Mechanics of Solid Polymers – Theory and Computational Modeling

Mechanics of Solid Polymers – Theory and Computational Modeling
اسم المؤلف
Jörgen Bergström
التاريخ
7 أغسطس 2023
المشاهدات
219
التقييم
(لا توجد تقييمات)
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Mechanics of Solid Polymers – Theory and Computational Modeling
Jörgen Bergström
Contents
Preface xiii
1 Introduction and Overview 1
1.1 Introduction 1
1.2 What Is a Polymer? 2
1.3 Types of Polymers 4
1.4 History of Polymers 7
1.5 Polymer Manufacturing and Processing 11
1.6 Polymer Mechanics 11
1.7 Exercises 15
References 16
2 Experimental Characterization Techniques 19
2.1 Introduction 20
2.2 Mechanical Testing for Material Model Calibration 22
2.2.1 Uniaxial Compression Testing 24
2.2.2 Uniaxial Tension Testing 29
2.2.3 Plane Strain Tension 33
2.2.4 Simple Shear Testing 37
2.2.5 Impact Testing 40
2.2.6 Dynamic Mechanical Analysis 43
2.2.7 Hardness and Indentation Testing 47
2.2.8 Split-Hopkinson Pressure Bar Testing 53
2.2.9 Bulk Modulus Testing 64
2.2.10 Other Common Mechanical Testing Modes 71
2.2.11 Testing for Failure Model Calibration 73
2.3 Mechanical Testing for Material Model Validation 73
2.3.1 Material Model Verification and Validation 75
2.3.2 Small Punch Testing 77
2.3.3 V-Notch Shear Testing 80
2.4 Surface Characterization Techniques 80
2.4.1 Optical Microscopy 81
2.4.2 Scanning Electron Microscopy 84
2.4.3 Atomic Force Microscopy 87
vvi Contents
2.5 Volume Characterization Techniques 89
2.5.1 Differential Scanning Calorimetry 89
2.5.2 Transmission Electron Microscopy 90
2.5.3 X-Ray Diffraction 92
2.5.4 Birefringence 95
2.5.5 Swell Testing 97
2.6 Chemical Characterization Techniques 99
2.6.1 Fourier Transform Infrared Spectroscopy 100
2.6.2 Energy Dispersive Spectroscopy 101
2.6.3 Size-Exclusion Chromatography 103
2.6.4 Thermogravimetric Analysis 107
2.6.5 Raman Spectroscopy 109
2.7 Exercises 110
References 112
3 Finite Element Analysis as an Engineering Tool 115
3.1 Introduction 115
3.1.1 Required Inputs for FEA 117
3.2 Types of FEA 119
3.3 Review of Modeling Techniques 120
3.3.1 Deformation Modeling 120
3.3.2 Failure Modeling 125
3.4 Exercises 130
References 130
4 Continuum Mechanics Foundations 131
4.1 Introduction 132
4.2 Classical Definitions of Stress and Strain 133
4.2.1 Uniaxial Loading 133
4.2.2 Multiaxial Loading 135
4.3 Large Strain Kinematics 137
4.4 Vector and Tensor Algebra 141
4.4.1 Vector Operations 141
4.4.2 The Dyadic Product 143
4.4.3 Tensor Operations 144
4.4.4 Derivatives of Scalar, Vector,
and Tensor Fields 147Contents vii
4.4.5 Coordinate Transformations 149
4.4.6 Invariants 150
4.5 Deformation Gradient 150
4.5.1 Eigenvalue and Spectral Decompositions 154
4.6 Strain, Stretch, and Rotation 157
4.7 Rates of Deformation 164
4.8 Stress Tensors 165
4.8.1 Stress Invariants 169
4.9 Balance Laws and Field Equations 171
4.9.1 Conservation of Mass 173
4.9.2 Balance of Linear Momentum 175
4.9.3 Balance of Angular Momentum 178
4.9.4 First Law of Thermodynamics 180
4.9.5 Second Law of Thermodynamics 183
4.10 Energy Balance and Stress Power 184
4.11 Constitutive Equations 187
4.11.1 Constitutive Equations for a
Thermoelastic Material 189
4.12 Observer Transformation 194
4.12.1 Objective Rates 198
4.13 Material Symmetry 198
4.14 List of Symbols 199
4.15 Exercises 202
References 206
5 Elasticity/Hyperelasticity 209
5.1 Introduction 210
5.2 Linear Elasticity 211
5.2.1 Isotropic Elasticity 211
5.2.2 Anisotropic Elasticity 215
5.2.3 Transversely Isotropic Elasticity 217
5.3 Isotropic Hyperelasticity 218
5.3.1 Continuum Mechanics Foundations 219
5.3.2 Similarity Between Uniaxial
Compression and Biaxial Tension 225
5.3.3 Similarity Between Pure Shear and
Planar Tension 226viii Contents
5.3.4 Dependence of Stored Energy on I1 and I2 229
5.3.5 Freely Jointed Chain Model 232
5.3.6 Neo-Hookean Model 236
5.3.7 Mooney-Rivlin Model 243
5.3.8 Yeoh Model 245
5.3.9 Polynomial in I1 and I2 Model 250
5.3.10 Eight-Chain Model 250
5.3.11 Ogden Model 259
5.3.12 Gent Model 263
5.3.13 Horgan and Saccomandi Model 265
5.3.14 Knowles Model 268
5.3.15 Response Function Hyperelasticity 270
5.3.16 Extended Tube Model 273
5.3.17 BAM Model 275
5.4 Summary of Predictive Capabilities of Isotropic
Hyperelastic Models 277
5.5 Anisotropic Hyperelasticity 281
5.5.1 Generalized Fung Model 282
5.5.2 Invariant Based Anisotropy 282
5.5.3 Bischoff Anisotropic Eight-Chain Model 283
5.5.4 Bergstrom Anisotropic Eight-Chain Model 285
5.5.5 Holzapfel-Gasser-Ogden Model 285
5.6 Hyperelastic Foam Models 287
5.6.1 Blatz-Ko Foam Model 289
5.6.2 Hyperfoam Model 290
5.7 Mullins Effect Models 292
5.7.1 Ogden-Roxburgh Mullins Effect Model 293
5.7.2 Qi-Boyce Mullins Effect Model 295
5.8 Use of Hyperelasticity in Polymer Modeling 295
5.8.1 Experimental Testing 296
5.8.2 Drucker Stability 297
5.8.3 Determination of Material Parameters 298
5.8.4 Limitations of Hyperelasticity 298
5.9 Hyperelastic Code Examples 299
5.10 Exercises 303
References 304Contents ix
6 Linear Viscoelasticity 309
6.1 Introduction 310
6.2 Small Strain Linear Viscoelasticity 310
6.2.1 Stress Relaxation Behavior 313
6.2.2 Monotonic Loading Response 314
6.2.3 Cyclic Loading Response 320
6.2.4 Experimental Determination of the
Storage and Loss Moduli 322
6.2.5 Representing Linear Viscoelasticity
Using Spectra 328
6.2.6 Computer Implementation 329
6.3 Large Strain Linear Viscoelasticity 331
6.3.1 Numerical Implementation 332
6.4 Creep Compliance Behavior 335
6.4.1 Relationships Between Creep Compliance
and Relaxation Modulus 336
6.5 Differential Form of Linear Viscoelasticity 337
6.5.1 Rheological Models 338
6.6 The Use of Shift Functions to Generalize
Linear Viscoelasticity Theory 340
6.6.1 Time-Temperature Equivalence 341
6.6.2 Vertical Shifts 345
6.7 Use of Linear Viscoelasticity in Polymer
Modeling 345
6.8 Exercises 349
References 350
7 Plasticity Models 353
7.1 Introduction 353
7.2 J2-Plasticity with Isotropic Hardening 354
7.2.1 Cyclic Loading 355
7.2.2 Matlab Implementation 357
7.2.3 Python Implementation 359
7.2.4 Application to Thermoplastics 361
7.3 Plasticity with Kinematic Hardening 362
7.4 Johnson-Cook Plasticity 365
7.5 Drucker Prager Plasticity 366x Contents
7.6 Use of Plasticity Models in Polymer Modeling 367
7.7 Exercises 368
References 369
8 Viscoplasticity Models 371
8.1 Introduction 372
8.2 Bergström-Boyce Model 372
8.2.1 Matlab Implementation of the BB-Model 382
8.2.2 Python Implementation of the BB-Model 384
8.2.3 Generic Numerical Implementation 386
8.2.4 Dynamic Loading Predictions 387
8.2.5 Use of the BB-Model for
Polymer Modeling 392
8.3 Arruda-Boyce Model 393
8.4 Dual Network Fluoropolymer Model 397
8.4.1 Matlab Implementation of the DNF Model 404
8.4.2 Use of the DNF Model for
Polymer Modeling 404
8.5 Hybrid Model 409
8.5.1 Matlab Implementation of the
Hybrid Model 413
8.5.2 Use of the Hybrid Model for
Polymer Modeling 414
8.6 Three Network Model 417
8.6.1 Matlab Implementation of the
Three Network Model 422
8.6.2 Python Implementation of the
Three Network Model 422
8.6.3 Use of the Three Network Model for
Polymer Modeling 426
8.7 Parallel Network Model 427
8.8 Use of Viscoplasticity in Polymer Modeling 431
8.9 Python Code Examples 432
8.10 Exercises 434
References 435Contents xi
9 Determination of Material Parameters from
Experimental Data 437
9.1 Introduction 437
9.2 Mathematics of Material Parameter Determination 438
9.3 Initial Guess of the Material Parameters 440
9.4 Error Measurement Functions 442
9.5 Algorithms for Parameter Extraction 444
9.6 Exercises 445
References 446
10 User Material Subroutines 447
10.1 Introduction 447
10.2 Abaqus/Explicit VUMAT for the
Neo-Hookean Model 448
10.3 Abaqus/Implicit UMAT for the
Neo-Hookean Model 450
Reference 454
11 Material Modeling Case Studies 455
11.1 Introduction 455
11.2 Acrylate-Butadiene Rubber 455
11.3 Chloroprene Rubber 460
11.4 Nitrile Rubber 464
11.5 Santoprene 468
11.6 High-Density Polyethylene 474
11.7 Polytetrafluoroethylene 479
11.8 Polyethylene Terephthalate 487
11.9 Polyether Ether Ketone 490
11.10 Exercises 496
References 497
Index 49
Index
Note: Page numbers followed by f and t refers to figures and tables
respectively.
A
Abaqus/Explicit VUMAT
subroutine, for NeoHookean model, 448–450
Abaqus/Implicit UMAT
subroutine, for NeoHookean model, 450–454
AB model. See Arruda–Boyce
(AB) model
Acrylate-Butadiene rubber (ABR)
Bergstrom–Boyce model, 458,
459–460, 459f
calibrations, 456, 458t
linear viscoelasticity model,
458, 459f
mechanical response, 455–456
stress-time response, 456,
457f
uniaxial compression data, 456,
456f
Yeoh hyperelastic model, 456,
457f
Addition polymerization, 11
Adiabatic thermoelastic material,
194
Almansi strain, 163
Amorphous polymers, 5–7, 6f
Anisotropic elasticity, 215–217
Anisotropic hyperelasticity
Bergstrom anisotropic eightchain model, 285
Bischoff anisotropic eightchain model, 283–285
generalized Fung model, 282
Holzapfel–Gasser–Ogden
model, 285–287
invariant based anisotropy,
282–283
Anisotropic material, 199
Arrhenius model, 344–345
Arruda–Boyce (AB) model, 283
athermal shear resistance, 396
deviatoric back stress,
394–395
glassy polymers, 393–394
linear elastic response, 394
plastic flow rate, 395–396
rheological representation, 394,
394f
stress-strain predictions, 396,
397f
Atomic force microscopy (AFM),
87–88
B
Balance law, 171–184
Balance of angular momentum,
178–180
Balance of linear momentum,
175–178
BAM model, 275–277
Barcol hardness testing, 50
BB model. See Bergstrom–Boyce
(BB) model
Bergstrom anisotropic eight-chain
model, 285
499500 Index
Bergstrom–Boyce (BB) model,
27–28
acrylate-butadiene rubber, 458,
459–460, 459f
applied strain history, 372–374,
373f
Brownian motion, 379, 380f
Cauchy stress, 376–377
chain stretch, 379–381
chloroprene rubber, 372–374,
373f , 462, 463f
creep experiment, 379–381
crosslinked polymer,
378, 379f
dynamic loading predictions,
387–392
eight-chain model, 376–377
elastic and viscoelastic
components, 376, 376f
elastomers, 375
equilibrium stress, 374–375,
374f
generic numerical
implementation, 386–
387
hyperelastic response, 377
hypothetical stress-strain curve,
374–375
Matlab implementation,
382–384
nitrile rubber, 466, 467f
non-linear viscoelastic flow
element, 375
polyether ether ketone, 491,
493f
polymer modeling, 392–393
Python implementation,
384–386
Rouse relaxation time, 379–381
santoprene, 470–474, 473f ,
474f
strain amplitude dependence,
372
time derivative, 378
viscous components, 377–378
viscous flow, 381–382
Biot strain, 163
Birefringence spectroscopy,
95–97
Bischoff anisotropic eight-chain
model, 283–285
Blatz–Ko foam model, 289
Boltzmann’s superposition
principle, 310
Bulk modulus, 64–71, 241
Buna-N. See Nitrile rubber
C
Cauchy stress theorem,
176–178
Cauchy surface tractions, 176
Chemical characterization
techniques
energy dispersive X-ray
spectroscopy, 101–103
Fourier transform infrared
spectroscopy, 100–101
Raman spectroscopy, 109–110
size-exclusion chromatography,
103–107
thermogravimetric analysis,
107–109
Chloroprene rubber (CR)
BB model with Mullins
damage, 462, 463f
calibrations, 461–462, 461t
linear viscoelastic model, 462,
463f
stress relaxation response, 460,
461f
uniaxial compression data, 460,
460fIndex 501
uniaxial tension, 440, 441f
Yeoh hyperelastic model,
461–462, 462f
Coefficient of determination, 444
Condensation polymerization, 11
Conductive polymers, 9
Confocal microscopy, 83
Conservation of mass, 173–175
Continuum mechanics
foundations, 219–224
balance laws and field
equations, 171–184
constitutive equations, 187–194
coordinate transformations,
149
deformation gradient, 150–157
derivatives of scalar, vector,
and tensor fields, 147–149
Dyadic product, 143–144
energy balance and stress
power, 184–186
invariants, 150
large strain kinematics,
137–141
material symmetry, 198–199
multiaxial loading, 135–137
observer transformation,
194–198
rates of deformation, 164–165
strain, stretch, and rotation,
157–164
stress tensors, 165–170
symbols, 199
tensor operations, 144–147
uniaxial loading, 133–135
vector operations, 141–143
Coordinate transformations, 149
Corrugated hose failure, 127
CR. See Chloroprene rubber (CR)
Creep compliance
definition, 335
vs. relaxation modulus,
336–337
D
Dark field microscopy, 83
Deformation
modeling, 120–125
simple shear, 152
undeformed state, 151
uniaxial tension, 151
volumetric deformation, 153
Dependence of stored energy,
229–232
Differential interference contrast
(DIC) microscopy, 83
Differential scanning calorimetry
(DSC), 89–90
Digital image correlation (DIC)
strain measurement
system, 66
DNF model. See Dual network
fluoropolymer (DNF)
model
Drucker Prager plasticity,
366–367, 367f
Drucker stability, 297
Dual network fluoropolymer
(DNF) model, 121–122
Cauchy stress,
398–400
constant viscosity, 402
deviatoric viscoelastic flow,
401–402
kinematics of deformation,
398, 399f
material parameters, 403
Matlab implementation, 404
plastic flow, 402–403
polymer modeling, 404
strain rates, 397–398
structure, 398, 399f502 Index
Dual network fluoropolymer
(DNF) model (Continued)
thermal expansion, 398–400
thermoplastics, 398
velocity gradient, 401
viscoelastic deformation
gradient, 400–401
volumetric viscoelastic flow,
401–402
Dyadic product, 143–144
Dynamic mechanical analysis
(DMA), 43–47, 347
E
Eigenvalue and spectral
decompositions, 154–
157
Eight-chain (EC) model,
250–259
Elastomers, 24, 25f
Energy balance and stress power,
184–186
Energy dispersive X-ray
spectroscopy (EDS),
101–103
Entropy, 183
Environmental SEM (ESEM), 86
Environmental stress cracking
(ESC), 126
Euler–Almansi strain, 164
Extended tube (ET) model,
273–275
F
Failure model calibration, 73
Failure modeling, 125–130
FEA. See Finite element analysis
(FEA)
Fiber-reinforced composite, 217,
218f
Finite element analysis (FEA)
deformation modeling, 120–
125
failure modeling, 125–130
polymer mechanics, 115
properties of polymers and
metals, 116–117
required inputs, 117–118
types, 119
First law of thermodynamics,
180–182
First Piola–Kirchhoff stress
tensor, 167
Flex circuit pressure sensor, 124
Fluorescence microscopy, 84
Fourier transform approach, 326
Fourier transform infrared
spectroscopy (FTIR),
100–101
Freely jointed chain (FJC) model,
232–236
G
Gaussian chains, 258
Gel permeation chromatography
(GPC), 103–107
Generalized Fung model, 282
Genetic algorithm, 445
Gent model, 263–265
Glass transition temperature, PET,
24
Green–Lagrange strain, 163
H
Hardness and indentation testing,
47–51
HDPE. See High-density
polyethylene (HDPE)
Heaviside step function, 310
Helmholtz free energy,
191–192
Hencky strain, 163, 164Index 503
High-density polyethylene
(HDPE)
Arruda–Boyce eight-chain
model, 477–478, 478f
calibrations, 477, 477t
elastic-plastic material model,
477–478, 478f
linear viscoelastic model, 479,
479f
PN model, 479, 480f
power-flow model,
479, 481f
stress relaxation data, 474–476,
476f
stress-strain response, 474–476
uniaxial tension data, 476f
Holzapfel–Gasser–Ogden (HGO)
model, 285–287
Hooke’s law, 67–68, 211–212
Horgan and Saccomandi model,
265–266
Hybrid model (HM)
backstress network, 411
deformation map, 409, 410f
energy activation approach, 412
isotropic linear elasticity
expression, 410–411
Matlab implementation,
413–414
polymer modeling, 414–416
relative stiffness, 411
rheological representation, 409,
410f
strain elastic constants, 412
ultra-high molecular weight
polyethylene, 409
viscoelastic deformation
gradient, 412
viscoplastic flow, 411–412
Hyperelastic foam models
Blatz–Ko foam model, 289
hyperfoam model, 290–291
Hyperelasticity
code examples, 299–303
Drucker stability, 297
experimental testing, 296–297
limitations, 298–299
material parameters, 298
Hyperfoam model, 290–291
II1
and I2 model, 250
Impact testing, 40–43
Incompressible biaxial
deformation, 237–238
Incompressible planar
deformation, 237–238
Incompressible uniaxial
deformation, 237–238
Interface friction, 27–28
Invariant based anisotropy,
282–283
Inverse Langevin function,
256–257
Isothermal thermoelastic material,
194
Isotropic elasticity, 211–215
Isotropic hardening plasticity
model. See J2-plasticity,
isotropic hardening
Isotropic hyperelasticity
BAM model, 275–277
continuum mechanics
foundations, 219–224
dependence of stored energy,
229–232
eight-chain model, 250–259
extended tube model,
273–275
freely jointed chain model,
232–236
Gent model, 263–265504 Index
Isotropic hyperelasticity
(Continued)
Horgan and Saccomandi model,
265–266
I1 and I2 model, 250
Knowles hyperelastic model,
268–270
Mooney–Rivlin model,
243–245
Neo–Hookean model,
236–242
Ogden model, 259–261
predictive capabilities,
277–281
pure shear vs. planar tension,
226–228
response function
hyperelasticity, 270–
272
uniaxial compression vs.
biaxial tension, 225–226
Yeoh model, 245–248
Isotropic material, 199
J
Johnson–Cook plasticity model,
365–366, 366f
J2-plasticity, isotropic hardening
abacus, 354
ANSYS, 354
cyclic loading, 355–357, 356f
Matlab implementation,
357–359
Python implementation,
359–360, 359f , 360f
stress-strain representation,
355, 355f
UHMWPE thermoplastic
material, 361–362, 361f ,
362f
K
Kinematic hardening plasticity
model
Abaqus material definition,
363, 364, 365
backstress network, 363, 363f ,
364f , 365
Chaboche type, 362–363
limitations, 365
MCalibration software, 363
Knowles hyperelastic model,
268–270
L
Lagrangian and Eulerian
Formulations, 139
Large strain kinematics, 137–141
Large strain linear viscoelasticity
generalization, 331–332
hyperelastic stress function,
332
numerical implementation,
332–334
Linear elasticity
anisotropic elasticity, 215–217
isotropic elasticity, 211–215
transversely isotropic elasticity,
217–218
Linear viscoelasticity
creep compliance, 335–337
differential form, 337–340
large strain, 331–334
polymer modeling, 345–349
shift functions, 340–345
small strain, 310–331
Loss modulus, 322–323
M
Material parameters, 437
determination, 438–440Index 505
error measurement functions,
442–444
extraction, 439, 439f
find_material_params, 444–445
initial guess, 440–442,
441f
mathematical minimization
problem, 439–440
Monte Carlo method, 442
optimization algorithm,
444–445
prior knowledge, 442
Matlab implementation
Bergstrom–Boyce model,
382–384
dual network fluoropolymer
model, 404
hybrid model, 413–414
J2-plasticity, isotropic
hardening, 357–359
small strain linear
viscoelasticity, 329,
330f
three network model, 422
Maxwell rheological model,
338–339, 339f
MCalibration software, 445
Mechanical stress, 134
Metal plasticity model, 353
Mises stress, 123, 123f , 170
Monte Carlo method, 442
Mooney–Rivlin (MR) model,
243–245
Mullins effect models
Ogden–Roxburgh,
293–295
Qi–Boyce, 295
Multiaxial loading, 135–137
Multi-network Maxwell model,
340f
N
Nanoindentation, 51
Nanson’s formula, 156
Natural polymers, 4–5, 5f
NBR. See Nitrile rubber
Near-field scanning optical
microscopy (NSOM), 83
Nelder–Mead simplex method,
444–445
Neo–Hookean hyperelastic
material model
Abaqus/Explicit VUMAT,
448–450
Abaqus/Implicit UMAT,
450–454
stress, 447–448
Neo–Hookean (NH) model,
236–242
Neoprene. See Chloroprene
rubber (CR)
Nitrile rubber
BB model, 466, 467f
calibrations, 464, 465t
linear viscoelastic model, 466,
467f
stress-time response, 464, 465f
uniaxial compression data, 464,
464f
Yeoh hyperelastic model,
465–466, 466f
Nominal strain, 164
Nominal traction vector, 167
Normalized mean absolute
difference, 444
Normalized root-mean square
difference, 444
O
Ogden model, 259–261
Ogden–Roxburgh Mullins effect
model, 293–295506 Index
Optical microscopy, 81–84
Orthotropic elasticity,
216–217
P
Parallel network (PN) model,
427–431, 459–460
high-density polyethylene, 479,
480f
polyether ether ketone,
491–492, 494f
Payne effect, 348–349
PEEK. See Polyether ether ketone
(PEEK)
Plane strain tension, 33–37
Plasticity theory. See J2-plasticity,
isotropic hardening
Polarized light microscopy, 82
Polyether ether ketone (PEEK)
BB model, 491, 493f
calibrations, 490, 492t
force-displacement results,
494–495, 495f
Johnson–Cook plasticity
model, 491, 493f
PN model, 491–492, 494f
spherical indentation test,
495–496
TN model, 492, 494f
uniaxial tension and
compression data,
490, 491f
Polyethylene terephthalate (PET),
487–489
Polylactic acid (PLA), 7
Polymers
description, 1, 2–3
history, 7–10
manufacturing and processing,
11
mechanics, 11–15
plasticity models, 367–368
types, 4–7
Polypropylene (PP), 10
Polytetrafluoroethylene (PTFE)
calibrations, 484t
dual network fluoropolymer
model, 483–484, 486f
elastic-plastic material model,
482–483, 485f
mechanical behavior, 479–481
microporosity, 479–481
TN model, 484, 487f
volumetric compression data,
483f
yield stress, 479–481
PolyUMod library, 447–448
Powell method, 445
Pressure-volume-temperature
(PVT) testing, 66
Prony series, 315–316, 317f ,
336–337, 345–346
PTFE. See Polytetrafluoroethylene (PTFE)
Pure shear vs. planar tension,
226–228
Python implementation
Bergstrom–Boyce model,
384–386
J2-plasticity, isotropic
hardening, 359–360,
359f , 360f
small strain linear
viscoelasticity, 330–
331, 331f
three network model, 422
viscoplasticity models, 432–
434
Q
Qi–Boyce Mullins effect model,
295Index 507
R
Raman spectroscopy, 109–110
Rates of deformation, 164–165
Relaxation time spectrum, 328
Residual error
strain-controlled experiment,
442, 443f
stress-controlled experiment,
443, 443f
Response function hyperelasticity,
270–272
Retardation time spectrum, 328
Rheologically simple material,
342
Rheological models, 338–340,
339f , 340f
Rockwell hardness testing,
47–48
S
Santoprene
BB model, 470–474, 473f ,
474f , 475f
calibrations, 468, 470t
elastic-plastic material model,
469–470, 472f , 473f
isotropic hardening plasticity
model with ratedependence, 469,
472f
linear viscoelastic model, 469,
471f
uniaxial tensile stress-strain
data, 468, 468f , 469f
Yeoh hyperelastic model,
468–469, 471f
Scanning electron microscopy
(SEM), 84–86
Second law of thermodynamics,
183–184
Semicrystalline polymers, 5–7, 6f
Shear and bulk relaxation moduli,
312–313
Shear modulus, 239
Shore (durometer) testing, 48–49
Simple anisotropic hyperelastic
model, 283
Simple shear, 37–39, 152
Size-exclusion chromatography
(SEC), 103–107
Small-angle X-ray diffraction, 95
Small punch testing, 77–79
Small-strain classical theory, 135
Small strain linear viscoelasticity
applied strain history,
311, 312f
Boltzmann’s superposition
principle, 310
characteristic relaxation time,
313
cyclic loading response,
320–322
Heaviside step function, 310
Matlab implementation, 329,
330f
mat_LVE( ) function, 329
monotonic loading response,
314–320, 317f
Prony series, 315–316, 317f
Python implementation,
330–331, 331f
relaxation time spectrum, 328
retardation time spectrum, 328
shear and bulk relaxation
moduli, 312–313
storage and loss modulus,
322–327
stress relaxation, 310, 311f ,
313, 314f
stretched exponential stress
relaxation modulus,
316–318, 318f , 319f508 Index
Small strain linear viscoelasticity
(Continued)
test_mat_LVE function, 329,
330f
Spatial velocity gradient,
164–165
Spin tensor, 164–165
Split-Hopkinson pressure bar
(SHPB) testing, 53–63
Stereo microscopy, 84
Storage modulus, 322–323
Strain matrix, 136
Stress invariants, 169–170
Stress-strain response, 24
Stress tensors, 165–170
Surface characterization
techniques
atomic force microscopy,
87–88
optical microscopy, 81–84
scanning electron microscopy,
84–86
Swell testing, 97–99
Synthetic polymers, 4–5, 5f
T
Tensor operations, 144–147
Thermoelastic material,
189–194
Thermogravimetric analysis
(TGA), 107–109
Thermomechanical deformations,
121
Thermoplastics, 5, 6f , 24, 26f
Thermoplastic vulcanizates
(TPV). See Santoprene
Thermosets, 5, 6f , 24, 27f
Threaded connection gasket, 121
Three network model (TNM),
459–460
arbitrary rigid body rotation,
420–421
Cauchy–Green deformation
tensor, 417–418
deformation gradient, 417–418
elastic and viscous components,
419–420
flow rate, 419–420
material parameters, 421, 421t
Matlab implementation, 422
plastic strain, 418–419
polyether ether ketone, 492,
494f
polymer modeling, 426
polytetrafluoroethylene, 484,
487f
Python implementation, 422
rheological representation, 417,
417f
shear modulus, 419
viscoelastic deformation
gradient, 418–419
Time shifts, 342
Time-temperature equivalence,
341–345, 341f , 343f , 344f
TNM. See Three network model
(TNM)
Transmission electron microscopy
(TEM), 90–91
Transversely isotropic elasticity,
217–218
Tresca stress, 170
U
Ultra-high molecular
weight polyethylene
(UHMWPE), 213–214,
214f
isotropic hardening plasticity
model, 361–362Index 509
Johnson–Cook model, 365–
366, 366f
kinematic hardening plasticity
model, 362–365
linear viscoelasticity
application, 346, 346f
Uniaxial compression
vs. biaxial tension,
225–226
testing, 24–29
Uniaxial loading, 133–135
Uniaxial tension, 29–33, 151
User material subroutines
Abaqus/Explicit VUMAT,
448–450
Abaqus/Implicit UMAT,
450–454
description, 447–448
purpose, 447–448
V
Vector and tensor algebra,
141–150
Vertical shifts, 345
Viscoplastic deformations, 130
Viscoplasticity models
Arruda–Boyce model, 393–396
Bergstrom–Boyce model,
372–393
dual network fluoropolymer
model, 397–404
hybrid model, 409–416
parallel network model,
427–431
polymer modeling, 431–432
Python code examples,
432–434
three network model,
417–426
V-notch shear testing, 80
Volume characterization
techniques
birefringence, 95–97
differential scanning
calorimetry, 89–90
swell testing, 97–99
transmission electron
microscopy, 90–91
X-ray diffraction, 92–95
Volumetric deformation, 153
Vulcanized natural rubber,
8–9
W
Water filter failure, 126
Wide-angle X-ray diffraction,
93–94
William–Landel–Ferry (WLF)
equation, 343, 344, 344f
Work conjugate stress, 185
X
X-ray diffraction (XRD), 92–95
Y
Yeoh hyperelastic model, 456,
457f
acrylate-butadiene rubber, 456
chloroprene rubber, 461–462,
462f
nitrile rubber, 465–466, 466f
santoprene, 468–469, 471f
Yeoh model, 245–248
Young’s modulus, 23, 23f

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