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Viscoelastic Materials Roderic Lakes 2009 Part 1-2.rar
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目录
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8 c- c& S& \ x* c- B* W2 q; OContents
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Preface page xvii/ _, Q& I6 h* o9 }
1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 g. R7 Z5 ~8 P
1.1 Viscoelastic Phenomena 1
6 W; f3 f- }! _. B2 }% b6 N3 O! l: d1.2 Motivations for Studying Viscoelasticity 3! Z# N5 L# i$ l3 ~9 S3 [
1.3 Transient Properties: Creep and Relaxation 3
' \' b% j9 @- B5 ?! n+ C, r" [) ^5 J1.3.1 Viscoelastic Functions J (t), E(t) 3' }8 N7 @& Q4 @1 G
1.3.2 Solids and Liquids 7- I+ t7 o7 s6 K/ `. a
1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
2 ?: |: |% @: s. ~- H7 g: O1.5 Demonstration of Viscoelastic Behavior 10
# p3 z; R# U6 T1.6 Historical Aspects 10
9 O8 f* L& v: U' N. t1 o1.7 Summary 11- @0 I5 F; M( l
1.8 Examples 11
$ y! d( `& k8 A9 X1.9 Problems 12
) H {% S- g# Z" g9 \+ W* tBibliography 12
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; M$ F4 q0 T. l' S2 D3 U2 a2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
: S8 v/ w1 V+ o: i2 L0 P2.1 Introduction 14
) k7 G/ F# n$ |$ q2.2 Prediction of the Response of Linearly Viscoelastic Materials 14
+ U, [, f0 C# F" k! C. ~# }2.2.1 Prediction of Recovery from Relaxation E(t) 14% F# a# @# F2 k. w* w* |- G
2.2.2 Prediction of Response to Arbitrary Strain History 15
$ V1 R G/ q; u% Q# Q; T; ?' K2.3 Restrictions on the Viscoelastic Functions 17' k+ u/ c1 m2 p& y
2.3.1 Roles of Energy and Passivity 17& h4 C% v$ s# M" T+ p
2.3.2 Fading Memory 18
% C4 j# Z/ |: p, A) `2.4 Relation between Creep and Relaxation 19
# g e5 ], b9 r) @7 v6 N' {" A2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19
1 `# Z. R" N3 G" @, V5 E/ K4 f% R2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20
& W, G: Y2 K3 g8 _/ h1 y! D. }2.5 Stress versus Strain for Constant Strain Rate 20) I [. b* q5 t
2.6 Particular Creep and Relaxation Functions 21
* I$ T: R; f5 o2.6.1 Exponentials and Mechanical Models 21- K. _2 g6 `* x+ X3 ?
2.6.2 Exponentials and Internal Causal Variables 26
& }. i- Z! r" e9 T2.6.3 Fractional Derivatives 27
0 t; a. Z" W, e9 O" h2.6.4 Power-Law Behavior 28; x/ }# y$ l- N: ?; m% Q
2.6.5 Stretched Exponential 29
6 n8 W" `- T4 \# W# F7 H0 l# {2.6.6 Logarithmic Creep; Kuhn Model 295 T& |) l) {2 O
2.6.7 Distinguishing among Viscoelastic Functions 30! ^* y1 X, z- }/ J) I
2.7 Effect of Temperature 30
7 E5 f3 s( S# k6 v- S. [9 X) p$ y9 M2.8 Three-Dimensional Linear Constitutive Equation 33/ f' \, n2 W. o7 U5 j
2.9 Aging Materials 35
+ X$ S. x2 |% V2.10 Dielectric and Other Forms of Relaxation 35( U1 \, @2 w3 { ?* s
2.11 Adaptive and “Smart” Materials 368 C+ f2 R9 q: f1 }6 S3 W
2.12 Effect of Nonlinearity 37! A4 s7 u7 J) t* T0 @9 P( O
2.12.1 Constitutive Equations 37
0 m' r5 Z0 Y- [( Q* Z7 @; _! |7 c2.12.2 Creep–Relaxation Interrelation: Nonlinear 407 Z5 d4 H. g; t- b6 z- c
2.13 Summary 43+ ?, r+ r! Z3 S5 i" ]2 B
2.14 Examples 430 |, y; I# y: K' M
2.15 Problems 51
3 t, z( T4 N G6 l# nBibliography 52
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3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 L5 |+ z. v2 k0 Q- A) P S
3.1 Introduction and Rationale 55
# b) G8 j0 m" y/ P9 m D- d3.2 The Linear Dynamic Response Functions E∗, tanδ 56% ^" @$ R0 m# Z: r4 F4 ^6 m: x
3.2.1 Response to Sinusoidal Input 57
$ o8 Z1 M! f) p& m3.2.2 Dynamic Stress–Strain Relation 59
4 l7 k, [. z$ f+ w# U3.2.3 Standard Linear Solid 62. C& g5 f3 h4 H2 b/ c
3.3 Kramers–Kronig Relations 63
) S- o# X, o* u' w( A, W3.4 Energy Storage and Dissipation 65# p. b( o" n4 X0 G% j$ R% f9 D
3.5 Resonance of Structural Members 67
) e! Q' X3 c) W" a( s) B |6 r* y4 j3.5.1 Resonance, Lumped System 678 L; k) F; A# S4 x
3.5.2 Resonance, Distributed System 71
+ n: d; q4 c3 ^9 W$ ~8 p2 N3.6 Decay of Resonant Vibration 74
5 @* n' q4 Q' y, K2 }) A/ q3.7 Wave Propagation and Attenuation 77
7 F' |. y3 {. h' y9 G4 {3.8 Measures of Damping 79
9 ^; r/ E) w' s& o b. W3.9 Nonlinear Materials 79 @2 W5 J2 J/ K0 {' `
3.10 Summary 812 W3 z) m; r" ~# Z
3.11 Examples 819 c# ]4 A7 l2 |5 x0 @
3.12 Problems 88- F5 c' K$ H3 ^; w9 }7 h
Bibliography 89
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. @1 i% I+ t& U5 Z* Q2 R) N+ I8 G3 M4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 911 T. p$ d5 G8 N& d# J% [
4.1 Introduction 91
: s9 W) g7 h% m, w# I7 U4.2 Spectra in Linear Viscoelasticity 92
1 j0 k7 W( ^9 M4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 92
8 j' J$ y$ |6 U2 T$ R8 o4.2.2 Particular Spectra 93: g& l: d& b. S; Q' k' T" f
4.3 Approximate Interrelations of Viscoelastic Functions 954 A5 S7 {/ K% C2 K, v, G) Z
4.3.1 Interrelations Involving the Spectra 95
! S& t5 l, S3 M' }: W4.3.2 Interrelations Involving Measurable Functions 98
' G3 ^- _$ c% M: P& x# D, g4.3.3 Summary, Approximate Relations 101: v2 N0 b+ n2 X2 x: v0 b. W, R9 h
4.4 Conceptual Organization of the Viscoelastic Functions 101
/ a/ X6 E/ E+ e+ A' ]4.5 Summary 104
' W6 C! K! V3 \; @( k, A4.6 Examples 104
; L$ t2 V9 ~$ r( t8 G$ q4.7 Problems 109* n. K2 F5 _ O* a6 j8 e2 P S1 x
Bibliography 109' V' O% b. m3 h6 w2 }
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111
* S3 _9 z1 _ l( b. C% _ R5.1 Introduction 111 X8 N- `; H$ H
5.2 Three-Dimensional Constitutive Equation 111
3 E! n4 u- @0 J( [5.3 Pure Bending by Direct Construction 112
# h4 y! T5 s6 n4 p5 W+ N3 P$ i' r5.4 Correspondence Principle 114
( t3 m* u9 s6 h9 b5.5 Pure Bending by Correspondence 1164 q& _) V x2 E, {* c$ j
5.6 Correspondence Principle in Three Dimensions 1164 u0 r" S; y/ X6 V
5.6.1 Constitutive Equations 116
- k* W9 _ V( E& }& c% a5.6.2 Rigid Indenter on a Semi-Infinite Solid 117; ]. M1 i+ o2 {
5.6.3 Viscoelastic Rod Held at Constant Extension 119
) Y8 D R( K8 p- @" h4 I! @5 |5.6.4 Stress Concentration 119* W% o/ o% H/ \ T% ~2 R& g
5.6.5 Saint Venant’s Principle 120
( ]: s. d! a F- ?/ \5 O' W5.7 Poisson’s Ratio ν(t) 121: z7 [8 }3 a$ P5 P- T2 o
5.7.1 Relaxation in Tension 121' [& @+ H: ]1 D' k# d
5.7.2 Creep in Tension 123$ B' ~8 e7 S4 N3 t) [. A; _1 W1 ~8 J; z
5.8 Dynamic Problems: Effects of Inertia 1242 V( y# C' ~1 w/ J4 R
5.8.1 Longitudinal Vibration and Waves in a Rod 124* v5 H' a1 f7 q2 _" V, A0 x
5.8.2 Torsional Waves and Vibration in a Rod 125& Q# z7 T) l; V& ]( a
5.8.3 Bending Waves and Vibration 1285 F$ F" Y9 S* [2 T% H) r
5.8.4 Waves in Three Dimensions 129
# }, y/ {9 W. Q4 ]+ K5.9 Noncorrespondence Problems 131
& j8 d( P% t; N* p8 E/ v5.9.1 Solution by Direct Construction: Example 131) e$ Q& E* U- u: y& F2 |
5.9.2 A Generalized Correspondence Principle 132/ _& B# \& @/ \% w. O; j/ s
5.9.3 Contact Problems 1321 T: `3 M- x- j! S. R
5.10 Bending in Nonlinear Viscoelasticity 133
# i# }$ `# o" m( C& l7 B" h8 I( l1 `5.11 Summary 134
1 |( ?3 k9 l+ R5 }# O5.12 Examples 1344 J5 H) W4 J7 l9 _! f
5.13 Problems 142. g# s- ^& X5 V! V R) K2 S* g' H+ _5 o
Bibliography 142* F" Z2 ~, j3 {$ o8 ^
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6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453 S1 v' J% V& h! C2 \+ q- V/ h- i
6.1 Introduction and General Requirements 145& w# _, q( q6 Y7 B5 [% l3 |
6.2 Creep 146
- k: ]2 f) v4 X A* c0 v9 M6.2.1 Creep: Simple Methods to Obtain J (t) 146
, ?* K/ F* A' \$ ?/ @# p& _! y6.2.2 Effect of Risetime in Transient Tests 146
: k+ Y+ h! q4 R# }6.2.3 Creep in Anisotropic Media 148! O/ H8 K' b2 t [. }6 b, ?
6.2.4 Creep in Nonlinear Media 148
3 y6 f' o9 E1 M; g! k6.3 Inference of Moduli 150
. g. b" E, y# K, S6.3.1 Use of Analytical Solutions 150
0 H. T8 k9 j) K5 x/ A6.3.2 Compression of a Block 151
% o& F3 h* ^/ W6 p5 H6.4 Displacement and Strain Measurement 152
. \: Q/ y! J+ O, Y2 I6.5 Force Measurement 156$ d% U! c1 X. \* u, f
6.6 Load Application 157
: t' Z/ h$ l& [' S. w; p; x+ w6.7 Environmental Control 157) b, l1 Q3 G" z/ U* _
6.8 Subresonant Dynamic Methods 158
0 a7 w5 v+ Q S W6.8.1 Phase Determination 158
* z* {% l. t1 V% t6.8.2 Nonlinear Materials 160
1 A+ N! C/ Z" J0 R' W( `2 Z6.8.3 Rebound Test 161
" I! B4 G9 m) Q" r6.9 Resonance Methods 1616 g8 g1 l, `8 z7 M# J
6.9.1 General Principles 1614 ]: L$ P" [" d4 {1 d6 p
6.9.2 Particular Resonance Methods 163
$ K( _+ T2 a5 }% Z0 s) U6.9.3 Methods for Low-Loss or High-Loss Materials 166
% h5 ?: @# h$ T6.9.4 Resonant Ultrasound Spectroscopy 168: z7 I, c1 r' z# P; f4 X
6.10 Achieving a Wide Range of Time or Frequency 171, `7 K8 \0 g, x, y
6.10.1 Rationale 171, a' f) u. I/ h& K, [7 m
6.10.2 Multiple Instruments and Long Creep 172
# _9 K; k E: b6.10.3 Time–Temperature Superposition 1725 y/ ^, v- K5 G* h/ W T9 ~
6.11 Test Instruments for Viscoelasticity 173
; b/ L, g% q5 H% x8 K$ m6.11.1 Servohydraulic Test Machines 173
# W/ z* ]# ^9 m0 p6 G# [- j6.11.2A Relaxation Instrument 174
3 x; w; m4 J8 F8 ~, F% N6.11.3 Driven Torsion Pendulum Devices 174
% p1 S8 u# A2 D2 H6.11.4 Commercial Viscoelastic Instrumentation 178" \# F" g2 y# _
6.11.5 Instruments for a Wide Range of Time and Frequency 179
' ^. L H Z/ h1 \1 f6.11.6 Fluctuation–Dissipation Relation 182
' P1 B1 B1 v) F% j5 H6 w/ z* c' m6.11.7 Mapping Properties by Indentation 183: S( v+ ?* g4 J% K a- E
6.12 Wave Methods 184
7 Y0 o$ }( {! s) A9 ?+ b0 q4 F! S6.13 Summary 188
) x" ~& y" q' C) |. u( ]6.14 Examples 188! r0 U J' j3 I+ \* C D' H' s3 ]
6.15 Problems 200
3 q5 J+ e/ o6 G2 S \8 BBibliography 201) f) H) C- c' v/ r
V a% _% a' y% b: {2 W0 F5 d2 M$ }
/ D1 j! ]" C' M7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207
8 d9 S+ f3 i) ~3 Y* r0 ^' U+ w( o7.1 Introduction 207
. W$ r+ g& Z/ w$ a B& r7.1.1 Rationale 207
d( ]" I8 E/ ^3 W" X! s7.1.2 Overview: Some Common Materials 207
% O- o, G% W' B! S2 `/ R7.2 Polymers 208
7 a& h1 J# O& _1 b' X7.2.1 Shear and Extension in Amorphous Polymers 208& E& y1 i+ _4 n( P$ u
7.2.2 Bulk Relaxation in Amorphous Polymers 2125 I. v7 V/ a' _5 S1 m( H F
7.2.3 Crystalline Polymers 2135 n' N) w( L* |& q0 O4 l! |* K
7.2.4 Aging and other Relaxations 214
[" U/ p3 b; s7.2.5 Piezoelectric Polymers 214. y+ J9 l% N" A$ t9 C v( Z. e
7.2.6 Asphalt 214
3 E$ Y- a3 d0 p5 J/ T% V7.3 Metals 215) T; \7 P" H2 n: I2 L$ H
7.3.1 Linear Regime of Metals 2156 F9 h: @: o. _
7.3.2 Nonlinear Regime of Metals 217
# V6 a6 q$ q: E' }/ [- d$ i; ^ y7.3.3 High-Damping Metals and Alloys 219
. y: e% M2 M1 H: ~. ~7.3.4 Creep-Resistant Alloys 224
2 m3 u7 O- Z1 d" I% `8 q7.3.5 Semiconductors and Amorphous Elements 225
: L( ~9 R' S' O% @! j5 V; _7.3.6 Semiconductors and Acoustic Amplification 226
# q6 L n1 r" q. L7.3.7 Nanoscale Properties 226
$ L+ @+ Y% ]: @2 F( o4 l7.4 Ceramics 227- R3 I# _! R( ~6 T; x$ g
7.4.1 Rocks 227
4 N0 A0 e0 B, `+ u, y3 G$ S% c: Z7.4.2 Concrete 2294 @+ h8 a& d1 I5 x' M& Z1 X1 O
7.4.3 Inorganic Glassy Materials 231
; K2 s* s: Q' C9 c0 v! z( K% T1 r7.4.4 Ice 231
# x0 C& e- M/ C1 F( l/ ? B% P7.4.5 Piezoelectric Ceramics 2325 E* {& C$ O$ p0 X8 V" b+ y2 ^4 z
7.5 Biological Composite Materials 233! r6 r( X% A% H$ O9 \" w+ g; U
7.5.1 Constitutive Equations 2349 o; z" f- b8 l5 b e
7.5.2 Hard Tissue: Bone 234) w1 B. [1 W) m1 }
7.5.3 Collagen, Elastin, Proteoglycans 2366 ] n1 n, h7 W9 F! a( M" ^
7.5.4 Ligament and Tendon 237
: a2 S) a3 m; E; F7 ]7.5.5 Muscle 240$ T2 X0 j$ v' W1 D% I
7.5.6 Fat 243" p |8 ]' L5 S/ |% W5 h
7.5.7 Brain 243- [9 A, t8 z7 _6 j1 g; L& [
7.5.8 Vocal Folds 2446 \9 I0 z1 J# _- A
7.5.9 Cartilage and Joints 244
* P1 \9 ]2 b; y; Q" o5 I7.5.10 Kidney and Liver 246' K M# Z5 e3 @4 M& Z9 b
7.5.11 Uterus and Cervix 246! Z( Z* w4 Z2 [' I8 i
7.5.12 Arteries 247
6 K9 X$ n: x2 n5 o# l' {8 L7.5.13 Lung 248
5 x# g: }$ S8 a0 @6 k7.5.14 The Ear 2484 w" b I, O0 A9 Y
7.5.15 The Eye 249
[- k( u0 P* r7.5.16 Tissue Comparison 2519 Q" A ~* \8 R
7.5.17 Plant Seeds 252
' @+ p0 l* I6 U+ u5 Y3 f4 |7.5.18 Wood 2527 y% q# _. `$ D
7.5.19 Soft Plant Tissue: Apple, Potato 2539 m( Y' w/ _3 p
7.6 Common Aspects 253
( q! m0 _. p. `9 d7 |& [9 s- h7.6.1 Temperature Dependence 253
+ L U' L5 q7 Q- \7.6.2 High-Temperature Background 254 S- Y& i" d$ B8 m: @( g3 _
7.6.3 Negative Damping and Acoustic Emission 255' t- D( w' _9 X: V# k# [5 i4 Q6 [
7.7 Summary 255
7 T, \$ k, k% o- g8 ?7.8 Examples 255+ u8 c' A! U; x. i* c. u" C
7.9 Problems 256
( q; r. L9 h! x% }Bibliography 2577 G. g3 ~( a+ V8 J6 b
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* X( [- Z; U. b H7 U- @8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271" ]( U; y) e& x m( \$ G
8.1 Introduction 271
! ~8 I# }/ W7 X/ P8 j" N3 Q8.1.1 Rationale 271; d, e3 M5 l8 ?$ W ?
8.1.2 Survey of Viscoelastic Mechanisms 271# t* T: ?2 B& n) g; w
8.1.3 Coupled Fields 273
+ |7 a# A7 i: G/ C' H8.2 Thermoelastic Relaxation 2743 U3 h2 Z* M5 h b/ y- S/ @
8.2.1 Thermoelasticity in One Dimension 274
, O2 g6 C Q9 C6 {# s8.2.2 Thermoelasticity in Three Dimensions 275
& I, F% T# i/ F; C) z8 Q8.2.3 Thermoelastic Relaxation Kinetics 276
- I! b4 e+ q8 \3 D' m+ o8.2.4 Heterogeneity and Thermoelastic Damping 278
+ E ]2 A: J" p1 N0 f# V7 _8.2.5 Material Properties and Thermoelastic Damping 280& t: {* i; X; d1 Q, ]+ _! A
8.3 Relaxation by Stress-Induced Fluid Motion 280- \ f: X6 E; d1 `
8.3.1 Fluid Motion in One Dimension 280
3 w% J* `: r# H8 X& k8 d8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281
# K) y* D z. ]2 P3 ]8.4 Relaxation by Molecular Rearrangement 2865 |. V* A w) `7 E' a! S
8.4.1 Glassy Region 286' D# K) q7 A0 |/ Q
8.4.2 Transition Region 2876 J. s4 r$ q( C# v0 c9 O$ E: z
8.4.3 Rubbery Behavior 2897 X4 M1 o$ p/ `5 P/ L. a& i4 _
8.4.4 Crystalline Polymers 291
4 A1 G# g4 c* Z6 [8.4.5 Biological Macromolecules 292
5 @9 n# K1 o R! {4 k8.4.6 Polymers and Metals 292
, f! r+ H/ m' O2 I0 h& d8.5 Relaxation by Interface Motion 292
& Q& q& ]3 N8 W, U9 z7 X8.5.1 Grain Boundary Slip in Metals 2928 E0 v1 H; I; Q0 }2 n2 O2 ^
8.5.2 Interface Motion in Composites 294
' t; y. L, \; E* \! e! M8.5.3 Structural Interface Motion 294
" K- {! v' a; R9 |3 I8.6 Relaxation Processes in Crystalline Materials 294, G* D. w. y. \; R, n7 |4 p% w
8.6.1 Snoek Relaxation: Interstitial Atoms 294
4 g6 D9 J0 F! N2 b0 o+ h6 k$ r. M0 }8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298
7 }& ^, C# [$ n3 S8.6.3 Gorsky Relaxation 2992 r; b# S$ Q: K: J0 t
8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 3007 Z/ j6 U. f+ x+ ]( H9 f
8.6.5 Bordoni Relaxation: Dislocation Kinks 303
$ k0 @1 J N& i5 c! L% P8.6.6 Relaxation Due to Phase Transformations 305
" P- u9 m. B+ a# T( \. O: U! a8.6.7 High-Temperature Background 314
) K6 Z- S4 t: b% r* @8.6.8 Nonremovable Relaxations 3154 P1 [; H8 z8 l/ q1 d! c4 m/ c
8.6.9 Damping Due to Wave Scattering 316" D1 u1 m1 E8 E
8.7 Magnetic and Piezoelectric Materials 316
) S! h9 k. W6 N$ l3 _7 _2 U0 a8.7.1 Relaxation in Magnetic Media 316
4 C5 g! z; Z. P1 k6 G8.7.2 Relaxation in Piezoelectric Materials 318% d: C& u7 r0 [5 H
8.8 Nonexponential Relaxation 3225 j5 { {- R0 R
8.9 Concepts for Material Design 323
( D6 ^; u. w% F9 b0 u8.9.1 Multiple Causes: Deformation Mechanism Maps 3231 q2 [+ o2 D. c5 {1 E' k7 w; a, {
8.9.2 Damping Mechanisms in High-Loss Alloys 3267 r% J2 W0 W" D+ l& a/ N
8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326
, Y5 ? u0 P9 {2 j8 D' `0 p) r3 t8.10 Relaxation at Very Long Times 327
5 @. j7 Q8 X @& Z9 o& c8.11 Summary 327
?! h' W& B9 ^8.12 Examples 3286 i* x V. n. Q! p
8.13 Problems and Questions 332
z8 G# c) U8 }) J' tBibliography 332$ G; U& x( t* C3 h/ p! o) E0 a
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$ c6 G2 ~' \1 m6 G5 k9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341
]2 `" h" _ A- ^3 o- B9.1 Introduction 341
% c, e4 u6 t4 ?* i9.2 Composite Structures and Properties 341
- q6 Y0 t9 t0 Y. h8 l1 g9.2.1 Ideal Structures 341
1 |: q6 i* I) d) t. x: A9.2.2 Anisotropy due to Structure 342 A6 k7 A! K9 f4 \4 A1 q% u/ a
9.3 Prediction of Elastic and Viscoelastic Properties 3449 \3 p' ^# {, W
9.3.1 Basic Structures: Correspondence Solutions 344* a6 Y7 I+ R2 \6 S+ ]" M
9.3.2 Voigt Composite 3457 M* X0 I) E- B1 ]9 ~
9.3.3 Reuss Composite 345
7 F$ A9 K4 j* G9.3.4 Hashin–Shtrikman Composite 3460 O* y8 ]# d: p& K% @% `* `
9.3.5 Spherical Particulate Inclusions 347
8 F ?: z; q, R' E) w/ u- Q9.3.6 Fiber Inclusions 349
# C& p* v7 h6 b9.3.7 Platelet Inclusions 3496 @4 w/ r: U; m" z. w% N
9.3.8 Stiffness-Loss Maps 3503 A n" t$ \, I& a/ t- ], f% c
9.4 Bounds on the Viscoelastic Properties 3532 S! @) S* q/ a2 V- B/ V
9.5 Extremal Composites 354, w$ H% G q8 B
9.6 Biological Composite Materials 356
" z) E3 t( P/ l, R9.7 Poisson’s Ratio of Viscoelastic Composites 357! ^; A) s, \8 E$ m8 H5 U
9.8 Particulate and Fibrous Composite Materials 358
, I/ N2 Q, l, ~" W* z9 q9.8.1 Structure 3589 a8 _' |4 o6 c* G8 M' J; u/ s
9.8.2 Particulate Polymer Matrix Composites 359! h4 [+ D: h$ D- y7 b+ v
9.8.3 Fibrous Polymer Matrix Composites 3619 J8 q) O& B. {
9.8.4 Metal–Matrix Composites 362
: c+ A2 k3 S7 ^; {0 p/ R9.9 Cellular Solids 363
3 T% Z" u: C9 @. W0 s( i9.10 Piezoelectric Composites 366
+ @5 S; S H; c& a& F" F9.11 Dispersion of Waves in Composites 366/ F3 b B( @; G: t0 t
9.12 Summary 367( d6 F/ \2 e! A, a
9.13 Examples 367$ E; o* s/ y: w r! N
9.14 Problems 370
3 \) Q) J7 u9 ?Bibliography 370
" Q: U/ z0 r7 t; r. u. x4 g
. A- u7 \0 O1 s5 x$ t I+ B/ ~" t M5 j3 g% n
. s" x4 n0 H6 X$ V9 }1 H
10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377: w6 n& D4 T+ J8 z% K! I
10.1 Introduction 377
o6 J; C; W6 Y5 \10.2 A Viscoelastic Earplug: Use of Recovery 377
; D& `2 t5 |& u5 D6 {8 r10.3 Creep and Relaxation of Materials and Structures 378$ U* ^. ^4 O2 n. k) I8 W' i1 \
10.3.1 Concrete 378
% s- [" |/ o; c2 u10.3.2 Wood 378+ i* l8 l# J- L" R2 g0 {
10.3.3 Power Lines 379
' q3 O% a2 K# ^& F& z10.3.4 Glass Sag: Flowing Window Panes 380 Z- w8 s J8 s; i. Q% U: ]
10.3.5 Indentation: Road Rutting 380
h# b% T' P7 j5 h- \7 i. o! G10.3.6 Leather 381
4 @$ \ Y) i) {: p" ~5 _! e10.3.7 Creep-Resistant Alloys and Turbine Blades 381# `- f7 C3 H9 u0 Z3 y4 ^
10.3.8 Loosening of Bolts and Screws 382
# t. b5 I: L! P) b& N: t9 C10.3.9 Computer Disk Drive: Case Study of Relaxation 384
/ F2 O! w# L& ^# Q+ `10.3.10 Earth, Rock, and Ice 385
" M6 T5 U! Z. k9 d1 Y7 C d10.3.11 Solder 386
: n* R# [: Y! G3 k! W2 g& f10.3.12 Filamentsi nL ight Bulbs and Other Devices 387" u4 r @* i* e( R( O# X; M6 A3 O
10.3.13Tires: Flat-Spotting and Swelling 3889 B2 N4 c0 P) @( c# d
10.3.14Cushionsfor Seats and Wheelchairs 388 k, }! G( Y8 Y+ u. v( T( t7 a
10.3.15 Artificial Joints 389
; Z" f4 [+ n O+ ~: m- F10.3.16 Dental Fillings 3898 F. }) x# _, S3 l
10.3.17 Food Products 389
( k* a3 j' Y$ R( m8 b! I10.3.18 Seals and Gaskets 390/ R9 M; H# C7 G2 `
10.3.19 Relaxationi nM usical Instrument Strings 390: ^4 t' ?; A- p/ a2 j. e
10.3.20 Winding of Tape 391, I8 z) y" X C: l: o4 w
10.4 Creep and Recovery in Human Tissue 391& L7 q2 x+ s, V) N9 o
10.4.1 Spinal Discs: Height Change 391) a) o3 a- ?8 ^4 ^! X1 n9 K8 m* P) C
10.4.2 The Nose 392! o3 h5 |: X( e+ ~
10.4.3 Skin 392
! k. T3 _) W- z10.4.4 The Head 3937 W- k; j7 k& A( b
10.5 Creep Damage and Creep Rupture 394
z# g/ V$ o: H10.5.1 Vajont Slide 394
# Z/ [" D( Y5 d3 s/ J10.5.2 Collapse of a Tunnel Segment 394
4 @. h8 c6 g$ V& _10.6 Vibration Control and Waves 3947 K0 ~! n" n& ?( J& t/ |
10.6.1 Analysis of Vibration Transmission 394& I) V' {* ~" S% b0 U3 D, f! t1 l+ ?
10.6.2 Resonant (Tuned) Damping 397
8 e2 B0 ~4 U1 Y. K10.6.3 Rotating Equipment Vibration 397
O5 @! @7 C6 u& B10.6.4 Large Structure Vibration: Bridges and Buildings 3980 |3 b0 b h# }4 I, a5 n2 z
10.6.5 Damping Layers for Plate and Beam Vibration 399" j, o* A+ W0 O( j" u v
10.6.6 Structural Damping Materials 400# ?0 ]3 F# \3 T5 j! V+ z7 b- X
10.6.7 Piezoelectric Transducers 402
8 C% m0 \2 k d7 H10.6.8 Aircraft Noise and Vibration 402) ? R, O) A r) ^- P7 D. Q# Y
10.6.9 Solid Fuel Rocket Vibration 404
% Z) C# {+ b# s% p5 z8 q( Z" c10.6.10 Sports Equipment Vibration 404. t* Q& w, g1 X* b
10.6.11 Seat Cushions and Automobiles: Protection of People 4045 }7 T- ^3 S$ f
10.6.12 Vibrationi n ScientificI nstruments 406) l% r$ b4 Y- r! {- Y6 a6 g! D- ~
10.6.13 Waves 4069 L" Z A3 ~) n% K2 @; M; g: g* Q8 X
10.7 “Smart” Materials and Structures 407
* p9 q/ x C* Q9 D1 \. Z0 Q, L10.7.1 “Smart” Materials 407
) m r2 s# w0 a/ d10.7.2 Shape Memory Materials 4080 @7 ^. [5 O# j6 i( r- u
10.7.3 Self-Healing Materials 409# @ p& s8 _! Y
10.7.4 Piezoelectric Solid Damping 409
! O/ ]3 ~, z; u3 G$ m$ n10.7.5 Active Vibration Control: “Smart” Structures 409
l6 S& w4 i: _4 r9 @+ p# ?10.8 Rolling Friction 4094 r" v6 |. ~) h
10.8.1 Rolling Analysis 410) H! _) k7 K( I5 g
10.8.2 Rolling of Tires 411
9 ?+ _7 w M4 G( C }10.9 Uses of Low-Loss Materials 412, E% @# |8 o5 q9 C% } w# S
10.9.1 Timepieces 412
8 c# F9 O8 r' d$ L2 C3 W10.9.2 Frequency Stabilization and Control 413. U; ? X' t3 o4 @
10.9.3 Gravitational Measurements 413
& c- A/ ^/ n8 ~& l+ p10.9.4 Nanoscale Resonators 4148 B; Z; s8 D% | _* W
10.10 Impulses, Rebound, and Impact Absorption 414+ z4 s. n$ j D; m2 m0 U" |
10.10.1 Rationale 4141 S3 K* t% Y6 e
10.10.2 Analysis 415' k3 [4 P- O5 j
10.10.3 Bumpers and Pads 418
6 [" x4 I7 ^9 D$ z- S10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 4197 i; n* d5 Z* c
10.10.5 Toughness of Materials 419+ B# ~: U( M; n ]9 j: U
10.10.6 Tissue Viscoelasticity in Medical Diagnosis 4206 S) \8 H' [8 R& _% o% h
10.11Rebound of a Ball 421& S; j) [' k6 k e
10.11.1 Analysis 421/ S' d& c1 M" W' H& W% R. R* s
10.11.2 Applications in Sports 422: i. U1 r1 d0 V5 j( @: Y& ~
10.12 Applications of Soft Materials 424
. e8 m- ]' K8 F10.12.1 Viscoelastic Gels in Surgery 4243 t4 p1 u4 ~. S4 F1 X
10.12.2 Hand Strength Exerciser 424. C' ?( v% |0 k5 {
10.12.3 Viscoelastic Toys 424
& K2 |: X! p. { } U" o10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425
: a" k2 V9 m, k* N7 _0 K10.13 Applications Involving Thermoviscoelasticity 425
0 ~1 w8 s" G/ U/ R V4 C9 @10.14 Satellite Dynamics and Stability 426
& ^' B7 [$ [ M2 a0 @* d, M2 b* y10.15 Summary 428( S2 H6 L6 ^. c" t! D( Q, Y+ U
10.16 Examples 4295 S$ E4 g# R) X. w: x0 m
10.17 Problems 4318 b# _5 J" F0 v
Bibliography 4317 }4 p, D/ c7 K
; V& j' `1 r' x! z" o7 S9 R: K1 K' U8 B8 T B; g4 \$ |
5 B+ c/ `: P6 ~2 r6 ?( i+ N2 yA: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
8 H; W& ~/ Q& PA.1 Mathematical Preliminaries 441
8 o1 X& c& I, O& MA.1.1 Introduction 441
) x1 f- O% T1 |8 Z" ^* GA.1.2 Functionals and Distributions 441
* H' ~9 F* h2 }. K- ?9 U, U2 F- s1 eA.1.3 Heaviside Unit Step Function 442/ d0 V! O) [) L5 j) {; w: X
A.1.4 Dirac Delta 442
9 J I% d' n3 ]# \A.1.5 Doublet 443; F9 I' a3 O7 J
A.1.6 Gamma Function 445
5 M2 _- B. s2 k3 J, ]6 O0 ^% SA.1.7 Liebnitz Rule 445
* F: f3 B3 L4 B% A. tA.2 Transforms 445' c( R/ o! r7 B: V+ w) L
A.2.1 Laplace Transform 4465 b" T, k; S3 w( s; K
A.2.2 Fourier Transform 446, a+ b2 a# V9 k1 d
A.2.3 Hartley Transform 4478 |3 d5 Z* g+ q$ p5 T6 R. s- I
A.2.4 Hilbert Transform 447, ?, Z& f$ a8 {2 C' F
A.3 Laplace Transform Properties 448
' t {( C9 \$ x+ x6 }, x/ Z3 V) R4 yA.4 Convolutions 449
G5 n2 I2 X$ N% B' u8 t uA.5 Interrelations in Elasticity Theory 451
* P; }' O3 ~% J/ GA.6 Other Works on Viscoelasticity 451- ^; H- z% r$ l- m& ~+ f
Bibliography 452. J' p5 S" Y K, H: j
; U2 G- B7 Z" ^# m7 N3 |7 @: }3 m" a2 Q* B9 c; z7 W$ S) O
B: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 C4 _% y9 j$ k, u3 q' S% {; X& G
B.1 Principal Symbols 4552 P' Q, q% ^ F$ F5 E$ m9 p
Index 457
. ^6 `. }" o5 q) n. ]; A
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