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目錄( Z" {( w9 s" k
2 Q6 n" U: q! m1 y9 B l- }6 PContents3 t9 |! x2 A6 U2 [
, Q: S' y+ V1 jPreface page xvii0 G5 C$ i* ]8 h2 Z; z4 a( O
1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
8 _/ t' h, V1 d- @4 F( v0 ~1.1 Viscoelastic Phenomena 1
0 |$ j4 @+ i$ B' I1.2 Motivations for Studying Viscoelasticity 3, b0 z) n" i5 j& N X m
1.3 Transient Properties: Creep and Relaxation 31 b& [2 J- v* D8 y) H6 e" |
1.3.1 Viscoelastic Functions J (t), E(t) 32 E, K% u1 k' |0 M% I d
1.3.2 Solids and Liquids 7
, O& O1 W9 c& H" Z8 u. m4 E# X) ?1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
# z" X: y# o" f' Q( @1.5 Demonstration of Viscoelastic Behavior 10
. T9 v/ p4 m2 v. z; E0 g1.6 Historical Aspects 105 l, V. a5 s; X; _
1.7 Summary 11+ F( }' R# O# W$ n: `: f
1.8 Examples 112 w) H* w5 W: I. r) |) F
1.9 Problems 12
; D+ C* c$ u( W* z: b" ]; sBibliography 12; H; J/ Q$ A6 o* s
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+ B4 G ?1 R( l) B3 N3 T2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 l7 I7 j) s: x2.1 Introduction 14
' z; T+ B8 s) D) R2.2 Prediction of the Response of Linearly Viscoelastic Materials 14% N& D9 T+ }6 }9 q3 a
2.2.1 Prediction of Recovery from Relaxation E(t) 14
! s9 `4 m3 _. S6 z: i1 B! l. n2.2.2 Prediction of Response to Arbitrary Strain History 15
$ J3 h; q; v- [! k1 _; K1 N2.3 Restrictions on the Viscoelastic Functions 178 |/ N& `. I5 g v
2.3.1 Roles of Energy and Passivity 17/ n4 b/ Z4 p0 ?
2.3.2 Fading Memory 18
, R& t; n( F1 C `; i) c2.4 Relation between Creep and Relaxation 19# K1 ]' [6 r% k1 f, ~5 ^" v4 p
2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19 t: N/ }1 L* k/ X0 ]+ n
2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20
' f; }* k0 J& z0 L2.5 Stress versus Strain for Constant Strain Rate 200 _3 O" a" S$ r$ J1 k# N
2.6 Particular Creep and Relaxation Functions 21$ \4 I# c$ p8 M9 \0 H4 U: A5 d
2.6.1 Exponentials and Mechanical Models 21+ U, T; p$ c8 @8 O+ i
2.6.2 Exponentials and Internal Causal Variables 26
% [+ i" L, B4 m% |8 m2.6.3 Fractional Derivatives 27
+ H- T4 L# [- h; {* q( A2.6.4 Power-Law Behavior 28, Y, E6 F7 B; ^
2.6.5 Stretched Exponential 29
& j' I Q0 n- @% g6 G3 ]3 c9 c2.6.6 Logarithmic Creep; Kuhn Model 29- S6 s& L4 M$ K- h/ l5 X: j; K3 G
2.6.7 Distinguishing among Viscoelastic Functions 30# H% r( k8 V) o# N
2.7 Effect of Temperature 30
G$ V9 ^& ~: E9 W2.8 Three-Dimensional Linear Constitutive Equation 33/ ]: K2 Z2 O' L) X& s9 D; Q1 Y
2.9 Aging Materials 35
: [- o7 c9 A' l* g$ w/ e) Z7 }" D2.10 Dielectric and Other Forms of Relaxation 351 N( r! M. t3 w/ F- z0 h4 y$ v
2.11 Adaptive and “Smart” Materials 36$ o' I: @9 ^( W, P( ?' n7 c4 M# s. `
2.12 Effect of Nonlinearity 37' i: g. d: b- k$ k: j+ K5 K7 q
2.12.1 Constitutive Equations 37
6 y9 t/ H0 _, e2 F3 C0 n6 [" y2.12.2 Creep–Relaxation Interrelation: Nonlinear 40
. y# X. a0 b: N$ J* z0 A4 I2.13 Summary 43/ R% {8 a* }* U0 r: x* Q5 e
2.14 Examples 43
2 t# ?; S4 ^6 M4 K( [2.15 Problems 51
( S& i4 b' o( a/ sBibliography 52
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3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
- U" W+ }4 a9 N" |3.1 Introduction and Rationale 557 L& }& A% b1 _
3.2 The Linear Dynamic Response Functions E∗, tanδ 56% j* r/ f5 s5 a. m3 `8 j6 y
3.2.1 Response to Sinusoidal Input 57! d) X4 q) V- o% d" b* w
3.2.2 Dynamic Stress–Strain Relation 59
! a7 S |1 [( ^2 R3 _3.2.3 Standard Linear Solid 62
) C5 x" n" @' X) N0 E) Y7 W3.3 Kramers–Kronig Relations 63
5 l: M5 L8 [6 i( ^3.4 Energy Storage and Dissipation 659 G0 I' a k* D' c8 r
3.5 Resonance of Structural Members 67
3 ~" V2 m6 G" K, e* h" g1 i3.5.1 Resonance, Lumped System 67 E" Z) ?6 K1 ]' y
3.5.2 Resonance, Distributed System 71
3 \, e; I- F( D4 E3 Q2 G3.6 Decay of Resonant Vibration 746 v* h7 Z/ m6 @6 d$ \! m8 N
3.7 Wave Propagation and Attenuation 77" l' b3 @" @6 O3 D; z* C2 l4 X
3.8 Measures of Damping 79
+ u% t3 m: K; P. G$ l k3.9 Nonlinear Materials 79 T7 V6 F: ^, z* h
3.10 Summary 81
' d6 x$ f0 Y# _# }* A+ I: d3.11 Examples 818 p3 F) I! _8 P+ |- k4 D$ Q
3.12 Problems 88, i, y# z6 l; `
Bibliography 89
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0 n$ O* S/ w* M4 q) j( z4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91+ b" h g! m! \" A/ A) I, d+ o. ^
4.1 Introduction 91+ A' b* Q% E2 N
4.2 Spectra in Linear Viscoelasticity 927 M( E1 \) a, E5 Y
4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 92
5 f0 D( b k$ M) c3 A4.2.2 Particular Spectra 93
9 {! s# E$ b4 K& F4.3 Approximate Interrelations of Viscoelastic Functions 957 t# D' H: v9 E5 u6 l9 i) Z5 T
4.3.1 Interrelations Involving the Spectra 95) ?, d4 n: s* {+ ]0 v- H
4.3.2 Interrelations Involving Measurable Functions 982 _) X6 `" |/ h7 B. V
4.3.3 Summary, Approximate Relations 1015 H# J* X5 u" J- }# l
4.4 Conceptual Organization of the Viscoelastic Functions 101+ l5 U; }$ n. b9 z$ G9 u0 w
4.5 Summary 1048 b3 d$ \* ?7 r% j
4.6 Examples 1049 @" `# D/ `: P4 U1 ?9 ]' s/ X
4.7 Problems 109" B8 P( N9 f6 A$ p; K+ J
Bibliography 109
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 1111 ~7 E/ N1 N9 E! C
5.1 Introduction 111
5 q. Y. c4 `( M$ W( D7 Q. A5.2 Three-Dimensional Constitutive Equation 1112 y& K6 C( \. k8 |5 E3 S
5.3 Pure Bending by Direct Construction 1124 \! V8 Z/ ?% v# s
5.4 Correspondence Principle 114
8 ^* [$ z- D6 p' g1 o5.5 Pure Bending by Correspondence 116- J$ K0 W" _: O1 k2 V
5.6 Correspondence Principle in Three Dimensions 1164 J Y7 r1 s6 W; `) W. D/ f! C
5.6.1 Constitutive Equations 116 E' q+ y1 Z2 ~+ r+ T
5.6.2 Rigid Indenter on a Semi-Infinite Solid 117
$ X# T9 E$ Z- @' A0 w5 J5 a% R5.6.3 Viscoelastic Rod Held at Constant Extension 119
% I1 W! b! i0 W8 c3 }- ^8 m9 F. [5.6.4 Stress Concentration 1197 }2 [+ S" g* h9 k
5.6.5 Saint Venant’s Principle 120- |. I4 C6 L+ X( h
5.7 Poisson’s Ratio ν(t) 121
0 m" |) {# N5 Y8 G8 O. ~! G5.7.1 Relaxation in Tension 121
* `% @3 L3 y0 J" _5.7.2 Creep in Tension 123
$ G# d3 o% x2 M) K5.8 Dynamic Problems: Effects of Inertia 124" ~" F: {0 Q& _: w9 g0 M
5.8.1 Longitudinal Vibration and Waves in a Rod 124
; y7 D5 W) {* D. D5.8.2 Torsional Waves and Vibration in a Rod 125
+ M% T$ B6 Z' J- [! N! k8 G5.8.3 Bending Waves and Vibration 128$ F' ?0 [3 F+ ^8 V8 F) W1 `0 g
5.8.4 Waves in Three Dimensions 129
5 R& P, E0 L; s! {5.9 Noncorrespondence Problems 131
8 z- D* h6 U! a4 }) F/ u5.9.1 Solution by Direct Construction: Example 131" r4 C, r9 P( Y. t! ?/ f: _
5.9.2 A Generalized Correspondence Principle 1323 X, C# A+ U" S" o8 m$ P
5.9.3 Contact Problems 132- M& _' s9 @! Y( H
5.10 Bending in Nonlinear Viscoelasticity 133
( c0 P @8 n- w t5.11 Summary 1349 d0 g( T* W' A/ M. ?4 U; ^
5.12 Examples 134
3 B! T; e4 b1 d; @- d! {! B5.13 Problems 142
7 V8 x+ V+ l& _Bibliography 142
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) ~. L" g1 z/ H* z/ Y M! @+ H D6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145; o7 C1 o- V0 F w
6.1 Introduction and General Requirements 145
5 u! h4 d$ H' h- d+ s% D; A6.2 Creep 146- i. Y% Q0 u0 \+ D. N8 m
6.2.1 Creep: Simple Methods to Obtain J (t) 146
4 m# _9 D( R: S2 m5 R# E6.2.2 Effect of Risetime in Transient Tests 146
5 |1 M/ f2 s/ H2 T6.2.3 Creep in Anisotropic Media 148, Z) p6 `- c- L8 F
6.2.4 Creep in Nonlinear Media 1483 `% [0 `# ~/ t3 u7 y
6.3 Inference of Moduli 150
: f5 ~- N1 E) S; O* ~8 `" G8 g6.3.1 Use of Analytical Solutions 150" I. A. D V/ ~& c$ s3 r ?- o
6.3.2 Compression of a Block 151
0 }9 H8 X9 |+ [+ `6.4 Displacement and Strain Measurement 152
" H# [; ~7 g! O7 L3 ]6.5 Force Measurement 156
) P6 |2 s; {) T, a6.6 Load Application 157
0 g) V# @/ O% c5 t- x# q6.7 Environmental Control 157
4 p: }+ P; g! e6.8 Subresonant Dynamic Methods 158# V; `: i+ f4 u
6.8.1 Phase Determination 158
" T G! j/ ~; e1 z7 C6.8.2 Nonlinear Materials 160
- l, U6 T; }' A1 M. k6.8.3 Rebound Test 161
8 \4 c( w6 |% s: t1 G) O0 F6.9 Resonance Methods 1611 x0 {0 N0 d, w1 ^$ p
6.9.1 General Principles 1610 R8 {. y( I- L9 |' w
6.9.2 Particular Resonance Methods 163
7 m8 i$ h* p, D/ \6 m6.9.3 Methods for Low-Loss or High-Loss Materials 166- T( z8 r1 K8 g) v4 j
6.9.4 Resonant Ultrasound Spectroscopy 168
6 `5 ~# o) `- J. s6.10 Achieving a Wide Range of Time or Frequency 171- T0 p. x4 u) n3 G* {* {. r
6.10.1 Rationale 171
1 n* @9 z- [) X% }% J: @6.10.2 Multiple Instruments and Long Creep 172
" N! o; M2 h1 P8 ~7 W' r0 A6.10.3 Time–Temperature Superposition 172' N* V4 l5 L8 m6 u6 b8 `$ t
6.11 Test Instruments for Viscoelasticity 173
0 ?7 i, d* X, I/ t6.11.1 Servohydraulic Test Machines 173
5 O! v5 h3 b5 `1 O$ c! X) U: X/ A- {6.11.2A Relaxation Instrument 174
- }5 B5 _5 V0 ?$ W+ b$ j6.11.3 Driven Torsion Pendulum Devices 174
+ V" v' F$ R( b. l6.11.4 Commercial Viscoelastic Instrumentation 178! }( T8 X+ i& R5 ?: j
6.11.5 Instruments for a Wide Range of Time and Frequency 179
8 j9 n" P- l$ E9 p9 P5 s' H8 p6.11.6 Fluctuation–Dissipation Relation 182
+ E- D' j/ o I$ y6 I6.11.7 Mapping Properties by Indentation 183, g7 N/ J, y% [& j# d4 }. K
6.12 Wave Methods 184
. \ V5 v( G" p, {# C6.13 Summary 188
& H5 ~7 d( ~& I4 m, v( |6.14 Examples 188
) Z( s$ U; j3 f/ T1 A6.15 Problems 200+ r5 c0 l1 i Z% U
Bibliography 201- w- c! E5 f" t$ M# g' e% E
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7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207. h( t% ~$ K" z' Y9 o5 v
7.1 Introduction 207
) b* O5 b) c) }- d4 _* V( R+ ^7.1.1 Rationale 207
3 o" Q) O* }4 D0 b7.1.2 Overview: Some Common Materials 207
! V9 `. |0 [# L) ~7.2 Polymers 2084 i# l) ], }( W! }. Z
7.2.1 Shear and Extension in Amorphous Polymers 2085 j: D& w$ j5 l5 u6 @; H# b; G& Q
7.2.2 Bulk Relaxation in Amorphous Polymers 212. A- M; p" v3 Y- P: ]) `' \
7.2.3 Crystalline Polymers 213
+ e+ w, j" P3 l( ?1 A4 j: m$ ?7.2.4 Aging and other Relaxations 214
1 H1 A5 u. B, f' T1 H7.2.5 Piezoelectric Polymers 214/ ]: E$ ^. y( y4 U" M
7.2.6 Asphalt 214
1 P r4 P. D* k, g4 W7.3 Metals 215
9 R. ?5 n1 W; Q& [8 \# E# R7 U$ o7.3.1 Linear Regime of Metals 215
& B) i$ N! C; V" n, p8 d7.3.2 Nonlinear Regime of Metals 217
# O( W4 r9 W' Z. J. T- x0 P7.3.3 High-Damping Metals and Alloys 219 ?1 g B+ M) ^
7.3.4 Creep-Resistant Alloys 2244 a" y% _8 f L5 d! p2 O
7.3.5 Semiconductors and Amorphous Elements 225
+ r0 l9 p- w2 v7.3.6 Semiconductors and Acoustic Amplification 226
! y% A1 Z9 H2 s7 C6 s4 R1 u! d7.3.7 Nanoscale Properties 226
) j$ g& I4 f o. H1 g. p4 Y7.4 Ceramics 2278 M/ E8 l" U4 e" p
7.4.1 Rocks 2279 I9 z- {9 k. K& M& t5 Q4 y
7.4.2 Concrete 229
0 W# v3 k6 ^* u8 [6 G6 W7.4.3 Inorganic Glassy Materials 231: @) h* [* q7 B, C# }1 U% P
7.4.4 Ice 231
8 r! h6 `; ?3 s4 w" E( ?$ f% f7.4.5 Piezoelectric Ceramics 232
. @% \, u+ {' h9 C( v' S7.5 Biological Composite Materials 233) M# k5 G9 o' d6 Y
7.5.1 Constitutive Equations 2345 |1 L& S& i3 l
7.5.2 Hard Tissue: Bone 234
5 |; |$ P* H9 |8 n, `6 r; p7.5.3 Collagen, Elastin, Proteoglycans 236, ?, M; w1 ]5 w5 R
7.5.4 Ligament and Tendon 237
' f' J4 Q& \/ s2 a- s1 z1 \. ^7.5.5 Muscle 240
- I* s j7 }8 i* j# y1 k: j) N7.5.6 Fat 2435 [2 R, d/ _7 G0 x- N, k/ c
7.5.7 Brain 2439 m# s5 S" v5 D" H( B# d" ]
7.5.8 Vocal Folds 244
2 G2 I B* s' ]& \, [7 k7.5.9 Cartilage and Joints 244( {: A/ D- ?: Q2 G8 L
7.5.10 Kidney and Liver 246% k# L+ D" r) m* n5 `9 d
7.5.11 Uterus and Cervix 2466 W9 o4 D B# C5 H) u0 X' p
7.5.12 Arteries 247
: @& V$ S5 j+ y7.5.13 Lung 248
1 c* G' n( ]* W; [7.5.14 The Ear 248) T) |; Z+ S0 s# A/ Z" ^
7.5.15 The Eye 249! Y5 p9 [8 y$ {1 a. B- r
7.5.16 Tissue Comparison 251
% b+ z, I" }+ ]; E" K# J4 q' _7 X3 I7.5.17 Plant Seeds 252, z+ B: P! F$ `( H5 _- k
7.5.18 Wood 252
: ?' B0 R& U" [4 m3 b7.5.19 Soft Plant Tissue: Apple, Potato 253" l, o' N* U: W, F: I2 k* s% m
7.6 Common Aspects 253
+ w, V: G( d3 a+ s8 R4 T7.6.1 Temperature Dependence 253
) X3 S- T; [# g7.6.2 High-Temperature Background 254: X; E" A" P& V& G8 T
7.6.3 Negative Damping and Acoustic Emission 255* `4 W0 T/ k% s1 Y) v# ~
7.7 Summary 255
/ b+ f, i! b* f9 V+ D: t N" j7.8 Examples 2555 B; H4 g* x2 r* g; q( g- d1 W, h
7.9 Problems 256
3 ?8 j) y6 c8 P7 H: x1 f) Z" qBibliography 257 P n; b* Y6 M) i; h+ n% z& }
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; P/ c" o/ y+ k) C, g1 v5 B1 M8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
- V5 M) E0 m1 H0 j* G) |8 @8.1 Introduction 271% ]' j1 h: }9 W) [0 \" [% G4 Q/ u& i
8.1.1 Rationale 2710 v' P' Y+ E2 R1 N( d6 c* q
8.1.2 Survey of Viscoelastic Mechanisms 271
; z/ p, B) U. a" J8 a8 B/ q8.1.3 Coupled Fields 273
' q! G; }7 M# g9 I4 ?9 s! y9 I8.2 Thermoelastic Relaxation 274
2 Z6 @/ O: k9 q/ \9 u% x8.2.1 Thermoelasticity in One Dimension 274
' `1 ^& `, ] I7 g; ]0 M% g8.2.2 Thermoelasticity in Three Dimensions 275
8 t% \6 Q1 o2 G* {) v% J5 a; \7 o8.2.3 Thermoelastic Relaxation Kinetics 276% n* W1 O: _( n9 c# Y
8.2.4 Heterogeneity and Thermoelastic Damping 278( b8 M& F7 }2 R$ h% ~$ Y% F6 d
8.2.5 Material Properties and Thermoelastic Damping 280
9 `- I" m4 c& b1 @: h8.3 Relaxation by Stress-Induced Fluid Motion 280
' Q" V; `% g% p# h7 ]& k3 O8.3.1 Fluid Motion in One Dimension 2808 t* j' I& h4 Q7 e9 g" F9 g
8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281/ u" b- V" I0 L1 N G1 E3 p7 L
8.4 Relaxation by Molecular Rearrangement 286; u$ ? @2 s) A2 r7 S R r; K
8.4.1 Glassy Region 286
& k& o0 C5 q. e' M, e) D' R8.4.2 Transition Region 287' ^+ k( ]) P' H9 Q6 y
8.4.3 Rubbery Behavior 289
. U/ S8 M* V: ~4 ^5 y8.4.4 Crystalline Polymers 2913 _! u+ R6 m0 l$ e
8.4.5 Biological Macromolecules 2925 m4 o! K" a& D0 S% i- } {
8.4.6 Polymers and Metals 292: r$ o$ b4 n e7 r7 w `# R$ T8 z' }
8.5 Relaxation by Interface Motion 292
7 N& E* S7 Q4 `5 Q7 j% M8.5.1 Grain Boundary Slip in Metals 292
" V! L" m$ S3 ^5 l) O8.5.2 Interface Motion in Composites 2941 @; Q$ {0 |2 S- v0 c/ _" }
8.5.3 Structural Interface Motion 2941 K8 x" B4 i- v. H
8.6 Relaxation Processes in Crystalline Materials 294
8 `! \' I6 O- V" V0 A6 r! X1 @8.6.1 Snoek Relaxation: Interstitial Atoms 294
& L2 [5 U* s4 u7 B" G. u8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298# d+ N+ C+ X+ P- i1 H1 T
8.6.3 Gorsky Relaxation 299
`) S8 ^) C3 h/ J" s8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300
/ t9 S8 x, P# |8.6.5 Bordoni Relaxation: Dislocation Kinks 303/ x$ w# K& s- A: T6 ~4 }
8.6.6 Relaxation Due to Phase Transformations 305
9 _, }: B' l: W: \8.6.7 High-Temperature Background 314
$ n" I6 k4 ?' M& @+ d' e. h8.6.8 Nonremovable Relaxations 315! _5 m% ^7 D0 f6 C6 U. {$ J
8.6.9 Damping Due to Wave Scattering 3161 [; j0 d- D& B" e( E
8.7 Magnetic and Piezoelectric Materials 316) M# a% ?$ O: o: U4 o7 E
8.7.1 Relaxation in Magnetic Media 316. K2 G6 N$ D/ D/ \. i' a
8.7.2 Relaxation in Piezoelectric Materials 3184 \. C: m( N1 M
8.8 Nonexponential Relaxation 322
& ?. `2 m, ]6 l6 R g1 k8.9 Concepts for Material Design 323
* W" t: C2 [+ ?) v8.9.1 Multiple Causes: Deformation Mechanism Maps 323
! l" Y2 E% d* b+ P4 c2 E. {8.9.2 Damping Mechanisms in High-Loss Alloys 326 c( ?5 U5 G/ Y; ]1 `: ^+ E9 I' r
8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326& o" K& d8 G% t
8.10 Relaxation at Very Long Times 327- v9 h$ h E U/ C x
8.11 Summary 327- p! W( Q5 K, y4 H# P' I
8.12 Examples 328
0 | ~8 X! }7 x8.13 Problems and Questions 332
% K% P+ L; B( l" ~ o: {Bibliography 3322 r& e- r2 z4 x4 v7 W
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' N. e5 j' t# \! O, e) v) ^ i9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341- m+ s1 b( _" x9 K8 `- N: |, P
9.1 Introduction 341
( t" a- V( @$ l" w" n1 k/ u+ H& c9 N9.2 Composite Structures and Properties 341+ k+ x* E3 h6 I0 d6 o* ?
9.2.1 Ideal Structures 341
7 z! A5 C) R, @9.2.2 Anisotropy due to Structure 342/ f3 u& O& `3 j
9.3 Prediction of Elastic and Viscoelastic Properties 3442 P4 a4 Z) L8 B) v
9.3.1 Basic Structures: Correspondence Solutions 344+ V' g- c# ]' ^ ]
9.3.2 Voigt Composite 345
8 h# c+ e2 D/ O/ U' E9.3.3 Reuss Composite 345
9 y6 ^6 B/ Q! O+ A9.3.4 Hashin–Shtrikman Composite 346
X( i1 e. \! y; A- u6 ?* [! i1 w' @0 A9.3.5 Spherical Particulate Inclusions 347/ y( \1 ~4 E5 ^/ p5 Z
9.3.6 Fiber Inclusions 349' p5 v; L p$ u+ u- Z3 S6 R
9.3.7 Platelet Inclusions 349! T' W2 _. y5 ^7 [* {
9.3.8 Stiffness-Loss Maps 350+ O( Q$ x2 a; G8 x3 p
9.4 Bounds on the Viscoelastic Properties 353
/ d: V* ^" R, b% }9 L9.5 Extremal Composites 354 D* @. S$ \% c# D
9.6 Biological Composite Materials 356
+ ~* E' m" O* w7 q9.7 Poisson’s Ratio of Viscoelastic Composites 3576 k/ ^4 B9 U$ K# O; f" o
9.8 Particulate and Fibrous Composite Materials 358- y; k9 a- Q! a- @7 i+ N# D' a* t
9.8.1 Structure 358
- q0 y( i* a" g& q- p# S9.8.2 Particulate Polymer Matrix Composites 3593 v0 H5 A5 u/ i. F z' x0 K0 I
9.8.3 Fibrous Polymer Matrix Composites 361
, ?" b+ }" R8 j% u# C X) B9 D1 C8 f( h9.8.4 Metal–Matrix Composites 362
% s; `* L0 [; s( Y1 r$ ~0 m9.9 Cellular Solids 363& j, ?- v6 _( C/ @5 ]6 q3 m
9.10 Piezoelectric Composites 366$ H& }0 T2 e0 n8 i1 l5 I6 C
9.11 Dispersion of Waves in Composites 366
, a M4 m Q# A9.12 Summary 367
) w+ y, C7 H! R9.13 Examples 367) H) f) u" u7 V
9.14 Problems 370
- G" F; C( [, G; ]4 u' j4 kBibliography 3705 o8 R& U6 O* D. c& M
" b( _+ P% w) E, E0 Y/ g# @
# H3 L- _$ i( d) @1 z6 l
% g+ ^1 A/ A( v1 L/ Q, v) C9 @10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377! o* i7 J- l. y" [$ x, V
10.1 Introduction 377$ }& h% B' D$ x" V" ~7 ? R; q
10.2 A Viscoelastic Earplug: Use of Recovery 377 A0 |5 R# i# v, |* U( M
10.3 Creep and Relaxation of Materials and Structures 3783 {$ t$ h( h7 v7 T, H. i, @ m
10.3.1 Concrete 378( u4 x: t* e( t: E" N
10.3.2 Wood 378
# x9 x' v; [$ q4 p1 ^6 {2 \9 `10.3.3 Power Lines 379
, C2 K* K9 y+ m! _. B5 v! X2 Z10.3.4 Glass Sag: Flowing Window Panes 3806 [& Z8 f' b9 q5 E0 _1 V
10.3.5 Indentation: Road Rutting 380
7 r; C3 v+ I ? j10.3.6 Leather 381* j0 T& `3 y$ e+ d. [
10.3.7 Creep-Resistant Alloys and Turbine Blades 381; M' C d# g/ }& e6 g( O
10.3.8 Loosening of Bolts and Screws 3823 s) c% B0 ]8 X
10.3.9 Computer Disk Drive: Case Study of Relaxation 384
1 e) _: ]& s; w10.3.10 Earth, Rock, and Ice 385
. G. ~# }9 Q0 J+ m8 j10.3.11 Solder 386 Z8 d( M0 i% T& A' A
10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
/ ?2 p, ^+ a2 x3 b8 ] y9 |10.3.13Tires: Flat-Spotting and Swelling 388' d4 a. M8 w& M, ], X, D7 ?! a' c
10.3.14Cushionsfor Seats and Wheelchairs 388
* _7 X/ E$ w" ]0 a/ g5 ^10.3.15 Artificial Joints 3896 @* R. O# r: W
10.3.16 Dental Fillings 3891 x8 m+ u3 \& F
10.3.17 Food Products 389- ]- T2 B) ?, d9 c9 T s6 N! U
10.3.18 Seals and Gaskets 390
. f6 O- {4 C e( i9 t' W10.3.19 Relaxationi nM usical Instrument Strings 390% [& i: Y% i( {, b. `% o
10.3.20 Winding of Tape 3913 ?+ S7 M8 t; M6 [8 [3 w& K
10.4 Creep and Recovery in Human Tissue 391
6 C+ k) F, p* a4 y& i10.4.1 Spinal Discs: Height Change 391! c: E! ]' a# Y N0 g' Y
10.4.2 The Nose 392
; D( N2 O$ Z9 ]" d10.4.3 Skin 392
" A9 Z7 w0 r. T3 e10.4.4 The Head 3939 N3 _' O6 ?5 m, v, t+ p
10.5 Creep Damage and Creep Rupture 394
$ h( M+ h) `8 L! r; P3 Z( m10.5.1 Vajont Slide 394
. \6 u% B: @: y- b9 I10.5.2 Collapse of a Tunnel Segment 394
; W3 m" w% U9 R6 p10.6 Vibration Control and Waves 394- e `, N! N/ y+ U" C3 |" Z+ Q
10.6.1 Analysis of Vibration Transmission 394
6 c4 X: I! D8 y" ~+ z* l' A) r10.6.2 Resonant (Tuned) Damping 397
+ u7 x* o1 q0 T6 |. b; |10.6.3 Rotating Equipment Vibration 397
7 K# s' h; u+ [( v% R6 d; X; `10.6.4 Large Structure Vibration: Bridges and Buildings 3984 P. {2 `9 r K n
10.6.5 Damping Layers for Plate and Beam Vibration 399: J/ H, `' H/ c
10.6.6 Structural Damping Materials 400
" ]7 D' J8 V! V ?! ^: [# Q10.6.7 Piezoelectric Transducers 402) a7 k" S5 ^" E8 h3 N* {) o
10.6.8 Aircraft Noise and Vibration 402
8 {* k. h% a J. q+ L+ ]10.6.9 Solid Fuel Rocket Vibration 404
, l [. M0 A) [- q( r9 C! z2 d10.6.10 Sports Equipment Vibration 404- A$ B q* p! Q. e0 T$ k
10.6.11 Seat Cushions and Automobiles: Protection of People 4049 Z2 V4 ?6 A! b
10.6.12 Vibrationi n ScientificI nstruments 4063 |% b3 Z; R4 o
10.6.13 Waves 406* Z7 G* ^$ U, ]& k
10.7 “Smart” Materials and Structures 407
+ O' M& j; K! a/ b- O10.7.1 “Smart” Materials 407; X4 n) }! c* z% t( r6 q9 B6 B( z. T
10.7.2 Shape Memory Materials 408 o8 o! d: @9 e% W4 w+ u
10.7.3 Self-Healing Materials 409
# {, {. x2 U, i/ \1 Z: I9 F10.7.4 Piezoelectric Solid Damping 4095 v; J. f1 r1 @$ X& C, j- Q8 A; l
10.7.5 Active Vibration Control: “Smart” Structures 409. A4 y$ i% E7 N% Y( d i
10.8 Rolling Friction 409
# j, K( @" Z7 J5 U10.8.1 Rolling Analysis 410
. F( w7 ~5 J }3 I8 h+ _8 |10.8.2 Rolling of Tires 411$ _) [5 h$ A" F( G+ E, G) D
10.9 Uses of Low-Loss Materials 412( t- P) H& y5 d; R/ R9 m6 ^
10.9.1 Timepieces 412
3 M" x0 S- L4 ?9 ^9 ~- z10.9.2 Frequency Stabilization and Control 4131 w# {# }1 q1 H9 Z( p) J
10.9.3 Gravitational Measurements 413) L! M5 c' h) e! v
10.9.4 Nanoscale Resonators 414! C7 G$ u9 t7 D3 `) t) s8 W
10.10 Impulses, Rebound, and Impact Absorption 414
5 E" d1 b7 E6 g8 O8 K, w$ K10.10.1 Rationale 414
' m' ]. x# N7 M: k! h! z! S10.10.2 Analysis 415) Y, y1 A% z: a" ?0 d
10.10.3 Bumpers and Pads 418& d& p' Q9 x: m% P) A6 G
10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 4198 H }. Y! {1 N' i" d7 l3 W+ Q
10.10.5 Toughness of Materials 419" P& ]: |/ N0 p7 O# D% r
10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420, c; y! ^8 e! I3 N6 w
10.11Rebound of a Ball 4219 e' _* M; [& e5 z, K. G f
10.11.1 Analysis 421
/ h( q6 k. \. k4 Q" j2 B! x10.11.2 Applications in Sports 422- G$ `1 R, }' u/ V' o8 Q. i6 P
10.12 Applications of Soft Materials 424
$ { b7 Y8 b X% H, R# e10.12.1 Viscoelastic Gels in Surgery 424: k( _* [- h6 C/ ~* L) D* R& s
10.12.2 Hand Strength Exerciser 424
" `; Q1 y$ n$ ~5 @7 [10.12.3 Viscoelastic Toys 4244 B. n+ T; e1 s7 E; D! {4 l) j5 X
10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425
: r! w: `% c+ c1 R( c10.13 Applications Involving Thermoviscoelasticity 425
9 K: G% o h' T10.14 Satellite Dynamics and Stability 4263 r4 I1 R& j" \6 f% A b
10.15 Summary 428 r( L2 y1 k' {$ ~" E; y4 L* C& p4 c
10.16 Examples 429
- X ?; {( T2 g& p+ Z10.17 Problems 431
3 j8 D d. X d$ ?Bibliography 4313 \+ x- ]' K F- w! h
$ Z4 J& y8 T& b# O: ^
/ I; Q7 n, h1 a8 h! [
5 x2 c9 |, k( Z& P4 {- ?3 l
A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412 ?* E+ a1 u1 R- {( z/ r; i7 p
A.1 Mathematical Preliminaries 441
) l) s& V n9 O5 [/ H0 WA.1.1 Introduction 441& p% g0 f3 ?/ U0 b
A.1.2 Functionals and Distributions 4414 S! h* y: S) p: H
A.1.3 Heaviside Unit Step Function 4423 C4 i8 Y! N: G. c
A.1.4 Dirac Delta 4422 U( {. \7 P! U) ?. E! R. p: ^" {% k, E
A.1.5 Doublet 443. b7 ~% L$ V8 A/ O+ D! o0 C. |
A.1.6 Gamma Function 445, W+ t' T4 j. C/ b$ I9 F8 g H
A.1.7 Liebnitz Rule 445" y9 y) q( k! ]8 ~6 p% z) E
A.2 Transforms 445
# }4 f. Z9 q) `A.2.1 Laplace Transform 446
- }1 r3 N L @; s7 v9 Y4 vA.2.2 Fourier Transform 446
8 B5 |2 V: q) F2 C/ kA.2.3 Hartley Transform 447 l) y' |' T, V- R4 e
A.2.4 Hilbert Transform 447. |6 I: r5 }& B6 p
A.3 Laplace Transform Properties 448
4 z8 u; i( k1 m( M4 i+ ZA.4 Convolutions 449+ M7 s2 ]) R8 B8 S
A.5 Interrelations in Elasticity Theory 451
: P+ z! G- K5 [0 oA.6 Other Works on Viscoelasticity 451
! `7 `! x" V4 pBibliography 452: y; f# g8 b1 u* G z
1 X) p3 X* u1 h
. b- f% p v# e, _ W% t) MB: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
6 {0 e$ H( E n# F9 l% s! \B.1 Principal Symbols 4554 s0 C; Z! Y$ R; R4 C
Index 457
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