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Viscoelastic Materials Roderic Lakes 2009 Part 1-2.rar
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Viscoelastic Materials Roderic Lakes 2009 Part 2-2.rar
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( a" ]" t$ y3 N8 A& c" n$ L) L目录9 s, d ?2 l% Y5 Q# s# Q6 C( d y; J
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Preface page xvii
9 T6 |0 ~% m% [, T1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1" t' U' b$ A/ Y s/ R* O
1.1 Viscoelastic Phenomena 13 h: \; H& b7 k/ ]
1.2 Motivations for Studying Viscoelasticity 3
0 q# t6 p% d0 ]1.3 Transient Properties: Creep and Relaxation 3+ i( t$ N5 A. T
1.3.1 Viscoelastic Functions J (t), E(t) 3
! \) [" F6 ^$ M- M- l1.3.2 Solids and Liquids 7# f3 r! j! J# [1 J4 B; y& w
1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 87 m8 V: `! [2 Y2 s
1.5 Demonstration of Viscoelastic Behavior 10# R$ |0 j( ?6 C5 L) P
1.6 Historical Aspects 10
, w R4 \9 h# ~7 F( z1.7 Summary 11
; D5 l3 D$ Y4 e1.8 Examples 11. [+ D, g' l* ?7 a1 Q1 C- C+ x
1.9 Problems 12' x/ F2 I5 q7 a) G- b8 |
Bibliography 12
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9 X6 ~* v4 Y, |) a* u$ U! ?2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
$ x' x. Y5 C! o4 F5 Y2.1 Introduction 14
3 n) I+ q& V" D6 K) [2 P$ L/ d5 W2.2 Prediction of the Response of Linearly Viscoelastic Materials 14
: P$ \0 r4 U7 ~& Z/ n) a- }% G7 r2.2.1 Prediction of Recovery from Relaxation E(t) 14
/ Y# h7 ~6 H' h2.2.2 Prediction of Response to Arbitrary Strain History 15
' C3 k% @ k9 s2.3 Restrictions on the Viscoelastic Functions 17
- a# t" b2 Q# d& ]0 c2.3.1 Roles of Energy and Passivity 17
; k {; R9 f7 W" Z8 c2.3.2 Fading Memory 18* a/ E* }5 k" N! } C8 V7 m3 P
2.4 Relation between Creep and Relaxation 19
$ H; Q; `. T; P& S- \. ~6 C# \2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19
' \" r! ]' [& y% Z0 i/ X2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20% q" d: j z3 z% v ^
2.5 Stress versus Strain for Constant Strain Rate 20+ O: v% n) g/ n) f5 V
2.6 Particular Creep and Relaxation Functions 21, {, N2 O; e7 L9 g. w
2.6.1 Exponentials and Mechanical Models 21% I- h7 O" p8 M h
2.6.2 Exponentials and Internal Causal Variables 26- ^3 y* b$ A" F8 y) n6 V1 _
2.6.3 Fractional Derivatives 27
0 F- G2 c# t1 }4 D- f0 Z6 U& f2.6.4 Power-Law Behavior 282 K7 J, A2 Y$ M! L/ G) ]$ j
2.6.5 Stretched Exponential 29
# ^8 D7 ^6 p# s1 F2.6.6 Logarithmic Creep; Kuhn Model 29! v9 Z* l: S2 C2 Z- F! T5 H1 C i
2.6.7 Distinguishing among Viscoelastic Functions 30
: E0 ]5 |( L0 p2.7 Effect of Temperature 30' ]& s- C* v/ J, Q
2.8 Three-Dimensional Linear Constitutive Equation 33$ p6 M) I+ L V0 v" p/ w1 z7 E7 [
2.9 Aging Materials 35
4 @) o, i" o1 y- M" q( v2.10 Dielectric and Other Forms of Relaxation 35
: S6 ?8 P: g% f: X2.11 Adaptive and “Smart” Materials 365 q# U+ ^" ]! W
2.12 Effect of Nonlinearity 37, h- l" ^& }$ s4 R; F0 c
2.12.1 Constitutive Equations 37$ a) i0 L: g' a4 |" u
2.12.2 Creep–Relaxation Interrelation: Nonlinear 40
% n7 O' M" e3 m+ b' `/ f5 _9 |1 `2.13 Summary 43
7 k7 D% a& V- t) ~9 W1 M1 r2 I2.14 Examples 43
* v8 J I! l, L/ ], Q2.15 Problems 51 i9 r$ E e0 Q5 Y; \+ D+ R; ~
Bibliography 52
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8 a5 u$ w) W& Y$ I3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 [$ u- [7 G! [* _7 d3.1 Introduction and Rationale 558 g h: P o2 T! y3 x
3.2 The Linear Dynamic Response Functions E∗, tanδ 563 B0 B1 _& {9 i! |, `4 C
3.2.1 Response to Sinusoidal Input 57! }: |- D6 M' N# J# G+ p
3.2.2 Dynamic Stress–Strain Relation 599 e7 g8 [; |. P1 b
3.2.3 Standard Linear Solid 62& l9 U7 y- @/ L
3.3 Kramers–Kronig Relations 63
. z5 }8 `+ e/ s! L# ^6 A4 B" @0 o3.4 Energy Storage and Dissipation 65; C, I4 h( E% c6 z4 D u
3.5 Resonance of Structural Members 67
5 _* u s# r2 |5 k3 @ T3 v3.5.1 Resonance, Lumped System 67
/ O# X s7 n7 s0 ]0 o3.5.2 Resonance, Distributed System 71
! l% `3 h; b0 t$ D3.6 Decay of Resonant Vibration 74
% T4 _& W; X& w9 l3.7 Wave Propagation and Attenuation 77
; \9 M+ v. k8 r! U# u1 _5 m5 }) l3.8 Measures of Damping 794 i3 M/ h( C T! `
3.9 Nonlinear Materials 79
5 Q0 g* x/ H y3.10 Summary 81
) N/ g5 @: I5 O& r; W3.11 Examples 81
' ]1 @8 B' \; z1 P: n; Q+ Y* V3.12 Problems 880 V: t1 K B) P) h+ [0 Y7 v! |( |
Bibliography 89
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/ Z& w% G4 r \4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91: E6 g3 y$ M) P7 u0 e
4.1 Introduction 91# H3 E |" A. ?7 [" N
4.2 Spectra in Linear Viscoelasticity 92( C/ U- {; i( S* d
4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 928 V5 P( a) ^6 J9 m0 P
4.2.2 Particular Spectra 93
3 x$ d) m' s# _- ^4.3 Approximate Interrelations of Viscoelastic Functions 95
5 w4 _$ D8 d$ l. x/ M( i1 }4 w4.3.1 Interrelations Involving the Spectra 958 L9 }6 A' d% n' N9 h X2 Y& _9 ?
4.3.2 Interrelations Involving Measurable Functions 98; y# S( F/ f% _2 H
4.3.3 Summary, Approximate Relations 101% v% V, U# h, R2 ]
4.4 Conceptual Organization of the Viscoelastic Functions 101
3 ?( d2 X' M; K/ e- C6 i4.5 Summary 104
, ^# X, v ?) l' N) e0 q1 V4.6 Examples 1044 M$ `$ |0 [8 ^' b9 z
4.7 Problems 1093 P0 Q% O) {4 o2 [: P1 Y. l* q
Bibliography 109
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& r: g! i' [, Z- e- z5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111
1 ^% k- w" h# _4 _% R4 t" n5.1 Introduction 111
& m# F& D: n' [8 ]" V4 D5.2 Three-Dimensional Constitutive Equation 1118 Z8 i& Y% H H i- w
5.3 Pure Bending by Direct Construction 112
8 z. w4 e7 F* {0 s- f8 K% u3 |5.4 Correspondence Principle 114* g$ H9 F3 }" n3 J- u! Z- L5 j
5.5 Pure Bending by Correspondence 116; n& C) g4 `5 Z2 z6 X
5.6 Correspondence Principle in Three Dimensions 116 H; [- ^: L, C! A; R2 A
5.6.1 Constitutive Equations 116
0 K& D4 M0 }4 N" t+ W# l* t. J5.6.2 Rigid Indenter on a Semi-Infinite Solid 1173 Z. j& D- ]9 L9 b. ~6 W9 [
5.6.3 Viscoelastic Rod Held at Constant Extension 119% h0 e$ @6 }! j( \1 q& N/ r
5.6.4 Stress Concentration 1195 f/ `' @: P+ |" a6 c& t# e
5.6.5 Saint Venant’s Principle 120+ `( i9 n: y- n" d) l
5.7 Poisson’s Ratio ν(t) 121; a1 u- Y* ?0 _. y( {
5.7.1 Relaxation in Tension 121
j7 a- s2 `! ?+ M. Z8 _5.7.2 Creep in Tension 123
5 o6 n, s8 d, D: \4 A' _) S8 B5.8 Dynamic Problems: Effects of Inertia 124
" c$ k! s' T) z! P5.8.1 Longitudinal Vibration and Waves in a Rod 1246 C8 d( ~4 h @) ?
5.8.2 Torsional Waves and Vibration in a Rod 125
2 y3 `' J. ^9 b. x& ^5.8.3 Bending Waves and Vibration 128
2 p3 @( M, w! m2 j2 o2 ^5.8.4 Waves in Three Dimensions 129
' i; D' t5 M: E5.9 Noncorrespondence Problems 131
; J" [- O4 B+ T0 w+ U5.9.1 Solution by Direct Construction: Example 131
1 h* {) A) H8 W7 b. F5.9.2 A Generalized Correspondence Principle 132
, }9 w: n( i2 P+ ~$ R: J5.9.3 Contact Problems 1325 D! T6 D/ m1 w/ U
5.10 Bending in Nonlinear Viscoelasticity 133
0 a5 u! I* L+ k- l5.11 Summary 134
, { N4 L# h: x# \5.12 Examples 134
5 U! h6 ~6 {# {5.13 Problems 142
P9 j& |8 M/ _% ]7 B* |Bibliography 142
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6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
?* a! B, ~+ {) t: j U6.1 Introduction and General Requirements 145
+ O2 j% n% }- S: ]6.2 Creep 146# o, }6 x1 Q; }7 O
6.2.1 Creep: Simple Methods to Obtain J (t) 146
; x5 B9 `7 g8 r3 s6.2.2 Effect of Risetime in Transient Tests 146) y7 o3 U* b% C, Y
6.2.3 Creep in Anisotropic Media 148% t- o0 w! r9 d/ D
6.2.4 Creep in Nonlinear Media 148+ u$ q: l5 j1 L1 q) v5 {6 Z! P
6.3 Inference of Moduli 150
0 }/ v+ f; W& A" ^. e% G% H0 b6.3.1 Use of Analytical Solutions 1505 l* [: ]5 b' u! B- V
6.3.2 Compression of a Block 151' T( I% {- k' B) d' N& ]) m6 D
6.4 Displacement and Strain Measurement 152) v# Q8 U( O& K( B! u; F, E
6.5 Force Measurement 156
! A2 n( i) Z4 `# P+ U) S6.6 Load Application 157
" v1 t: }0 R: X3 A8 S' o% L# N6.7 Environmental Control 157# q* m+ H* b9 H; I8 G
6.8 Subresonant Dynamic Methods 158
0 q2 ^0 d8 @8 K4 g8 {6.8.1 Phase Determination 158
: a z1 a% K8 B, D* @- h3 b# ^' P6.8.2 Nonlinear Materials 160- d6 @6 i' V. A' H1 s9 @5 R
6.8.3 Rebound Test 161
* L* V8 w( E h6 Z6.9 Resonance Methods 161
! A8 {! `9 ~ n6.9.1 General Principles 161
" ?5 |8 D5 V& ^- p% I+ e6.9.2 Particular Resonance Methods 163
4 U# }' _: G" s2 x' ^" B# @6.9.3 Methods for Low-Loss or High-Loss Materials 166
+ C* Y, O. k+ T8 d1 @* H6 Z% j6.9.4 Resonant Ultrasound Spectroscopy 168: [; g" j( E" J6 l$ ` R
6.10 Achieving a Wide Range of Time or Frequency 171
" m( [) @ t# K" s F6.10.1 Rationale 171
2 j1 E$ g0 `- n* L3 P6.10.2 Multiple Instruments and Long Creep 172, y' T3 i) g! l+ W
6.10.3 Time–Temperature Superposition 1727 G R2 S' i% R; o' l# ]6 n5 d
6.11 Test Instruments for Viscoelasticity 1735 `% J3 p9 ~8 P- m
6.11.1 Servohydraulic Test Machines 173
8 p+ l8 x4 v4 } L L6 c" q3 x4 c6.11.2A Relaxation Instrument 174
R! i1 [! a, `6 l& F( |+ o0 \6.11.3 Driven Torsion Pendulum Devices 1740 H' c, e& ]) Z2 J; F
6.11.4 Commercial Viscoelastic Instrumentation 178
5 c7 l+ {* {3 p- e9 z6.11.5 Instruments for a Wide Range of Time and Frequency 179
% s, u) [! j" i0 H' K% p6.11.6 Fluctuation–Dissipation Relation 182
. Q0 n& i9 o, q6.11.7 Mapping Properties by Indentation 183* p# C1 J& ?8 E3 D$ D: x( y
6.12 Wave Methods 184
( R7 b/ a( s2 S8 T+ n( N+ }6.13 Summary 188
$ o4 }5 y* u1 \; ^- M6.14 Examples 1881 [7 \' T- r0 }3 V7 W
6.15 Problems 2001 A2 n& F8 H8 B0 w& o. z# G
Bibliography 201
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7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207
, A( H+ K8 @. n) m& W7.1 Introduction 207
, V i G3 q/ u+ s# ?% B7.1.1 Rationale 207( ]9 _, Y# P2 g9 c0 o) N
7.1.2 Overview: Some Common Materials 207
, g" M) ]9 Y2 k2 ?5 [7.2 Polymers 208: F3 C! B& k7 A- C9 L+ G7 C
7.2.1 Shear and Extension in Amorphous Polymers 208
: U: _% J+ r/ Y- R* ^3 q' U7.2.2 Bulk Relaxation in Amorphous Polymers 212
8 d, B; a2 M5 f7.2.3 Crystalline Polymers 2135 y. g0 O5 D6 w$ Y) l% ?( ]
7.2.4 Aging and other Relaxations 214! x3 k# p- Q" @1 h/ @5 @
7.2.5 Piezoelectric Polymers 214! K# y; l& E j/ C% L. J2 M, Z
7.2.6 Asphalt 214! Y! ]% Q: w3 t+ i6 D
7.3 Metals 215
, c4 q' B, v0 Q9 s( i4 ^4 B7.3.1 Linear Regime of Metals 215, {2 @/ U; i6 O2 G2 I3 \9 K
7.3.2 Nonlinear Regime of Metals 2172 S \$ U8 a2 A9 T) r0 Z9 Z5 G
7.3.3 High-Damping Metals and Alloys 219
% o' E9 G% m! \( j& u7.3.4 Creep-Resistant Alloys 224% U: r; p& G0 J+ J( A* } \: j
7.3.5 Semiconductors and Amorphous Elements 225
9 q0 c1 u- W$ p: V8 O3 i$ I7.3.6 Semiconductors and Acoustic Amplification 226
# d% o& Z# j" u* L7.3.7 Nanoscale Properties 226
4 i) [+ t& b7 l9 K7 u3 J) h1 y7.4 Ceramics 227( h0 G8 u1 D/ m0 I: |
7.4.1 Rocks 227
/ W: Z' T" S, O5 a0 y4 s7.4.2 Concrete 229$ u" X2 N, e/ M# G6 p
7.4.3 Inorganic Glassy Materials 231. j- I) o# {0 ]/ ?2 k/ Z
7.4.4 Ice 231
?! u2 G* ]8 i6 `7.4.5 Piezoelectric Ceramics 232$ M3 k% z1 M4 ]4 e
7.5 Biological Composite Materials 233) V+ _! }2 _$ |$ p+ l* m! Q# @: R
7.5.1 Constitutive Equations 234
9 k1 _, {+ z, Y7.5.2 Hard Tissue: Bone 234; P8 b2 q7 v" i7 Y( P' _5 E$ c B
7.5.3 Collagen, Elastin, Proteoglycans 236% [+ @& u1 j1 k3 B6 b
7.5.4 Ligament and Tendon 237' }* O9 f K8 y$ }
7.5.5 Muscle 240
9 i( ~* m5 `7 @2 c) a7.5.6 Fat 243
# r, v. w; \. g" j- S5 @8 A7.5.7 Brain 243
8 w, R* i x& G1 Y7.5.8 Vocal Folds 244' i7 ~2 [2 [" K+ E# T# y% V& |* |
7.5.9 Cartilage and Joints 244. ^1 K9 {, O; l
7.5.10 Kidney and Liver 246# l( G. J, ], O0 X$ R# a
7.5.11 Uterus and Cervix 246& ^7 d5 {( b5 D
7.5.12 Arteries 247. Q, B3 _0 n% W9 j
7.5.13 Lung 248
: a* P5 S% G$ ]$ L* N7 k3 ^. ?7.5.14 The Ear 248
3 ]* t8 I) N$ ~7.5.15 The Eye 249: u9 W+ W2 C) L& @" }* h+ V
7.5.16 Tissue Comparison 251+ d% P" r R5 V f0 c
7.5.17 Plant Seeds 252! ^- q3 `2 x( S2 \4 V
7.5.18 Wood 252
4 X y# w) n0 I! ~7.5.19 Soft Plant Tissue: Apple, Potato 253% `* B0 c% c; }+ m& M$ e d
7.6 Common Aspects 253
+ p2 p2 p: N* B* l, n7.6.1 Temperature Dependence 2530 X p" I2 I) o, w- |8 V/ f, J$ {
7.6.2 High-Temperature Background 2542 y) i: `0 a4 [) i
7.6.3 Negative Damping and Acoustic Emission 255
2 ?. t! R: @" m$ A7.7 Summary 2558 r1 F7 ~/ ~+ \( V+ A
7.8 Examples 255. @* B. @9 y, q4 c& K1 x0 v' F
7.9 Problems 256
+ {5 ]$ ^8 o$ @0 _Bibliography 257
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8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271, F" u* M& X: M/ K
8.1 Introduction 2711 F8 Q, o1 d' p' k: B B. Z! g$ ?3 w
8.1.1 Rationale 271' x8 I$ _! U A) t0 ]
8.1.2 Survey of Viscoelastic Mechanisms 271
4 I; ]- D. Y7 Z( _8.1.3 Coupled Fields 273
6 Z4 P* R1 h2 ^% \+ y6 U7 {/ O8.2 Thermoelastic Relaxation 274
- b: U4 H2 m' N' @6 `; {3 s, Q8.2.1 Thermoelasticity in One Dimension 274
+ _! N- r. |. P, k8.2.2 Thermoelasticity in Three Dimensions 275, D4 ~. I7 R. M& B! J( r$ V1 P$ ~9 d
8.2.3 Thermoelastic Relaxation Kinetics 276/ M" h, `) F/ [, t6 Y- ?. Z; u1 N) K
8.2.4 Heterogeneity and Thermoelastic Damping 278
6 `2 p8 P* J) p. }8.2.5 Material Properties and Thermoelastic Damping 280
6 |* B" Y7 Q6 o% n8.3 Relaxation by Stress-Induced Fluid Motion 280
. @# t0 l2 Q7 v8 U# @" U9 F2 Z, o8.3.1 Fluid Motion in One Dimension 280
9 l9 Z9 m7 J2 u: k3 `. O! H6 V) L8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281
/ c, U& w3 d' u. ~8.4 Relaxation by Molecular Rearrangement 286
: k6 d; y$ i9 u9 X6 Q8 S2 O8.4.1 Glassy Region 286
+ s4 Q6 u! w( r- z- b( ]8.4.2 Transition Region 287, N* C0 f9 {. {' D$ w w
8.4.3 Rubbery Behavior 2892 p9 l1 s7 t. P2 I6 }% j: O
8.4.4 Crystalline Polymers 291
! D! q5 O( [1 h# C8.4.5 Biological Macromolecules 292; ]7 v, X/ @' h0 l1 L% Z$ {( l2 d
8.4.6 Polymers and Metals 292
3 j% a: e0 c4 Y8 U% Y/ t2 s/ E8.5 Relaxation by Interface Motion 292
6 q/ L, q0 z4 Q( \8 t+ l: {8.5.1 Grain Boundary Slip in Metals 292, X2 [/ v% v1 b ]9 F3 X
8.5.2 Interface Motion in Composites 294' D/ k6 K! \7 S8 b2 s# o/ C" g
8.5.3 Structural Interface Motion 294
6 y" s+ c2 V' U8.6 Relaxation Processes in Crystalline Materials 294) i4 O- C, [1 ~, i7 F; H Q2 V
8.6.1 Snoek Relaxation: Interstitial Atoms 294( ^- P: Q+ `. n U. v* D
8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298
8 u- K1 N0 K' R0 z" s8.6.3 Gorsky Relaxation 299
9 \) B- B" x5 b- [" ?' t8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300
1 b+ V7 c+ L3 f* a; r+ c8.6.5 Bordoni Relaxation: Dislocation Kinks 303
; W! x6 J* A7 X8.6.6 Relaxation Due to Phase Transformations 3056 B1 S( w a: o+ K. D4 B0 @( f2 W
8.6.7 High-Temperature Background 3147 r$ D' B J' E4 N
8.6.8 Nonremovable Relaxations 3154 Z [4 x# L0 \& _
8.6.9 Damping Due to Wave Scattering 316
_* H9 G9 \0 m* [6 [! A8.7 Magnetic and Piezoelectric Materials 316
' ]5 P0 _1 T& ]- g5 ^3 B/ j% d8 ~) F1 B8.7.1 Relaxation in Magnetic Media 316
7 l% k$ O- `( {8.7.2 Relaxation in Piezoelectric Materials 3186 X3 Q+ T3 B0 d; D
8.8 Nonexponential Relaxation 322+ }( O( d/ U" v7 R
8.9 Concepts for Material Design 323
5 e- g. G: H' t) @( [0 \+ G8.9.1 Multiple Causes: Deformation Mechanism Maps 323) x/ C T, X3 s7 i
8.9.2 Damping Mechanisms in High-Loss Alloys 326
, C0 @" v J, R$ H8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326
0 |# ?) \) I8 l; c0 [+ b8.10 Relaxation at Very Long Times 3272 q- j6 q! o# w& q4 b! q
8.11 Summary 3273 Q& Y& ]8 d% x5 z/ M6 {% x
8.12 Examples 328
) L. c, |1 ^' j0 m2 g3 e8.13 Problems and Questions 332
: J$ N6 Z1 E' EBibliography 332& q# D: v" P: i
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9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341; M+ J7 ~3 n8 \ C
9.1 Introduction 341
# j/ \- b G% d3 }# @9 Q% m9.2 Composite Structures and Properties 341
( y, i* [# n/ i9 O* q9.2.1 Ideal Structures 3419 F1 r4 A" F Z, `6 u" d- f
9.2.2 Anisotropy due to Structure 342
1 E0 H. X2 l( h9.3 Prediction of Elastic and Viscoelastic Properties 344
! I D" O- L) K h( D: g9.3.1 Basic Structures: Correspondence Solutions 344
7 n4 a% [ b; Q9.3.2 Voigt Composite 345
4 N9 U) A' e; l9.3.3 Reuss Composite 345; a$ T: e3 m/ ^8 K, i5 s' w, A" O$ u
9.3.4 Hashin–Shtrikman Composite 346
, b4 D/ q7 `6 v7 ?9.3.5 Spherical Particulate Inclusions 347
1 T% U4 j$ g$ m9.3.6 Fiber Inclusions 349
7 R$ t1 e2 A& J* f# M9.3.7 Platelet Inclusions 349! r' Y, p, g# E2 b8 a
9.3.8 Stiffness-Loss Maps 3505 B9 v% T; \! O; y4 C6 X
9.4 Bounds on the Viscoelastic Properties 353& E, d: i% Y4 u9 B
9.5 Extremal Composites 354
+ r9 q% |0 V# C% S9.6 Biological Composite Materials 356! O8 a) h; z" `; ]# H: g0 P& M8 o
9.7 Poisson’s Ratio of Viscoelastic Composites 3579 u: i+ u9 g& ]; d- u* m- }) |6 d4 b
9.8 Particulate and Fibrous Composite Materials 358
3 ^; Y, _+ M: h, T4 B9.8.1 Structure 358' r- e1 x0 w9 F. x
9.8.2 Particulate Polymer Matrix Composites 3596 e7 F" L. r4 }% z2 z" q
9.8.3 Fibrous Polymer Matrix Composites 361
$ `# V/ T4 l8 S; k O9.8.4 Metal–Matrix Composites 3623 Y1 E. o. O" Z" t: S
9.9 Cellular Solids 363& S3 g) H4 z, E8 Z: {
9.10 Piezoelectric Composites 366
5 T% |4 n5 h8 W' u; {9.11 Dispersion of Waves in Composites 366- _3 L, g8 a: D s! [
9.12 Summary 3673 C& s5 j$ q/ G1 G8 C
9.13 Examples 367: x. G; }6 {5 D2 I
9.14 Problems 370 M% Y- w1 c; D* G" o: B# k8 g
Bibliography 370 I4 ?- I" m' A! `, y* i4 A7 Q
% }- e# X [0 `* Y8 d, d4 k k2 w& Z6 t5 F% \( n" K0 Q$ |
( }9 q7 l( V7 |- p
10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377 a- R: v" R8 [8 y* `6 d
10.1 Introduction 377- |3 o$ Q1 n! l/ P5 K: V: ?
10.2 A Viscoelastic Earplug: Use of Recovery 377( j" ]# M1 P% f5 v0 w0 @! C) F$ m
10.3 Creep and Relaxation of Materials and Structures 3780 F4 g9 L6 r/ _" T B
10.3.1 Concrete 378! B, E% @& M, T5 @8 w# {
10.3.2 Wood 3788 v; q' k4 p2 j7 h; @0 `& {( N
10.3.3 Power Lines 379
+ _; K. e2 C% Z5 A8 C10.3.4 Glass Sag: Flowing Window Panes 380
1 M `3 \- h6 }10.3.5 Indentation: Road Rutting 3800 a! ?$ c7 U, i/ E w3 z
10.3.6 Leather 381: l& L. g; f- ] N( ^. x0 ~- w! g- L J
10.3.7 Creep-Resistant Alloys and Turbine Blades 381
$ \# B6 T6 p2 [# T( o9 K10.3.8 Loosening of Bolts and Screws 382
0 Y, q, I: o' H" Z10.3.9 Computer Disk Drive: Case Study of Relaxation 3843 m6 C& r" _6 ?3 v ]- F
10.3.10 Earth, Rock, and Ice 385& s% g; @# \# ?. F3 |* z5 O3 j a
10.3.11 Solder 386
P% r; w- o# I X) z10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
, n0 x& ~ X0 i# y. V# T4 u8 m10.3.13Tires: Flat-Spotting and Swelling 388- E" Z2 g$ V& m: `9 U! d3 ?- P, \6 Z
10.3.14Cushionsfor Seats and Wheelchairs 388. g( r4 J! m$ C0 c, A
10.3.15 Artificial Joints 389
" E$ R3 e3 U$ G5 n b10.3.16 Dental Fillings 389
" A# E' I U) P4 i+ v. H10.3.17 Food Products 389
% t u( u# S. u' R10.3.18 Seals and Gaskets 390
; ]; V& ]6 T! M! a2 @6 T; u10.3.19 Relaxationi nM usical Instrument Strings 390
8 Z* v9 {: `. ~: M# O" o4 B10.3.20 Winding of Tape 391
( O" [/ i# L' _. Z10.4 Creep and Recovery in Human Tissue 391
$ T: d2 @" t+ s# l' R1 ^/ [4 }. U10.4.1 Spinal Discs: Height Change 391
) o* d! N3 W( y10.4.2 The Nose 3925 e+ g4 j- Y8 j* H8 O1 u7 Y
10.4.3 Skin 392' L: j- E1 a( o7 h6 l8 Q
10.4.4 The Head 393
5 W2 X! V1 H9 ~4 n. K. V10.5 Creep Damage and Creep Rupture 394
" \: I. o: D# q* s) z# D* N10.5.1 Vajont Slide 394 p: y7 o: ?* e
10.5.2 Collapse of a Tunnel Segment 3944 n( I) F3 o1 |3 G
10.6 Vibration Control and Waves 3949 v$ E9 ?2 I7 @% l0 u' X
10.6.1 Analysis of Vibration Transmission 394
& n s8 N3 m6 V: @10.6.2 Resonant (Tuned) Damping 397! K. Z3 l3 m, A) s( ~
10.6.3 Rotating Equipment Vibration 397
+ j. e- Q9 d4 z* [! v. x10.6.4 Large Structure Vibration: Bridges and Buildings 3989 c8 P, j) A7 P! ^, d
10.6.5 Damping Layers for Plate and Beam Vibration 399
U7 k3 m- K. ~2 @* F5 T10.6.6 Structural Damping Materials 400; V% `) R- Y: ~9 D
10.6.7 Piezoelectric Transducers 4026 R& O: Z; E# m P5 m! R
10.6.8 Aircraft Noise and Vibration 402( p m/ K/ Y) |# W) j
10.6.9 Solid Fuel Rocket Vibration 404
: r4 m9 }3 x$ P' \10.6.10 Sports Equipment Vibration 404
7 W+ r7 u& T w6 O, Q! {' q10.6.11 Seat Cushions and Automobiles: Protection of People 404
1 v' H5 e2 ?$ T! N t10.6.12 Vibrationi n ScientificI nstruments 4065 n5 R% p8 |/ s7 M
10.6.13 Waves 406* Z- ` ?& F* h# x% A/ q
10.7 “Smart” Materials and Structures 4078 X+ w) D1 Y* `6 Z% B/ M+ N2 c7 N
10.7.1 “Smart” Materials 407: e% x' o6 b4 r
10.7.2 Shape Memory Materials 4081 `+ y" z- d5 W# e3 z* _9 ^
10.7.3 Self-Healing Materials 409
6 J% U0 K! t8 r5 z' |% f& O10.7.4 Piezoelectric Solid Damping 409( t+ y6 h8 B0 A9 D- X
10.7.5 Active Vibration Control: “Smart” Structures 409
3 K) V2 E3 _% m1 k2 L6 B10.8 Rolling Friction 409
8 o }3 ?+ A, \, d$ C& Q' m10.8.1 Rolling Analysis 410
% d. K" D4 M6 b9 \10.8.2 Rolling of Tires 411# k! v5 |6 X# S" X- x Z
10.9 Uses of Low-Loss Materials 412 J& }8 x2 e% N7 l# S
10.9.1 Timepieces 412
, ~) v! p. C9 { T# ?5 j3 C) j3 K/ V% Q10.9.2 Frequency Stabilization and Control 4133 r# h2 _( x2 ?, z
10.9.3 Gravitational Measurements 413- `3 i) N1 l" M0 q/ i
10.9.4 Nanoscale Resonators 414+ v* }3 d- R- P4 f0 c2 x4 ~& Z$ A
10.10 Impulses, Rebound, and Impact Absorption 414# B9 c% l2 P3 Z2 B# M% m
10.10.1 Rationale 4147 m! V# }) D) n0 Z6 T: d% f
10.10.2 Analysis 415
8 O% s' n9 X( H. ~10.10.3 Bumpers and Pads 418' z1 J- M6 \: I, Y( w
10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 419" M2 @, s4 `/ G" c* A1 b( f- P
10.10.5 Toughness of Materials 419
8 k8 Y+ P Q) m4 D& {3 @10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420
, s$ J" X: ]( {! O+ z2 l$ S* J10.11Rebound of a Ball 421* v1 z/ s( H9 F2 c, I3 M' ?) O
10.11.1 Analysis 421$ f9 C- f, ?7 y6 N
10.11.2 Applications in Sports 422
% c% |* u- w! m3 Q3 o+ u0 L, j( I10.12 Applications of Soft Materials 424' F5 l& j! b! ]! @3 V
10.12.1 Viscoelastic Gels in Surgery 424
' B( K: G" Z- f/ N1 o) [10.12.2 Hand Strength Exerciser 424
, X- Y7 G. D" n5 y4 q2 b4 x$ m10.12.3 Viscoelastic Toys 4242 e9 G- h% o# A/ [" e7 r6 O% o4 s
10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425
! n h: U1 f& z, s* r) o10.13 Applications Involving Thermoviscoelasticity 425
- d' V' l' p& Y w& h* ^( d10.14 Satellite Dynamics and Stability 426* T [7 f3 J+ \; P- d3 O6 k, [
10.15 Summary 428
! G! [6 N2 }. @# V: ~9 e10.16 Examples 429+ @7 w& G; s( u2 t6 z/ B5 {/ l* b8 P q
10.17 Problems 431
" @" E' v9 r% Z' h" @# @Bibliography 431
( u& e9 ]- e: s" p4 Q! s* J3 w& `8 E; u4 U, h% Q; o
* p- F0 [, t* X
1 _# u2 I: M) w2 |( T: B& u* f& n
A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
: k+ g0 S- V( d7 SA.1 Mathematical Preliminaries 441
" D2 `0 \3 {- @1 M- R6 wA.1.1 Introduction 441% ~+ t- z0 h0 O/ g( s2 p$ Q
A.1.2 Functionals and Distributions 441
: h5 ^) O9 F* W9 PA.1.3 Heaviside Unit Step Function 4429 o: q: p$ M; S8 M- {$ ?1 Q* o
A.1.4 Dirac Delta 442% ?8 C0 A9 g( J( x0 ?+ Q+ q& H; }! L1 n
A.1.5 Doublet 443
& @0 l5 y: }8 Z- ZA.1.6 Gamma Function 445& M0 n7 q/ r+ Q. c% e
A.1.7 Liebnitz Rule 4456 z' p4 d- [; V2 B- X: {* j
A.2 Transforms 445
/ j, D& U& q- ]- B8 ~ _2 o1 oA.2.1 Laplace Transform 446
7 O+ n0 h! W5 E: i' z" ^9 ~: LA.2.2 Fourier Transform 446
( E2 V7 Y9 A5 aA.2.3 Hartley Transform 447
8 P! T0 W- t5 B6 u; A- t* ?A.2.4 Hilbert Transform 447
. N3 d1 k: {& E, {# MA.3 Laplace Transform Properties 448: ]7 [* O0 t3 o' E m
A.4 Convolutions 449
# M# m. X ?$ [A.5 Interrelations in Elasticity Theory 4519 r7 s4 \; c, v' g1 C8 M0 W
A.6 Other Works on Viscoelasticity 451
. I! U m* w% t: }* d ?Bibliography 452; ?/ B' _/ }: T( e9 Z2 I) M
3 M( N" l8 {* C+ s6 `
7 E* V' J6 U! {8 |# O' kB: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4550 N6 I! t9 T$ n9 ~7 D7 t
B.1 Principal Symbols 455& [6 |5 S4 ]+ V& d7 l+ L) m
Index 457- T& y* o4 _/ ]3 V7 y5 M
/ o% Y9 J5 M) o6 V T
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