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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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3 ~5 `& z8 E# e1 d( f目录
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Contents5 P; a+ {3 L# {( J, S; n
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Preface page xvii
; w0 H1 l( S* W7 S! v, n1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 l7 Y; L8 @7 a8 r5 }/ B
1.1 Viscoelastic Phenomena 1
3 O( u" |# v. z1.2 Motivations for Studying Viscoelasticity 3
& u0 t# A) S3 _3 |6 N' |% S1.3 Transient Properties: Creep and Relaxation 3
- s# u/ T2 b0 o9 e1.3.1 Viscoelastic Functions J (t), E(t) 3; @4 q2 ?. q6 K5 i: q K! t
1.3.2 Solids and Liquids 7
4 b k! E- G. Q% u8 o1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8! l3 x3 S5 H; ?% C0 n ]. l: G
1.5 Demonstration of Viscoelastic Behavior 10
3 T( M; p2 M7 }" V3 L6 v/ b9 X1.6 Historical Aspects 10
% N% E: x0 H4 C3 g1.7 Summary 117 m8 G1 z. o5 j, E
1.8 Examples 11& ^2 j: p& Y9 A4 E# y/ O0 l
1.9 Problems 12; E# z# T4 k+ o" N
Bibliography 12
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2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
( O2 {( r" ~3 u$ w) m2.1 Introduction 14( Z0 N U' j* C1 V
2.2 Prediction of the Response of Linearly Viscoelastic Materials 14
! Y+ |/ }% @/ \$ `) U# G2.2.1 Prediction of Recovery from Relaxation E(t) 14
. b Q1 p/ J; c2 l6 v3 W2.2.2 Prediction of Response to Arbitrary Strain History 15
0 b- J1 a: t0 t$ L$ ^8 f2.3 Restrictions on the Viscoelastic Functions 17" L1 x) L" O& D8 |; v8 i
2.3.1 Roles of Energy and Passivity 17
8 L( X ~8 a- T& S, ~2.3.2 Fading Memory 18* u, z& i' y2 J' b2 M
2.4 Relation between Creep and Relaxation 19' R7 q4 p6 W4 L* [2 F
2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19( S0 v8 c L2 m5 F
2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20; C# q8 }2 S& A* [
2.5 Stress versus Strain for Constant Strain Rate 20
0 ? ^0 h4 F |# G3 K7 ^2.6 Particular Creep and Relaxation Functions 21
\ w( |& _' x- R, {5 |" j3 N2.6.1 Exponentials and Mechanical Models 21; S" f4 W1 @4 w, F6 o
2.6.2 Exponentials and Internal Causal Variables 267 b7 L2 V3 p. c2 _8 n* S% X
2.6.3 Fractional Derivatives 27
. g3 x, c. S& a( o+ m+ q+ V4 P2.6.4 Power-Law Behavior 28
5 u% y& M) z9 ?- ]5 O: O# N2.6.5 Stretched Exponential 297 M$ P; a6 Z1 L3 t4 G
2.6.6 Logarithmic Creep; Kuhn Model 29) W; ]9 N$ ^/ K8 J9 r2 x
2.6.7 Distinguishing among Viscoelastic Functions 30
) e* F! D& I2 V8 c+ A2.7 Effect of Temperature 30/ E1 m8 {% m' c* X5 x0 A
2.8 Three-Dimensional Linear Constitutive Equation 33
: |! B% r# J- d. L2.9 Aging Materials 35
b4 U1 |1 i- l" t0 n: t2.10 Dielectric and Other Forms of Relaxation 35! M6 ^0 @* j6 b. K+ o
2.11 Adaptive and “Smart” Materials 36
$ Q; k0 K' t5 b5 ], n' q7 L5 O4 Y7 l* Y2.12 Effect of Nonlinearity 37. V# o6 }# K2 x- d5 V# O
2.12.1 Constitutive Equations 37
% i9 d! q8 Y: O1 d2.12.2 Creep–Relaxation Interrelation: Nonlinear 40- q* m Z( \* D' \
2.13 Summary 43
/ P z: m* m0 b5 U. f7 j: X5 @# ]1 r2.14 Examples 43
0 M" i0 i4 W- }# i5 _. E1 ~2.15 Problems 51
1 B" @5 Q4 }2 i: HBibliography 528 C V+ n& F4 }3 Z0 ^! g
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+ u- J. j( _2 R( V7 ~* y5 L- d: q3 f7 ~3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
- {1 \' b0 d( X' S3.1 Introduction and Rationale 55
* R {: a" C d" P4 l( _* L3.2 The Linear Dynamic Response Functions E∗, tanδ 56
! l) Q) R3 P2 c3.2.1 Response to Sinusoidal Input 57
4 R( o0 s; I0 X4 C9 I3 I7 ^5 k4 m3.2.2 Dynamic Stress–Strain Relation 593 I, A( e9 I/ I5 d7 _4 T0 A# U) V
3.2.3 Standard Linear Solid 62$ @. C$ ]8 l$ B7 F' O2 k/ ? o
3.3 Kramers–Kronig Relations 63% I8 }8 c( q6 u# g" h
3.4 Energy Storage and Dissipation 65
; p \9 i2 I0 q! w5 T% n2 H R. ]3.5 Resonance of Structural Members 67
7 L2 H4 Z8 `. {8 H) _3.5.1 Resonance, Lumped System 67
7 c6 x( x) S1 i4 J5 Z. b' q3 i, J! ]3.5.2 Resonance, Distributed System 71- v7 H- t j6 K
3.6 Decay of Resonant Vibration 741 Z; @. w, H$ A* p# B! ^0 ^
3.7 Wave Propagation and Attenuation 77% G* v! y9 Y( |; I( S% H
3.8 Measures of Damping 79/ T0 j- M" D& L! g4 d3 @4 c
3.9 Nonlinear Materials 79
/ ?2 I) O% T" l/ F+ D- Z; e3.10 Summary 81. y! J0 m0 \) e* \5 }* U$ m. T, h
3.11 Examples 81
5 d8 e' B# d0 z; L5 ^3.12 Problems 88/ C. i$ k3 t' \% v1 H- ^+ H4 V
Bibliography 89# L% l% ~ k' ~' ]: v
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4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91 ?5 ^& i) u& G$ v
4.1 Introduction 91
, m/ }: H6 j0 X6 V# x4.2 Spectra in Linear Viscoelasticity 92
! u( c+ z; f, G* O2 C+ Y N4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 929 W9 v' e8 L. [* q/ Z8 A' s0 z
4.2.2 Particular Spectra 93, u4 N# Y: I9 x! E0 r& r1 y
4.3 Approximate Interrelations of Viscoelastic Functions 958 G) @( s. P! v. M- [
4.3.1 Interrelations Involving the Spectra 95( _: W. l8 i# m
4.3.2 Interrelations Involving Measurable Functions 98" n, ^4 J2 J0 z" I9 b4 R) Y# d
4.3.3 Summary, Approximate Relations 101
: X' l e) E# P$ B8 S1 X; y4.4 Conceptual Organization of the Viscoelastic Functions 101+ G7 A! d. G- a
4.5 Summary 1041 k6 L# O# u4 H, @8 V/ c8 `. n
4.6 Examples 104 U- h( k/ G8 ^( k; x
4.7 Problems 109# W; a9 N8 j, x9 m ?) B- S
Bibliography 109
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 1114 V3 F& O2 H. I8 W* n7 { a
5.1 Introduction 111 l: o9 a- [: v4 M& w; D
5.2 Three-Dimensional Constitutive Equation 1119 l9 \: o+ u G
5.3 Pure Bending by Direct Construction 112" `3 H9 ]& ~% ~9 o* K1 ]& Y
5.4 Correspondence Principle 114
9 d L" S# p4 s: e" u- Q: ~2 F5.5 Pure Bending by Correspondence 1167 ^/ k' b/ K+ e
5.6 Correspondence Principle in Three Dimensions 116
& l( C2 B1 l% o2 l/ l5.6.1 Constitutive Equations 1166 D6 t; a/ h Z6 @0 R9 r z
5.6.2 Rigid Indenter on a Semi-Infinite Solid 117
; E2 i M8 x# L) W* l; u, I7 e) E5.6.3 Viscoelastic Rod Held at Constant Extension 119
; e. }7 v8 R8 v3 }5.6.4 Stress Concentration 119
5 R b6 Y! B+ J6 L/ `0 w+ a5.6.5 Saint Venant’s Principle 120
; @% M. f n( }! a6 A5.7 Poisson’s Ratio ν(t) 121
6 V; P# u8 v7 I% k# a- k# j5.7.1 Relaxation in Tension 121
6 s' V- Q: B1 l5.7.2 Creep in Tension 123" O' k9 y& r+ W$ K) m
5.8 Dynamic Problems: Effects of Inertia 124) Y8 }- `. d* [ b* K1 j6 [& |( c
5.8.1 Longitudinal Vibration and Waves in a Rod 124, j8 @: n2 [1 |& A" J
5.8.2 Torsional Waves and Vibration in a Rod 125
. X: I0 P1 ~9 l! N, D( F5.8.3 Bending Waves and Vibration 128
: Q& P7 z$ Z( F, a5.8.4 Waves in Three Dimensions 129
f. ^, ]' R& M( |% d5 A5.9 Noncorrespondence Problems 131
4 e0 W' m' _0 I4 G& O' x, E5.9.1 Solution by Direct Construction: Example 131
9 w2 S( d2 { h/ y! y, B5.9.2 A Generalized Correspondence Principle 1323 T) o, z3 I4 l; P" x, V
5.9.3 Contact Problems 132. T' h1 s% R0 M0 F: N
5.10 Bending in Nonlinear Viscoelasticity 133
* z" l- n' ?: }% F3 b. x1 G5.11 Summary 134) @, B- F: C" l
5.12 Examples 134
; s: |, L- h0 k" @- H4 a5.13 Problems 142
4 v: W! |# ]0 Y5 Y- {2 ?Bibliography 142
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) F2 \9 g P# N/ N( Y6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145' h3 m9 @, M9 |% z4 b
6.1 Introduction and General Requirements 145" |" M3 _( `! P" G
6.2 Creep 146
0 [; e+ O f1 z2 {0 Y6.2.1 Creep: Simple Methods to Obtain J (t) 1466 N; b R7 i; t6 R! U6 A
6.2.2 Effect of Risetime in Transient Tests 146+ I. {7 H3 A" ~* ^8 T( [' O
6.2.3 Creep in Anisotropic Media 148
8 y) _5 u2 b+ B1 D6.2.4 Creep in Nonlinear Media 148
' ]/ y0 Z: H6 ?7 | c8 ]( R- \8 e6.3 Inference of Moduli 150) J3 h+ ~" ?* ~/ j6 |
6.3.1 Use of Analytical Solutions 150% K6 N; b9 N, }; X7 v) ], O3 N2 z
6.3.2 Compression of a Block 1512 L' W" e% S& E) K
6.4 Displacement and Strain Measurement 1524 f0 ]: [% Y3 C1 ]$ O9 T* A: ]* Z
6.5 Force Measurement 156. W \9 k, a, B# F/ `
6.6 Load Application 157$ e5 H% J% w* F4 J8 D
6.7 Environmental Control 1575 E3 f' v; J' d5 d
6.8 Subresonant Dynamic Methods 1588 c/ b' o! G* R7 n1 k
6.8.1 Phase Determination 158 p# p9 m. i* R9 d
6.8.2 Nonlinear Materials 160
% V" q( M: E) ?$ z+ F6.8.3 Rebound Test 161% y; T2 m0 ^+ }) l: e
6.9 Resonance Methods 161
2 x l, Q! t' v z, o6.9.1 General Principles 161! s# Q. h# Q8 T S! g
6.9.2 Particular Resonance Methods 1634 F+ D7 ^& ~8 b
6.9.3 Methods for Low-Loss or High-Loss Materials 166( U) t; e3 _, `$ _! i, E
6.9.4 Resonant Ultrasound Spectroscopy 168
1 g! x3 L- W; ?% y6.10 Achieving a Wide Range of Time or Frequency 171
6 w# d5 B# `7 c( I6.10.1 Rationale 171
2 R* d$ t# W/ M3 |; B& i8 O8 }6.10.2 Multiple Instruments and Long Creep 172
* C3 f! o: Z, z- p6.10.3 Time–Temperature Superposition 172
9 q) H. E" |8 K; k9 ]9 Z6.11 Test Instruments for Viscoelasticity 1734 n2 g% }7 M# Q1 V% Z
6.11.1 Servohydraulic Test Machines 173
O# ~! a7 Y0 U1 `6.11.2A Relaxation Instrument 1747 R! R6 W* l! l3 _
6.11.3 Driven Torsion Pendulum Devices 1748 w9 `' ]. j! ]/ [
6.11.4 Commercial Viscoelastic Instrumentation 178
+ |0 c& V7 k$ K) A& B+ e$ f6.11.5 Instruments for a Wide Range of Time and Frequency 179
" t/ a3 g7 R9 s/ D( f; Z6 P1 M6.11.6 Fluctuation–Dissipation Relation 182
+ B" |6 a0 i: H( O6.11.7 Mapping Properties by Indentation 183. m8 }- {: A) k' F% N- I
6.12 Wave Methods 184$ }7 q7 i8 m% ^+ V* i& `1 _+ b
6.13 Summary 188! b. t$ ^/ |) O& `
6.14 Examples 188$ {2 }4 B! Y6 T0 [
6.15 Problems 200- `# [/ G2 p' s7 G5 l
Bibliography 2019 G( j4 {( J! M; k. a; ^0 g
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6 E* x2 ~) [8 J; _7 O: R- w( c7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207' ^- J/ P' ?6 f2 Q8 A b& R1 I5 b
7.1 Introduction 207" K" N3 j' R& i
7.1.1 Rationale 207
+ d8 m: F/ J8 E7.1.2 Overview: Some Common Materials 207
, Z4 J" T( M* _0 b8 E7.2 Polymers 2081 y9 h! T6 n5 [, S( C
7.2.1 Shear and Extension in Amorphous Polymers 208& X6 W I S3 ]0 p1 ?
7.2.2 Bulk Relaxation in Amorphous Polymers 212
3 ?6 l% B2 _! v' e3 L6 y* F7.2.3 Crystalline Polymers 213
+ y" Y# X& V* i7.2.4 Aging and other Relaxations 214, Z- |& @* x+ z2 I, P
7.2.5 Piezoelectric Polymers 214, j: \5 U9 k, c" A
7.2.6 Asphalt 2147 x: V- x- K2 P1 ~/ N# X
7.3 Metals 215
" r3 J3 U2 u, I! B7.3.1 Linear Regime of Metals 215
; f2 i& H7 s1 }, v9 y7.3.2 Nonlinear Regime of Metals 217
' C3 I' Q; n0 ?7.3.3 High-Damping Metals and Alloys 219
3 P w2 ]) [+ u* v7.3.4 Creep-Resistant Alloys 224# i: Q: j0 V n2 ^
7.3.5 Semiconductors and Amorphous Elements 225
- ?* X: k5 p( I7.3.6 Semiconductors and Acoustic Amplification 226* C( a* D9 e* T4 l
7.3.7 Nanoscale Properties 226
+ [$ i' n% g P' b) j7.4 Ceramics 2273 `4 f$ a4 }0 k% A4 f, ^- R- G
7.4.1 Rocks 227, B( Z" m, P5 h' T
7.4.2 Concrete 2290 Z5 Y( r/ g- }8 H) ]7 j7 F$ A0 f
7.4.3 Inorganic Glassy Materials 231
; s5 n% m( K9 P7.4.4 Ice 2315 e' G6 V' r, `5 B
7.4.5 Piezoelectric Ceramics 232
5 G- r, N* [3 k2 p4 \7.5 Biological Composite Materials 233
8 x" @( X+ Z. E7 s& @" W( P7.5.1 Constitutive Equations 234+ @2 x" e& _, L! L( M
7.5.2 Hard Tissue: Bone 2344 V; m' y* z( n- |7 t& R. H- m
7.5.3 Collagen, Elastin, Proteoglycans 236
2 w3 W" W8 B% ]7.5.4 Ligament and Tendon 237
3 ^! B8 E8 w5 K7.5.5 Muscle 240) u" P3 m; W5 A! x
7.5.6 Fat 243
) m( S9 w( [3 O/ n4 j. h9 R" ?7.5.7 Brain 243
- ?, C# y/ B# H; f' ?7.5.8 Vocal Folds 244
7 U# e7 m6 K8 U& _% }; G7.5.9 Cartilage and Joints 244% d4 Y3 X9 n5 k! ?0 B) K
7.5.10 Kidney and Liver 246+ ]. x* i% r. W: d0 V( m
7.5.11 Uterus and Cervix 246
6 q$ v& o2 ^" q5 G2 a7.5.12 Arteries 247 ]( j: A0 T6 c( Y
7.5.13 Lung 248, w8 `$ R/ X9 U/ q& Z
7.5.14 The Ear 248( c- U2 [0 L/ z2 {) a, {( S4 H
7.5.15 The Eye 249
, w9 T" `; m1 R1 _ u7.5.16 Tissue Comparison 251. q4 C& S1 t4 n- w, D9 R
7.5.17 Plant Seeds 252
$ t0 m4 T; n0 C! [1 l! s) W. [7.5.18 Wood 252! i( R- Z- p& K# N
7.5.19 Soft Plant Tissue: Apple, Potato 253. Q1 [0 V( w% [
7.6 Common Aspects 253
+ c- G% l. w2 I. d3 E$ V" c7.6.1 Temperature Dependence 253
- f8 h8 A% ?; e6 v/ z$ P; F F7.6.2 High-Temperature Background 254
0 I3 s& }4 P) k( w7.6.3 Negative Damping and Acoustic Emission 255/ o9 Y) b5 n) k
7.7 Summary 255
) _( Q" X" b8 Q' Z, A7 o) B7.8 Examples 255# Y" O0 \& B( s( t7 p( r
7.9 Problems 2563 |- Q5 J2 {* j! u: m
Bibliography 257
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. }3 v; }; h! H. Z) I, I8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718 ^& g, F9 B: X6 w3 h: X$ F, _! \
8.1 Introduction 271
7 I. Y ^! v1 Q( }$ J$ L( [ Q2 d& P8.1.1 Rationale 271
0 [! c4 I3 D' i& u8.1.2 Survey of Viscoelastic Mechanisms 271% r( o Q( I, `4 x' J+ A
8.1.3 Coupled Fields 273
9 N) l' M+ Z$ g* I8.2 Thermoelastic Relaxation 274
* U7 y" O6 r9 Y; a8.2.1 Thermoelasticity in One Dimension 274& E4 U0 [* x" n: i0 }
8.2.2 Thermoelasticity in Three Dimensions 275% n0 @3 \5 W. W( P7 m3 D/ q
8.2.3 Thermoelastic Relaxation Kinetics 276! I5 n/ ^* |0 o) g. z$ u r8 ^2 ?
8.2.4 Heterogeneity and Thermoelastic Damping 278
$ ~, ?6 v6 D% ?2 o5 j8.2.5 Material Properties and Thermoelastic Damping 280
8 C7 @6 ~& X( M0 Q; @8.3 Relaxation by Stress-Induced Fluid Motion 280' E/ c* Y. N4 H! H
8.3.1 Fluid Motion in One Dimension 2802 @* l; Q6 y9 q6 B$ d* t0 [% D
8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281
$ p* C+ Y7 S: I) i6 {7 N3 C& F6 E8.4 Relaxation by Molecular Rearrangement 2869 [4 g% W. L+ l1 f0 { V
8.4.1 Glassy Region 286
- q- C' k! J I% r o8.4.2 Transition Region 2875 ?/ k+ a0 c: h y/ n/ F. [ E: s3 H" F0 l
8.4.3 Rubbery Behavior 2897 ]& Y* o4 n7 }& q' I# g
8.4.4 Crystalline Polymers 291
8 f0 }- A2 c$ }& g' k8.4.5 Biological Macromolecules 292
: \ a3 W3 ~4 B; |0 c. g' U/ v8.4.6 Polymers and Metals 292
6 j1 R D. \' P' b8 Z8.5 Relaxation by Interface Motion 292
$ U- W9 S! `9 g7 c: d$ d8.5.1 Grain Boundary Slip in Metals 292
0 E$ c* g" @) I& U3 d% N; l8.5.2 Interface Motion in Composites 294, ^* i& t4 [% C$ X1 g& J6 B
8.5.3 Structural Interface Motion 2946 o3 Z* d# Z, W1 l3 L
8.6 Relaxation Processes in Crystalline Materials 2948 \3 ?; w7 M* J" E: b
8.6.1 Snoek Relaxation: Interstitial Atoms 294
; Q. Q. |" V$ l$ p' I! k7 s/ e8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 2984 s' Z' o- ~* I. O, Y
8.6.3 Gorsky Relaxation 299
$ E1 \! j5 p/ O) X. H' e p1 c8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 3009 a- }& w4 x0 L% l. L
8.6.5 Bordoni Relaxation: Dislocation Kinks 303
5 f: w$ ]( g: b9 @, C( _& _8.6.6 Relaxation Due to Phase Transformations 3059 \0 F# K. m" r0 W5 \$ y8 p$ j6 I
8.6.7 High-Temperature Background 314
2 B$ a2 s- U) z. b5 @6 [: @8.6.8 Nonremovable Relaxations 315
$ P+ y6 @5 d9 }% f2 ^8.6.9 Damping Due to Wave Scattering 316* m4 X6 Y' g) p( T
8.7 Magnetic and Piezoelectric Materials 316, i" E: [) U/ h5 j, R
8.7.1 Relaxation in Magnetic Media 316
' N9 y( t/ X! o8.7.2 Relaxation in Piezoelectric Materials 318
, @( z5 G% h; ?+ L$ _- M8.8 Nonexponential Relaxation 322, p( o" G8 p; g
8.9 Concepts for Material Design 323* o( }; A9 P$ s7 m
8.9.1 Multiple Causes: Deformation Mechanism Maps 323
9 T! e& H1 @! W U8.9.2 Damping Mechanisms in High-Loss Alloys 326$ u# p! P6 y. W* Q0 \/ x
8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326( Q9 V: Q3 r! T- c) K; S. z1 N
8.10 Relaxation at Very Long Times 327
% Q% V+ v$ R9 x+ l/ j8.11 Summary 327
- ~% W6 T! S& B% ?# A% n8.12 Examples 328* z k! E3 L7 a+ c7 c
8.13 Problems and Questions 332& Z4 H& I$ a3 Q9 @' C/ z
Bibliography 3320 U* D0 r% T3 O2 f
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9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 3415 P# c/ D' b2 ]. C' y
9.1 Introduction 341
- n8 N) |2 J$ ~/ x P9.2 Composite Structures and Properties 341( v: i5 x% @: v6 W! X: g2 _
9.2.1 Ideal Structures 341
, J, s( c) J0 O8 S. c9.2.2 Anisotropy due to Structure 342
% `4 r7 C% T3 Q# i. f, c! Z" M: g9.3 Prediction of Elastic and Viscoelastic Properties 344
6 T9 v( O5 N! i2 ]; X9.3.1 Basic Structures: Correspondence Solutions 344
, d1 r0 o E& T) R7 }3 s0 U2 Z( I9.3.2 Voigt Composite 345
& X8 a1 o1 }1 l9.3.3 Reuss Composite 3452 P# y" I [# t( R( m! x! V
9.3.4 Hashin–Shtrikman Composite 346- [$ a( I2 r/ `
9.3.5 Spherical Particulate Inclusions 347* `' L% G$ O2 S- N
9.3.6 Fiber Inclusions 3490 z* m- A* W) `1 e. ^% i
9.3.7 Platelet Inclusions 349
) H9 S. k2 F3 a4 \/ {/ T9.3.8 Stiffness-Loss Maps 350
3 {1 _9 r# B) l8 l) Q: ^1 Q3 x! e8 i9.4 Bounds on the Viscoelastic Properties 3532 @" R# \) G- H1 p: u( ?. S' s
9.5 Extremal Composites 354
4 B- M+ f: d, ^4 B0 x2 }9.6 Biological Composite Materials 356
8 D0 P0 V4 f3 T1 E# ? d7 h5 i9.7 Poisson’s Ratio of Viscoelastic Composites 357
: d+ {+ Z3 x) `0 [: N# f5 a9.8 Particulate and Fibrous Composite Materials 358
/ @# Q7 V/ t$ J3 d9.8.1 Structure 3588 |; d) E7 ]- _) \- P
9.8.2 Particulate Polymer Matrix Composites 3599 p5 D S1 S/ s8 n) R9 P1 U- h
9.8.3 Fibrous Polymer Matrix Composites 361
) c& E* n8 Y& `5 H! x9.8.4 Metal–Matrix Composites 3624 P K' D+ t5 M* J2 L$ K
9.9 Cellular Solids 363# I6 e9 ]3 e" _" ]
9.10 Piezoelectric Composites 3660 ~5 y1 t7 r& _" Z" U
9.11 Dispersion of Waves in Composites 3669 [9 R, p2 d% ?1 b" U+ ^ M
9.12 Summary 3679 T7 m8 p/ V+ i( p
9.13 Examples 367
' h, j6 L' a! S9.14 Problems 370/ h, l' d& O+ H4 c M
Bibliography 370( ?* N) }7 Q5 A, z, u
8 c- {* [1 ]: L: T* n
4 u; x* M) R8 Y3 O0 l) P$ O3 u8 v
, ^1 W& [: B/ a5 p& c! T3 ?10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377 j2 V/ P! Y- b( I0 y- L
10.1 Introduction 3773 p$ W' W! \# D% r% e
10.2 A Viscoelastic Earplug: Use of Recovery 377
" K3 e/ M) V! T' w* X, K10.3 Creep and Relaxation of Materials and Structures 378
# P, T7 M M% W; j- |7 t10.3.1 Concrete 378- k( R+ K# s9 H3 G' y1 ]
10.3.2 Wood 378
) ?4 j2 `, v) ^; a9 s; S/ z10.3.3 Power Lines 379
" w5 M/ ~3 |, {10.3.4 Glass Sag: Flowing Window Panes 3800 J4 \1 M3 t2 G- q
10.3.5 Indentation: Road Rutting 380
8 H& e6 B/ n. O2 `10.3.6 Leather 381
! ]. [0 g3 M$ L1 A1 q* L10.3.7 Creep-Resistant Alloys and Turbine Blades 381# p" {8 F# K5 B5 s* X q
10.3.8 Loosening of Bolts and Screws 382
8 X* Y1 S9 e+ j10.3.9 Computer Disk Drive: Case Study of Relaxation 384' ]* C$ m) \9 L D, F" O& {; ]
10.3.10 Earth, Rock, and Ice 385
! D2 R( w& f! v/ U" n* p10.3.11 Solder 386* Q: C1 g' _" }* q' ?
10.3.12 Filamentsi nL ight Bulbs and Other Devices 3871 z; ? J, l, p+ b% N# K+ c5 N( |
10.3.13Tires: Flat-Spotting and Swelling 388
9 ^* K) s/ f) b: F9 _( w10.3.14Cushionsfor Seats and Wheelchairs 388+ y( n$ u) G5 i8 z0 R! o. S% `; @+ {
10.3.15 Artificial Joints 389
4 f* c1 p8 @2 p# }# D10.3.16 Dental Fillings 389
2 m/ [- R- }+ s10.3.17 Food Products 389/ G1 j+ u6 F& ~% }% V
10.3.18 Seals and Gaskets 390
& H X% v8 @* E6 W! q10.3.19 Relaxationi nM usical Instrument Strings 390
* T6 g8 Z3 u- Z* b10.3.20 Winding of Tape 391
' j( y4 }% R$ C2 I% W; R8 q10.4 Creep and Recovery in Human Tissue 391- `: F5 Y' M) v3 y
10.4.1 Spinal Discs: Height Change 391
, u2 @ ?- r% u- v# { \( b10.4.2 The Nose 392
- ? i* U, B4 w10.4.3 Skin 392 G! \" g6 D1 {: ]& Z d! \- G
10.4.4 The Head 393
% H( W! A S+ ~. [6 i10.5 Creep Damage and Creep Rupture 394
, ~, S3 p/ ~ R& \8 `% X s% y10.5.1 Vajont Slide 394. d6 T5 ~4 }" N* c$ k1 Y; ~9 S
10.5.2 Collapse of a Tunnel Segment 3948 q6 {, s! X h8 F" t
10.6 Vibration Control and Waves 3947 M; P3 u5 v" y- O! Z
10.6.1 Analysis of Vibration Transmission 394
3 Z. J; \& ^ v |10.6.2 Resonant (Tuned) Damping 3970 F2 E( I, A: e
10.6.3 Rotating Equipment Vibration 397
6 B7 h1 B# @( u8 [" k- J10.6.4 Large Structure Vibration: Bridges and Buildings 398
/ P9 d/ u6 y& \5 v10.6.5 Damping Layers for Plate and Beam Vibration 399
/ f- J& ~# h: c1 T) ]+ B7 _' `' B9 ]10.6.6 Structural Damping Materials 4005 A7 j P! K) c, y8 U7 I4 |
10.6.7 Piezoelectric Transducers 402; f) |6 c# [: Q: r
10.6.8 Aircraft Noise and Vibration 402" T/ A* c( z! E" [
10.6.9 Solid Fuel Rocket Vibration 404' P! A4 b# `/ h! W" [
10.6.10 Sports Equipment Vibration 404
4 l2 f3 {$ H! m! ?* S% R10.6.11 Seat Cushions and Automobiles: Protection of People 4043 y5 P+ ~! U/ Z; M A
10.6.12 Vibrationi n ScientificI nstruments 406
7 ?" y, c0 h) B, ^' z. _10.6.13 Waves 406
- m2 j9 `: K6 o6 ~) O10.7 “Smart” Materials and Structures 407
% C" }; Z; o b10.7.1 “Smart” Materials 407$ {" C$ m$ a- s" l7 s5 w( O
10.7.2 Shape Memory Materials 408; x( T+ r1 M4 }" l" }
10.7.3 Self-Healing Materials 4095 B; Q& ~0 q. u- L1 E
10.7.4 Piezoelectric Solid Damping 409
! P" q+ W$ \4 D* }10.7.5 Active Vibration Control: “Smart” Structures 409
6 q1 O* O6 x( R" ^" ~10.8 Rolling Friction 409
9 H0 @% m( Q4 [" M10.8.1 Rolling Analysis 410
4 s* o$ i% B3 K) O10.8.2 Rolling of Tires 4111 f1 \, Q& ? E, c! n! k5 d) w
10.9 Uses of Low-Loss Materials 412) v& R4 ]+ F9 C5 B
10.9.1 Timepieces 412! J( @. f' `0 z& q- M! A7 f
10.9.2 Frequency Stabilization and Control 413& G) t; W) M# b0 R P& C; W
10.9.3 Gravitational Measurements 413
1 u9 O' v% y: H$ B0 V( q9 i10.9.4 Nanoscale Resonators 414! Z) G* `( ~4 [+ I/ t8 L
10.10 Impulses, Rebound, and Impact Absorption 414$ _. p. ^( U, N; c1 n5 [! i
10.10.1 Rationale 414; B8 Z n+ q9 Q# a
10.10.2 Analysis 415+ u; V3 w# v6 X& Q. P" O% ~# f2 F
10.10.3 Bumpers and Pads 418
# C8 [/ b" Y7 K4 c10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 419
|0 X) M0 W2 w$ x9 J! v+ \10.10.5 Toughness of Materials 419
6 J6 i8 B7 f3 M" c10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420
5 B2 T, w3 D4 ~. k& o$ W10.11Rebound of a Ball 421
0 w) `" \2 B N. u10.11.1 Analysis 421- [9 H+ J' v$ h+ H$ \) A& c
10.11.2 Applications in Sports 422, ~5 o' O/ K+ X& t% R* l
10.12 Applications of Soft Materials 4248 g* R* H: U7 b1 f
10.12.1 Viscoelastic Gels in Surgery 424 J @ g b+ T' q& f( s2 Q7 f' T! p; T
10.12.2 Hand Strength Exerciser 424
9 r* i* o! B9 z w! G10.12.3 Viscoelastic Toys 424
. S& ]" K: T+ ]) ]+ x. o5 P10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425
. l* E/ |, k) p/ d" r$ [3 H10.13 Applications Involving Thermoviscoelasticity 425
% @3 Z2 l1 p2 |( t) k0 \/ C+ q10.14 Satellite Dynamics and Stability 426
( [) _7 ]% T; B10.15 Summary 428" m4 k y* T |3 b/ ?$ a
10.16 Examples 429
3 H) j' r4 |4 q9 o- u10.17 Problems 4319 [! D/ _; P& h" m4 q. k, g
Bibliography 4315 M4 T9 k. q' a# X; C5 J) ]& d
7 A8 y) x- v( o' {5 `
' A3 I9 c6 b+ h( i+ G
% L. k( c, e& q, P8 m
A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
% x9 |& {* Y% R2 `( ]A.1 Mathematical Preliminaries 441
% ?9 S/ O7 P7 l- q$ UA.1.1 Introduction 441
. W5 p& }- m: t" zA.1.2 Functionals and Distributions 441 u: }! P$ V- w# Y
A.1.3 Heaviside Unit Step Function 442% l5 `/ R& L: s3 u
A.1.4 Dirac Delta 442
# U9 h/ y7 V$ I' `" v8 M2 c( cA.1.5 Doublet 443/ ?) [& T# g9 T
A.1.6 Gamma Function 445
5 t" H4 @, v. G2 `) WA.1.7 Liebnitz Rule 445
( n$ h' h9 ]; r; X& W) `- }! sA.2 Transforms 4458 Z, l* B9 A5 k3 U4 U2 J" @
A.2.1 Laplace Transform 446
1 o- `6 d3 D) N/ X% w# B, {A.2.2 Fourier Transform 4463 B% I6 e$ c+ N" e
A.2.3 Hartley Transform 447: H" K* A1 |# w8 v/ M7 t( j$ L
A.2.4 Hilbert Transform 447
# c. R) \8 U% M9 a% tA.3 Laplace Transform Properties 448
! f# }- t; E- [" ~6 S0 a. FA.4 Convolutions 449
& c$ ~% @" A- W; b! J$ TA.5 Interrelations in Elasticity Theory 4514 V. V9 [3 a6 w: ]' V' P
A.6 Other Works on Viscoelasticity 451! P& J' ~) N) K8 X. a8 e
Bibliography 452
! m( D P; i! S' M
5 j! q' n8 M# y/ B5 X) ]% b3 S, q' ]. b: B9 F! d2 v* g
B: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455. O* t/ H, U& W, l8 o( L( T- H% U
B.1 Principal Symbols 455# ?$ C* N- G9 L4 M
Index 457
4 a; G5 I$ u1 X& B
9 a( E3 v6 B5 v% ]. y! y( Z: D& q" ?
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