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Contents

Preface page xvii
0 G/ k! K# W+ l9 e( {- V# x1 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
9 p& d7 L& \, ]" R! T  Z4 z1.3.2 Solids and Liquids 7
7 n7 n" O9 V3 D( _1 p7 l- x* \1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
; N5 l3 t2 Y7 \; M1 X1.5 Demonstration of Viscoelastic Behavior 10
1.6 Historical Aspects 10
1.7 Summary 11
# M& k' a* R( ^6 G1.8 Examples 11
: t% p( N5 q( C% |6 r8 N3 I1.9 Problems 12
Bibliography 12


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2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Introduction 14
2.2 Prediction of the Response of Linearly Viscoelastic Materials 14
/ F  ]) e# E! p  A2.2.1 Prediction of Recovery from Relaxation E(t) 14
' [* F$ I5 Y  d* @9 b4 [9 P2.2.2 Prediction of Response to Arbitrary Strain History 15
: L) R' u8 ~: u# E( b2.3 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
' c4 q; x. h7 S1 S* ?2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20
2.5 Stress versus Strain for Constant Strain Rate 20
7 }' I. E* q! _7 j4 J: J, ]2.6 Particular Creep and Relaxation Functions 21
2.6.1 Exponentials and Mechanical Models 21
2.6.2 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
( E9 M3 x  m$ _+ ]2 L  s, u2 }2.6.7 Distinguishing among Viscoelastic Functions 30
2.7 Effect of Temperature 30
2.8 Three-Dimensional Linear Constitutive Equation 33
1 R2 d# S9 d$ a2.9 Aging Materials 35
9 q' `" _% l- Q+ |2.10 Dielectric and Other Forms of Relaxation 35
0 L0 x; G: o. S8 v. F  \2.11 Adaptive and “Smart” Materials 36
6 e) ]. ^6 [2 z8 w6 V2.12 Effect of Nonlinearity 37
2.12.1 Constitutive Equations 37
) C8 ~9 X7 g; m5 R4 C$ @2.12.2 Creep–Relaxation Interrelation: Nonlinear 40
2.13 Summary 43
1 r7 Q5 s' k5 w4 M2.14 Examples 43
2.15 Problems 51
1 z  L! H" V8 w9 X! hBibliography 52

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* Z, e5 d, |: ]; R) W* v$ m9 g3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 d# a) ?) v# _) f; @5 D/ N3.1 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 t# \6 X( v: ]/ A8 j! \3.2.3 Standard Linear Solid 62
3.3 Kramers–Kronig Relations 63
3.4 Energy Storage and Dissipation 65
+ z- K( ~8 N$ K! ]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 y6 {5 {; E* A3 M; G1 [- O3.8 Measures of Damping 79
' x# A7 P$ s" R% Q9 y7 w3.9 Nonlinear Materials 79
7 U' Y3 M% p+ \1 x+ |3.10 Summary 81
# v3 Z: t( b2 n: U: Z. B. V% k8 y2 ?3.11 Examples 81
3.12 Problems 88
! G3 p! d  m5 c& `% [* r% f) Z0 Y+ aBibliography 89



( v* y  Z2 X1 F$ O+ c2 {4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91
" A. b# G/ N8 j  E; E/ b& `4.1 Introduction 91
) s) X: n( |: u! V4.2 Spectra in Linear Viscoelasticity 92
4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 92
4.2.2 Particular Spectra 93
- ?; |7 m! K) F- E% N5 o4.3 Approximate Interrelations of Viscoelastic Functions 95
4.3.1 Interrelations Involving the Spectra 95
4.3.2 Interrelations Involving Measurable Functions 98
  R/ n3 o) J; n- k4.3.3 Summary, Approximate Relations 101
4.4 Conceptual Organization of the Viscoelastic Functions 101
! K% ]" o. W# t0 I4.5 Summary 104
) k6 p+ K0 d& A+ Y( h4.6 Examples 104
/ J! Y) c) `+ ^+ U4.7 Problems 109
# z; ?' l6 B6 ^* c. Z4 gBibliography 109
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111
0 f* Y+ d6 C9 z- f/ E8 _5.1 Introduction 111
5.2 Three-Dimensional Constitutive Equation 111
! X# X  [/ q2 x0 D5.3 Pure Bending by Direct Construction 112
5.4 Correspondence Principle 114
5.5 Pure Bending by Correspondence 116
+ N: s) ]) ?9 e- v+ [- e8 P5.6 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
' y( D+ N3 X7 y* r7 a# k* l5.7.1 Relaxation in Tension 121
5.7.2 Creep in Tension 123
5.8 Dynamic Problems: Effects of Inertia 124
3 @) C5 Z+ k  x1 v7 C: j0 h$ k! ^5.8.1 Longitudinal Vibration and Waves in a Rod 124
5.8.2 Torsional Waves and Vibration in a Rod 125
" I6 L( d& u/ R8 p( l* i2 f5.8.3 Bending Waves and Vibration 128
# s+ j7 ]8 V& V- E5.8.4 Waves in Three Dimensions 129
5.9 Noncorrespondence Problems 131
5.9.1 Solution by Direct Construction: Example 131
- v# _  g, H% t8 c1 U5.9.2 A Generalized Correspondence Principle 132
+ X/ G5 h: @1 ~+ r% T1 `5.9.3 Contact Problems 132
5.10 Bending in Nonlinear Viscoelasticity 133
5.11 Summary 134
$ S% e7 ~1 I5 s/ q8 E/ p; K5.12 Examples 134
5.13 Problems 142
! c; h& e4 L/ m3 g% _/ F% @# pBibliography 142



& }' r" k) f1 L1 l3 F6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
# U4 S. L  Q# B* c5 G) `  u6.1 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
- W% j( e% H1 m1 f% {0 B6.2.3 Creep in Anisotropic Media 148
- F3 W9 l5 d' n1 p( X: B6.2.4 Creep in Nonlinear Media 148
6 g3 _1 x- Q: j& w9 s0 B6.3 Inference of Moduli 150
5 d2 S; i% z1 Z- T6.3.1 Use of Analytical Solutions 150
4 M2 e7 q" K0 u$ Y1 n* S6.3.2 Compression of a Block 151
6.4 Displacement and Strain Measurement 152
6.5 Force Measurement 156
6.6 Load Application 157
5 U+ ^8 h: M' D6 W7 s& {6.7 Environmental Control 157
6.8 Subresonant Dynamic Methods 158
6.8.1 Phase Determination 158
" ?7 S1 O7 y6 G- I0 A6.8.2 Nonlinear Materials 160
6.8.3 Rebound Test 161
. h; W! u* F" [7 ?' k, F6.9 Resonance Methods 161
6.9.1 General Principles 161
) {+ M- C6 z1 D6.9.2 Particular Resonance Methods 163
8 Y7 h* s8 a4 m) Z4 K: O6.9.3 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
. z) q9 F$ w& d- e; L6.10.3 Time–Temperature Superposition 172
0 C- r, `& e# ~) g+ c6.11 Test Instruments for Viscoelasticity 173
! p  F5 L0 @! T9 Q* T& D6.11.1 Servohydraulic Test Machines 173
/ H9 W' O' r: N7 D( ~5 V6.11.2A Relaxation Instrument 174
6.11.3 Driven Torsion Pendulum Devices 174
0 T' z* Y( I5 }6.11.4 Commercial Viscoelastic Instrumentation 178
2 F9 v& t6 f+ H4 ]1 _1 I% c# [- E* R& ^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
) L5 O8 ^: C# H9 _1 H" W0 N& g2 g6.13 Summary 188
6.14 Examples 188
/ N, ?, `( @7 L9 m, G2 M9 M! g& ~0 I6.15 Problems 200
Bibliography 201



7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207
7.1 Introduction 207
. [' D% Z" N! ^+ R7.1.1 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
0 t, H- l& M5 B+ X7.3 Metals 215
8 U# C2 l" X, a4 j* e5 w$ d" ^7.3.1 Linear Regime of Metals 215
7.3.2 Nonlinear Regime of Metals 217
8 F* Z, {1 v' m/ }4 G$ c/ e7.3.3 High-Damping Metals and Alloys 219
7.3.4 Creep-Resistant Alloys 224
7.3.5 Semiconductors and Amorphous Elements 225
" @+ L9 Q! s" i' |7.3.6 Semiconductors and Acoustic Amplification 226
7.3.7 Nanoscale Properties 226
8 N8 y0 h0 g$ D7.4 Ceramics 227
6 _" d- D$ z& I1 ^. A" U. V  m7.4.1 Rocks 227
7.4.2 Concrete 229
7.4.3 Inorganic Glassy Materials 231
& ^% \$ E7 a7 j  G7.4.4 Ice 231
6 h3 G' d0 V$ Y$ y- L1 K8 ~# p: \7.4.5 Piezoelectric Ceramics 232
7.5 Biological Composite Materials 233
/ `) E% o3 F/ [7 r, j& y. m3 g7.5.1 Constitutive Equations 234
7.5.2 Hard Tissue: Bone 234
( w0 a8 f* A% K6 l4 U( U; [. j7.5.3 Collagen, Elastin, Proteoglycans 236
7.5.4 Ligament and Tendon 237
# f& ]1 N5 }! c# M2 O: S7.5.5 Muscle 240
7.5.6 Fat 243
' K1 w, z% ^& R" b# J7.5.7 Brain 243
" m, w8 y+ M% W3 _! I" ]1 s7.5.8 Vocal Folds 244
7 z2 E0 T. d: F  w6 o2 E7.5.9 Cartilage and Joints 244
7.5.10 Kidney and Liver 246
9 v+ Z5 i# D& X1 Q) k7.5.11 Uterus and Cervix 246
  s) a% @, q+ k( N' |3 X+ v7.5.12 Arteries 247
7.5.13 Lung 248
1 j) u3 P% U- P' G' N7.5.14 The Ear 248
+ ~8 L, {7 s  Z2 }, d* C7.5.15 The Eye 249
7.5.16 Tissue Comparison 251
7.5.17 Plant Seeds 252
7.5.18 Wood 252
; z; }/ `1 j8 A0 r8 o& ^7.5.19 Soft Plant Tissue: Apple, Potato 253
- w& ^( ]) \+ B: m7.6 Common Aspects 253
1 d6 S( z% s; s; _2 i; T. D7.6.1 Temperature Dependence 253
7.6.2 High-Temperature Background 254
% E! N, T% p8 r5 l) D, w5 \7.6.3 Negative Damping and Acoustic Emission 255
7.7 Summary 255
+ b2 ?4 J" R' {* b2 m3 i. j, B7.8 Examples 255
, p5 {& x! Q5 W5 t2 {& V7.9 Problems 256
9 y, x6 F6 x% O& L. d4 BBibliography 257

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8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
' M8 I6 E& G' d: ?8.1 Introduction 271
8.1.1 Rationale 271
, G& I' D# e% Z8.1.2 Survey of Viscoelastic Mechanisms 271
8.1.3 Coupled Fields 273
: x5 W# K: t5 c  ^8.2 Thermoelastic Relaxation 274
2 X$ Q1 j/ |. \  A& A8.2.1 Thermoelasticity in One Dimension 274
8.2.2 Thermoelasticity in Three Dimensions 275
1 T% ~' C# I: M/ T: z; w8 K  r8.2.3 Thermoelastic Relaxation Kinetics 276
8.2.4 Heterogeneity and Thermoelastic Damping 278
/ \2 x3 m2 `2 C8.2.5 Material Properties and Thermoelastic Damping 280
3 n1 Q+ h$ a" c0 k& z8.3 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
& k3 d1 d9 n9 B- v2 O& c8.4 Relaxation by Molecular Rearrangement 286
7 f% z9 B5 x- h$ p: U# g8.4.1 Glassy Region 286
8.4.2 Transition Region 287
/ J+ E8 W! r7 C8.4.3 Rubbery Behavior 289
8.4.4 Crystalline Polymers 291
8.4.5 Biological Macromolecules 292
7 J# P' F+ {8 _2 e% J7 H8.4.6 Polymers and Metals 292
" ^/ a: r+ ]. c; O% W8.5 Relaxation by Interface Motion 292
, c8 Q  ~) n" h* X" q& D. F8.5.1 Grain Boundary Slip in Metals 292
6 U- Z( x5 A8 P# y( J/ S) c8.5.2 Interface Motion in Composites 294
8.5.3 Structural Interface Motion 294
! S  x, S' ~) L" H. n$ u: ?8.6 Relaxation Processes in Crystalline Materials 294
8.6.1 Snoek Relaxation: Interstitial Atoms 294
2 K- L5 @8 x& G6 J! _- f6 e2 f8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298
8.6.3 Gorsky Relaxation 299
8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300
! C3 f0 N: C6 j4 V$ j9 [0 s5 k( }8.6.5 Bordoni Relaxation: Dislocation Kinks 303
0 b+ A- z4 }- K8.6.6 Relaxation Due to Phase Transformations 305
1 f) r! ]7 d$ C+ ~- R8.6.7 High-Temperature Background 314
/ Z0 L  |! t% W& [' p# K. ]8.6.8 Nonremovable Relaxations 315
, _  \/ D2 Q4 y8.6.9 Damping Due to Wave Scattering 316
4 V  z+ v( i0 Q% ~8.7 Magnetic and Piezoelectric Materials 316
8.7.1 Relaxation in Magnetic Media 316
1 N! W) H! G4 g8.7.2 Relaxation in Piezoelectric Materials 318
8.8 Nonexponential Relaxation 322
8.9 Concepts for Material Design 323
9 w. C, B, h. X. R& V9 S8.9.1 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
0 _. r" U1 ^( Y+ C0 f8.10 Relaxation at Very Long Times 327
8.11 Summary 327
% i+ [" h) y& e7 W+ q2 x8.12 Examples 328
8.13 Problems and Questions 332
* Y0 _" |1 |5 D7 n) C, QBibliography 332
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9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341
9.1 Introduction 341
9.2 Composite Structures and Properties 341
, \1 @* Y! y3 f, Q; U8 \, r$ E9.2.1 Ideal Structures 341
/ N+ H! l9 {; `2 q6 o1 |9.2.2 Anisotropy due to Structure 342
8 `& [9 d3 Q1 F& U2 M8 G9 Y9.3 Prediction of Elastic and Viscoelastic Properties 344
, N' H0 z) a  R  u% E; R: s9.3.1 Basic Structures: Correspondence Solutions 344
9.3.2 Voigt Composite 345
9.3.3 Reuss Composite 345
9 I! S! D6 y6 x" Y* [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
: m" u/ Z! A  G: R$ w7 z7 o9.3.8 Stiffness-Loss Maps 350
9.4 Bounds on the Viscoelastic Properties 353
, d6 j  ?% i# I  a9.5 Extremal Composites 354
  C# ~. n" w0 Q8 z* F; Q9.6 Biological Composite Materials 356
: a* f, q7 X2 O9.7 Poisson’s Ratio of Viscoelastic Composites 357
0 Q$ P- \" W8 `0 }& r3 c9 o9.8 Particulate and Fibrous Composite Materials 358
% T% V0 [4 j# C0 j9.8.1 Structure 358
9.8.2 Particulate Polymer Matrix Composites 359
7 C% b4 N7 |* L5 O" v9.8.3 Fibrous Polymer Matrix Composites 361
- j7 b6 D* F$ L" d3 C7 ]4 J% [9.8.4 Metal–Matrix Composites 362
. L, m) {/ z3 t) E1 E9.9 Cellular Solids 363
3 ~1 P: v( k8 q: U2 _* |/ C# j9.10 Piezoelectric Composites 366
9.11 Dispersion of Waves in Composites 366
9.12 Summary 367
9.13 Examples 367
  E5 {$ N2 a' |, L/ ?" ^9.14 Problems 370
Bibliography 370
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4 [( ?4 b0 j7 o( N6 s0 e, n* v- D10 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
- G" N% C1 }7 s10.3.2 Wood 378
: A. x/ i& o- l7 M* l0 |% f8 `10.3.3 Power Lines 379
: B( P1 R0 T; U6 b9 a10.3.4 Glass Sag: Flowing Window Panes 380
) E8 D0 H& a4 I3 t. U9 a  X' V10.3.5 Indentation: Road Rutting 380
( ~% |% j6 M8 c# t/ o1 s- z10.3.6 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
- }6 ~1 Q8 r" B# @10.3.10 Earth, Rock, and Ice 385
10.3.11 Solder 386
7 u6 q7 c: ^( P- l& n1 {5 a2 ]3 a10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
: j" r5 ?& M3 R' e. f10.3.13Tires: Flat-Spotting and Swelling 388
10.3.14Cushionsfor Seats and Wheelchairs 388
10.3.15 Artificial Joints 389
' q- I5 s0 O- B) u' w1 e0 o+ C; ]10.3.16 Dental Fillings 389
10.3.17 Food Products 389
10.3.18 Seals and Gaskets 390
0 y: V1 q* J1 ?10.3.19 Relaxationi nM usical Instrument Strings 390
3 C. s* r2 N: T10.3.20 Winding of Tape 391
5 E& ^6 n! c  A+ P10.4 Creep and Recovery in Human Tissue 391
10.4.1 Spinal Discs: Height Change 391
. d+ f$ Z# D4 Y, F6 C# w& |10.4.2 The Nose 392
: a$ \( n6 o+ G" `9 g10.4.3 Skin 392
10.4.4 The Head 393
10.5 Creep Damage and Creep Rupture 394
& Q! O6 ~2 K  W( T" _6 C. E% L% i10.5.1 Vajont Slide 394
! e$ X  m6 i7 [$ }6 ]2 L10.5.2 Collapse of a Tunnel Segment 394
; R( g' @$ k/ V$ b& S! v3 {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
* x0 R2 L$ V. f: V9 u; _7 Q/ m; F10.6.6 Structural Damping Materials 400
10.6.7 Piezoelectric Transducers 402
0 I* Z' v9 o/ E' m2 y; g10.6.8 Aircraft Noise and Vibration 402
6 s0 m) b$ Q/ ~+ }8 E  M10.6.9 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 Vibrationi n ScientificI nstruments 406
1 Z( s9 P+ K5 p10.6.13 Waves 406
2 x" c2 X: T: z7 k$ w: [) _$ _10.7 “Smart” Materials and Structures 407
10.7.1 “Smart” Materials 407
& {9 l& N  l& ~" B9 L8 z10.7.2 Shape Memory Materials 408
10.7.3 Self-Healing Materials 409
1 b. k5 p$ S. b# z) z/ |1 ^10.7.4 Piezoelectric Solid Damping 409
) ^% q) w# e' ]+ |! N10.7.5 Active Vibration Control: “Smart” Structures 409
: Z- t8 u& ~# R! A" v. \' w10.8 Rolling Friction 409
; b4 I9 b, }% z4 U10.8.1 Rolling Analysis 410
) Q9 d) M" [1 [( G10.8.2 Rolling of Tires 411
10.9 Uses of Low-Loss Materials 412
" h3 v5 n1 Q- ^. ~. B/ j10.9.1 Timepieces 412
" ^" G$ |8 u: x9 h! f  Q( P* n10.9.2 Frequency Stabilization and Control 413
6 W, i, Q4 m6 j8 [2 {10.9.3 Gravitational Measurements 413
; l9 O! i* W7 K% B* A) Z. C10.9.4 Nanoscale Resonators 414
2 |# x7 L9 K. r* e10.10 Impulses, Rebound, and Impact Absorption 414
* P, Z4 B$ p' H+ |4 W9 `4 J10.10.1 Rationale 414
7 e$ r0 C% j) d* [% `3 p" t: F! E10.10.2 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
  P3 P/ a& i+ T% u  m" }7 n10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420
) }* G* V$ m- D10.11Rebound of a Ball 421
' ^% Z8 i- ^) J. m3 }3 D: ^10.11.1 Analysis 421
10.11.2 Applications in Sports 422
10.12 Applications of Soft Materials 424
$ Y6 k; H$ P5 V8 M# r0 G3 ^: y2 P10.12.1 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
; ~5 @* v! W$ A0 k10.13 Applications Involving Thermoviscoelasticity 425
10.14 Satellite Dynamics and Stability 426
' c) M" P5 W( h$ f: X10.15 Summary 428
10.16 Examples 429
10.17 Problems 431
Bibliography 431



A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
A.1 Mathematical Preliminaries 441
9 N  Q& a5 B3 o1 l2 o0 P+ |A.1.1 Introduction 441
0 U0 M  U6 \3 W. o! L2 \A.1.2 Functionals and Distributions 441
A.1.3 Heaviside Unit Step Function 442
A.1.4 Dirac Delta 442
% u% p" k9 l5 |8 l) H& t( _A.1.5 Doublet 443
( {+ _: L' _  v5 g8 OA.1.6 Gamma Function 445
+ K) I& `! D% T1 uA.1.7 Liebnitz Rule 445
. h* Q: n  l: h/ @+ K. YA.2 Transforms 445
6 ~% K$ W  {" k: t2 T6 c, l5 ]2 EA.2.1 Laplace Transform 446
: p4 K1 H) C! T$ }$ b- L3 nA.2.2 Fourier Transform 446
0 B3 g$ S( d1 h2 I. L9 u& H! aA.2.3 Hartley Transform 447
; I; l& Z5 F: k$ |. S; nA.2.4 Hilbert Transform 447
A.3 Laplace Transform Properties 448
1 j* D- O& z1 r- dA.4 Convolutions 449
A.5 Interrelations in Elasticity Theory 451
- K4 n% F& \- s8 a* K0 B* tA.6 Other Works on Viscoelasticity 451
Bibliography 452

2 t3 v9 I5 F3 y4 ]( t% g
" `& {: l5 w3 |% V1 ]! TB: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
+ C$ ~2 C* k* }8 ~9 aB.1 Principal Symbols 455
  s/ ~; y; a6 Z! F+ HIndex 457
  K1 L- P3 ~% u1 a/ z" j
5 |/ N6 [; j2 p9 d$ b