7%
100
:b
c
d
a
f
9
-~-
200
300
400
Time [ps]
Figure 6-17 Evolution of kinetic energy and deformation profile in Taylor impact tests for a
variety of "model" polymers (a) evolution of normalized kinetic energy in copolymeric, glassy
and rubbery constitutive model (IJD ~ 2, V
-
245 m/s) (b) deformation profiles of the "model"
hyperelastic rubbery polymer at local minima and maxima of kinetic energy
145
(a)
(b)
1.0
b
d
>. 0.8
Figure
100
W'
c I
b
C
b
0i.6 -
d
200300o400500
d
.....
dP.....
(c)
b
b'
C
d
d'
Fiue6-18 Evolution of kinetic energy and deformation profile in Taylor impact tests for the
"model" hyperelastic rubbery polymer (L/D ~4) (a) evolution of normalized kinetic energy at
velocities of 245 m/s and 330 m/s (b) deformation profiles at 245 m/s (c) deformation profiles at
330 m/s
ata
engy
o mattet
oh
temperature
okeis
fomuae
pofil
Deformation
due to Inelastic
Heating rom
and thdefosration
Temperature Rise
6.5.
EffectEvolution
of Adiabatic
Adiabatic heating during deformation may lead to a substantial thermal softening in the
response of glassy and rubbery polymers. To quantify the adiabatic heating incurred during an
extreme straining, thermomechanical effects have been analyzed. A governing field equation for
where Ois the temperature,
is the specific heat per a unit reference volume, qis the heat flux,
and 4 is the rate of heat generation per a unit reference volume. The heat flux is determined by
the Fourier's law,
q= -K.-V0 6,
(2)
146
where K is the thermal conductivity tensor. Here, the thermal transport is assumed to be
isotropic and the conductivity tensor is hence reduced to a scalar thermal conductivity, kI .
The heat dissipated due to the inelastic straining is therefore determined via,
q ='81 T': fy
=p
|T'l
a
iJ
(3)
where T,' is thea deviatoric stress, fnga is the rate of inelastic stretching, nP is the magnitude of
inelastic straining of a component in the micro-rheological constitutive model, and 6 is the
converting ratio of inelastic dissipation to heat generation. The adiabatic heating due to inelastic
flows is then determined via ao= #
at
rp I
C
-Ta. Here the spatial
events is neglected since the thermal diffusion length (
fo,,fu,,on
thermal transport during the
) is much shorter than the
characteristic length ( fe,,,,,, ) of the rod.
The thermal diffusion length is estimated via,
e
diffusion
-ITA - t
(4)
where At is the characteristic time of the event (At - o[102ps]) ; and > is the thermal diffusivity
of polyurea which is estimated via,
0.02[
A==
C
]
~1x10-[M2 /s]
smK
1.997x10 6 [
M3
]
K
(5)
Here the thermal conductivity and the specific heat were taken from Lee at al. 17 and Amirkhizi
et al. 18 respectively. The characteristic length
magnitude as the length of rods
(le
,
geonet,
is assumed to have the same orders of
o10-2M]) which are subject to impact. The thermal diffusion
length is hence found to be much less than the characteristic length by four orders of magnitude
as follows,
147
e
~ffusion
O[TXj4l <<I.
geo netry
(6)
(1X10-4]
In addition, work during inelastic straining is assumed to convert entirely into heat
generation (#8=1). Temperature is hence updated at every material point simultaneously with the
constitutive model and the viscoelastic-viscoplastic kinematics during finite element procedures.
Figure 6-19 shows the evolution of temperature contours in a polyurea rod of L/D ~ 4/3 under
an impact velocity of 245 m/s. The level of temperature rise was relatively low during
deformation. The average level of temperature rise through the body was found to be 15 K. As
shown in the DMA curves in Figure 1, there is no significant change in the storage modulus in a
temperature range of 298 K and 315 K, which results in a "negligible" change in elastic moduli
of the material. The effect of temperature rise due to inelastic straining was further investigated
in uniaxial tests. Temperature increases due to inelastic straining via Equation (A23) during
deformation under an adiabatic condition. The thermally-activated viscoplasticity was also
19 2 0
captured by a Ree-Eyring viscoplastic flow model. '
) =f'exp -
AGa
in
si
AGa -r
k
is the reference viscoplastic rate, A G , is the activation free energy against inelastic
where ro
flow, k, is Boltzmann's constant, 6 is the absolute temperature, 3a is the athermal shear strength,
and
a
=
2
T
aa
is a magnitude of the deviatoric stress tensor. Temperature rise during the
adiabatic, uniaxial tests was found to be under -7 K at high strain rates up to 104 /s, which leads
to negligible thermal softening in the post-yield regime as evidenced by comparison to stressstrain curves under an isothermal condition at room temperature. In addition, temperature rise
was found to be under ~ 15 K, in particular, at an upper limit of dissipation, where all of the
dissipative sources involving the inelastic flows as well as the stretch-induced softening is
assumed to entirely convert into heat generation. This also supports that temperature rise due to
adiabatic heating does not significantly affect the overall mechanical response of the material
under large deformation at high strain rates.
148
Temperature [K]
303
320
316
298
298
298
45ps
lops
95Ps
348
32S
298
180ps
275ps
Figure 6-19 Contours of temperature evolution at various stages in the Taylor impact test with
L/D
6.6.
-
4/3 and V
-
245 m/s
Taylor Impact Behavior of "Model" Linear Viscoelastic Polyurea
Here we investigate the Taylor impact behavior of polyurea rods with a linear
viscoelastic constitutive model by comparison to the viscoelastic-viscoplastic constitutive model
proposed in this study. Over the past decade, a number of numerical studies have been performed
to examine the high strain rate behavior of elastomeric copolymer polyureas using linear
viscoelastic constitutive modelsis,21-23 which employed a Prony series representation for a wide
range of viscoelastic relaxation processes as expressed in Equation (8).
G(t) =G 1+
pi exp
J,
(8)
where, G(t) is the time dependent elastic shear modulus; G. is the elastic shear modulus at
steady state (t -+>oo); p, is the weighting factor for each relaxation process; and i is the time
constant for each relaxation process. Detailed information on the viscoelastic constitutive model
of polyurea can be found in Amirkhizi et al.18
Figure 6-20 shows the model predictions of stress-strain behavior of polyurea at low and
high strain rate using the linear viscoelastic constitutive model. Here, the Prony series
representation was used in conjunction with a simple hyperelastic model. By comparison to those
in our constitutive model discussed in Chapter 3, the model predictions were found to be
capable of capturing the stress-strain behavior up to only a moderate strain and strain rate.
149
Additionally, the stress response at high strain rate was highly overestimated in the viscoelastic
model without any stress-rollover behavior; the hysteresis under cyclic compression testing was
also much less than those found in experiments and our constitutive models as described in
Chapter 3. Though the viscoelastic model used a Prony series of "five" distinct time constants,
the transition in the rate-sensitivity was not captured well.
I
- I
125
-Viscoelastic
-Viscoelastic
100
I
I
I
model, 1000 Is
model, 0.1 Is
CL
7550
25-
,
.0
0.2
0.4
0.6
0.8
1.0
True Strain
Figure 6-20 Stress-strain behavior in viscoelastic constitutive model 8at low to high strain rate
Figure 6-21 shows the simulated deformation profiles under Taylor impact testing (an
impact velocity of 245 m/s and ID
-
4/3) using the viscoelastic constitutive model under an
isothermal condition. By comparison to experiments and numerical simulations using our
viscoelastic-viscoplastic constitutive model (See Figure 6-4), the model predictions did not
capture well the localized and overall deformation under impact. In particular, the level of
maximum spreading at the impact surface was found to be much less than those in experiments
and numerical simulations in our model. Additionally, as shown in in Figure 6-22 , the
separation time after impact loading and unloading was dramatically reduced in the viscoelastic
constitutive model as confirmed in the stress-strain behavior at high strain rates, i.e. the stiffer
behavior at post-yield regime leaded to much less maximum spreading at the impact surface and
consequently resulted in the much shorter separation time.
150
In conclusion, the linear viscoelastic constitutive model was found to be capable of
capturing the mechanical behavior of polyurea copolymers at a relatively narrow range of strains
and strain rates. Though it spanned a wide spectrum of the elastic modulus at short to long
timescales via multiple relaxation processes, the model predictions were found not to capture the
major features of constitutive responses of materials including yield-like stress rollover, stretchinduced softening, and nonlinear unloading that accompanies a substantial amount of hysteresis.
Consequently, the simple viscoelastic model was not able to capture well the resilient yet
dissipative features of materials under a high speed impact, where an extreme deformation and
deformation rates were incurred.
LOADING
0 gs
15 ps
30 ps
45ps
60pgs
UNLOADING
75 ps
90 ps
105 ps
120 ps
135 ps
Figure 6-21 Deformation profiles during a Taylor impact test with IJD - 4/3 and V ~ 245 m/s
using the viscoelastic constitutive model; all of the numerical details are the same as those used
in Figure 4 including finite element mesh, time-step and boundary conditions
151
3.0
@
2W
0
2.54.
exp, D(t)
exp, L(t)
PU model, D(t)
0 PU model, L(t)
.
viscoelastic, D(t)
U vlscolastc, D~t)
2.01.5
0.5
0.0
iP 0 EN
0
50
100
150
200
250
Time [gs]
Figure 6-22 Evolution of selected geometric dimensions in the Taylor impact test with IJD ~ 4/3
and V
6.7.
-
245 m/ in viscoelastic model, PU model (this study) and experiment
Discussion and Future Work
In this chapter, the mechanics of extreme strain rate behavior of elastomeric segmented
copolymers was addressed by examining Taylor impact testing on polyurea rods in experiments
and computational modeling. The two-phase copolymeric constitutive model was found to
predictively capture the dynamic and inhomogeneous deformation processes of the exemplar
copolymer polyurea under extreme strain rate events reaching over 10 5 /s upon high speed impact
loading. The simulated evolution of overall shape and localized deformation profiles was in good
agreement with experiments, revealing the ability of the constitutive model to capture highly
nonlinear behavior during impact together with the substantial energy dissipation as well as the
significant and immediate shape recovery. Additionally, energy dissipation and storage pathways
in glassy and rubbery polymeric constitutive behavior and their combinations were quantified by
modeling the hyperelastic and the viscoplastic constitutive laws taken from the original
copolymeric constitutive components. This investigation offers insight into tailoring the multiple
"dissipation" and "storage" mechanisms as well as the deformation features involving shape
152
recovery and separation time in the segmented copolymeric materials through their multi-phase
morphology which travels between the "glassy" and "rubbery" phases.
Our constitutive modeling framework was found to capture well the features of resilience
and dissipation of polyurea rods including overall and localized deformation profiles and level of
dissipated energy during such extreme and ultrafast deformation. Though overall behavior
observed in experiments was captured nicely in models, there was difference between
experiments and numerical simulations, especially in the unloading stages as shown in the
figures on the deformation profiles. Additionally, the damped behavior after impact unloading
was found to be underestimated in the numerical simulations. Though we have no experimental
data for long time behavior over 300 microseconds after impact unloading, the highly damped
oscillation can be expected with no significant free oscillation of the impacted rods. A more
damped oscillation may be available by a simple modification in the viscous mechanisms in the
constitutive model. The model prediction may be improved with further parametric studies of the
constitutive models. However, the main purpose of present work is to address how we can apply
our constitutive modeling framework constructed based on stress-strain data under simple
deformation conditions for capturing such a complicated deformation process as the Taylor
impact behavior without any major modification of constitutive modeling framework and
material parameters. The constitutive model may lack more detailed information on the
microstructural changes such as localized melting and cavitation due to the extreme loading
conditions by which another critical dissipation source can be supplied. In particular, the cavityinduced fracture and relevant propagation of radial cracks were exclusively observed in polyurea
rods over an impact velocity of 300 ~ 400 m/s as shown in Figure 6-23, which was not covered
in this research. In addition, the effects of interphases of hard and soft segments may become
more significant in the macroscopic mechanical responses against such extreme loading
conditions as ballistic impact and penetration. We may be able to improve the predictive
capability of our models by further studies on the microstructural evolution of hard, soft and
interphase under extreme environments. Furthermore, the extreme rate behavior of PU650
should be addressed under Taylor impact testing in experiments and numerical simulations to
quantify the roles of "mixed" and "inter-phase" at ultrafast events in resilience and dissipation.
As discussed in Chapter 5, PU650 exhibited dramatic stress amplification in conjunction with
enhanced dissipation under deformation in comparison to PU1000 behavior since PU650
153
possesses greater weight fraction of hard phase and hence the degree of phase-separation of hard
and soft components decreases, which leads to "phase-mixing". In future research, the Taylor
impact behavior of PU650 can be investigated using the modified constitutive modeling
framework of PU650 composed of hard, soft and mixed phase mechanisms provided in Chapter
6. The greater "glassy" nature of PU650 will affect the macroscopic features of resilience and
dissipation of materials and the microstructural features including cavitation and cavity-induce
failure.
Figure 6-23 Cavitation and radial crack pattern in polyurea rod under an impact velocity of 450
Finally, the proposed modeling framework on high strain rate behavior of the elastomeric
materials in Chapter 3 and 5 (and Chapter 8 on ethylene-based ionic elastomers) can be further
used for investigations on the elastomer-blast interaction
and the pressure-shear impact
testing 26 which are of great interest for the predictive design of protective polymeric composite
systems. Furthermore, deformation-induced microstructural evolution should be investigated to
quantify changes of internal energy storage mechanisms of the materials at extreme strain rate
conditions.
154
6.8.
Reference
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J. C. F. Millett, N. K. Bourne and G. S. Stevens, International Journal of Impact
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12.
P. J. Rae, E. N. Brown and E. B. Orler, Polymer 48 (2), 598-615 (2007).
13.
W. Mock and W. H. Holt, 2007.
14.
H. Cho, R. G. Rinaldi and M. C. Boyce, Soft Matter 9 (27), 6319-6330 (2013).
15.
R. G. Rinaldi, M. C. Boyce, S. J. Weigand, D. J. Londono and M. W. Guise, Journal of
Polymer Science Part B: Polymer Physics 49 (23), 1660-1671 (2011).
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R. B. Dupaix, Ph.D., Massachusetts Institute of Technology, 2003.
17.
J. Lee, G. Gould and W. Rhine, Journal of Sol-Gel Science and Technology 49 (2), 209220 (2009).
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A. V. Amirkhizi, J. Isaacs, J. McGee and S. Nemat-Nasser, Philosophical Magazine 86
(36), 5847-5866 (2006).
19.
A. S. Argon, Philosophical Magazine 28 (4), 839-865 (1973).
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M. C. Boyce, D. M. Parks and A. S. Argon, Mechanics of Materials 7 (1), 15-33 (1988).
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M. R. Amini, J. Simon and S. Nemat-Nasser, Mechanics of Materials 42 (6), 615-627
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G. Chevellard, K. Ravi-Chandar and K. Liechti, Mechanics of Time-Dependent Materials,
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W. Mock Jr, (Naval Surface Warfare Center, Dahlgren Division, 2012).
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156
157
Chapter 7
Resilience, Dissipation and Shape Recovery of
Elastomeric Segmented Copolymers under
Microindentation
Portions of this chapter will be submitted to a journal paper, H. Cho and M. C. Boyce,
"Resilience, Dissipation and Shape Recovery of Elastomeric Segmented Copolymers under
Microindentation",in preparation,2013
7.1.
Introduction
Resilience and dissipation have been attractive features for a myriad of applications
involving highly recoverable and protective elastomeric architectures. Elastic resilience often
provides a moderate shape recovery in materials upon unloading, by which stored deformation
energy is exerted without any loss. Inelastic power due to viscous flows has been found to be
dissipated via multiple pathways producing heating of the material in high rate conditions when
there is insufficient time for heat transfer process. In addition, viscoelastic and viscoplastic flows
usually lead to a residual deformation upon unloading in tandem with energy dissipation.
However, in a variety of natural and synthetic elastomers, the energetic nature of resilience and
dissipation was found to often result in a remarkable shape recovery upon unloading. In this class
of materials, a substantial shape recovery can be achieved beyond elastic resilience without any
further physical treatment. The presence of multiple resistances comprising intermolecular and
network mechanisms of elastomeric copolymers was found to be essential to address the
underlying mechanisms of shape recovery behavior, where a mutual balancing between the
micro-rheological components provides a major driving force for the substantial shape recovery
beyond elastic resilience during unloading. The elastically- and inelastically-driven recovery
158
behavior is hence of central importance for a better understanding of the physically-sound shape
memory' and self-healing mechanisms 2 of elastomeric materials, which are recently utilized for
material architectures under severe mechanical environments such as localized fracture and
damage. In particular, the mechanical behavior of elastomeric copolymers at small scales is of
central importance in terms of a combination of resilience, dissipation and shape recovery to
enable a robust design of soft material architectures in bio-inspired robotics 3-, mechanicallytunable omniphobic surfaces
'
and self-healing microcapsules for controlled drug delivery and
release.8'9 Additionally, a class of polyurethane-urea elastomers with encapsulated isocyanate
was found recently to exhibit dramatic self-healing behavior when damaged.10' "
In situ indentation tests have been extensively employed for a quantitative characterization
of localized deformation mechanisms of natural and engineered materials such as crystalline
metals12, 13, bulk metallic glasses
biological tissues2 1
23
-
' 1,
amorphous polymers 16-1, carbon nanotube forests
19,20
and
at a wide range of length-scales. Various instrumented micro-indentation
techniques have been developed to investigate the mechanical properties of materials ranging
from adhesion' 7 and elastic-plastic behavior24 to fracture and fatigue 2 5, 2 6 in polymeric materials.
Recent studies on the microindentation behavior of amorphous glassy polymers revealed highly
rate-dependent elastic-plastic features of polycarbonate (PC) and polymethylmethacrylate
(PMMA) at microscales.16, 27 In particular, the time-dependent creep27 and relaxation properties'
6
of amorphous polymeric materials were quantified using an instrumented sharp indenter.
However, at present, a few studies have demonstrated the microindentation behavior of
elastomeric polymers focusing on the characterization and mapping of their time-dependent
resilient properties.28-32 In this chapter the dissipative yet resilient microindentation behavior of
elastomeric segmented copolymers is examined via in situ indentation testing in a combination
of novel modeling, experimentation and computational approaches, by which physically-sound
deformation mechanisms at microscales are quantitatively characterized with a variety of
indentation loading histories. Force-displacement behavior was characterized with an increasing
level of maximum applied force involving cyclic loading conditions, revealing a highly rate- and
deformation-dependent creep flows in the materials. In addition, a temporal evolution of
indented surfaces was monitored via a physical contact to an indentation tip revealing a
substantial shape recovery of the materials under localized inhomogeneous deformation within a
few minutes. Numerical simulations were found to capture well load-displacement curves and
159
nonlinear shape recovery localized under indentation, revealing a three-dimensional predictive
capability of proposed models under complicated loading scenarios composed of elastic-plastic
loading, creeping, unloading, and creep-assisted recovery at small length scales.
Keyword: polyurea 1000 (PU1000), polyurea 650 (PU650), resilience, dissipation, shape
recovery, microindentation, creep, finite element simulation
7.2.
Shape Recovery Mechanism in Elastomeric Materials
In many polymeric materials, the highly nonlinear elastic behavior can provide a
significant level of elastic shape recovery upon unloading. Beyond the elastic resilience, a
marked shape recovery of samples has been extensively reported in cyclic deformation tests of
elastomeric copolymers. 33-35 In Figure 3-5 in Chapter 3, we observed a perfect shape recovery
of the polyurea samples under cyclic compression in both experiment and numerical simulation.
Here, we further demonstrate a simple yet intuitive physical mechanism of inelastically-driven
shape recovery of elastomeric copolymers that exhibit a substantial residual strain under cyclic
deformation.
Figure 7-1 schematically shows a shape recovery process under cyclic compression. A
remarkable shape recovery is achieved in a few minute after unloading as illustrated in Figure
7-1b. The shape recovery mechanism can be explained by a total stress (Figure 7-1c) breakdown
into intermolecular (Figure 7-1d) and network (Figure 7-ie) stress component; though the total
stress is zero at the end of unloading, an individual stress component of the intermolecular
mechanism has a "finite" magnitude with an opposite sign to the network stress component; the
inelasticity in the intermolecular resistance hence drives additional flow which leads to shape
recovery. During recovery, the total stress is maintained in zero with a mutual-balancing between
intermolecular and network resistance as shown in and Figure 7-1f. Shape recovery was also
found to be strongly dependent upon an imposed deformation.
160
(b)
(a)
(c)
___________
__________
artota
~I
+ON
loading
IE f
FA
UN-.M
At
Loading
6
VI
unloading
Recovery
Unloading
Time
(f)
(e)
(d)
0NC
Strain
_
W
0.
C
W
IA
recovery
E
-
08
Strain
Strain
Time
Figure 7-1 Shape recovery mechanism in elastomeric materials: (a) schematics of shape
recovery under cyclic compression; (b) residual strain-time curve during recovery; (c) total
stress-strain curve; (d) intermolecular stress component during loading-unloading; (e) network
stress component during loading-unloading; (f) individual stress component-time curves during
recovery
Figure 7-2 shows simulated shape recovery in order of increasing imposed strains of 0.1
and 0.5. There was no significant recovery after unloading at an imposed strain of 0.1 since
viscoplastic flow due to yield was not activated under the small deformation. This observation
was supported by a stress breakdown into the intermolecular and network component during
loading, unloading and zero-force creep. As found in Figure 7-2b, at the end of unloading, the
intermolecular stress was found to be very small and insufficient for the inelastic flow while
balanced the network stress. Figure 7-2c shows shape recovery at an imposed strain of 0.5,
where viscoplastic flow was fully activated, resulting in substantial shape recovery after
unloading, where a rapid shape recovery was achieved in the initial transient regime, followed by
a steady-state behavior. As seen in Figure 7-2d, a magnitude of intermolecular stress was found
to be much greater than that found at an imposed strain of 0.1. This observation supports that the
viscoplastic flow in the intermolecular mechanism is of central importance for shape recovery as
illustrated such that the intermolecular resistance provides a driving force for inelastic shape
161
recovery; the level of shape recovery was highly enhanced beyond the stress-rollover regime at
which the viscoplastic flow is accommodated in addition to the linear viscoelastic flow.
Additionally, the activation free energy against viscoplasticity in the model was found to
significantly affect a shape recovery supporting it was mainly driven via viscoplastic flows when
beyond the initial linear viscoelastic regime.
(a)
(b)
0.
-
1
a 4-5
0.0 8
= 0.0 6
C0
3
0
0.0 4
True Strain
E
0
0.5-
C.
0.0 2
(c)
200
300
Time [s]
400
500
I1
100
10
(d)_
~.1
0.6
N
01
U)
100
2
200
400
300
Time [s]
-
*
500
-
1.
0.3
4
5 3
U0.2
0
True Strain
I-
EO-I
0.1
1)
500
1000
Time [s]
1500
2000
II'~
I I
I I
0
ON
500
1000
Time [s]
1500
2000
Figure 7-2 Shape recovery at increasing imposed strains of 0.1 and 0.5: (a) Strain vs time at an
imposed strain of 0.1 (inset: total stress-strain behavior); (b) Individual stress component vs time
in intermolecular and network mechanism at an imposed strain of 0.1; (c) Strain vs time at an
imposed strain of 0.5 (inset: total stress-strain behavior); (d) Intermolecular/network stress
component vs time at an imposed strain of 0.5; (black lines: loading-unloading, red lines:
recovery after unloading)
7.3.
Microindentation Behavior of Elastomeric Copolymer Polyureas
162
Here, we demonstrate in situ micro-indentation testing of the polyurea copolymers
(PU1000 and PU650) in experiments and numerical simulations involving complicated loading
scenarios comprising elastic-plastic loading-creep-unloading and creep-assisted recovery. In
particular, the shape recovery mechanisms of PU1000 and PU650 are elucidated via elastic
resilience and inelastically-driven recovery under micro-indentation. In addition, a threedimensional capability of proposed constitutive models is evaluated by the nonlinear finite
element simulations of PUlOOG and PU650 samples under inhomogeneous,
localized
deformation. This investigation provides a physical insight into a precursor of shape memory and
healing processes, which have been extensively reported in other elastomeric copolymers
including thermoplastic polyurethanes' 36 and ethylene-based ionic elastomers2,
37, 38,
whose
mechanical properties are very similar to those in the polyurea copolymers. (See Chapter 3, 5
and 8)
7.3.1. In situ Micro-Indentation Test
We employed a microindenter (Hysitron Triboindenter, Nanomechanical Laboratory,
Massachusetts Institute of Technology) which is instrumented via an electronic force control for
desired load functions. A spherical diamond tip (radius: -10.25 pm ) was used for indentation
procedures, allowing for axisymmetric computations of each experimental set. We designed load
functions comprising loading, creep at a constant peak force, unloading, and creep at a (nearly)
zero force with a force control mode. By maintaining a continuous contact between the tip and
the samples, a displacement at the contact point was recorded to quantify a time evolution of
recovery of the indented surfaces. Figure 7-3 shows an exemplar load-displacement curve of
polyurea under microindentation with a load function (inset of Figure 7-3a). As found in Figure
7-3a, the load-displacement curve is characterized in highly nonlinear elastic-inelastic loading,
viscoplastic creep at a constant force, and nonlinear elastic unloading behavior involved to the
resilience of materials. After the unloading, the zero-force creep leads to a substantial shape
recovery (red symbols) of indented surfaces as shown in Figure 7-3a and Figure 7-3b, where
shape recovery exhibited a transient behavior followed by a steady state recovery. In the
163
following section, the microindentation behavior of PU650 and PU1000 is examined under a
variety of load functions in experiments and numerical simulations.
(b)
* loading
creep
*
0
0
tim.
0
U.
~
gO
f-0
S
0
0
E
unloading
recovery
0
S
0
0
0
S
*
S
S
S
S
0
S.
Displacement
Time
Figure 7-3 Schematics of microindentation testing: (a) load-displacement curve with an inset of
load function composed of loading, creep, unloading and zero-force creep (a zero force
maintained to monitor shape recovery after unloading); (b) displacement-time curve
7.3.2. Load-Displacement Behavior under Microindentation
Figure 7-4 shows force-displacement curves for PU650 samples in order of increasing
peak forces (Figure 7-4d: displacement-time curves). Over the three loading scenarios, the
force-displacement behavior exhibited nonlinear loading and unloading with a substantial elastic
recovery as well as a moderate residual deformation. In particular, a creep at a constant peak
force was observed between loading and unloading as seen in Figure 7-4a, b and c; here the
holding period at creep was taken to be 10 s with a constant peak force; the magnitude of
accumulated creep flows at the peak force increased as the peak force increased in experiments
and numerical simulations. Dissipated work was found to dramatically increase revealing greater
viscous flows with the increased peak force as seen in Figure 7-14. Finite element simulations
captured well the main features of force-displacement behavior and displacement-time curves
164
including the time-dependent creep and the highly nonlinear loading-unloading revealing threedimensional predictive capabilities of the model.
(a)
(b)
0.004
u.uuq
0.003
---
experiment
-
simulation
0.003[
0 0.002
0.002
0
LL
I
U
0.001
2SmN
mN
'
I
I
0.001
0.0000
0.002
s30s
10s
0.003
f
,'I
0.004
0O00
0.001
Displacement [mm]
t
I
0.002
0.003
0.004
Displacement [mm]
(d)
(c)
0.003
0
E
0.002
0
2L
J.-
, ,
'
5-
8
0.0025 F
a
0.001
tn
00
0.001
0.002
0.003
Displacement [mm]
I
0.004
u.uuw~.
0
5
10
15
20
25
30
Time [s]
Figure 7-4 Microindentation behavior of PU650 in experiment and numerical simulation: (a-c)
Force-displacement curves at increasing peak forces of 1.5, 2.5 and 3.5 mN (inset: load functions
of loading, creep and unloading); (d) Displace-time curves (dashed line: experiment; solid line:
simulation)
Spatial strain field in the loading direction at each stage of load functions is provided for
the three cases in Figure 7-5, Figure 7-6 and Figure 7-7. The maximum axial strain was found
to be over ~ 0.32 at a peak force of 3.5 mN (0.18 and 0.25 for 1.5mN and 2.5mN, respectively),
where deformation under indentation was located in inelastic regime beyond yield-like stressrollover. As found in the strain field, there was a remarkable elastic recovery over the spherically
deformed region underneath the indenter with a residual strain at the end of unloading. Figure
7-8 shows the magnitude of inelastic strain rates at the end of unloading. As expected, a finite
165
magnitude of inelastic flow in the elastic-plastic mechanism is still sustained with no additional
force input after unloading. Consequently, the inelastic flow lead to additional creep-assisted
shape recovery as detailed in the next section.
t=30sec
t=20sec
t=10sec
0.0
0.0
I-0.17
)
0.0
I-0.07
I-0.18
Figure 7-5 Contours of axial strain field of PU650 under microindentation at a peak force of 1.5
mN: loading, creep and unloading
/I
0.0
t=30sec
t=20sec
t=10sec
0.0
0.0
I-0.22 II01
E.
125
Figure 7-6 Contours of axial strain field of PU650 under microindentation at a peak force of 2.5
mN: loading, creep and unloading
0.0
I0.28
t=30sec
t=20sec
t=10sec
0.0
11-0.32
0.0
E-0.13
166
Figure 7-7 Contours of axial strain field of PU650 under microindentation at a peak force of 3.5
mN: loading, creep and unloading
(a)
f.=1.5mN
(
2.5mN
(C)
f=3.5mN
0.002
0.004
0.009
0.0
0.0
0.0
Figure 7-8 Contours of inelastic strain rate of PU650 under microindentation at the end of
unloading (t=30s) in order of increasing peak forces: (a) 1.5 inN; (b) 2.5 mN; (c) 3.5 mN
Load-displacement behavior under three different loading histories was additionally
examined for PU1000 in experiments and numerical simulations as presented in Figure 7-9.
Overall force-displacement of PU1000 in Figure 7-9 was very similar to that found in PU650,
exhibiting nonlinear loading, creep and unloading; as seen in displacement-time curves in Figure
7-9d, creep at a constant peak force (hold period: 10 sec) was found to be strongly dependent
upon the peak force, followed by a marked elastic recovery and a moderate residual
displacement. However, PU1000 exhibited much softer behavior by comparison to PU650 under
the same peak force of 2.5 mN as seen in Figure 7-4c as informed from the constitutive
responses of PU1000 and PU650 in Chapter 3 and Chapter 5. Additionally, a residual
displacement upon unloading was found to dramatically decrease in PU1000 indentation
behavior supporting the greater rubbery nature of PU1000. Figure 7-14 shows dissipated work
under microindentation in PU1000 and PU650 revealing greater dissipation capability in PU650
in order of increasing maximum displacements. The dissipative nature under indentation was
found to be very consistent with that observed in the uniaxial constitutive behavior of PU1000
and PU650, supporting the greater glassy behavior in PU650 due to a higher weight fraction of
hard domain and the presence of phase-mixing.
167
(a) 0.003
0.003
---
experiment
simulation
0.002
0.002-
Z
V
Z
0.001
'1
2.mN
'.
7s20s 30s
00
2. o
0 0.00'1 -
-
-
L0.00
.5
0
0.001 0.002 0.003 0.004 0.006
0
1000 0.001 0.002 0.003 0.004 0.006
Displacement [mm]
Displacement [mm]
0.003
(d)
0.006
E
E.004
0.002-
PU650
,
0f
U- 0.001-
0.00
0.000
,
25m
0000100
0.002 0.003 0.004 0.006
Displacement [mm]
-
0.002-
0
5
10
20
16
lime
26
30
[s]
Figure 7-9 Microindentation behavior of PU1000 in experiment and numerical simulation: (a-c)
Force-displacement curves at increasing peak forces of 1.5, 2.0 and 2.5 mN (inset: load functions
of loading, creep and unloading); (d) Displace-time curves (dashed line: experiment; solid line:
simulation)
Contours of axial strain field in PUlIOO under microindentation are presented in Figure
7-10, Figure 7-11 and Figure 7-12. Axial strain (max: 0.39) under microindentation at a peak
force of 2.5 mN (Figure 7-12) was found to be much greater than that (max: 0.25) observed in
PU650 indentation under the same peak force supporting substantially softer features in PU1000
under indentation. Figure 7-13 shows the magnitude of inelastic strain rates at the end of
unloading. The inelastic flow rate in the intermolecular component still sustained with no
additional force input after unloading as seen in the contours. Through the time-dependent loaddisplacement behavior of PU650 and PUlO0O under microindentation, the modeling framework
we rationalized in Chapter 3 and 5 were found to have predictive capabilities to capture the
complicated time-dependent, multi-dimensional features under inhomogeneous deformation field
at small scales.
168
I
t=30sec
t=20sec
t=10sec
0.0
0.0
-0.23
-0.26
0.0
I
-0.11
Figure 7-10 Contours of axial strain field of PUOOG under microindentation at a peak force of
1.5 mN: loading, creep and unloading
I
0.0
I
-0.28
t=30sec
t=20sec
t=10sec
0.0
0.0
I
-0.12
-0.32
Figure 7-11 Contours of axial strain field of PU1000 under microindentation at a peak force of
2.0 mN: loading, creep and unloading
I
t=30sec
t=20sec
t=10sec
0.0
0.0
-0.33
-0.38
0.0
I
-0.14
Figure 7-12 Contours of axial strain field of PU1000 under microindentation at a peak force of
2.5 mN: loading, creep and unloading
169
(a)
(b)
f 1.5mNV
0.012
f
(c)
=2.OmN
f.. =2.5mV
0.014
I0
0.017
I0
0.0
I
0.0
0.0
Figure 7-13 Contours of inelastic strain rate of PU1000 under microindentation at the end of
unloading (t=30s) in order of increasing peak forces: (a) 1.5 mN; (b) 2.0 mN; (c) 2.5 mN
0.004
* PU1000, exp
0
0.003
.9
o PU1000, sim
-
Q
PU650, exp
o PU650, sim
: 0.002 0.
(I)
0.001-
5
0.000 0.001 0.002 0.003 0.004 0.005
Displacement [mm]
Figure 7-14 Dissipation in microindentation testing of PU650 and PU1000 at increasing
indentation displacement (See Figure 7-4 and 7-9 for load-displacement curves)
7.3.3. Shape Recovery under Microindentation
As shown in the load-displacement behavior in the previous section, the highly nonlinear
unloading provided a substantial elastic recovery of the indented surfaces. Over the
170
microindentation testing, the indented surfaces were found to be recovered over
-
60% of the
total indentation depth just upon unloading. In this class of materials, the elastic behavior in
network and intermolecular mechanisms provide the marked shape recovery just upon unloading.
Shape recovery further develops beyond unloading without any further physical treatment as we
addressed the inelastically-driven mechanism in Figure 7-1 and Figure 7-2. Here the shape
recovery of indented surfaces was further monitored via a contact mode between the samples and
the indentation tip by maintaining a nearly zero force. Time evolution of displacement of
indented surfaces was recorded to quantify the shape recovery of materials under the localized
deformation.
Figure 7-15 shows shape recovery of a PU650 surface, which was indented to a peak force
of 1.5 mN. As seen in a force-displacement curve in Figure 7-15a, a marked shape recovery was
achieved beyond the elastic unloading; here, a nearly zero force contact mode was maintained
for a few hundreds of seconds as illustrated in a load function in Figure 7-15b. A marked
residual displacement was observed (- 0.7 /m) upon elastic unloading as seen in Figure 7-15c.
Figure 7-15d shows a displacement-time curve from the end of unloading. As seen in the
temporal evolution of the indented surface, a substantial recovery (-
60% of residual
displacement) was achieved during the additional creep mode. Most of shape recovery was
achieved at the initial transient regime; after the initial transient regime, shape recovery was
found to be saturated to a steady state recovery at a long time scale. A numerical simulation was
also presented with experimental data. As seen in Figure 7-15d, the simulated displacement-time
curve well agreed with the experimental data, reasonably predicting the overall behavior of
inelastically-driven shape recover. However, the transient behavior of recovery in experiment
was found to be stiffer than simulated recovery revealing a shorter characteristic time (r ~125 s)
than in the numerical simulation (r- 200s ); i.e. the model is not capturing precisely this initial
stiff response with a very rapid molecular relaxation. The disparity between experiment and
numerical simulation may be improved by employing additional distinct "shorter" viscoelastic
processes in the constitutive model to capture a very rapid relaxation; additionally, an electronic
force drift in the indentation direction may affect the "stiffer" transient behavior in experiments
under a "zero" force mode in the instrumentation of the tester.
171
(b)
(a)
4
3-
exp
simulation
f
1.5mN
2
U_
1
recovery @ff O
,
loading
10s
0
0
recovery
1. 25
A
exp
----- simulation
3
1. 00 C
C
8.
t
30s
(d)
(c)
E
eL
W
ZOs
4
3
2
1
[jm]
Displacement
E
2
simulation
0. 750. 50
I
0
5
10
20
15
Time [s]
25
30
0. 00
0
100
200
Time [s]
300
400
Figure 7-15 Shape recovery of PU650 surface at a peak force of 1.5 mN: (a) load-displacement
curve; (b) load function composed of loading-creep-unloading and "recovery"; (c) timedisplacement curve during loading and unloading; (d) time-displacement curve during recovery
(here, time was zeroed from the end of unloading)
172
(a)
I/JMI
0.0
Unloading
Creep
Loading
0.0
-1.7
3
/I
..
0.0
2.
002
-0.6
-1.91
simulation:
-
3
Displacement [m)
t=10s
t=5Os
t=100s
t=250s
0.0
0.0
0.0
0.0
-0.4
-0.31
-0.26
-0.19
Figure 7-16 Contours of displacement field of PU650 under microindentation at a peak force of
1.5 mN (inset: load-displacement curve); (a) displacement field at loading-creep-unloading; (b)
displacement field during recovery (after unloading)
Figure 7-16 shows the simulated contours of displacement field underneath the indentation tip as
a function of time at each stage of load function composed loading-creep-unloading and recovery
with the corresponding load-displacement curve at a peak force of 1.5 mN. Hemispherical
residual displacement field under the tip was gradually recovered to a steady-state; most of
indented displacement was found to be recovered at the initial regime due to the transient viscous
creep flow as found in experimental data.
173
(b)
(a)
--- exp
3
f
simulation
-
2.5mN
2-
loading,
0
U_
recovery @f 0
1Os
0
2
1
0
30s
t
4
3
Displacement [pm]
recovery
(d)
3
20s
1.5
r-- exp
-- simulation
--
-
exp
simulation
1.0
I~p
E
2
E
0.5
CL
0.
II 0
.
5
10
15 20
Time [s]
25
0.0
30
0
100
200
Time [s]
300
400
Figure 7-17 Shape recovery of PU650 surface at a peak force of 2.5 mN: (a) load-displacement
curve; (b) load function composed of loading-creep-unloading and "recovery"; (c) timedisplacement curve during loading and unloading; (d) time-displacement curve during recovery
(here, time was zeroed from the end of unloading)
The creep-assisted shape recovery was further examined for PU650 surfaces which were
indented to greater peak forces of 2.5 and 3.5 mN as shown in Figure 7-17 (displacement fields
during recovery in Figure 7-18) and Figure 7-19 (displacement fields during recovery in Figure
7-20). The overall recovery behavior in this case was found to be very similar to that observed at
a peak force of 1.5 mN from short to long timescale with a disparity between experiment and
numerical simulation at the transient regime. The "absolute" level of shape recovery was found
to increase as the peak force (or imposed maximum displacement) increases; the shape recovery
was found to be
-
0.4 pn,
-
0.7 phm and
-
0.9 /#n at peak forces of 1.5 mN, 2.5 mN and 3.5
mN, respectively. As seen in Figure 7-8 on the magnitude of inelastic strain rate at the end of
unloading for both cases, the inelastic flow was greater at a peak force of 3.5mN, which resulted
in the greater shape recovery.
174
[un
Loading
0.0
Unloading
IPsi
Creep
Lu'E
0.0
.0
pj
-
2-
-- U
-1.00
-2.9
-2.5
0
.
*i
1
2
3
Displacement [pmn]
(b)
t=250s
t=lO0s
t=50s
t=l0s
0.0
0.0
0.0
0.0
-0.7
-0.5
-0.4
-0.28
Figure 7-18 Contours of displacement field of PU650 under microindentation at a peak force of
2.5 mN (a) displacement field at loading-creep-unloading (inset: load-displacement curve); (b)
displacement field during recovery (after unloading)
In addition to the disparity at the transient regime, the numerical simulations were found to
overestimate the final level of shape recovery by 10 ~ 15% in comparison to experiments over
the three loading cases presented in Figure 7-15
-
Figure 7-20. In the microindentation testing,
the drift effect in the electronic motor in the loading direction (z axis) was found to increase as
the total indentation time increases. After the initial loading-creep-unloading cycle (30 s), we
maintained the zero-force contact mode to monitor the temporal evolution of indented surfaces
via inelastic shape recovery for hundreds of seconds. Due to the increasing drift effect for a long
time indentation, the final level of recovered displacements measured in experiments may have
accumulated errors.
175
(a)
4
*-
3
--
---
(b)
exp
f
fl'ln
simulation
3.5mN
0
2
loading
U.
1
10 s
A
0
20 s
t
30 s
4
3
2
1
[pLm]
Displacement
recovery
recovery @f ~0
i
0
(d)
(c)
1.75
-exP
simulation
1.503
1.25
C
E
8cc
0.
LO
8
C
0
1.00
0
0.75
E
2
%
1
-
---
simulation
.
0.50
-- - -
-
0.25
00
5
10
15
Time [s]
20
25
30
0.00
0
100
200
Time [s]
300
400
Figure 7-19 Shape recovery of PU650 surface at a peak force of 3.5 mN: (a) load-displacement
timecurve; (b) load function composed of loading-creep-unloading and "recovery"; (c)
recovery
displacement curve during loading and unloading; (d) time-displacement curve during
(here, time was zeroed from the end of unloading)
176
(a)
4
Unloading
Creep
Loading
. exp
Simulation
....
3
0.0
.0
0.0
-3.1
-3.7
-1.3
r
E
0
1
2
3
4
Displacement [pm]
(b)
t=10s
0.0
t=50s
0.0
I8.I.
0.8
-0.6
t=1O0s
t=250s
IIEIU
0.0
0.0
-0.5
-0.35
Figure 7-20 Contours of displacement field of PU650 under microindentation at a peak force of
3.5 mN (a) displacement field at loading-creep-unloading (inset: load-displacement curve); (b)
displacement field during recovery (after unloading)
7.4.
Conclusion
Dissipation due to inelastic flow has been considered to be against resilience and shape
recovery in polymeric materials. We demonstrated energy storage and dissipation can lead to
pathways towards highly recoverable elastomeric architectures taking advantages of glassy and
rubbery features in elastomeric copolymers. We examined elastically- and inelastically-driven
shape recovery of elastomeric "segmented" copolymers under localized large deformation via
microindentation testing. Microindentation behavior of the materials was addressed in terms of
load-displacement responses under complicated loading scenarios comprising loading, creep,
unloading and creep (at a zero force) revealing a predictive capability our modeling framework
at micro-scale. Additionally, remarkable shape recovery behavior under microindentation was
rationalized in experiments and numerical simulations in conjunction with a simple physical
mechanism for the creep-assisted shape recovery beyond elastic resilience. The exemplar
polyurea copolymers exhibited a marked shape recovery upon unloading, followed by
inelastically-driven recovery up to -70% of residual deformation. Our observation may support
177
that substantial dissipation can lead to additional shape recovery in tandem with elastic resilience;
i.e. dissipation due to viscoelastic and viscoplastic flows was found to be exploited for further
shape recovery beyond elastic resilience in this class of materials.
In future, the remarkable resilience and dissipation of copolymer polyureas may be further
investigated to provide a physical foundation of shape recovery, memory and healing in other
segmented copolymers such as thermoplastic polyurethane and transparent polyurethane-urea. In
particular, shape memory in polymeric materials was found to be a thermomechanically-driven
process as detailed in a number of previous studies. 39-41 However, in most of previous studies,
the inelastically-driven shape recovery without any further thermomechanical treatment was not
examined well. For a better understanding of physically-sound shape memory mechanisms in
diverse elastomeric materials, the shape recovery capability of materials should be first
quantified in tandem with the effect of temperature4 0 and moisture treatment.4 2 Furthermore, a
marked shape recovery under severe deformation was found to be essential in self-healing
polymers.37,
38
Modeling the self-healing process that couples mechanical deformation to heat
and mass transport may require a sound understanding of the deformation-dependent shape
recovery of materials upon and beyond unloading. Additionally, the shape recovery may be
highly rate-dependent as evidenced in Chapter 6 and Chapter 7. The rate-dependent shape
recovery mechanisms should be further investigated in this class of elastomeric materials
operating at a broad range of deformation rates.
178
7.5.
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182
Chapter 8
Mechanics of Elastomeric Ionic Copolymers
Portions of this chapter will be submitted to journal, H. Cho and M. C. Boyce, "Mechanics of
Ionic Elastomers: resilience,dissipationand constitutive models ", in preparation,2013
8.1.
Introduction
An ethylene methacrylic acid copolymer is an ionic elastomer of critical importance for a
broad variety of engineering applications involving their highly resilient yet dissipative
mechanical
features
under
large
deformation.1-3
The
thermodynamically-incompatible
microstructures of ionic aggregates pendent with crystalline and amorphous phases of branched
and unbranched ethylene chains and methacrylic acid groups have received great attention for
elastomeric architectures of tunable resilience and dissipation, which provide new avenues
towards highly recoverable and protective multi-functional materials via an outstanding
combination of mechanical and thermal properties of constituents.4-6 The remarkable shape
recovery in the ionic elastomers was found to often lead to self-healing, where the resilience and
dissipation play key roles to drive a healing process, immediately following localized fracture
and damage.7'
The self-healing was found to be due to shape recovery furthered by thermally-
activated chain diffusion at fractured interfaces, at which the substantial dissipation due to
inelastic flows and other hysteresis sources supplies localized heating at and around the failed
surfaces. Additionally, an order-disorder transition in ionic clusters was found to provide another
driving force for healing. The self-healing behavior has been extensively reported for diverse
ethylene-based ionic elastomers and the relevant physical interpretations were suggested to
account for healing mechanisms. 9'
10
In tandem with the thermally-activated healing mechanisms,
the complicated elastic and inelastic shape recovery behavior also posed challenges for a better
understanding of initiation of healing in the materials. The substantial shape recovery should be
183
accompanied for healing since it cannot be initiated without a physical contact of fractured
interfaces. Ethylene methacrylic acid (EMAA) copolymers can be also chemically modified to
provide a remarkable range of mechanical behavior. In particular, ethylene methacrylic acid
butyl acrylate (EMAABA) terpolymers can be derived by adding butyl acrylates to EMAA
copolymers."'
1
The chemical modification was found to offer substantial changes in
mechanical stiffness and relaxation structures. The methacrylic acid groups in EMAA and
EMAABA can be additionally neutralized with metallic salt cations such as sodium (Na'), zinc
(Zn*) and magnesium (Mg'), by which remarkable increases in the mechanical stiffness and
11
viscous dissipation can be achieved. , 13
In this chapter, we investigate the underlying mechanical principles for resilience,
dissipation and shape recovery of various ionic elastomers including EMAA, EMAABA and
their chemically-modified counterparts, especially neutralized with sodium cations under large
deformation. A constitutive modeling framework is rationalized to capture the key deformation
mechanisms of ionic elastomers following the viscoelastic-viscoplastic constitutive model of
elastomeric materials whose mechanical behavior and microstructural separation are very similar
to those in the elastomeric "segmented" elastomers. In particular, a simple modeling framework
for sodium-neutralized EMAABA is addressed to provide a physical intuition for chemical
modification. Furthermore, the mechanical behavior of materials under localized, threedimensional deformation at small scales was quantified via in situ micro-indentation testing in
experiments and numerical simulations.
Keyword: ionomers, EMAA, EMAABA, neutralization, resilience, dissipation, shape recovery,
microindentation,
viscoelastic-viscoplastic
constitutive
model,
nonlinear
finite
element
simulation
8.2.
Mechanical Behavior of Ethylene-based Ionic Copolymers: Constitutive Modeling
Framework
184
In this research, the EMAA copolymer contained 9 wt% of methacrylic acids. The
EMAABA terpolymer contained 9 wt% of methacrylic acids and 23 wt% n-butyl acrylate (nBA).
The EMAABA terpolymer was partially neutralized by sodium ions; EMAABA-Na was formed
by neutralization of 53% of the acid groups with Na'.
The microstructures of ethylene methacrylic acid copolymers (EMAA) comprise ion-rich
and ion-poor domains with crystalline and amorphous phases at which the methacrylic acid
groups are connected to the neutralized or un-neutralized methacrylic groups via linear or
branched ethylene chains. Figure 8-1a presents the dynamic mechanical properties representing
multiple relaxations in EMAA. Here the r relaxation is essentially the same as that in ethylene
homopolymers ; the Y' peak is due to segmental motion at the glass transition of this material';
as the acid groups are neutralized, the intensity of the f' peak diminishes and a new relaxation,
j6
appears at slightly lower temperature, which is affiliated with the relaxation of amorphous,
branched ethylene chains.' 11'
'3
The macroscopic mechanical behavior of materials is strongly
dependent upon the microstructures and it can be characterized by highly nonlinear elasticity,
rate-dependent inelasticity, nonlinear hardening due to the alignment of chain molecules and
highly nonlinear unloading with substantial amount of hysteresis and recovery. In particular, the
presence of phase-separated microstructures leads to the multiple relaxation processes
responsible for a transition in the stress-strain behavior as shown in Figure 8-la and b. The
macroscopic stress contribution from the soft, amorphous phases is negligible at low strain rates
since they are fully relaxed at long time scales. On the other hand, it is non-negligible at high
strain rates; the soft phases provide additional stiffness at shot time scales.
(a)
(b)
400
101~r
Swa
SrgModus
-%
30
LOU ModUts,
-Lo" Tangeot
~40
0~!
15I1
(Cc).
70
50.
60
50
.
3
P30
20
-100
50
0
TempeTftrrrC)
10
0.0
0
0
201
10 0-
10
10
=03
C0
0.2
0.4
0.6
ue Strain
0.8
1.0
a 10 1
10 1
Strain Rate [/s]
Figure 8-1 Mechanical behavior of EMAA: (a) dynamic mechanical analysis data: storage
modulus and loss factor as a function of increasing temperature; (b) stress-strain behavior at
185
strain rates of 0.0016, 0.016, 0.16, 16, 115, 1300 and 6000 /s (black: low rate, gray: intermediate
rate, light gray: high rate); (c) flow stress as a function of strain rate at increasing strains of 0.3
and 0.8 (from Deschanel et al.")
The rate-dependent yield and inelastic deformation of EMAA were also found to be
tailored via cationic neutralization of the methacrylic acid groups such that the increased
neutralization with sodium leads to a substantial increase in the yield stress.
There have been a
few studies on constitutive modeling of the stress-strain behavior of EMAA and its chemicallymodified counterparts. Scogna and Register predicted the rate-dependent yield and its
relationship with the degree of sodium neutralization using a modified Ree-Eyring viscoplastic
flow model and a micromechanical analysis. 12 ' 15-17 Deschanel et al. 1 demonstrated a constitutive
model composed of viscoplastic components for hard and soft phases providing a simple yet
physically intuitive framework for multiple-rheological mechanisms for constituents. However,
these models were not capable of capturing the detailed features of mechanical behavior at large
strains including the highly resilient yet dissipative loading-unloading properties. In recent
studies by Cho at al.18' 19, a microstructurally-informed constitutive theory was proposed to
capture the rate-dependent resilient yet dissipative large deformation of elastomeric segmented
copolymer polyurea possessing multi-phase morphology. We employ the constitutive modeling
framework to address the mechanical behavior of ionic elastomers as illustrated in Figure 8-2a
since the stress-strain behavior in polyurea is very similar to that found in EMAA and its
counterparts.
Multiple micro-rheological mechanisms are employed to capture the macroscopic stress
contributions from amorphous and crystalline phases in the ionic elastomers. The elastic-plastic
mechanism (I) captures the intermolecular shear and the thermally-activated molecular motions
responsible for the inelastic flows; the elastic mechanism (N) capture the entropic elasticity due
to the orientation and locking of molecular chain networks. From the notion of deformation
compatibility with a homogenized motion in the multi-phase, we have a kinematic constraint for
each constitutive element as follows,
F =Fa*
(1)
186
Here, F is the total deformation gradient, which is defined via F =
that maps the undeformed
reference configuration ( x ) to the deformed spatial configuration (x ); a stands for the
intermolecular (IH, IS) and network (NH, NS) components for amorphous (soft) and crystalline
(hard) phases.
(b)
F = F FP
(a)
oo
C
.
. .............
Hard Domain
FPK
Soft Domain
Figure 8-2 Schematic of constitutive models of EMAA: (a) multiple micro-rheological
mechanisms (b) kinematics of elastic-plastic deformation
Following the kinematics framework of viscoplastic solids undergoing finite straining 20 which
was discussed in Chapter 3, we employ a multiplicative decomposition of elasticity and
plasticity in the total deformation gradient as described in Figure 8-2b,
(2)
F = FeF,
where Fe and F; represent the elastic and inelastic part of total deformation gradient in a
mechanism, respectively. The elastic and plastic deformation gradients are decomposed into the
right stretch (U) and rotation (R) or the left stretch (V) and rotation (R) terms following a polar
decomposition: Fe = R U=
VeR' ; FP, = RPUP =VR
through the spatial velocity gradient, La grad v = ).
. The deformation rate is examined
The velocity gradient is decomposed into
elastic and plastic contributions,
La~
!X~
L~+LF7
1
+)PFIe-I
e-F
a
aa
=Ua
IU
(3)
187
(4)
aD
fa =b+WS,
where J) and W, represent the rate of plastic stretching and spin, which are the symmetric and
skew part of LP., respectively. With no loss on generality, the viscoplastic flow in the current
configuration is taken to be irrotational, giving
#=F ,
(5)
-
fPFFa.
The rate of plastic deformation gradientPF, is then numerically integrated to obtain the plastic
deformation gradient FP and, consequently, the elastic deformation gradient is obtained via
F =FFP-' . The rate of plastic stretching DP under a given stress state is constitutively
prescribed:
DP = 'NP,
(6)
where the plastic flow is taken to be co-axial to the deviatoric stress tensor, and hence NP is the
normalized deviatoric stress tensor,
N
(7)
a
1
where T, =T, -- tr(Ta)I.
3
The rate-dependent yielding event is considered to be a thermally-activated process, whereby the
energy barrier to intermolecular interactions must be overcome. The thermally-activated inelastic
process can be expressed in terms of the thermodynamic parameters such as the activation free
energy, the activation volume and the intrinsic shear resistance against inelastic flow. The
magnitude of inelastic flow f is constitutively prescribed by,
=
+
t
- AG)t'
exp KK
kBO
sinh
G.J
kB
(8)
- ia
where Y0, is the reference viscoplastic rate for the magnitude of flow, A G, is the activation free
energy of inelastic flow, kB is Boltzmann's constant, 0 is the absolute temperature, sa is the
188
athermal shear strength, r is the initial shear viscosity that captures the initial linear nature of the
viscoelastic behavior,
and
:
T.
is a magnitude of the deviatoric stress tensor. The
shear resistance may be further modified to account for the pressure sensitivity via,
+a- p, where p = - I trT and a is the pressure sensitivity coefficient2o.
S7
=S
3
The intermolecular resistance is represented by the linear elasticity model with the
Hencky's logarithmic strain measure 21 in Equation (8).
T. =
J
(9)
C-eLE*E,
where the subscript a stands for "IH" or "IS", J = det F, = det F. det FP = det F, is the elastic
volume change; E
=In Ve is the Hencky's strain tensor;
, = 2 pI
+ 3B -2p" I
3
I is the fourth
order tensor of elastic constants, where p, is the elastic shear modulus; B is the initial bulk
modulus of the material; n is the fourth order identity tensor; and i is the second order identity
tensor. Here, the bulk contribution was lumped into the intermolecular hard component for
convenience. The network stretching and rotation are captured by the Arruda-Boyce eight-chain
The network stress is taken to be deviatoric and the Cauchy stress is given by
elasticity model.
Ta
"a (B
ca
-a)
where the subscript a stands for "NH" or "NS",
function; VlNa
Alock4
a
(10)
I),
is the limiting chain extensibility,
"ca denotes the inverse Langevin
" =
jtr(B a) is an "average" chain
stretch in the eight chain network, and p, is the initial elastic shear modulus of the network.
A phenomenological evolution which develops with inelastic straining is employed to
capture the elastic-plastic softening in the intermolecular component. The athermal shear
strength evolves to a preferred state with plastic straining,
189
=h
=a
-
(11)
n
Sss'
0.077pu,
(12)
l-Va
SO'a
where ha is the softening slope, s,,, is the steady state shear strength, va is the initial Poisson's
ratio, and ma controls the rate of evolution to the steady state. The intermolecular elastic shear
modulus p, also evolves with plastic straining,
=h, I-
Ea
(13)
where u,,, is the steady state elastic shear modulus.
To capture the stretch-induced softening extensively reported in this class of ionic
elastomers, we employed an evolution rule for the limiting chain extensibility and the network
shear modulus
18,23
=p t
hhNO,
(14)
e(t)eNst
The chain limiting extensibility is taken to follow a single evolution rulet",
2ckNH
c,NH In, (
=
SlockNH = SONH
c(4k, NH /
I'lockNH
(Alock.NH
IH
2"'NH
Ass~lock,NH
where
=
Ass.IockNH
-r~tanh
ss, lock, NH
Ar2
I
2
))sinh
5 NH ),
Slock,NH
(15)
(16)
(0))n1
(17)
lockNH
*
is a maximum achievable chain limiting extensibility dependent upon the rate of
chain stretch, slock,NI is the internal resistance to the breakdown of chain networks which also
evolves during deformation, and FNH
-NH
2
:TNH is a magnitude of the current stress tensor in
190
the hard network. The evolution of the chain limiting extensibility also captures the ratedependence of the network structural breakdown.
Nonlinear viscoelasticity is employed to capture the time dependent relaxation of the
network response of the soft domains which is observed to be strongly dependent on strain rate.
Following Bergstrom and Boyce 24 and Dupaix and Boyce25 , nonlinear viscoelasticity in the soft
network component is captured as follows:
Ks
(18)
=(D(0)$i;NS) ln
where 1 /D(G) is the reference shear viscosity that may be taken to be a thermally-activated
NS*
2Ns
process,
NS
is a magnitude of the stress tensor, and # is an orientation parameter,
which provides a quantitative assessment of molecular chain alignment. The orientation
parameter # is determined via:
r
#=---cos
2
,,
min{,,
_Cos.tr
(19)
,
(BNS)
where {A} is a set of principal stretches in the soft network.
(b)
(a)
%RW
0.
U
75
(U
60-
50
. 0 0 0*
40-
0)
I-
C')
ci,
20
25
0
U-
-0
* e=0.4, exp
Se=0.8, exp
o e=0.4, model
c3 e=0.8, model
I I
.
0.2
.
0.4
0.6
True Strain
0.8
1
00
0
103 10-2 10-1 100 101 102 103
4
Strain Rate [s]
Figure 8-3 Stress-strain behavior of EMAA in experiments and simulations: (a) stress-strain
curves under compression at strain rates of 0.0016, 0.016, 0.16, 16, 115, 1300 and 6000 /s (solid
191
line: simulation, symbol: experiment) (b) flow stress as a function of strain rate at strains of 0.4
and 0.8 (experimental data reproduced from Deschanel et al.")
Figure 8-3 shows the stress-strain behavior of EMAA at low to high strain rates in
experiments and simulations. The simulated curves agreed well with those in experimental data.
As shown in Figure 8-3b on flow stress as a function of strain rate, we only have a single
relaxation process that results in a single rate-sensitivity at low strain rates. At high strain rate
behavior, there was a dramatic stress hardening, where stress contribution from soft components
dramatically increased. Figure 8-3b also shows flow stress levels at increasing strains of 0.4 and
0.8 at low to high strain rate revealing the "transition" in the rate-sensitivity. The constitutive
model was found to naturally capture the transition behavior due to the multiple relaxation
processes in hard and soft mechanisms. In particular, the degree of hardening at a strain of 0.8 at
high strain rates was found to been relatively small by comparison to that in PU1000 and PU650
we discussed in Chapter 3 and Chapter 5, respectively. The dramatic hardening at large strains
at high strain rates in PU1000 and PU650 was mainly due to the effect of cessation of molecular
relaxation which is strongly dependent upon chain alignment in soft phases. Thus, we introduced
an orientation-dependent nonlinear viscoelastic flow model in the soft network in the polyurea
models. However, at EMAA data, we have no evidence of orientation-dependent relaxation at
high strain rates. Therefore, we modified a simple power-law viscoelasticity model via
fis = (D (0)TNS )1/n for EMAA model not including the orientation parameter.
The constitutive models and the kinematics of elastic-plasticity of EMAA were
numerically implemented for use in finite element simulations. In the following section, the
microindentation behavior of EMAA will be discussed in experiments and numerical simulations
incorporating the EMAA constitutive models. The material parameters of EMAA constitutive
models are provided in 8.6.
8.3.
Microindentation Behavior of EMAA
192
Microindentation of polymeric materials has been found to provide quantitative
information about viscoelastic-viscoplastic properties at small scales.26-28 We used the
constitutive modeling framework to simulate the inhomogeneous deformation fields under
microindentation testing of EMAA. To quantify the microindentation behavior of EMAA, we
designed load functions involving loading, creeping and unloading, at which the maximum load
increases to quantify the deformation-dependent indentation behavior and the time-dependent
creep at the peak load; here we used a diamond spherical tip of a radius of 10.25 pm .
Figure 8-4 and Figure 8-5 show microindentation behavior of EMAA (force-displacement
curves and displacement-time curves) in experiments and numerical simulations. Here, we used a
load function comprised of loading (10 sec), creep (10 sec) at a constant force and unloading (10
sec) in order of increasing peak forces as shown in the insets. Numerical simulations captured
nicely the force-displacement curves including the nonlinear loading and the creep deformation
at the peak force. Creep flow was found to be strongly dependent upon the applied force; i.e. the
level of creep displacement at a peak force of 3.0 mN was found to be much greater than that
found at a peak force of 1.5 mN. This was also confirmed in contours of inelastic strain rates
during creep at t = 10 sec (at the end of loading) in order of increasing peak forces in Figure 8-6.
A magnitude of inelastic strain rate was found to be much greater at a peak force of 3.0 iN.
Additionally, the finite element simulations provided detailed spatial strain field in Figure 8-7,
Figure 8-8, Figure 8-9 and Figure 8-10. At a peak force of 3.0 mN, the maximum axial strain
was found to be
-
0.34 (-0.22 at 1.5 mN), revealing deformation at the end of creep was located
in the viscoplastic regime beyond the initial stress-rollover. The highly nonlinear unloading was
found to lead to a substantial elastic shape recovery over a wide range of peak forces as shown in
Figure 8-7d, Figure 8-8d, Figure 8-9d and Figure 8-10d. The nonlinear unloading behavior
accompanied by the elastic shape recovery was captured well in the finite element simulations as
shown in Figure 8-4 and Figure 8-5 over the four loading scenarios. Additionally, the highly
resilient yet dissipative microindentation behavior in experiments and numerical simulations
supports that the constitutive modeling framework of EMAA built on the simple uniaxial stressstrain data has a three-dimensional predictive capability of capturing the inhomogeneous, timedependent deformation fields under localized indentation testing at microscales.
193
(b)
(a)
- --
0.002
ftI
-mdel
0.0031
U.
exp
10s
-
0.004.
1.smN
Z1O 30s
.
0.003
0.002 -
-
0
-2.OmN
'ftt
2ks 3Os
-L 10sJ.\.
0.001
0.001
T.100
-
1-
o
0.0
0.001
0.002
0.003
0.000
0.A00
0.0 04
0.001
0.002
0.003
0.1 04
Displacement [mm]
Displacement [mm]
(d)
(c)
0.004
I
30s
2.5mN
0.003
0.003
10s
0.002
U.
.
20s 30s
0.002
0.001
-
Os ZOs
30s
0.001
1
u.000
0.001
0.002
0.003
0. 004
Displacement [mm]
U .000
0.001
0.002
0.003
0.004
Displacement [mm]
Figure 8-4 Force-displacement behavior of EMAA under microindentation in experiments and
numerical simulations: force-displacement behavior (inset: load functions) at increasing peak
forces of (a) 1.5; (b) 2.0; (c) 2.5; and 3.0 mN
194
0.005
E 0.004
91"
.
.. .
,
.
d
E 0.003
E
0.
ca
6
0.002
0
0.001
0 .0 0 0C
5
10
20
15
25
30
Time [s]
Figure 8-5 Displacement-time curves of EMAA under microindentation in experiments and
numerical simulations at increasing peak forces of 1.5, 2.0, 2.5 and 3.0 mN (dashed line:
experiment, solid line: simulation)
(a)
0.005
'I
II
(b)
0.006
I0.0
I0.0
P I/s]
(c)
0.007
0.0
)
(d)
0.009
I
I0.0
Figure 8-6 Contours of inelastic strain rates of EMAA under microindentation at t = 10 sec (a)
Fmax=1.5 mN; (b) Fmax=2.0 mN; (c) Fmax=2.5 mN; (b) Fmax=3.0 mN
195
t = 10s
(a) 0
I
i
t = 20s
t = 30s
(b) 0.0
(c)
I -0.22
-0.2
0.0
j
I -0.09
Figure 8-7 Contours of axial strain field of EMAA under microindentation at Fmax=1.5 mN in
indentation direction (a) loading; (b) creep; (c) unloading
(a) 0.0
t = 10s
j
I-0.23
(b) 0.0
t = 20s
(c)
I
t = 30s
0.0
I -0.12
-0.25
8zz
Figure 8-8 Contours of axial strain field of EMAA under microindentation at Fmax=2.0 mN in
indentation direction (a) loading; (b) creep; (c) unloading
t = 10s
t = 20s
(a) 0.0
(b) 0.0
I -0.26
-0.29
j
t = 30s
(c)
0.0
I-0.15
Ezz
Figure 8-9 Contours of axial strain field of EMAA under microindentation at Fmax=2.5 mN in
indentation direction (a) loading; (b) creep; (c) unloading
196
(a) 00(b)
-0.29
t =los
t=20s
t = 30s
(c)
I I
-0.34
0.0
I
-0.18
Figure 8-10 Contours of axial strain field of EMAA under microindentation at Fmax=3.0 mN in
indentation direction (a) loading; (b) creep; (c) unloading
8.4.
Resilience, Dissipation and Shape Recovery of Chemically-modified Counterparts:
EMAABA and EMAABA-Na+
In this section, we examine the mechanical behavior of EMAABA (EMAA modified with
butyl acrylate) and EMAABA-Na (EMAABA neutralized with sodium cations) in terms of their
resilience, dissipation and shape recovery under deformation. As shown in Figure 8-11, the
stress-strain behavior of EMAABA and EMAABA-Na was found to be substantially softened by
comparison to that in EMAA by the presence of butyl acrylate regions. Additionally, the sodiumneutralization in EMAABA was found to result in a substantial stiffening effect that
accompanies greater dissipation and residual strain in the stress-strain curves. However, the
structure of dynamic mechanical properties of EMAABA-Na+ was found to be very similar to
that found in EMAABA; r and q peaks in EMAABA-Na were considered to be affiliated with
the same segmental motion and relaxation as for EMAABA. (See Figure 6 in Chapter 2) A
simple chemical modification with butyl acrylates was found to lead to a substantial change in
both resilience and dissipation; in EMAABA and EMAABA-Na, the resilience was found to
increase significantly, which resulted in a decreased residual strain while the dissipation and
hysteresis decreased. The base constitutive modeling framework of EMAA should be furthered
to capture the main features of resilience and dissipation of EMAABA and EMAABA-Na by the
197
simple chemical modification. Additionally, the modified features of resilience and dissipation
should be physically addressed in the constitutive framework of EMAABA and EMAABA-Na+.
25
20
EMAA
EMAABA
---EMAABANa+
-
1510-
p-
5
0
0.0
0.2
0.4
0.6
0.8
1.0
True Strain
Figure 8-11 Stress-strain data of EMAABA and EMAABA-Na under cyclic compression at a
strain rate of 0.0016 /s (data reproduced from Deschanel et al. 13 )
First, the material parameters for EMAABA were determined using the stress-strain data at low
to high strain rate under monotonic and cyclic deformation conditions. Figure 8-12a shows the
stress-strain behavior of EMAABA at low to high strain rates in experiments and models, where
the modeling results agreed well to the experimental data with the new material parameter set for
EMAABA. To model the stress-strain behavior of EMAABA-Na, a simple scaling rule was
employed for the constitutive model parameters. The ratio of initial elastic moduli in EMAABA
and EMAABA-Na was directly used to determine a new set of material parameters related to
nonlinear elasticity and inelastic flows since the rate-dependence in EMAABA-Na at increasing
strains was found to be very consistent with that in EMAABA. The magnitude of stress levels
was found to be scaled simply in EMAABA-Na via the ratio of initial elastic moduli. These
observations lead to simple scaling rules for constitutive material parameters of EMAABA-Na
such that:
0-
PEMAABA-Na
-2.05,
(20)
P EMAABA
198
(21)
1SEM)
SEMAABA-Na
AGEMAABA-Na
g|EMAABA-Na
=
(22)
AGFmAABA
(EMAABA
(23)
P
where p is the elastic modulus; s is the shear strength; AG is the activation free energy against
viscoplastic flow; and K is the reference inelastic strain rate. Without any further parametric
studies, the simple scaling rule was found to be able to capture the dramatically enhanced stress
in EMAABA-Na at low to high strain rates as shown in Figure 8-12 and Figure 8-13.
The constitutive models of EMAABA and EMAABA-Na were further examined for their
resilience and dissipation capabilities under cyclic compression test. Figure 8-14 shows a
stretch-induced softening known as Mullins effect in EMAABA that provides significantly
softened behavior in the second cycle. The stretch-induced softening due to the irreversible
microstructural change provides another major dissipation mechanism in addition to the viscous
dissipation mechanisms involving viscoelasticity and viscoplasticity as we discussed in detail in
previous chapters on the elastomeric segmented copolymers. The constitutive models should be
hence able to capture the structural change of materials during deformation via the evolution
model for the corresponding internal variables. The stretch-induced softening was found to be
captured via the single evolution rule for the chain extensibility and the elastic modulus in the
network resistance in Equation (14)
Na was scaled again via SEMMBA-Na
=O
-
(17). Here the resistance against softening in EMAABAFigure 8-14 and Figure 8-15 show the stress-strain
behavior of EMAABA and EMAABA-Na under cyclic compression tests at increasing imposed
strains of 0.7 and 1.0. In both materials, substantially decreased stress was observed in the
second cycle due to the stretch-induced softening during the first cycle. Also, a residual strain
was found to substantially decrease in the second cycle while it increased as an imposed
maximum strain increased. In particular, there was a remarkable shape recovery between the first
and the second cycle, which was also found to be proportional to the imposed maximum strain in
the first cycle. Furthermore, as the imposed strain increases, the level of dissipation was found to
increase in both cycles of N=1 and N=2. Furthermore, the energy dissipation and hysteresis in
EMAABA were found to increase by sodium-neutralization as illustrated in Figure 8-16. This
199
observation also supports a simple chemical modification of ionic elastomers can provide
"tunable" capability of energy dissipation without any major changes in microstructural features.
The overall features of cyclic behavior were well captured in the models of EMAABA and
EMAABA-Na with the material parameters scaled from EMAABA involving stretch-induced
softening and deformation-dependent shape recovery. Though the material parameters for
EMAABA-Na were determined using the simple scaling rules informed from the "elastic"
properties of EMAABA and EMAABA-Na, the increased energy dissipation and hysteresis
accompanied by the increased residual strains were captured very well; i.e. the inelastic features
of EMAABA-Na arising in the intermolecular mechanisms were found to be well examined in
the model with the simple scaling rules. However, there was disparity between experiments and
simulations in the unloading behavior as shown in Figure 8-14 and Figure 8-15 since stress
relaxation was inevitable in experiments due to a time lag in instrumentation when loadingunloading direction changes. Consequently, the dissipated work during cyclic deformation was
underestimated in simulations in both cycles for EMAABA and EMAABA-Na copolymers
(Figure 8-16).
(a)
(b)
_
_
_
_
_
_
_
_
_
_
_
_
_
80
40
60
~3O
0 4
0
20
00
00
0)0
0 .2
0.6
0.4
True Strain
0.8
1
0.2
0.6
0.4
True Strain
0.8
1
Figure 8-12 Stress-strain behavior of EMAABA and EMAABA-Na under compression at low to
high strain rates (a) EMAABA at strain rates of 0.001, 0.01, 0.1, and 3000 /s (solid lines:
simulations, symbols: experiments); (b) EMAABA-Na at strain rates of 0.001, 0.01, 0.1, 68, 600,
and 6500 /s (line: simulations, symbol: experiments); experimental data reproduced from
Greviskes et al.' 3
200
75
0
e=0.36, exp
e0.7, Sexp
EMAABA
o e=0.3, model
o v--0.7, model
1- 8=0.36,eXp
0 e=0*7, exp
EMAABA-Na
*
50
TEMAABA-Na
0
o eMO.35 , modelJ
o a0.7, model
4)
-
I
0-
L, 251
TEMAABA =Thard
0
2
-1 100
+ Tsoft,EMAABA
0
(0
103
a(Thrd,EMAABA
10 102
3
+Tsft
Strain Rate [/s]
Figure 8-13 Flow stress as a function of strain rate at strains of 0.35 and 0.7 in EMAABA and
EMAABA-Na
(b)
(a)
10
,..,
-N=1,
model
8- --- N=2, model
o N=1, exp
6-
(U 8
0~
N=2, exp
(U
(U
a,
L.
.0 0*
4-
4.1
U)
a,
I-
I-
0
0.2
recovery
0.4
0.6
True Strain
0.8
1
6j
00
4
aa
0
2
0
0 0
a
0.2
recovery
a
0.4
0.6
True Strain
0.8
1
Figure 8-14 Stress-strain behavior of EMAABA under cyclic compression at a strain rate of 0.01
/s (a) maximum imposed strain of 0.7 (b) maximum imposed strain of 1.0 (experimental data
reproduced from Greviskes et al. 13)
201
(b)
(a)
2C
-N=1,
--
(U
model
N=2, model
SN=2,
10o
5
0
10
0
.0-
expco
0a
00
0
I-
0
7'15 0 N=1, exp
15
*
Cn
2
E li
*
*0
-*
I'
2recovery
0.4
0.6
0.4
-0.2
True Strain
recovery
1
0.8
0.6
True Strain
0.8
1
Figure 8-15 Stress-strain behavior of EMAABA-Na under cyclic compression at a strain rate of
0.01 /s (a) maximum imposed strain of 0.7 (b) maximum imposed strain of 1.0 (experimental
data reproduced from Greviskes et al.13)
(b)
(a)
emaaba-Na
emaaba
100
100
80
80
60
* exp
40
model
20
60
* exp
model
4020
0-
0
1
2
1
2
Figure 8-16 Dissipated work during cyclic deformation at an imposed strain of 1.0 in (a)
EMAABA; (b) EMAABA-Na
Material parameters of EMAABA and EMAABA-Na constitutive models are provided in 8.6.
202
8.5.
Conclusion
Motivated from the mechanical principles of elastomeric "segmented" copolymers
addressed in the previous chapters, we explored the mechanics of resilience, dissipation and
shape recovery of elastomeric "ionic" copolymers in this chapter. The constitutive modeling
framework we developed in the previous chapters was directly used to model the stress-strain
behavior of various ionic elastomers and their chemically-modified counterparts revealing a
predictive capability of models capturing the main features of large deformation behavior of
ionic elastomers. To our best knowledge, this investigation is the first demonstration of a
predictive constitutive model of ethylene-based ionic elastomers under a variety of loading and
unloading conditions at a wide range of strain rates.
Microindentation testing also revealed the highly nonlinear load-displacement behavior of
EMAA including creep and shape recovery, which were well captured in the nonlinear finite
element simulations incorporating the newly constructed constitutive models. Additionally, we
aimed at the development of constitutive models of EMAABA and its sodium neutralized
counterpart. A simple scaling rule was able to capture the main features of chemically-modified
counterpart of EMAABA in terms of resilience and dissipation without any further complicated
parametric studies.
This work can be furthered to address other mechanochemical29 phenomena such as
fracture and self-healing of central importance in this class of ionic elastomers. Specifically, our
investigations on resilience and inelastically-driven shape recovery under large mechanical
deformation may constitute the basis for a better understanding of the self-healing behavior in
EMAA and its chemically-modified counterparts. In future, the proposed modeling framework
may be furthered to capture coupled multi-physical phenomena of mechanical deformation,
thermal and mass transport via chain diffusion during self-healing processes in conjunction with
appropriate failure mechanisms in this class of materials. In addition, this work may be furthered
to address a physical foundation of deformation mechanisms of bio-macromolecular assemblies
found in mussel and spider threads that exhibit very similar resilient yet dissipative features as
well as self-healing under large deformation and catastrophic molecular rupture. 23, 30-33
203
8.6.
Material Parameters for EMAA and EMAABA
We followed the procedures detailed in Chapter 3 to determine the material parameters of
ionic elastomers.
Table 4 Material parameter for intermolecular components in EMAA
Intermolecular
Hard
Soft
Intermolecular Stress
= 21.8 [MPa]
=23.7[MPa]
eIH
E
T,
.W
E______
= -IY
B =1I.0(GPa].
Viscoelastic-Viscoplastic Flow
P~
i
y7 =-a+g
q
+
p
O
exp
"
(ra AGa>
sinh
-
kB9
~AGIH = 4.04[10
(AGh
aT
kB
Sa
20
AGIs = 2.04 [10- 2 0 j]
J]
?IS =397.I['-
IH=0.022
q=2000.0[MPa-s]
*
The shear resistance sll may include the pressure sensitivity via
0.075;
iH
=s
+ap ,
where a is
the stress in the soft intermolecular component is taken to be deviatoric, i.e.
2
TIS=I Es
Is
204
Table 5 Material parameter for network components in EMAA
Network
Hard
Soft
Network Stress
a
3
I
-
"
NH
e
J Aca"\IlNNH
=15.1[MPa]
pNS =13.4[MPa]
=5
NNS =10.0
Viscoelastic Flow
Kivs
1/D=1.2[105 Pa . so.33
= (D rNs )/n
n=0.33
Stretch-induced Softening
,LNH
(0)NNH
(0)=uNH
=co
=ockNH
C
(t)NNH
(t);IOCkNH (t)
. 5 (1H (OCkNH
ssNH
I
SNH = SONH
Ass.lock,NH
=
r
lock,NH
/
(0)
2
lockNH
C0 =.O[S. 5]
'7NH
SNH
(2lockNII
2
))sinh
NNH (t)
s
= 0.4[MPa]
())7.5
r= 1.6
205
Table 6 Material parameter for intermolecular components in EMAABA
Intermolecular
Hard
Soft
Intermolecular Stress
/IH
e
=4.1[MPa]
T, =-Ea - Ea
Iis
=6.5 [MPa]
.
B=1.O[GPa]
Viscoelastic-Viscoplastic Flow
AGIH
kap
=-
+ Xe ,i
AGa sinh A7IH
kB
exp
t9
= 2.01[10
20
J]
=0.02[s1]
kB9 Sa
AGIs =8.35[10~ 2 1 j]
KIs =12.72[s-1]
7 =1200.0[MPa-s]
Elastic-Plastic Softening
IH
=hIH
1
-
SIH
hIH =2.5 [MPa]
Sssv,IH
2')
ji=hI1H( 1_
*
AH
s,i
fJJ
ssIH
Iss,IH /
IH
=0.8
IoH = 0.4
The shear resistance s111 may include the pressure sensitivity via
0.075
-
sH
=sIH + ap, where a is
0.1, in particular for the high strain rate behavior; the stress in the soft intermolecular
component is taken to be deviatoric, i.e. Ts,= 2'E'
J
206
Table 7 Material parameter for network components in EMAABA
Network
Hard
Soft
Network Stress
B,
"
f = "
NH
3JAca",
NNH
=5.65[MPa]
=4.5
/pNS
= 2.93[MPa]
NNS =10.0
Viscoelastic Flow
1/D=1.2[0
ivS =4(DrNs)
=I
0=--cos
-
1 min{A,A 2 ,2
3
2
5
Pa .s0 33 ]
}
tr(BNS
n=0.33
Stretch-induced Softening
/,NH
(0)NNH (0)=,Nn (t)NNH (t);IOCkNH (t)
AockNH
SNH
CO
=cN
=cNH
= SONH
cNH
0.5 (1
(1
(AlockNH
/
lock.NH
(
~
-
2N =
Ass.lock,NH
,anh
=l.0[S-05
rSNH
('HNH))Snh
2
C0
so
ss N/
A(NH
,N
=VNNH (t)
[
2.0
.))
75
r 2 =0.7
(I1e
lock.NH
= 0.07 [ MPa]
(0)
A, =20[s1 ]
The material parameters for EMAABA-Na can be estimated by scaling all of the
parameters used in EMAABA via Equation (20) - (23). However, the viscoelastic flow
parameters in the soft network component may be used in EMAABA-Na without any further
modification.
207
8.7.
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H. Cho, S. Bartyczak, W. Mock Jr and M. C. Boyce, Polymer 54 (21), 5952-5964 (2013).
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21.
L. Anand, Journal of Applied Mechanics 46 (1) (1979).
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E. M. Arruda and M. C. Boyce, Journal of the Mechanics and Physics of Solids 41 (2),
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B. P. Greviskes, SM, Massachusetts Institute of Technology, 2007.
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209
210
Chapter 9
Concluding Remark and Future Work
9.1.
Summary and General Conclusion
Elastomeric copolymers have been versatile soft materials attractive to numerous scientific,
engineering and defense applications over the past several decades. In this thesis, a broad variety
of the aspects of mechanics and physics of elastomeric copolymers have been studied to
understand the physically-sound deformation mechanisms responsible for the highly resilient yet
dissipative features of large deformation ranging from "segmented" copolymer polyureas to
"ionic" copolymer ethylene methacrylic acids.
Large deformation viscoelastic-viscoplastic behavior of various elastomeric copolymers
has been addressed in terms of their resilience and dissipation. Microstructurally-informed
constitutive models have been developed to provide a simple yet physically-sound framework to
model the resilient yet dissipative finite deformation behavior of materials. The constitutive
models were found to successfully capture the stress-strain behavior of an exemplar polyurea and
the relevant physical features under a variety of loading conditions at a wide range of strain rates
over seven orders of magnitude. The preliminary constitutive model of PUOGO was then
furthered to capture the effects of weight fractions and segmental dynamics of hard and soft
microstructures to the macroscopic mechanical response in PU650 in conjunction with a simple
yet intuitive micromechanical example for the morphological effects of co-continuous networks
often found in various elastomeric copolymers.
Nature of resilience and dissipation was also examined to understand the mechanics of
elastomeric copolymers under "extreme" mechanical environments via Taylor impact testing.
The Taylor impact behavior of materials was found to be well captured by nonlinear finite
element simulations imparting the constitutive models for polyureas and "model" glassy/rubbery
211
polymers. The copolymeric nature of polyurea was found to enable taking advantages from
glassy and rubbery polymers in terms of shape recovery and energy dissipation under such an
extreme impact event, where an ultrafast deformation over a strain rate of 105 s-1 was incurred.
This investigation enables to provide a predictive design principle of copolymeric composites to
mitigate blast and ballistic penetration.
Resilience, dissipation and shape recovery of polyurea copolymers were also examined
under microindentation testing with a complicated loading history comprised of creep and creepassisted inelastic recovery. In particular, the micro-indentation behavior of materials was
characterized
in experiments
and numerical simulations
revealing
the time-dependent
mechanical features under localized, inhomogeneous deformation. Additionally, a marked shape
recovery of polyureas was quantified via a creep-assisted mode in micro-indentation providing a
physical insight into shape recovery, memory and healing in this class of materials.
Lastly, the mechanics of resilience, dissipation and shape recovery of elastomeric "ionic"
copolymers was addressed involving ethylene methacrylic acid, ethylene methacrylic acid butyl
acrylate and their chemically-modified counterparts, which are recently finding new avenues
towards highly recoverable and self-healing material architectures under severe deformation
environments. Our modeling framework was also found to have predictive capabilities to capture
the stress-strain behavior under a variety loading conditions including microindentation based on
the least information of material properties.
9.2.
Future Work
Some aspects of critical importance of the mechanics and physics of elastomeric
copolymers are suggested for future work, to which the present work may contribute directly.
9.2.1. Micromechanics of various co-continuous or occluded morphologies for energy
dissipation and shape recovery
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Morphological features in copolymeric materials can be tailored to manipulate the
mechanical, thermal and chemical properties through the thermodynamic incompatibility of
constituents. In Chapter 5, we introduced a simple micromechanical model of a bi-continuous
morphology and revealed a combination of stretching and bending in the eight-chain type hard
phase can significantly enhance the mechanical stiffness beyond weight fractional stiffening. The
interpenetrating co-continuous network structures arising in copolymeric materials have been
active areas of research for a broad variety of tunable mechanical behavior including elasticplastic properties, energy storage and dissipation."
2
In addition to the co-continuous
microstructures, micromechanical models for different morphologies including occluded and
isolated hard particles randomly distributed in soft matrices (or occluded soft particles in hard
matrices 3,4) will provide critical insight into a better understanding of the tunable properties of
copolymeric materials.
9.2.2. Multi-scale mechanics of elastomeric copolymers: atomistic, coarse-grained and
continuum models and their coupling
Numerous research efforts have been focused on the atomistic or the continuum level
investigation of elastomeric copolymers in parallel over the past decade. However, up to date, a
few multi-scale studies have been reported to address key features of mechanical and thermal
properties, where the macroscopic responses are strongly governed by the molecular behavior of
materials. As an example for the multi-scale theoretical modeling of multi-component polymers,
Sheng et al. 5 rationalized a simple yet intuitive theoretical model to incorporate the multi-scale
phenomena found in polymer-clay nanocomposites based on the hierarchical information ranging
from the molecular to the macroscopic structures; where atomistic simulation results were used
to parameterize a physically intuitive constitutive model whose form was developed first through
micromechanical modeling and validation against experimental data at the continuum level. The
model has been widely accepted and successful for many nanocomposites containing a variety of
"effective" particles with different geometries and volume fractions. In addition to such static
multi-scale modeling, dynamic multi-scale analysis may be possible if computational issues in
bridging the scales are resolved, where serious challenges have been posed due to a huge gap
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between the molecular and the macroscopic timescale for major deformation events.
Additionally the molecular computations may provide useful insight into the molecular origin of
Mullins effect in this class of materials. Though the stretch-induced softening observed in
polyurea looks very similar to that found in polyurethane and polyurethane-urea (and ethylenebased ionic elastomers in Chapter 8) in terms of the macroscopic stress-strain behavior, its
molecular origins may be different in each elastomeric materials. For example, we have
introduced the microstructural breakdown to account for the Mullins effect in polyurea; however
the Mullins effect in polyurethane in Qi and Boyce has been explained by a different mechanism
such that the softening was originated from the evolution of "effective" volume fractions of hard
and soft phases under deformation. In future research, the stretch-induced softening can be
detailed more explicitly using the molecular computations with appropriate force fields that can
account for molecular breakdown.
9.2.3.
Mechanics of other elastomeric copolymers including thermoplastic polyurethane
and polyurethane-urea
As described in Chapter 2, the thermoplastic polyurethane (TPU)6 chemistry is essentially
the same as that found in the polyurea copolymers. The thermoplastic polyurethanes and their
chemical derivatives have been attractive elastomeric materials for a myriad of applications. In
particular, TPU was found to exhibit a remarkable "shape memory" upon a simple
thermomechanical treatment. TPU has been hence used for a variety of stimuli-responsive soft
material architectures ranging from tunable adhesives to biomedical devices and actuators via its
unique mechanical and thermal properties. Furthermore, polyurea-urethane copolymers can be
derived via bulk polymerization used in the synthesis of polyurea and polyurethane alternating
urethane-urea links. The transparent polyurethane-urea (TPUU) 7 '8 was found to provide marked
optical properties with outstanding transparency in addition to the remarkable resilience and
dissipation. Chemical modifications via molecular weight controls of hard and soft segments
and urethane-urea links were also found to be highly flexible to provide an outstanding
combination of desired mechanical properties. The overall mechanical behavior and the
underlying deformation mechanisms in TPU and TPUU are very similar to those in the polyurea
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copolymers studied in this research. We expect the constitutive modeling framework and the
relevant deformation mechanisms revealed through this research will be directly applied for
further studies of TPU and TPUU. The rate-dependent softening behavior at low to high strain
rates in this class of copolymeric elastomers should be more rationalized in experiments and
constitutive models. Though our constitutive modeling framework provides the rate-dependent
elastic-plastic and stretch-induced softening via a simple phenomenological model, many
ingredients in our models have relied upon the monotonic stress-strain data at high strain rate and
the Taylor impact data to best fit the rate-dependent softening. In future research, the stretchinduced softening should be further studied at the intermediate to high strain rates using more
experimental data under reloading to quantify how the stretch-induced softening changes at
short-time scales. Additionally, the multiscale approaches using molecular computations may
provide an insight into the highly rate-dependent stretch-induced softening of materials.
9.2.4.
Crazing, cavitation and failure in elastomeric copolymers under high strain rates
Failure of elastomeric copolymers has been an active research topic over the past several
decades since the relevant synthesis and chemical principles were available. Cavitation and
fracture propagation in the materials have been extensively reported; in particular, we clearly
observed the radial crack patterns and the cavitation event in the Taylor impact testing over an
impact velocity of 330 m/s
-
400 m/s.9 A number of experimental and computational studies
have been performed to address initiation of cavitation and fracture criteria of the materialso, 11
but the main features of those phenomena remain unknown at present. Once the cavitation and
fracture behavior in the materials are addressed based on physically-sound mechanisms, the
much faster deformation events of materials not covered in this research can be characterized for
a better design of composites that employ the elastomeric copolymers as protective coatings of
primary structures. Recently, a mesh-free computation was employed to capture the cavitation
and the fracture in elastomeric materials. The mesh-free method was found to efficiently
compute the failure paths, where an extreme distortion often occurs near to the fractured region
and hence a huge number of elements are needed in finite element simulations. However, to date,
we have no report to demonstrate mesh-free
simulations that successfully captured
215
experimentally verified cavitation and fracture in this class of materials. In particular, such
continuum-level computations may not capture cavitation which has been considered to be a
micro- or molecular-level event. Additionally, the mechanical behavior of elastomeric
copolymers at intermediate strain rates of 1 - 100 m/s should be more detailed in future research.
Though we discussed the low to high strain rate behavior of materials exclusively in Chapter 3,
Chapter 5 and Chapter 8, the investigations still lack experimental data at the intermediate
strain rates. In future, more experimental studies at the intermediate strain rates should be
conducted to address detailed information on the mechanical behavior of materials under
"moderate" impact conditions.
9.2.5. Modeling of healing behavior of elastomeric copolymers
The present constitutive models have been constructed under isothermal conditions.
However, as shown in Chapter 6, localized heating due to inelastic deformation may result in
changes in material properties. In particular, the cavitation and failure may be strongly dependent
upon temperature rise under adiabatic conditions. Furthermore, the shape recovery and relevant
self-healing behavior have been found to be highly dependent upon the thermomechanical effects.
In future, a thermomechanically-coupled constitutive model should be constructed based on
appropriate stress-strain data under a wide range of temperature spanning a glassy transition of
hard phases. The thermomechanically-coupled constitutive models of elastomeric copolymers
should be highly useful to address deformation mechanisms at transient regimes, where the
temporal evolution of temperature is dramatic due to the viscous dissipation. In addition, a
simple yet robust coupled theory should be addressed in terms of mechanical deformation,
thermal transport and chain diffusion for modeling of healing processes. In particular, the
mobility of chain transport at interfaces was found to be strongly dependent on mechanical stress
(pressure), inelastic deformation and temperature in many literatures
on joining and healing
processes of polymeric interfaces.
216
9.3.
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Thank you.
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