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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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目录
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Contents$ b8 h9 j9 P w) e- r
% j/ a; _+ d1 o6 XPreface page xvii# T( a2 O' ]8 }) x9 f6 V! C
1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
! e' U' ]& t9 e; E+ ]1.1 Viscoelastic Phenomena 1
% q* B! m- h' Y5 j. }/ O% N j1.2 Motivations for Studying Viscoelasticity 3* y7 L: i! M% e3 L }
1.3 Transient Properties: Creep and Relaxation 3* M/ y) N9 e1 y
1.3.1 Viscoelastic Functions J (t), E(t) 3
) L3 }1 M& x( ^2 j0 H1.3.2 Solids and Liquids 7
" F" E+ F; a- d' y" F) X" P: y0 e1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
2 ]. l& C" n1 q1.5 Demonstration of Viscoelastic Behavior 10
( h0 r; g& \0 I: k( l$ p1.6 Historical Aspects 10
1 N" a& N8 l W* h- ]1.7 Summary 11& |: |5 r. Q$ }; {- t+ Y1 B; `
1.8 Examples 11' g. I X! F5 ?) S
1.9 Problems 122 e0 V- @0 `" B6 c7 U# o
Bibliography 127 H; A' `% n1 f
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2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 t* U W5 j* Y; p+ a
2.1 Introduction 14" }0 ~5 v; z' m* ]* D
2.2 Prediction of the Response of Linearly Viscoelastic Materials 14# F+ }" x$ I+ u) a, Z7 N
2.2.1 Prediction of Recovery from Relaxation E(t) 14% L: f3 f2 V% \+ m, G
2.2.2 Prediction of Response to Arbitrary Strain History 156 z r' j7 F7 Q2 Z% O* u
2.3 Restrictions on the Viscoelastic Functions 171 F. V+ c4 T8 u* x% Q6 w1 i
2.3.1 Roles of Energy and Passivity 17+ M+ N' z0 Y& i1 Y. E# e
2.3.2 Fading Memory 18
3 X0 h& a+ p$ w% B' t: |2.4 Relation between Creep and Relaxation 19! }$ u8 ]" A% O. o
2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19
; q" B4 a9 [8 U0 q4 q9 I2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20
3 M) U& h0 D5 N7 z7 h* m" C2.5 Stress versus Strain for Constant Strain Rate 20) O* |1 ~! B2 h9 A# h
2.6 Particular Creep and Relaxation Functions 21
+ z% f @* I0 s/ _2.6.1 Exponentials and Mechanical Models 21# B8 @5 t$ h- `# [
2.6.2 Exponentials and Internal Causal Variables 263 h( C9 p( f' ], [
2.6.3 Fractional Derivatives 277 \/ l, J5 V. K- L
2.6.4 Power-Law Behavior 28
5 R3 ~1 E/ W- z2 k6 W! h* }2.6.5 Stretched Exponential 290 q, y5 y1 [3 H, |+ w! Y4 s
2.6.6 Logarithmic Creep; Kuhn Model 29
0 e7 w. O8 L; M: H2.6.7 Distinguishing among Viscoelastic Functions 302 E5 |& L2 M" {) L0 x# m9 i
2.7 Effect of Temperature 30
" @4 h- Q7 e3 O3 h8 f2.8 Three-Dimensional Linear Constitutive Equation 33/ R4 B1 P0 c. H( D! E) I9 R% z
2.9 Aging Materials 35
% w& K# s" y, @" {2 `2.10 Dielectric and Other Forms of Relaxation 35
& q4 J2 \$ c" T9 Y4 A( a2.11 Adaptive and “Smart” Materials 369 ]" ]8 ~! R6 |
2.12 Effect of Nonlinearity 37* S f9 r/ }; [" @) \" V2 o( J
2.12.1 Constitutive Equations 37
1 L. g9 `1 t5 z8 r2.12.2 Creep–Relaxation Interrelation: Nonlinear 40
3 i& y- f1 e) a/ B7 v C: a2.13 Summary 43
/ G8 W% a. b1 [2.14 Examples 43/ Y1 z4 Q! r8 ^! ^
2.15 Problems 51
1 J: w( {% A& p" k% vBibliography 52
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4 `$ B! O# h% F+ D; I& v! W3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 [* ]! Z3 O! v8 g: m
3.1 Introduction and Rationale 55' o! S& g f# \) B
3.2 The Linear Dynamic Response Functions E∗, tanδ 56
% J1 i( L/ |9 T+ ?+ }* }3.2.1 Response to Sinusoidal Input 57% b; N# E$ A0 v* l; A
3.2.2 Dynamic Stress–Strain Relation 59" m, L* L( y R+ j) G
3.2.3 Standard Linear Solid 62) u! Y& y1 b+ p' ] z; K
3.3 Kramers–Kronig Relations 634 R$ J% M4 o/ D, t F& V% \
3.4 Energy Storage and Dissipation 65
R+ g8 h+ M$ \3 Z2 Z' d( R& ^1 ~3.5 Resonance of Structural Members 67) f' V! h9 m/ P0 L$ O) Q/ h2 t5 M
3.5.1 Resonance, Lumped System 67/ a; X% t. ]; z
3.5.2 Resonance, Distributed System 71
' X( O$ J) E8 Z2 F3 ^3.6 Decay of Resonant Vibration 743 j5 p$ O" [2 Q7 v0 d
3.7 Wave Propagation and Attenuation 77- r& U, s. ^1 @6 R" q
3.8 Measures of Damping 79
& Y1 U- h2 `" h. R. i) Z8 H3.9 Nonlinear Materials 79- _' j* H; Q8 T' D* j# w
3.10 Summary 81. @( X. t/ h& r) U
3.11 Examples 816 A" x4 b7 s( R( {, B
3.12 Problems 88+ S/ s0 A8 q% z! ]8 G
Bibliography 89 B8 J: }# }- d% v& l
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- z( q6 S; o; J( N( b4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 917 Z' q9 a& D2 D& Q* H1 b3 ^7 q& C0 B
4.1 Introduction 91( L# \! \' b- c- C* h# L
4.2 Spectra in Linear Viscoelasticity 92
& p* D* p/ e4 ?" [) l0 @4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 924 K: P0 f, k5 \9 `8 A
4.2.2 Particular Spectra 93
% x0 |, G3 i. O) B4.3 Approximate Interrelations of Viscoelastic Functions 95
! m8 g3 D1 H& O! n" k' V) m4.3.1 Interrelations Involving the Spectra 95
4 j p9 g/ @- O i4.3.2 Interrelations Involving Measurable Functions 98
/ R% E7 b: @2 i7 C" b6 x4 X& U4.3.3 Summary, Approximate Relations 101; N- q; q. H) P. R( S4 |$ d9 w
4.4 Conceptual Organization of the Viscoelastic Functions 101* l p( s5 m6 }% O2 j8 K3 f' ^( A
4.5 Summary 104! K" ]; k* S, ~# ^7 J2 V2 X
4.6 Examples 104
- ^8 x8 u. T0 i7 s5 X, M4.7 Problems 1091 J4 m: a' h2 Z' u7 u3 j! i
Bibliography 109
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111) l+ `5 w* x1 |! m+ z2 S9 B! x
5.1 Introduction 111* ~0 m* @% t9 l( @; I' w
5.2 Three-Dimensional Constitutive Equation 111( n+ |1 r- C9 m7 `
5.3 Pure Bending by Direct Construction 1125 x9 ]; c3 R1 o; \ g
5.4 Correspondence Principle 114" ?/ n$ j+ {! H" _0 K
5.5 Pure Bending by Correspondence 116
5 |( v3 Q3 i' W {5.6 Correspondence Principle in Three Dimensions 116& r1 Q4 T" @6 b4 R M
5.6.1 Constitutive Equations 116) N+ B4 R: n$ H4 s; a. @& z
5.6.2 Rigid Indenter on a Semi-Infinite Solid 117" r8 A& ^+ f O. V$ Q1 w: {
5.6.3 Viscoelastic Rod Held at Constant Extension 119
$ K1 U! E: o( g( m; `' D* s( ?5.6.4 Stress Concentration 119
1 ]2 l8 \4 L; A- g4 G" e$ v5.6.5 Saint Venant’s Principle 120' F" B) C) r8 N) G" l' E
5.7 Poisson’s Ratio ν(t) 121
5 Q3 i2 n7 N) V! B5 a0 a) ~5.7.1 Relaxation in Tension 121
% p0 o) N* o' I; Q5.7.2 Creep in Tension 123, X0 p( l4 m6 c
5.8 Dynamic Problems: Effects of Inertia 124
8 W2 e- M* m8 ~5.8.1 Longitudinal Vibration and Waves in a Rod 1246 L5 q; i1 w! ^4 P# `9 S" i, w
5.8.2 Torsional Waves and Vibration in a Rod 125! o2 q% e, A" W3 g9 _
5.8.3 Bending Waves and Vibration 128 O2 L. [7 Y4 J0 o
5.8.4 Waves in Three Dimensions 129( x0 K( V* t5 z2 m9 h; ]
5.9 Noncorrespondence Problems 131' r0 K9 c, H$ I
5.9.1 Solution by Direct Construction: Example 131 Y8 Y& D: k9 s3 R% B
5.9.2 A Generalized Correspondence Principle 132+ f5 f1 v* C4 H) H4 D
5.9.3 Contact Problems 132
$ E" M+ w, U7 s' t+ P s* a3 v5.10 Bending in Nonlinear Viscoelasticity 133" C1 j3 O' ?2 l! S1 Z% `+ j
5.11 Summary 134
- k/ x* B" ]$ f1 k. S5 P5 W5.12 Examples 1349 K$ J3 v" c7 J
5.13 Problems 142
8 s" J7 B5 J3 _9 _- @7 NBibliography 142
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4 n8 j4 k( J* y6 f6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145/ G J/ P0 ?( M$ |
6.1 Introduction and General Requirements 145# D$ C6 o# q! J0 P, L& y- P6 T
6.2 Creep 146
) Q$ `3 N3 [6 |/ e6.2.1 Creep: Simple Methods to Obtain J (t) 146$ @8 e! K* E+ X7 B; x5 I
6.2.2 Effect of Risetime in Transient Tests 146
. g/ \: V& X' M1 E- ^* Q6.2.3 Creep in Anisotropic Media 148
0 G, ]+ X+ N" h- u2 r: ^3 X1 j6.2.4 Creep in Nonlinear Media 148* ]: I5 I( O, s6 h
6.3 Inference of Moduli 150
. w. O" ]% [# q6.3.1 Use of Analytical Solutions 150
. y" a9 C3 Y' C# P& E6.3.2 Compression of a Block 151
$ [2 U$ ~7 e/ e6 [8 s6.4 Displacement and Strain Measurement 1528 [. P- W+ l4 L' K, O# o9 v' ]
6.5 Force Measurement 156
7 _$ c8 T9 y1 J- ?1 S6 ~6.6 Load Application 157
t! B4 U- K; j, ?4 L6 P5 K1 _6.7 Environmental Control 157
5 b/ r, T5 ~( ~9 n" Z4 G8 t# {$ F6.8 Subresonant Dynamic Methods 158
! Q2 O# k$ o$ F3 K- Q1 W/ b6.8.1 Phase Determination 158+ C5 h5 D) f- t0 f& S
6.8.2 Nonlinear Materials 160% V; |2 P$ ^4 y" F! }
6.8.3 Rebound Test 161
" K6 U0 T5 S w- z& ]" i6.9 Resonance Methods 161
' O) d x2 o T" Q8 p' d6.9.1 General Principles 161
7 Y7 W2 q$ u3 T6 T6.9.2 Particular Resonance Methods 1638 I" g! K) ^9 G* h
6.9.3 Methods for Low-Loss or High-Loss Materials 1662 x, S/ F$ s3 }8 }0 m$ Q
6.9.4 Resonant Ultrasound Spectroscopy 1682 T. ^1 u( X Y" c/ d
6.10 Achieving a Wide Range of Time or Frequency 171$ u% A: W# I! |) u$ |* ~0 k! O
6.10.1 Rationale 171
- ^* g0 m& x" Q$ K6.10.2 Multiple Instruments and Long Creep 172( _" s+ @& @& m. F4 G
6.10.3 Time–Temperature Superposition 172
& E& _$ A1 g" ]" \) N6.11 Test Instruments for Viscoelasticity 173
. \+ C) t/ p3 u2 ~, P( ]6.11.1 Servohydraulic Test Machines 173* t& v3 u( _) M* {9 t* A
6.11.2A Relaxation Instrument 174, B' i9 ~1 S$ W# h
6.11.3 Driven Torsion Pendulum Devices 1742 ~+ k' t$ P# m
6.11.4 Commercial Viscoelastic Instrumentation 178
# e& j' Y' V) S, S6.11.5 Instruments for a Wide Range of Time and Frequency 179- V4 `$ m: ^3 s' c8 F
6.11.6 Fluctuation–Dissipation Relation 1827 I6 |: o4 ]+ u/ U9 c0 U9 ]. m: U% `
6.11.7 Mapping Properties by Indentation 1835 g. T' K1 {# _/ C
6.12 Wave Methods 184
( x f Z: \' f" E9 v7 y0 W6.13 Summary 188 D" u9 h( V9 w" T. a. G R# b
6.14 Examples 1889 b/ h: k7 d# u- F f! b! J- X
6.15 Problems 200
' \' V, `, \. A3 a _; R6 |) LBibliography 201
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7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207
1 p9 L% n2 m1 ?# F' V7.1 Introduction 207
$ B, z5 M" t3 |7.1.1 Rationale 207
4 n& n% H2 [* `% ?6 ~8 ?7 B$ T7.1.2 Overview: Some Common Materials 207
7 Z# ]; v; A, f! o/ ~7.2 Polymers 2081 f- o+ V/ u. T1 q6 g+ l
7.2.1 Shear and Extension in Amorphous Polymers 208
$ j: M9 z. ^( H1 @: Z+ d r3 I' i7.2.2 Bulk Relaxation in Amorphous Polymers 2122 b8 w$ L# P6 Z" E |% k
7.2.3 Crystalline Polymers 213
" ~+ \ C, C" Z; ]7.2.4 Aging and other Relaxations 214) v( O4 F5 l( W6 i) K* g
7.2.5 Piezoelectric Polymers 214, I- Y" ]7 p) n+ k. E9 D2 r
7.2.6 Asphalt 214
2 E: p5 K" o8 g7.3 Metals 215
5 J E' ~# P! Z& t0 M# w G7.3.1 Linear Regime of Metals 215; b2 b! d6 r) b5 ?$ @
7.3.2 Nonlinear Regime of Metals 217! O- y( F3 i0 i( B* `9 H* N& q% Q6 f
7.3.3 High-Damping Metals and Alloys 2191 J$ R% j( H: h- S! M
7.3.4 Creep-Resistant Alloys 224
. l8 m# y0 m. p& x; {# h7.3.5 Semiconductors and Amorphous Elements 225" s* q e7 ~& w# K! {4 B
7.3.6 Semiconductors and Acoustic Amplification 2260 z0 y; T c$ x1 B/ b2 d
7.3.7 Nanoscale Properties 226
7 O9 R/ V3 `& B- H2 _! f) h: |7.4 Ceramics 227: B1 b u7 j$ z, y& B3 p8 C. k
7.4.1 Rocks 227
( o# E7 b/ D6 Y" A5 l- P" L7.4.2 Concrete 2298 J m8 m; Z8 N( g1 h& q
7.4.3 Inorganic Glassy Materials 2316 d( Z8 t# d+ w
7.4.4 Ice 231
+ W5 V* y; @9 B1 T7.4.5 Piezoelectric Ceramics 232
- g: V/ D3 Z9 B6 `! h( c. N7.5 Biological Composite Materials 233
4 O, ]5 ]/ P. v) u: v6 _; t( G7.5.1 Constitutive Equations 234
1 L# q; N( L) f' a7.5.2 Hard Tissue: Bone 234
0 J+ F3 L' i, O5 j: E" n7.5.3 Collagen, Elastin, Proteoglycans 236
" z5 v) M+ @9 X7.5.4 Ligament and Tendon 237
1 e3 V; `1 V& [( g7.5.5 Muscle 240
( d, w9 i( B g4 G7 ?$ ~' I( h$ A+ T7.5.6 Fat 243
+ B I+ n% q4 o7.5.7 Brain 243& {' f- U0 o* Q, o5 g& m/ X
7.5.8 Vocal Folds 244
4 H/ x A$ }. D$ [7.5.9 Cartilage and Joints 244
( f+ n9 Y! ^, [. D8 @2 N7.5.10 Kidney and Liver 246
$ R1 B. x$ |+ Z+ J7.5.11 Uterus and Cervix 246; U* \" A. x1 q. j6 j' G, R
7.5.12 Arteries 247
1 c# u, p9 i" q+ H( v7.5.13 Lung 248/ u3 P" ?! L' h
7.5.14 The Ear 248) O- Z- d1 T# w" t
7.5.15 The Eye 249
& K& s! ]$ P \/ Z Y; \( Q* q) W7.5.16 Tissue Comparison 251
. Q; A3 Q7 \8 C1 C7.5.17 Plant Seeds 2528 e; `& n. i/ _/ `0 ]8 {: J! z: o% a1 L: h
7.5.18 Wood 252
3 O( i, n6 @+ q5 r$ p* d7.5.19 Soft Plant Tissue: Apple, Potato 253/ N @: B6 O- H& d. t
7.6 Common Aspects 253
& L A4 V2 E5 H* y/ C5 z7.6.1 Temperature Dependence 253* X+ z' S4 D7 Q6 V* ^
7.6.2 High-Temperature Background 254
5 l1 ]& q5 {! f3 w/ Y/ [. r* B8 }7.6.3 Negative Damping and Acoustic Emission 255
5 _. e8 L* K, Z7 B- s7.7 Summary 255! \: W1 a" E3 W
7.8 Examples 255% ?6 V' X4 _0 S$ W
7.9 Problems 256) b' e; G; V: @8 h& [+ v! _( \
Bibliography 257, ]4 _) v/ I. c8 ]* [$ e
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4 n6 I3 v& g' T7 w+ q8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713 E* }: E+ K: u+ {/ b
8.1 Introduction 271
4 q# {& M! v0 c. k8.1.1 Rationale 271
/ E& Z; b2 C, L6 `1 ?9 d( T6 e8.1.2 Survey of Viscoelastic Mechanisms 2712 d1 v( s. @; G
8.1.3 Coupled Fields 273. U7 |& |" L- h
8.2 Thermoelastic Relaxation 274; i" f. F: n# Y( f- j2 K& m( C
8.2.1 Thermoelasticity in One Dimension 274# x) i, u! [4 L" A0 ~
8.2.2 Thermoelasticity in Three Dimensions 2754 h9 h$ w9 _; @. U8 ]* _# F$ k; ~ n
8.2.3 Thermoelastic Relaxation Kinetics 276( z& O ]/ w! P% K3 _' D( }2 i
8.2.4 Heterogeneity and Thermoelastic Damping 2787 e) t+ B+ n* \
8.2.5 Material Properties and Thermoelastic Damping 280# Q) ]. u( O0 ~: A8 h A
8.3 Relaxation by Stress-Induced Fluid Motion 280' I2 Z$ |# S. d( C8 a) W
8.3.1 Fluid Motion in One Dimension 280
7 T5 J& T# X( _: b" R* k8.3.2 Biot Theory: Fluid Motion in Three Dimensions 2816 k5 x% n6 M$ o8 U. g* d$ o
8.4 Relaxation by Molecular Rearrangement 286
; q5 _/ I3 }' J9 T+ T3 ]8.4.1 Glassy Region 2869 @0 ?5 W" Z$ Z) V
8.4.2 Transition Region 287
( n! ?6 [" r( e+ v: \8.4.3 Rubbery Behavior 289* J2 U6 v: s& a; Z
8.4.4 Crystalline Polymers 291( k2 L1 D7 _2 h) } y1 c
8.4.5 Biological Macromolecules 292
+ }1 k3 H, l" g; f4 p2 E8.4.6 Polymers and Metals 2922 j6 m2 z8 N4 {6 N
8.5 Relaxation by Interface Motion 292" m o9 x; L5 q( p9 c, |! d
8.5.1 Grain Boundary Slip in Metals 292
# a+ G& q: Z* n, X# i* |- J' w8.5.2 Interface Motion in Composites 294
7 f0 l5 p& W$ K' J8.5.3 Structural Interface Motion 294
5 n* [0 _) m% }8.6 Relaxation Processes in Crystalline Materials 2942 V/ l" p) y: I/ I
8.6.1 Snoek Relaxation: Interstitial Atoms 294" D: P5 S: \' Q( T
8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298; F+ M* C, Q! ?. d+ Q
8.6.3 Gorsky Relaxation 2990 M1 \. _$ ~6 t! r ^. K
8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300* }6 o" a/ I' M/ c: p
8.6.5 Bordoni Relaxation: Dislocation Kinks 303% E5 K) p# H( I) W8 Z/ H
8.6.6 Relaxation Due to Phase Transformations 305
$ y1 p2 a( q8 ?# c) S# c, C2 q8.6.7 High-Temperature Background 314
9 s" a3 W& F" u+ J) T& Q, @8.6.8 Nonremovable Relaxations 315- J9 o; I0 M: Z" H0 M+ Z
8.6.9 Damping Due to Wave Scattering 316, @. \( _% R5 e
8.7 Magnetic and Piezoelectric Materials 316, R8 R: n0 k! _
8.7.1 Relaxation in Magnetic Media 316
9 h6 x: Q4 c' @' @7 {; u% w8.7.2 Relaxation in Piezoelectric Materials 318( {" W& `/ ~) c! ` l5 z; i. O! n$ S
8.8 Nonexponential Relaxation 322* G% F7 s! W& X* I
8.9 Concepts for Material Design 323- P. _1 s: u5 K8 z( r( y6 e4 u3 W; r: ~
8.9.1 Multiple Causes: Deformation Mechanism Maps 323
; A) T; }& w/ c: ]/ m, D4 ~8.9.2 Damping Mechanisms in High-Loss Alloys 326
1 X$ P- V' E- o6 \8.9.3 Creep Mechanisms in Creep-Resistant Alloys 3268 j% x5 e* n8 D1 W* y, v
8.10 Relaxation at Very Long Times 327
& L8 [% O8 V" u; H8.11 Summary 327
& V# _0 q7 S: N& T u8 c8.12 Examples 328
3 r: _' L3 r i" ~" ?) X- V! e8.13 Problems and Questions 332+ y" L- P( H) _) c4 a) T% K% a% P( I
Bibliography 332) c6 g Y: R# X$ R1 P
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+ n1 |% ?1 I [! W3 g9 @0 N9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341( W2 [" g- {$ o: O
9.1 Introduction 341
+ Z- P/ r: g( k) Q ]9.2 Composite Structures and Properties 3413 o# X3 `6 V( e! r! A
9.2.1 Ideal Structures 3413 B4 G" F9 U. ~( ^
9.2.2 Anisotropy due to Structure 342
Q- ?/ u0 M5 G* `+ n/ C4 F( H9.3 Prediction of Elastic and Viscoelastic Properties 3446 f' r- Y$ l2 c/ \% \
9.3.1 Basic Structures: Correspondence Solutions 344
# u2 H. O. O$ W+ Y9.3.2 Voigt Composite 3454 |3 W2 y) [$ M0 ~
9.3.3 Reuss Composite 345
" t# e) A. D# Y+ r4 X$ X9 F9.3.4 Hashin–Shtrikman Composite 346
1 w$ s& l, S ]4 _* e7 F$ b, M/ j% k9.3.5 Spherical Particulate Inclusions 347
8 H& ?( t+ s0 a: x9.3.6 Fiber Inclusions 349; O! R0 q# L3 H( H; J% A$ z: r$ k
9.3.7 Platelet Inclusions 349
5 |0 D6 f: `4 M; }- W& V9 ^9.3.8 Stiffness-Loss Maps 350
6 O Y7 h7 B" ]% A" F9.4 Bounds on the Viscoelastic Properties 353' A# D$ Y" b2 D5 W) v @) Y
9.5 Extremal Composites 354 Z. T1 [5 z# o C+ E# {. ?
9.6 Biological Composite Materials 356" v8 d* Y" W/ A) ]7 r( u5 W
9.7 Poisson’s Ratio of Viscoelastic Composites 357' F6 `4 j4 ]- d* V+ P
9.8 Particulate and Fibrous Composite Materials 358( ~+ ~; ]" l7 f7 U/ [$ v) f( h' V
9.8.1 Structure 3588 l* q' x! r& g/ J T
9.8.2 Particulate Polymer Matrix Composites 359
# W# G& \0 Z* X& H9.8.3 Fibrous Polymer Matrix Composites 3614 `& I* o# O3 l" s2 W
9.8.4 Metal–Matrix Composites 362
+ w! i; j1 L9 |& t" }9.9 Cellular Solids 363
& a9 Q) Q: O5 Q6 x) W+ b$ [: Y) A9.10 Piezoelectric Composites 3660 D4 G1 l$ A* F% S5 s# d- W
9.11 Dispersion of Waves in Composites 366
4 _ H1 K$ e, w; y2 J9.12 Summary 367
! Y( z: \4 h; L& H( y; k9.13 Examples 367' r9 w. x1 q+ m1 ?+ K
9.14 Problems 370
: T6 ]# U8 O8 V3 cBibliography 370
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2 `3 t- ^/ a5 N6 u5 y6 R" F% m, m1 T' U6 W: a) J
10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377
; ]% P, U( @: x" q9 ~6 ~" `# ?10.1 Introduction 377
0 E5 J8 o% S. X- z10.2 A Viscoelastic Earplug: Use of Recovery 377
/ Y+ G; U; l0 F+ x" J7 t6 D10.3 Creep and Relaxation of Materials and Structures 378
7 F( ]% @6 B- o" j( S& b10.3.1 Concrete 378' u6 L( U' t& n* r, |# E
10.3.2 Wood 3784 U, D, I3 u0 L ^
10.3.3 Power Lines 379
2 N2 k" o1 q9 i- O R6 I. K+ G* C" H10.3.4 Glass Sag: Flowing Window Panes 380! b+ U& }, _- S+ X! L( S# V& }* x
10.3.5 Indentation: Road Rutting 380$ M% ~6 a' T9 k1 @% e: w$ u3 ]
10.3.6 Leather 381* c' h8 J/ f7 B7 V- i! C3 {
10.3.7 Creep-Resistant Alloys and Turbine Blades 3817 f7 s& m, M1 b
10.3.8 Loosening of Bolts and Screws 382 y6 ~0 R5 I$ d* S9 Q7 M/ t2 I
10.3.9 Computer Disk Drive: Case Study of Relaxation 384 E+ {" K' `# N( g6 j: Z3 N# D8 b6 j
10.3.10 Earth, Rock, and Ice 385
7 S/ V) g! F, V; I10.3.11 Solder 3865 W5 W# E3 B* R! B$ g
10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
8 m+ u d- w, S: l- F; ]/ j5 b9 ]10.3.13Tires: Flat-Spotting and Swelling 388
7 r: E6 r/ Q9 F! m% w10.3.14Cushionsfor Seats and Wheelchairs 388
# r$ ^5 X9 ^" g& M2 @3 R6 \10.3.15 Artificial Joints 389
( C6 J2 O; O4 |! y7 T10.3.16 Dental Fillings 389
4 e- B2 w7 Q% x7 H6 T% f5 }10.3.17 Food Products 389
- b2 Z' w) ~6 N8 m0 w9 {10.3.18 Seals and Gaskets 390+ [4 H5 l+ g; O# A1 `9 O4 J
10.3.19 Relaxationi nM usical Instrument Strings 390
) R8 o* C' G. L) m, F10.3.20 Winding of Tape 391. M& e6 J. R! Q
10.4 Creep and Recovery in Human Tissue 391
0 F+ Y3 U9 I/ f" c1 m% C10.4.1 Spinal Discs: Height Change 3917 ?1 s& t' H8 D# h
10.4.2 The Nose 392
1 m6 E- J% E {4 _+ R10.4.3 Skin 392
4 o7 j( z' C B4 I1 q- A6 d10.4.4 The Head 393
7 E2 _+ t/ E1 v6 c9 [10.5 Creep Damage and Creep Rupture 3941 Y: Y) C# J6 c" L# u
10.5.1 Vajont Slide 394% V9 r9 j" l$ {) G3 M) R
10.5.2 Collapse of a Tunnel Segment 394
" S, [4 a& m9 B( H1 |$ r10.6 Vibration Control and Waves 394
6 u0 O, p% a# M# E! O1 y B10.6.1 Analysis of Vibration Transmission 394 x+ |3 T1 A1 o' ~' C
10.6.2 Resonant (Tuned) Damping 397
5 K B* F4 j9 V10.6.3 Rotating Equipment Vibration 397# }5 r' j/ y0 \ w
10.6.4 Large Structure Vibration: Bridges and Buildings 3989 e% }& Z) a! \4 E2 C- U0 U
10.6.5 Damping Layers for Plate and Beam Vibration 399- V3 Y8 G' T/ [! X r H
10.6.6 Structural Damping Materials 400
- c+ `# _8 e, \4 ? B6 L2 U7 H2 b9 v2 R10.6.7 Piezoelectric Transducers 402) y+ K7 N. @( ]
10.6.8 Aircraft Noise and Vibration 402
5 |4 z {# z. @0 M10.6.9 Solid Fuel Rocket Vibration 404
% B! _8 c4 P% D: B+ O, w10.6.10 Sports Equipment Vibration 404* o, |, N1 ^1 b# a. ^
10.6.11 Seat Cushions and Automobiles: Protection of People 404
/ @. J" E0 g ?5 ^" r4 k10.6.12 Vibrationi n ScientificI nstruments 4062 D1 U) f, T5 f
10.6.13 Waves 406 f; j9 X- S/ K" k
10.7 “Smart” Materials and Structures 407' n, v" G5 I4 ^
10.7.1 “Smart” Materials 4077 S6 U. q; P4 o8 `, q
10.7.2 Shape Memory Materials 408; ?/ D. D* ~8 |2 _
10.7.3 Self-Healing Materials 409. e3 u9 t. h8 z
10.7.4 Piezoelectric Solid Damping 409
, V1 w; T9 q8 D& l |3 Z1 w" r10.7.5 Active Vibration Control: “Smart” Structures 409
) j1 H: h6 K* k: F. T' J$ Q/ t10.8 Rolling Friction 4095 r+ m5 O0 J/ h+ [
10.8.1 Rolling Analysis 410: {, X& [' u8 i \1 O
10.8.2 Rolling of Tires 4119 w2 H, t% N8 |, L( P7 S
10.9 Uses of Low-Loss Materials 412" Q! A5 B4 }1 @$ b$ ~6 `
10.9.1 Timepieces 412/ E7 M& O- d* u
10.9.2 Frequency Stabilization and Control 413# y% H. r: W' `; Y+ _
10.9.3 Gravitational Measurements 4133 x1 | J! ]8 c5 c8 b
10.9.4 Nanoscale Resonators 414
' V1 C% D6 X" c10.10 Impulses, Rebound, and Impact Absorption 414
% U4 j0 J2 j6 a10.10.1 Rationale 414& B' v d+ G- l+ u( b: D7 e
10.10.2 Analysis 415
; S5 y( N" a* g# k- R% Z0 u10.10.3 Bumpers and Pads 418& }# Q# l( L& n; Z6 k
10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 419
$ } v0 z! l2 i) I6 b# w: u4 y8 }7 M10.10.5 Toughness of Materials 419
/ S% c" X, x. ?) u- u$ g# _10.10.6 Tissue Viscoelasticity in Medical Diagnosis 4201 e0 O* [- S1 y1 ^7 E
10.11Rebound of a Ball 421
, Y0 \" c, r8 q- k# F2 w4 k. m10.11.1 Analysis 421. r w* Z( H/ u6 x0 n
10.11.2 Applications in Sports 422. ^+ b0 M! U. z2 V
10.12 Applications of Soft Materials 424- @3 H: f" C" a: J3 I
10.12.1 Viscoelastic Gels in Surgery 424# F; R0 C3 N! r. n& n& i
10.12.2 Hand Strength Exerciser 424
1 m, A) j7 Y6 B% ], [+ l* O$ X10.12.3 Viscoelastic Toys 424
+ Y; ?' r5 g# F. `& V. r; N% \10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425* w$ O; d, a: D. y- G
10.13 Applications Involving Thermoviscoelasticity 425
: u2 L$ S% n4 ?2 ?$ ]* p& H10.14 Satellite Dynamics and Stability 4265 w; K' R' \; z) q' C9 h6 i
10.15 Summary 428! r& D5 `. |8 Z* W3 W' b
10.16 Examples 429. L/ F: {9 ?" w& J6 t( p8 R
10.17 Problems 431$ @$ W5 K' f8 Z' w( W: o
Bibliography 431
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1 |4 s; {* u. U. [
A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
: |; V4 y! {) i+ z, S* {# `A.1 Mathematical Preliminaries 4413 c( @( Z' _. h k: `1 d7 s& u" A# \
A.1.1 Introduction 441
6 U/ }* s( i. { w# x# HA.1.2 Functionals and Distributions 441$ c. [! ^# k5 i+ Z8 U/ G) x
A.1.3 Heaviside Unit Step Function 4425 L' C8 U/ s1 K) X1 S; M
A.1.4 Dirac Delta 442, s3 Z( t% { i6 I( p2 Z2 Q/ P2 G
A.1.5 Doublet 4430 D3 f0 N: G9 S# @- y6 [
A.1.6 Gamma Function 445
1 l( l' G# C, dA.1.7 Liebnitz Rule 4451 ~3 N, r) g: ^
A.2 Transforms 4455 ] l U. ~ M6 ~8 V- [/ c
A.2.1 Laplace Transform 446! {0 L/ ?3 H+ M# a) b; c+ H, f
A.2.2 Fourier Transform 446% x+ {0 y+ I% D4 n2 b u
A.2.3 Hartley Transform 447% H# ~% T$ I E9 ^' u
A.2.4 Hilbert Transform 4471 W( K2 w$ j( e/ z
A.3 Laplace Transform Properties 448
# x7 x c5 v- ]6 B t$ p \3 l7 hA.4 Convolutions 4494 k5 g4 o5 w. M V- z
A.5 Interrelations in Elasticity Theory 451
a; J" t' _- V0 QA.6 Other Works on Viscoelasticity 451
6 I8 q0 B: X$ T0 jBibliography 452/ ]7 D. }- c$ I7 P
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% |' X8 Y7 m8 _$ f3 iB: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455( v, r* L0 | Z1 l" O
B.1 Principal Symbols 455
, h6 G1 {( i6 v2 PIndex 457
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