Preface ....................................................... xvii
1 Introduction: Phenomena ....................................... 1
1.1 Viscoelastic Phenomena ................................... 1
1.2 Motivations for Studying Viscoelasticity ................. 3
1.3 Transient Properties: Creep and Relaxation ............... 3
1.3.1 Viscoelastic Functions J(t), E(t) ................. 3
1.3.2 Solids and Liquids ................................ 7
1.4 Dynamic Response to Sinusoidal Load: E*, tanδ ............ 8
1.5 Demonstration of Viscoelastic Behavior .................. 10
1.6 Historical Aspects ...................................... 10
1.7 Summary ................................................. 11
1.8 Examples ................................................ 11
1.9 Problems ................................................ 12
Bibliography ................................................. 12
2 Constitutive Relations ....................................... 14
2.1 Introduction ............................................ 14
2.2 Prediction of the Response of Linearly Viscoelastic
Materials ............................................... 14
2.2.1 Prediction of Recovery from Relaxation E{t) ...... 14
2.2.2 Prediction of Response to Arbitrary Strain
History .......................................... 15
2.3 Restrictions on the Viscoelastic Functions .............. 17
2.3.1 Roles of Energy and Passivity .................... 17
2.3.2 Fading Memory .................................... 18
2.4 Relation between Creep and Relaxation ................... 19
2.4.1 Analysis by Laplace Transforms: J(t) ↔ E(t) ...... 19
2.4.2 Analysis by Direct Construction: J(t) ↔ E(t) ..... 20
2.5 Stress versus Strain for Constant Strain Rate ........... 20
2.6 Particular Creep and Relaxation Functions ............... 21
2.6.1 Exponentials and Mechanical Models ............... 21
2.5 Exponentials and Internal Causal Variables .............. 26
2.6.3 Fractional Derivatives ........................... 27
2.6.4 Power-Law Behavior ............................... 28
2.6.5 Stretched Exponential ............................ 29
2.6.6 Logarithmic Creep; Kuhn Model .................... 29
2.6.7 Distinguishing among Viscoelastic Functions ...... 30
2.7 Effect of Temperature ................................... 30
2.8 Three-Dimensional Linear Constitutive Equation .......... 33
2.9 Aging Materials ......................................... 35
2.10 Dielectric and Other Forms of Relaxation ................ 35
2.11 Adaptive and "Smart" Materials .......................... 36
2.12 Effect of Nonlinearity .................................. 37
2.12.1 Constitutive Equations ........................... 37
2.12.2 Creep-Relaxation Interrelation: Nonlinear ........ 40
2.13 Summary ................................................. 43
2.14 Examples ................................................ 43
2.15 Problems ................................................ 51
Bibliography ................................................. 52
3 Dynamic Behavior ............................................. 55
3.1 Introduction and Rationale .............................. 55
3.2 The Linear Dynamic Response Functions E*, tanδ .......... 56
3.2.1 Response to Sinusoidal Input ..................... 57
3.2.2 Dynamic Stress-Strain Relation ................... 59
3.2.3 Standard Linear Solid ............................ 62
3.3 Kramers-Kronig Relations ................................ 63
3.4 Energy Storage and Dissipation .......................... 65
3.5 Resonance of Structural Members ......................... 67
3.5.1 Resonance, Lumped System ......................... 67
3.5.2 Resonance, Distributed System .................... 71
3.6 Decay of Resonant Vibration ............................. 74
3.7 Wave Propagation and Attenuation ........................ 77
3.8 Measures of Damping ..................................... 79
3.9 Nonlinear Materials ..................................... 79
3.10 Summary ................................................. 81
3.11 Examples ................................................ 81
3.12 Problems ................................................ 88
Bibliography ................................................. 89
4 Conceptual Structure of Linear Viscoelasticity ............... 91
4.1 Introduction ............................................ 91
4.2 Spectra in Linear Viscoelasticity ....................... 92
4.2.1 Definitions H(τ), L(τ) and Exact Interrelations .. 92
4.2.2 Particular Spectra ............................... 93
4.3 Approximate Interrelations of Viscoelastic Functions .... 95
4.3.1 Interrelations Involving the Spectra ............. 95
4.3.2 Interrelations Involving Measurable Functions .... 98
4.3.3 Summary, Approximate Relations .................. 101
4.4 Conceptual Organization of the Viscoelastic Functions .. 101
4.5 Summary ................................................ 104
4.6 Examples ............................................... 104
4.7 Problems ............................................... 109
Bibliography ................................................ 109
5 Viscoelastic Stress and Deformation Analysis ................ 111
5.1 Introduction ........................................... 111
5.2 Three-Dimensional Constitutive Equation ................ 111
5.3 Pure Bending by Direct Construction .................... 112
5.4 Correspondence Principle ............................... 114
5.5 Pure Bending by Correspondence ......................... 116
5.6 Correspondence Principle in Three Dimensions ........... 116
5.6.1 Constitutive Equations .......................... 116
5.6.2 Rigid Indenter on a Semi-Infinite Solid ......... 117
5.6.3 Viscoelastic Rod Held at Constant Extension ..... 119
5.6.4 Stress Concentration ............................ 119
5.6.5 Saint Venant's Principle ........................ 120
5.7 Poisson's Ratio ν(t) ................................... 121
5.7.1 Relaxation in Tension ........................... 121
5.7.2 Creep in Tension ................................ 123
5.8 Dynamic Problems: Effects of Inertia ................... 124
5.8.1 Longitudinal Vibration and Waves in a Rod ....... 124
5.8.2 Torsional Waves and Vibration in a Rod .......... 125
5.8.3 Bending Waves and Vibration ..................... 128
5.8.4 Waves in Three Dimensions ....................... 129
5.9 Noncorrespondence Problems ............................. 131
5.9.1 Solution by Direct Construction: Example ........ 131
5.9.2 A Generalized Correspondence Principle .......... 132
5.9.3 Contact Problems ................................ 132
5.10 Bending inJNonlinear Viscoelasticity ................... 133
5.11 Summary ................................................ 134
5.12 Examples ............................................... 134
5.13 Problems ............................................... 142
Bibliography ................................................ 142
6 Experimental Methods ........................................ 145
6.1 Introduction and General Requirements .................. 145
6.2 Creep .................................................. 146
6.2.1 Creep: Simple Methods to Obtain J(t) ............ 146
6.2.2 Effect of Risetime in Transient Tests ........... 146
6.2.3 Creep in Anisotropic Media ...................... 148
6.2.4 Creep in Nonlinear Media ........................ 148
6.3 Inference of Moduli .................................... 150
6.3.1 Use of Analytical Solutions ..................... 150
6.3.2 Compression of a Block .......................... 151
6.4 Displacement and Strain Measurement .................... 152
6.5 Force Measurement ...................................... 156
6.6 Load Application ....................................... 157
6.7 Environmental Control .................................. 157
6.8 Subresonant Dynamic Methods ............................ 158
6.8.1 Phase Determination ............................. 158
6.8.2 Nonlinear Materials ............................. 160
6.8.3 Rebound Test .................................... 161
6.9 Resonance Methods ...................................... 161
6.9.1 General Principles .............................. 161
6.9.2 Particular Resonance Methods .................... 163
6.9.3 Methods for Low-Loss or High-Loss Materials ..... 166
6.9.4 Resonant Ultrasound Spectroscopy ................ 168
6.10 Achieving a Wide Range of Time or Frequency ............ 171
6.10.1 Rationale ....................................... 171
6.10.2 Multiple Instruments and Long Creep ............. 172
6.10.3 Time-Temperature Superposition .................. 172
6.11 Test Instruments for Viscoelasticity ................... 173
6.11.1 Servohydraulic Test Machines .................... 173
6.11.2 A Relaxation Instrument ......................... 174
6.11.3 Driven Torsion Pendulum Devices ................. 174
6.11.4 Commercial Viscoelastic Instrumentation ......... 178
6.11.5 Instruments for a Wide Range of Time and
Frequency ....................................... 179
6.11.6 Fluctuation-Dissipation Relation ................ 182
6.11.7 Mapping Properties by Indentation ............... 183
6.12 Wave Methods ........................................... 184
6.13 Summary ................................................ 188
6.14 Examples ............................................... 188
6.15 Problems ............................................... 200
Bibliography ................................................ 201
7 Viscoelastic Properties of Materials ........................ 207
7.1 Introduction ........................................... 207
7.1.1 Rationale ....................................... 207
7.1.2 Overview: Some Common Materials ................. 207
7.2 Polymers ............................................... 208
7.2.1 Shear and Extension in Amorphous Polymers ....... 208
7.2.2 Bulk Relaxation in Amorphous Polymers ........... 212
7.2.3 Crystalline Polymers ............................ 213
7.2.4 Aging and other Relaxations ..................... 214
7.2.5 Piezoelectric Polymers .......................... 214
7.2.6 Asphalt ......................................... 214
7.3 Metals ................................................. 215
7.3.1 Linear Regime of Metals ......................... 215
7.3.2 Nonlinear Regime of Metals ...................... 217
7.3.3 High-Damping Metals and Alloys .................. 219
7.3.4 Creep-Resistant Alloys .......................... 224
7.3.5 Semiconductors and Amorphous Elements ........... 225
7.3.6 Semiconductors and Acoustic Amplification ....... 226
7.3.7 Nanoscale Properties ............................ 226
7.4 Ceramics ............................................... 227
7.4.1 Rocks ........................................... 227
7.4.2 Concrete ........................................ 229
7.4.3 Inorganic Glassy Materials ...................... 231
7.4.4 Ice ............................................. 231
7.4.5 Piezoelectric Ceramics .......................... 232
7.5 Biological Composite Materials ......................... 233
7.5.1 Constitutive Equations .......................... 234
7.5.2 Hard Tissue: Bone ............................... 234
7.5.3 Collagen, Elastin, Proteoglycans ................ 236
7.5.4 Ligament and Tendon ............................. 237
7.5.5 Muscle .......................................... 240
7.5.6 Fat ............................................. 243
7.5.7 Brain ........................................... 243
7.5.8 Vocal Folds ..................................... 244
7.5.9 Cartilage and Joints ............................ 244
7.5.10 Kidney and Liver ................................ 246
7.5.11 Uterus and Cervix ............................... 246
7.5.12 Arteries ........................................ 247
7.5.13 Lung ............................................ 248
7.5.14 The Ear ......................................... 248
7.5.15 The Eye ......................................... 249
7.5.16 Tissue Comparison ............................... 251
7.5.17 Plant Seeds ..................................... 252
7.5.18 Wood ............................................ 252
7.5.19 Soft Plant Tissue: Apple, Potato ................ 253
7.6 Common Aspects ......................................... 253
7.6.1 Temperature Dependence .......................... 253
7.6.2 High-Temperature Background ..................... 254
7.6.3 Negative Damping and Acoustic Emission .......... 255
7.7 Summary ................................................ 255
7.8 Examples ............................................... 255
7.9 Problems ................................................ 256
Bibliography ................................................ 257
8 Causal Mechanisms ........................................... 271
8.1 Introduction ........................................... 271
8.1.1 Rationale ....................................... 271
8.1.2 Survey of Viscoelastic Mechanisms ............... 271
8.1.3 Coupled Fields .................................. 273
8.2 Thermoelastic Relaxation ............................... 274
8.2.1 Thermoelasticity in One Dimension ............... 274
8.2.2 Thermoelasticity in Three Dimensions ............ 275
8.2.3 Thermoelastic Relaxation Kinetics ............... 276
8.2.4 Heterogeneity and Thermoelastic Damping ......... 278
8.2.5 Material Properties and Thermoelastic Damping ... 280
8.3 Relaxation by Stress-Induced Fluid Motion .............. 280
8.3.1 Fluid Motion in One Dimension ................... 280
8.3.2 Biot Theory: Fluid Motion in Three Dimensions ... 281
8.4 Relaxation by Molecular Rearrangement .................. 286
8.4.1 Glassy Region ................................... 286
8.4.2 Transition Region ............................... 287
8.4.3 Rubbery Behavior ................................ 289
8.4.4 Crystalline Polymers ............................ 291
8.4.5 Biological Macromolecules ....................... 292
8.4.6 Polymers and Metals ............................. 292
8.5 Relaxation by Interface Motion ......................... 292
8.5.1 Grain Boundary Slip in Metals ................... 292
8.5.2 Interface Motion in Composites .................. 294
8.5.3 Structural Interface Motion ..................... 294
8.6 Relaxation Processes in Crystalline Materials .......... 294
8.6.1 Snoek Relaxation: Interstitial Atoms ............ 294
8.6.2 Zener Relaxation in Alloys: Pairs of Atoms ...... 298
8.6.3 Gorsky Relaxation ............................... 299
8.6.4 Granato-Lücke Relaxation: Dislocations .......... 300
8.6.5 Bordoni Relaxation: Dislocation Kinks ........... 303
8.6.6 Relaxation Due to Phase Transformations ......... 305
8.6.7 High-Temperature Background ..................... 314
8.6.8 Nonremovable Relaxations ........................ 315
8.6.9 Damping Due to Wave Scattering .................. 316
8.7 Magnetic and Piezoelectric Materials ................... 316
8.7.1 Relaxation in Magnetic Media .................... 316
8.7.2 Relaxation in Piezoelectric Materials ........... 318
8.8 Nonexponential Relaxation .............................. 322
8.9 Concepts for Material Design ........................... 323
8.9.1 Multiple Causes: Deformation Mechanism Maps ...... 323
8.9.2 Damping Mechanisms in High-Loss Alloys .......... 326
8.9.3 Creep Mechanisms in Creep-Resistant Alloys ...... 326
8.10 Relaxation at Very Long Times .......................... 327
8.11 Summary ................................................ 327
8.12 Examples ............................................... 328
8.13 Problems and Questions ................................. 332
Bibliography ................................................ 332
9 Viscoelastic Composite Materials ............................ 341
9.1 Introduction ........................................... 341
9.2 Composite Structures and Properties .................... 341
9.2.1 Ideal Structures ................................ 341
9.2.2 Anisotropy due to Structure ..................... 342
9.3 Prediction of Elastic and Viscoelastic Properties ...... 344
9.3.1 Basic Structures: Correspondence Solutions ...... 344
9.3.2 Voigt Composite ................................. 345
9.3.3 Reuss Composite ................................. 345
9.3.4 Hashin-Shtrikman Composite ...................... 346
9.3.5 Spherical Particulate Inclusions ................ 347
9.3.6 Fiber Inclusions ................................ 349
9.3.7 Platelet Inclusions ............................. 349
9.3.8 Stiffness-Loss Maps ............................. 350
9.4 Bounds on the Viscoelastic Properties .................. 353
9.5 Extremal Composites .................................... 354
9.6 Biological Composite Materials ......................... 356
9.7 Poisson's Ratio of Viscoelastic Composites ............. 357
9.8 Particulate and Fibrous Composite Materials ............ 358
9.8.1 Structure ....................................... 358
9.8.2 Particulate Polymer Matrix Composites ........... 359
9.8.3 Fibrous Polymer Matrix Composites ............... 361
9.8.4 Metal-Matrix Composites ......................... 362
9.9 Cellular Solids ........................................ 363
9.10 Piezoelectric Composites ............................... 366
9.11 Dispersion of Waves in Composites ...................... 366
9.12 Summary ................................................ 367
9.13 Examples ............................................... 367
9.14 Problems ............................................... 370
Bibliography ................................................ 370
10 Applications and Case Studies ............................... 377
10.1 Introduction ........................................... 377
10.2 A Viscoelastic Earplug: Use of Recovery ................ 377
10.3 Creep and Relaxation of Materials and Structures ....... 378
10.3.1 Concrete ........................................ 378
10.3.2 Wood ............................................ 378
10.3.3 Power Lines ..................................... 379
10.3.4 Glass Sag: Flowing Window Panes ................. 380
10.3.5 Indentation: Road Rutting ....................... 380
10.3.6 Leather ......................................... 381
10.3.7 Creep-Resistant Alloys and Turbine Blades ....... 381
10.3.8 Loosening of Bolts and Screws ................... 382
10.3.9 Computer Disk Drive: Case Study of Relaxation ... 384
10.3.10 Earth, Rock, and Ice ........................... 385
10.3.11 Solder ......................................... 386
10.3.12 Filaments in Light Bulbs and Other Devices ..... 387
10.3.13 Tires: Flat-Spotting and Swelling .............. 388
10.3.14 Cushions for Seats and Wheelchairs ............. 388
10.3.15 Artificial Joints .............................. 389
10.3.16 Dental Fillings ................................ 389
10.3.17 Food Products .................................. 389
10.3.18 Seals and Gaskets .............................. 390
10.3.19 Relaxation in Musical Instrument Strings ....... 390
10.3.20 Winding of Tape ................................ 391
10.4 Creep and Recovery in Human Tissue ..................... 391
10.4.1 Spinal Discs: Height Change ..................... 391
10.4.2 The Nose ........................................ 392
10.4.3 Skin ............................................ 392
10.4.4 The Head ........................................ 393
10.5 Creep Damage and Creep Rupture ........................ 394
10.5.1 Vajont Slide .................................... 394
10.5.2 Collapse of a Tunnel Segment .................... 394
10.6 Vibration Control and Waves ........................... 394
10.6.1 Analysis of Vibration Transmission .............. 394
10.6.2 Resonant (Tuned) Damping ........................ 397
10.6.3 Rotating Equipment Vibration .................... 397
10.6.4 Large Structure Vibration: Bridges and
Buildings ....................................... 398
10.6.5 Damping Layers for Plate and Beam Vibration ..... 399
10.6.6 Structural Damping Materials .................... 400
10.6.7 Piezoelectric Transducers ....................... 402
10.6.8 Aircraft Noise and Vibration .................... 402
10.6.9 Solid Fuel Rocket Vibration ..................... 404
10.6.10 Sports Equipment Vibration ..................... 404
10.6.11 Seat Cushions and Automobiles: Protection of
People ......................................... 404
10.6.12 Vibration in Scientific Instruments ............ 406
10.6.13 Waves .......................................... 406
10.7 "Smart" Materials and Structures ....................... 407
10.7.1 "Smart" Materials ............................... 407
10.7.2 Shape Memory Materials .......................... 408
10.7.3 Self-Healing Materials .......................... 409
10.7.4 Piezoelectric Solid Damping ..................... 409
10.7.5 Active Vibration Control: "Smart" Structures .... 409
10.8 Rolling Friction ....................................... 409
10.8.1 Rolling Analysis ................................ 410
10.8.2 Rolling of Tires ................................ 411
10.9 Uses of Low-Loss Materials ............................. 412
10.9.1 Timepieces ...................................... 412
10.9.2 Frequency Stabilization and Control ............. 413
10.9.3 Gravitational Measurements ...................... 413
10.9.4 Nanoscale Resonators ............................ 414
10.10 Impulses, Rebound, and Impact Absorption .............. 414
10.10.1 Rationale ...................................... 414
10.10.2 Analysis ....................................... 415
10.10.3 Bumpers and Pads ............................... 418
10.10.4 Shoe Insoles, Athletic Tracks, and Glove
Liners ......................................... 419
10.10.5 Toughness of Materials ......................... 419
10.10.6 Tissue Viscoelasticity in Medical Diagnosis .... 420
10.11 Rebound of a Ball ..................................... 421
10.11.1 Analysis ....................................... 421
10.11.2 Applications in Sports ......................... 422
10.12 Applications of Soft Materials ........................ 424
10.12.1 Viscoelastic Gels in Surgery ................... 424
10.12.2 Hand Strength Exerciser ........................ 424
10.12.3 Viscoelastic Toys .............................. 424
10.12.4 No-Slip Flooring, Mats, and Shoe Soles ......... 425
10.13 Applications Involving Thermoviscoelasticity .......... 425
10.14 Satellite Dynamics and Stability ...................... 426
10.15 Summary ............................................... 428
10.16 Examples .............................................. 429
10.17 Problems .............................................. 431
Bibliography ................................................ 431
A Appendix .................................................... 441
A.1 Mathematical Preliminaries ............................. 441
A.l.l Introduction .................................... 441
A.1.2 Functionals and Distributions ................... 441
A.1.3 Heaviside Unit Step Function .................... 442
A.1.4 Dirac Delta ..................................... 442
A.1.5 Doublet ......................................... 443
A.1.6 Gamma Function .................................. 445
A.1.7 Liebnitz Rule ................................... 445
A.2 Transforms ............................................. 445
A.2.1 Laplace Transform ............................... 446
A.2.2 Fourier Transform ............................... 446
A.2.3 Hartley Transform ............................... 447
A.2.4 Hilbert Transform ............................... 447
A.3 Laplace Transform Properties ........................... 448
A.4 Convolutions ........................................... 449
A.5 Interrelations in Elasticity Theory .................... 451
A.6 Other Works on Viscoelasticity ......................... 451
Bibliography ................................................ 452
B Symbols ..................................................... 455
B.l Principal Symbols ...................................... 455
Index .......................................................... 457
|
