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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
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! b3 M1 ^0 P( v0 |( W! T' [$ XPreface page xvii
0 r0 ~. N; V3 v9 F1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1! k9 }6 ~" a. s# v
1.1 Viscoelastic Phenomena 1
# _5 g3 l) [. b: v# U1.2 Motivations for Studying Viscoelasticity 3
6 P( s0 t) R0 W* l1.3 Transient Properties: Creep and Relaxation 3% |" f3 F3 [1 j% n
1.3.1 Viscoelastic Functions J (t), E(t) 3# W. [+ x# @2 r. W1 u
1.3.2 Solids and Liquids 7
* M' a! r; x7 t" P1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
. I5 j3 h# T$ r% e1 G! b# V, G1.5 Demonstration of Viscoelastic Behavior 10/ P _5 P% I4 R
1.6 Historical Aspects 101 Y/ c: r9 W! P$ m2 U6 ^
1.7 Summary 11
G; T$ g& p! P" \" w# h9 U+ C1.8 Examples 11
$ J' Z3 k6 ^4 H+ |8 I- O) q1.9 Problems 12
1 ]1 O6 H, R) S( t* XBibliography 12
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' T9 {6 I, i L# i2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
$ r% M5 \1 |' `3 t3 e2.1 Introduction 14
0 g2 G. E" R/ G! J/ i8 o7 n) g2.2 Prediction of the Response of Linearly Viscoelastic Materials 144 f) _* B" {1 T& M- O* Y
2.2.1 Prediction of Recovery from Relaxation E(t) 14: E+ L& Y* y3 S' C9 Y& y9 b3 A" l
2.2.2 Prediction of Response to Arbitrary Strain History 15* f+ g4 K: c3 N |3 f5 Q+ m
2.3 Restrictions on the Viscoelastic Functions 17
6 O2 |9 Z$ H/ H- f# c: v2.3.1 Roles of Energy and Passivity 17
3 s5 K: m' ~5 K7 y2.3.2 Fading Memory 180 @1 X) J9 {. g7 H
2.4 Relation between Creep and Relaxation 191 B& h$ y) L u1 C# [+ W* H
2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19
9 P7 r- ~9 \- V9 u0 ~: g0 ~2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20! ~+ P4 s3 t! I1 [1 l+ B( M
2.5 Stress versus Strain for Constant Strain Rate 20
0 `5 ?! S6 \3 B9 b2.6 Particular Creep and Relaxation Functions 217 @$ v7 s7 u6 R+ {9 P$ I+ }
2.6.1 Exponentials and Mechanical Models 21
' r1 Q$ `6 t# y9 V H2.6.2 Exponentials and Internal Causal Variables 26
q3 s0 `2 u1 H! W+ F; p2.6.3 Fractional Derivatives 27" s, R. \# R( U3 n* M, b G
2.6.4 Power-Law Behavior 28
, o5 Z- E/ Q! n; c3 z$ M2.6.5 Stretched Exponential 29
) E& W) n: j! A) l5 a2 w5 z2.6.6 Logarithmic Creep; Kuhn Model 29
8 U+ y# z3 f9 z: V2.6.7 Distinguishing among Viscoelastic Functions 30
6 i" V$ Z* N5 L3 s% q6 T2.7 Effect of Temperature 30
2 s2 J, \* d6 L8 E" R Q2.8 Three-Dimensional Linear Constitutive Equation 33
0 e2 ~: B5 I) r" {2.9 Aging Materials 35
, Q3 ~1 s' I, |$ ]) `2.10 Dielectric and Other Forms of Relaxation 35/ v' v. T3 ?5 K) U3 m- ?
2.11 Adaptive and “Smart” Materials 36
3 @2 a. C* N& S" A; `7 f2.12 Effect of Nonlinearity 37
7 u. L- U) h d3 r( w2.12.1 Constitutive Equations 37
0 k2 ^- K& i* |2.12.2 Creep–Relaxation Interrelation: Nonlinear 40, ]* x. M# p% Y
2.13 Summary 43
5 t1 L9 z5 [0 a7 t7 W# W% P e5 M2.14 Examples 43. k& Y4 u( B2 w
2.15 Problems 51
3 g+ C! `, o/ }, U8 h' q: hBibliography 52- N- m7 F5 @1 |, c" p3 R1 F/ M. v" d
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3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 t- A1 v/ {, _8 U
3.1 Introduction and Rationale 55
0 j. C6 @& }1 K2 L/ c! ^( a- V3.2 The Linear Dynamic Response Functions E∗, tanδ 56( n6 \) L& m) d6 `9 O5 x1 _' Y, t
3.2.1 Response to Sinusoidal Input 57
: w* h# F' B( Q# f3.2.2 Dynamic Stress–Strain Relation 59
8 o$ D( a3 C! c" X, e3.2.3 Standard Linear Solid 62
& C5 T/ X( e" Y. Z# Z3.3 Kramers–Kronig Relations 63
5 q" y, t% S- q/ O1 P' h$ E* F3.4 Energy Storage and Dissipation 65$ N" t3 l0 m3 V( R
3.5 Resonance of Structural Members 67; C2 P# X A9 y" Q$ M* p
3.5.1 Resonance, Lumped System 670 O3 U. {2 u, S. C0 O( F% v4 l/ ^. G
3.5.2 Resonance, Distributed System 71
$ [; ?5 k. Q4 _" N2 t- j K6 z3.6 Decay of Resonant Vibration 74$ [) i z1 h# D/ \" n: E ~6 [& E
3.7 Wave Propagation and Attenuation 77& F# r( z% N) j
3.8 Measures of Damping 79% g0 u6 I# o0 @* h# ^) u4 ]& C
3.9 Nonlinear Materials 79
% F7 ?# k- |3 l! Y# V7 J3.10 Summary 81$ F# ?& z" x8 L# V
3.11 Examples 815 r& Z' P5 X/ f2 E. I& N4 ?
3.12 Problems 88
4 b6 @- B8 E' e$ ?, cBibliography 89) ^- N/ C, X9 M% j4 n7 Y) @: |
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4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91
/ o% e1 v3 ~3 z# u! T) Z4.1 Introduction 91% W* t+ o% X1 Q7 c' O4 i
4.2 Spectra in Linear Viscoelasticity 92* K1 h1 Y4 Y$ @ A) j" q7 K% Q
4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 92 [& w- O+ | Z5 c1 c
4.2.2 Particular Spectra 93
0 N- X5 F" h7 o1 q7 Q4.3 Approximate Interrelations of Viscoelastic Functions 95 m! I8 h+ X( T j0 P6 r O7 _! _
4.3.1 Interrelations Involving the Spectra 95
6 A& [% C- W1 P& t |- W4.3.2 Interrelations Involving Measurable Functions 983 w4 h8 l; f- `0 o6 j
4.3.3 Summary, Approximate Relations 101% c+ k" t; d& P: s# P
4.4 Conceptual Organization of the Viscoelastic Functions 101
9 y8 V+ x& e9 `% `# t; K$ ]3 c4.5 Summary 104
: e' F0 h2 |: w+ B% L4.6 Examples 104
" q" p$ u' e2 |% O* r4.7 Problems 109
6 p0 w5 Y7 q8 ?6 e% ]% ABibliography 109
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/ K5 E7 r7 z- ^' Y5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111+ }+ t' V3 }- ^) l& Q
5.1 Introduction 111
* S& \' |9 S% f0 l) }. q5.2 Three-Dimensional Constitutive Equation 111+ h" {! G, ]: b* ]
5.3 Pure Bending by Direct Construction 112
U5 h9 b) F1 G5.4 Correspondence Principle 114
' c0 E5 j4 q" l, }% p' W# C0 G5.5 Pure Bending by Correspondence 116
& o1 T9 d+ P! k6 Z$ ^5.6 Correspondence Principle in Three Dimensions 116
' Z$ { ?' O& u9 t: ~+ }+ h5.6.1 Constitutive Equations 116
) n, N. R9 |8 Y; H5.6.2 Rigid Indenter on a Semi-Infinite Solid 1173 q2 l3 z$ A' Y9 s# T
5.6.3 Viscoelastic Rod Held at Constant Extension 119
$ n+ E2 `9 v1 u5.6.4 Stress Concentration 1193 X! @- u3 R5 o d
5.6.5 Saint Venant’s Principle 1204 `% x5 [/ y. U
5.7 Poisson’s Ratio ν(t) 121
! A5 L- K* ]8 `: K% }5.7.1 Relaxation in Tension 1216 P7 e8 @# C& S3 m# D7 [- Z
5.7.2 Creep in Tension 123! L9 H5 x7 ~& V. ^
5.8 Dynamic Problems: Effects of Inertia 124
1 C) ?" i3 q4 X) Q; {5.8.1 Longitudinal Vibration and Waves in a Rod 124
& `0 g5 N$ J, B8 `4 M) j5.8.2 Torsional Waves and Vibration in a Rod 125
* o0 @; Z: {( e" R( S5.8.3 Bending Waves and Vibration 1280 W/ u6 p' h0 p" b; [
5.8.4 Waves in Three Dimensions 129! _5 ]7 Z% Y& Y9 S
5.9 Noncorrespondence Problems 131
2 w: Z' C: x6 K/ E& j5.9.1 Solution by Direct Construction: Example 131
9 `% V/ r' Q/ P3 q/ A5.9.2 A Generalized Correspondence Principle 132' g7 @$ f+ o4 d" T' @( m: E: A |
5.9.3 Contact Problems 132
% q$ w4 Y/ y ~5 d- K3 N2 x5.10 Bending in Nonlinear Viscoelasticity 133
8 e% i* Y2 I9 F1 ^! u$ _5.11 Summary 134
( a) b4 N( v V5.12 Examples 134
4 ^& X6 r& D+ _2 p& s$ c5 D/ w3 Y5.13 Problems 142/ k% n3 E5 h9 B4 f
Bibliography 142
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6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9 P+ u9 w: T" e$ T7 m! H6.1 Introduction and General Requirements 145& \0 I- @, T. g8 {" q; g
6.2 Creep 146 \2 p6 S0 {5 M$ C% ?+ O
6.2.1 Creep: Simple Methods to Obtain J (t) 146* D0 v' W3 a* t
6.2.2 Effect of Risetime in Transient Tests 146
+ {& W$ C5 b b0 I" j! h6.2.3 Creep in Anisotropic Media 148* D8 m: u, ~7 @
6.2.4 Creep in Nonlinear Media 148! O4 p; |% a8 M; U- G+ T
6.3 Inference of Moduli 150
0 k4 p4 a+ t$ J+ n' P6.3.1 Use of Analytical Solutions 150
2 r8 N8 m1 O8 X& O7 V9 s6.3.2 Compression of a Block 1511 ~+ ?* |) f/ w c0 t
6.4 Displacement and Strain Measurement 152
* U7 @' `' w9 Y6.5 Force Measurement 156
) o; {$ O" N. D4 f8 Y$ S# g, j6.6 Load Application 157
' W# q' T+ u# j7 y+ c! J2 ?6.7 Environmental Control 157
; }. J- _, d7 h1 y8 v% Z* V I6.8 Subresonant Dynamic Methods 158
4 f% V% }4 j* h0 |4 [/ i" K6.8.1 Phase Determination 1586 D9 { G$ ~% j0 U5 F( f% h
6.8.2 Nonlinear Materials 160
2 s" k/ K% U, _4 V9 o' i6.8.3 Rebound Test 161
' k6 a& I# v% L" S! `6.9 Resonance Methods 1610 D) y5 o: T1 d7 p9 }$ r* X
6.9.1 General Principles 161
% R6 U" U, a) a6.9.2 Particular Resonance Methods 163. x/ x! ?: z; ~% _- n2 u5 j# X
6.9.3 Methods for Low-Loss or High-Loss Materials 1665 }- \4 z- T& _$ U* ~6 f. N
6.9.4 Resonant Ultrasound Spectroscopy 168 R! S4 A: f6 X! O3 n6 s4 ]
6.10 Achieving a Wide Range of Time or Frequency 171
6 W- ~3 ^ r" ]1 }6.10.1 Rationale 171+ V6 p' {7 h. |" [
6.10.2 Multiple Instruments and Long Creep 172
9 u) \' c! S; |6.10.3 Time–Temperature Superposition 172. u ~; h: B k) X5 P
6.11 Test Instruments for Viscoelasticity 173
; U% [4 T' G* e- H6.11.1 Servohydraulic Test Machines 173$ T' C- l3 K$ @6 T$ m/ q: m! Q
6.11.2A Relaxation Instrument 174
: _9 Z# d) v# I) v2 C! a& r6.11.3 Driven Torsion Pendulum Devices 174
! M# @7 f/ _% V6 T, p3 n6.11.4 Commercial Viscoelastic Instrumentation 178
, c3 {6 L \$ g' n1 A o6.11.5 Instruments for a Wide Range of Time and Frequency 179
( Q1 x% p S2 A" i: N6.11.6 Fluctuation–Dissipation Relation 182
* `# p/ ~/ J/ @7 c6.11.7 Mapping Properties by Indentation 183, ?' H& H6 W4 w: ^
6.12 Wave Methods 184
# _2 O) r( j! E5 t7 t7 W c6.13 Summary 188
, }: F$ J. b7 f$ y6.14 Examples 188; O9 N: j0 v3 T2 V
6.15 Problems 200. H7 @% ^$ U h8 A7 V
Bibliography 201
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2 W( @1 K2 E. |- B1 D: N$ M# e7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207$ {" a$ Q. p2 w: O1 V! S9 G4 k) A% u2 I
7.1 Introduction 207/ G# A6 d& x, l# J2 Q
7.1.1 Rationale 207+ S r& j7 B: o9 t/ n3 S( P" s
7.1.2 Overview: Some Common Materials 207
" K5 |5 ]8 p7 w L" q4 l7.2 Polymers 2086 q8 ]/ O$ C& |& E
7.2.1 Shear and Extension in Amorphous Polymers 208, j1 e' |2 p7 L
7.2.2 Bulk Relaxation in Amorphous Polymers 212, d- k1 p: w7 A1 ^
7.2.3 Crystalline Polymers 2139 m% d! W, d" o
7.2.4 Aging and other Relaxations 214- Y( }* O8 k3 a
7.2.5 Piezoelectric Polymers 2144 J# X- j1 w! K P3 h; S
7.2.6 Asphalt 214- I3 ^+ G0 s7 I9 ?% D4 A
7.3 Metals 215% w5 m0 J3 G* M8 }5 T! t
7.3.1 Linear Regime of Metals 215# k5 x2 r! y6 U; W0 N$ ]/ L/ \: i
7.3.2 Nonlinear Regime of Metals 2176 K8 b* i( e5 [" b3 K9 C
7.3.3 High-Damping Metals and Alloys 219) J0 ^6 f/ A3 Q) G) `" {7 Q/ Z3 ?
7.3.4 Creep-Resistant Alloys 2245 S; Z5 P0 f$ ?" t! K4 q, w- I
7.3.5 Semiconductors and Amorphous Elements 225
K' T( ` g- s& [9 n! D) m; ~- m7.3.6 Semiconductors and Acoustic Amplification 226; s) F. i8 C" H
7.3.7 Nanoscale Properties 226
9 f4 t: W2 a! w* k% M( S+ h6 Y" D$ M7.4 Ceramics 227
) I, I2 |4 m5 a. _7.4.1 Rocks 227 C- E& K+ U/ i1 h/ ]* y
7.4.2 Concrete 229
( c- _# j2 D% M. l9 s& Y: o7.4.3 Inorganic Glassy Materials 231
Z/ ~' P5 f( C7 G4 F, H1 _7.4.4 Ice 231& C: J7 S# I. A% J3 b
7.4.5 Piezoelectric Ceramics 2321 h) g9 e* `7 x/ `
7.5 Biological Composite Materials 2338 _8 V- P& G9 F1 C ^
7.5.1 Constitutive Equations 234" d; ^, U; ^/ x% k
7.5.2 Hard Tissue: Bone 234. i% U1 ~! R9 G
7.5.3 Collagen, Elastin, Proteoglycans 2364 d9 `# i; ^8 j9 P# [$ [/ L) d3 ]5 L
7.5.4 Ligament and Tendon 237, L m- @/ ?# r
7.5.5 Muscle 240
3 I3 W# X; S4 U) J- q3 @, c7.5.6 Fat 243
' g- E4 [. V6 {/ N( ?! C* r7.5.7 Brain 2431 @+ p# B. r6 w: V+ g% p
7.5.8 Vocal Folds 244. t$ F3 s3 P. l( `4 T$ R- `
7.5.9 Cartilage and Joints 244
# y' H# H9 a: L8 a7.5.10 Kidney and Liver 2464 S: q9 \- c, w1 @) I8 m1 _
7.5.11 Uterus and Cervix 246) ~4 u o2 N2 |, L2 z
7.5.12 Arteries 2474 a6 b3 C7 g: p2 ]
7.5.13 Lung 2480 x/ m' q: l( l9 l' E5 g
7.5.14 The Ear 248
. U: R/ ?0 s( T# b7.5.15 The Eye 249
" I9 a, g0 I/ y$ W C7.5.16 Tissue Comparison 251
8 y6 s I- G; f) W* r2 x7.5.17 Plant Seeds 252 t' p$ n- p( Z7 R: d6 s
7.5.18 Wood 252
/ q" U' x7 W0 O4 K6 ~% C4 M' Z8 F5 p7.5.19 Soft Plant Tissue: Apple, Potato 253
! Z$ C, B4 E5 i+ ]: G' m, S7.6 Common Aspects 253
+ K6 i5 \2 _; L' _7.6.1 Temperature Dependence 253
; s: b4 g2 `4 x* {# M7.6.2 High-Temperature Background 254
" E6 d6 i7 l/ s# I3 L7.6.3 Negative Damping and Acoustic Emission 255
6 s( ~1 ]% ]( s) M2 X7.7 Summary 255
* I0 `1 |4 b6 t0 r7.8 Examples 255# C1 m" B) I1 L) _: R' q0 Q" {
7.9 Problems 256
* y; \; ~ S5 x! P% yBibliography 257( G1 J% u A$ Y" L( v
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8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5 S$ g; K7 t/ a" k% n0 g8.1 Introduction 271
7 [ C5 K h$ D8.1.1 Rationale 271
5 ]/ b$ ^& H& `1 _' z8.1.2 Survey of Viscoelastic Mechanisms 271
8 k+ y- @1 |3 C0 M, g8.1.3 Coupled Fields 273
5 V5 M5 F& J. M% }' P" i8.2 Thermoelastic Relaxation 274
( ]4 x, r0 V+ _8.2.1 Thermoelasticity in One Dimension 274: ?1 P( @ ^/ W5 G# w
8.2.2 Thermoelasticity in Three Dimensions 275
5 K' |# N( X8 r) b5 Z- E* O8.2.3 Thermoelastic Relaxation Kinetics 276" {- N' h u w- h/ R
8.2.4 Heterogeneity and Thermoelastic Damping 278
; Q# \" M3 n; R' J8.2.5 Material Properties and Thermoelastic Damping 280
# U* d5 f. Y! a |- U$ _3 f8.3 Relaxation by Stress-Induced Fluid Motion 2808 |' @0 C4 v5 v% M: i
8.3.1 Fluid Motion in One Dimension 2804 p0 g" x; {% z/ M1 N: V& n* }
8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281
* B; s5 z4 u2 I& Z8.4 Relaxation by Molecular Rearrangement 286
8 {' Q" a% S0 m! Y( B3 d! S6 S8.4.1 Glassy Region 2869 L- ~% M6 Y- f5 w
8.4.2 Transition Region 287# j3 ~9 p7 X% N3 d
8.4.3 Rubbery Behavior 289" Z$ u) S7 A- L# M! A
8.4.4 Crystalline Polymers 291& F+ W& z! W/ a. f E5 j0 w
8.4.5 Biological Macromolecules 2920 m/ Z- `7 q) b9 s* d4 D
8.4.6 Polymers and Metals 2928 J- j' S7 m1 [ o) v! Y# T$ S
8.5 Relaxation by Interface Motion 292
9 P3 N- g. @: P7 x4 ]8.5.1 Grain Boundary Slip in Metals 292
- T1 J- q5 S$ g7 z8.5.2 Interface Motion in Composites 294
- O, H, ^3 \* w; a# g8.5.3 Structural Interface Motion 294
, E7 ]+ ?. |' _. y8.6 Relaxation Processes in Crystalline Materials 294
1 ]$ }" _3 V0 R" ~8.6.1 Snoek Relaxation: Interstitial Atoms 294
, g. [8 P* @8 E; j! K. d! u8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 2981 h0 a8 Z' u2 {" K
8.6.3 Gorsky Relaxation 299
/ }% J7 n- \- {( L8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300
0 A+ b; p5 k$ W: {1 C* z* W8 p' E1 i1 I8.6.5 Bordoni Relaxation: Dislocation Kinks 303, @" Z% K' w# Y' E0 }
8.6.6 Relaxation Due to Phase Transformations 305
+ _; L- J# z( B" E) H8.6.7 High-Temperature Background 314
+ _3 h9 A8 A }+ b ^; C1 I8.6.8 Nonremovable Relaxations 315
6 l# a6 n/ _% a4 W8.6.9 Damping Due to Wave Scattering 316
8 y, A2 g8 \7 Z7 x) }8.7 Magnetic and Piezoelectric Materials 316
+ y G+ `% e3 V& _6 \- i/ ~/ i8.7.1 Relaxation in Magnetic Media 316/ s# t+ d9 y$ R0 [
8.7.2 Relaxation in Piezoelectric Materials 318
8 u+ c5 U+ I2 I; O* }' Q8.8 Nonexponential Relaxation 3229 p4 G% J1 N: t. l5 }( o% S
8.9 Concepts for Material Design 3232 l: V) H+ }$ n8 q4 ]8 x! K. z
8.9.1 Multiple Causes: Deformation Mechanism Maps 323: i- ?- a% q5 H9 s# ?9 M
8.9.2 Damping Mechanisms in High-Loss Alloys 326
/ ]( d X# ?2 F) X- r3 [! i5 B! N8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326' o4 e. H) C: c
8.10 Relaxation at Very Long Times 327
/ ^) q" C0 l$ J4 f8.11 Summary 327
( b$ b/ t1 i/ [$ L- Z! m k8.12 Examples 328
. f. u9 G- b) ]) ^6 K" ]4 G8.13 Problems and Questions 332
c/ B( k' y Q0 P9 I% SBibliography 332
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9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341
- `2 @2 N2 J ]+ p9.1 Introduction 341. Q/ z! [1 x' N# e
9.2 Composite Structures and Properties 341
# X- Q7 c }% a2 Q9.2.1 Ideal Structures 341
1 ^, W; [# ], ^, y9.2.2 Anisotropy due to Structure 342
( Y* B- _' e' K% W% n- w& S: d9.3 Prediction of Elastic and Viscoelastic Properties 344: x- i0 Q& V) J6 V6 C: Y8 ]; g
9.3.1 Basic Structures: Correspondence Solutions 344
: A% f& a) w$ U9.3.2 Voigt Composite 345
' K, Z& C. b, J5 L# F: J" k8 C i( _9.3.3 Reuss Composite 345; m3 g7 w- d1 V+ X! b
9.3.4 Hashin–Shtrikman Composite 346- ?8 o1 ?8 y k9 t/ _" U6 E/ o# q
9.3.5 Spherical Particulate Inclusions 347# m. V2 X9 B2 X" }8 c8 x
9.3.6 Fiber Inclusions 349: k2 s) s# E9 X! I* N5 N
9.3.7 Platelet Inclusions 3496 K# |0 o( ]4 ?7 _0 f
9.3.8 Stiffness-Loss Maps 350% H1 H: R. ]) g0 p
9.4 Bounds on the Viscoelastic Properties 353
. r% w, d7 h% a1 s. I- c9.5 Extremal Composites 3549 A- M* @+ _: g' U
9.6 Biological Composite Materials 356
' l& r0 i2 N, H* P" ^9.7 Poisson’s Ratio of Viscoelastic Composites 357' y3 z2 B2 m! f2 J! l1 T
9.8 Particulate and Fibrous Composite Materials 3583 Z7 K; g) s& L p$ ^6 W1 \1 F
9.8.1 Structure 3589 J+ q/ ~+ z& e- x
9.8.2 Particulate Polymer Matrix Composites 359
+ \# q, u7 o& ]6 A: I, C! f: X' @9.8.3 Fibrous Polymer Matrix Composites 361& G% Z% y# R" p% a8 ?
9.8.4 Metal–Matrix Composites 3627 N1 _6 d5 z* ^% ]/ Y$ b+ ^
9.9 Cellular Solids 3637 H8 W0 v6 d7 [% k, {$ e
9.10 Piezoelectric Composites 366+ u$ p& R7 N" y2 }
9.11 Dispersion of Waves in Composites 366
# M! b/ u5 r5 B7 P2 X9.12 Summary 367
7 Z4 U1 I* ~1 i( I- t5 k! P9.13 Examples 367/ h0 s5 B5 X3 [( U3 }
9.14 Problems 370
: J# F+ h2 G7 c \- h# iBibliography 370
5 i; E# v5 w3 a A6 ^$ B8 |4 @
% K" g4 x1 G' W; Q, G
) u J" H' e3 P/ w' R! X9 |+ K2 @- [+ i+ ?
10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377/ W: z3 r- p# L2 `* x& o
10.1 Introduction 377* \5 L& f* m* x: G7 v g6 k
10.2 A Viscoelastic Earplug: Use of Recovery 377
5 _* }, I% C) D) B: l% D10.3 Creep and Relaxation of Materials and Structures 378" v% }5 u3 r7 c7 M! b2 G! i, A' F
10.3.1 Concrete 378! M. ^- T% g3 q3 ?3 y8 `& W0 h7 Q! s
10.3.2 Wood 378
' r: l8 D( P1 M5 O" m9 f10.3.3 Power Lines 379
@2 m0 k2 G( H' W( J0 I9 a. F10.3.4 Glass Sag: Flowing Window Panes 380, ~. {0 ^& W" n8 S
10.3.5 Indentation: Road Rutting 3807 |) r7 H* I" c- P1 @
10.3.6 Leather 381
) K: g& o- w% ^4 I3 ?10.3.7 Creep-Resistant Alloys and Turbine Blades 381
$ Y: c! u0 t" b6 t% {10.3.8 Loosening of Bolts and Screws 382
& M$ q- B( q- w/ @/ d4 G* Z10.3.9 Computer Disk Drive: Case Study of Relaxation 3842 @& c8 a- a" }! b5 v3 S
10.3.10 Earth, Rock, and Ice 3857 B. f4 ]4 E6 ]6 K) ~
10.3.11 Solder 386
: p+ G2 C' G3 D5 C9 n+ v; _4 f x10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
7 o4 X: f0 z" d X, ?10.3.13Tires: Flat-Spotting and Swelling 388/ |6 [* T4 q0 J! m) l& y
10.3.14Cushionsfor Seats and Wheelchairs 388* R& Y5 e" t( {' u. M) D' X3 u. {
10.3.15 Artificial Joints 389
7 C8 R' ~9 \& p4 [; @10.3.16 Dental Fillings 389
% P f+ [ k9 Y2 Y10.3.17 Food Products 389# r3 l3 o# \0 N
10.3.18 Seals and Gaskets 390- z, @. z$ m% V6 a r. H5 |6 |
10.3.19 Relaxationi nM usical Instrument Strings 390
) R- E3 l1 Q) A% f! a4 C10.3.20 Winding of Tape 3916 ~/ q8 b* u" |
10.4 Creep and Recovery in Human Tissue 391
$ U) A) D/ D% {: _& T: m K# e# J10.4.1 Spinal Discs: Height Change 391
+ T* ]( U$ t, d/ }/ t8 \9 e6 v10.4.2 The Nose 392
4 H; E1 Q+ Z0 q# f, T7 [10.4.3 Skin 392
% ~: T/ c' F% H10.4.4 The Head 393
J/ c5 T5 s( y6 P1 X10.5 Creep Damage and Creep Rupture 394
% D' s* I7 ^( q8 b* V6 E: @10.5.1 Vajont Slide 394) i' q! w# s% y6 {+ x
10.5.2 Collapse of a Tunnel Segment 394# x# E8 d1 C) u9 H: b3 e' e& K
10.6 Vibration Control and Waves 394
# Q7 B6 p- P* L4 K, M10.6.1 Analysis of Vibration Transmission 394' |4 Q# k/ q; E, M s, F
10.6.2 Resonant (Tuned) Damping 397
& e/ Q; Q. ^# \7 I5 I6 E8 J10.6.3 Rotating Equipment Vibration 397
. s" g4 n. _2 j5 @10.6.4 Large Structure Vibration: Bridges and Buildings 398
3 Z% ~: l4 C5 W! p {$ t9 \10.6.5 Damping Layers for Plate and Beam Vibration 399$ k' `8 T5 ]9 d4 W8 F
10.6.6 Structural Damping Materials 400' D7 }" ^% b5 _. r; G* ?
10.6.7 Piezoelectric Transducers 402. }. T/ F7 z/ h. T& F" F
10.6.8 Aircraft Noise and Vibration 402. p8 z6 s$ B! R
10.6.9 Solid Fuel Rocket Vibration 404; J0 ]9 m9 s: x. \& C1 S
10.6.10 Sports Equipment Vibration 404, w+ S! C* G9 ~3 |3 r+ O* b/ q
10.6.11 Seat Cushions and Automobiles: Protection of People 404
+ G3 z6 x. Y4 ]# R4 \10.6.12 Vibrationi n ScientificI nstruments 406
* u' P2 }0 N0 ]10.6.13 Waves 406
5 u" M7 a) n4 U10.7 “Smart” Materials and Structures 4079 n! J0 C6 d! z6 |4 N c+ v9 J( m
10.7.1 “Smart” Materials 407
1 p1 o" O" h7 T+ V0 {0 t0 |10.7.2 Shape Memory Materials 408
) x! D; }2 U" ~5 S. q$ e10.7.3 Self-Healing Materials 4095 D8 A1 L* V& v
10.7.4 Piezoelectric Solid Damping 409
. Y3 U8 `5 m% m K10.7.5 Active Vibration Control: “Smart” Structures 409" l" `- h1 {+ U1 a) V
10.8 Rolling Friction 409
/ I) p, d5 f% @8 G% x* \10.8.1 Rolling Analysis 4101 P: A) @2 P, }/ R5 H5 X' y
10.8.2 Rolling of Tires 4112 r4 g7 ]! k% h- x
10.9 Uses of Low-Loss Materials 4125 \. @: `! \9 x! }6 @; F- c. K
10.9.1 Timepieces 412
6 j8 O! A2 b0 D( J/ Y3 U" Q10.9.2 Frequency Stabilization and Control 413) |+ t+ \! b* q/ X) z
10.9.3 Gravitational Measurements 413# z) b! }" [( a# o& _: ^
10.9.4 Nanoscale Resonators 414
* Q; v; \. V% d3 O: y. u0 s10.10 Impulses, Rebound, and Impact Absorption 414( z$ A9 q) Q. @# C9 b1 Z% c% w
10.10.1 Rationale 414
" \; q' R& ]" n. t1 o7 p10.10.2 Analysis 4154 ~& z" I' F% |% K
10.10.3 Bumpers and Pads 418
2 l6 q* L0 Y' q5 {& j' x4 A10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 419
: b" j4 {5 Y- k10.10.5 Toughness of Materials 419
/ o, Z- E1 ^( l$ L: T: i% @- L% s4 F10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420+ k+ _7 ]4 P. b' x
10.11Rebound of a Ball 4216 W, z# X7 C$ n9 {' `! p- Y
10.11.1 Analysis 4212 G* U% c4 h( S+ t, u( j$ t8 ]
10.11.2 Applications in Sports 422 X+ r1 z! [& ~/ i3 _- }6 v
10.12 Applications of Soft Materials 424$ [: x1 g! z. q6 v) S( l% g u/ [
10.12.1 Viscoelastic Gels in Surgery 424; T0 q; x' }0 L+ m5 D( k1 k c! {* G
10.12.2 Hand Strength Exerciser 424% G5 C) Z) L, K6 f* ` c& _
10.12.3 Viscoelastic Toys 4249 f! r" |, U4 u& N5 P
10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425( f [7 |1 B! n3 T. @
10.13 Applications Involving Thermoviscoelasticity 425
/ j" w1 V- x6 Z7 z10.14 Satellite Dynamics and Stability 426+ y, l% c8 t" ]; R1 p# {. X
10.15 Summary 428' W( G/ Z* u0 N2 i4 y0 T: A
10.16 Examples 429& @: {8 i6 S; l$ @+ u5 |: r! g# J; a
10.17 Problems 431
; D8 G: ^5 Z2 r0 nBibliography 431
" k5 H0 z3 N" y; ?3 r
W2 T1 E" ^& s; r9 R
) y, X) U, p. P8 x. r$ r' n3 H0 `$ i
/ Q: D- c& a' _A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411 e1 H7 [7 `' w9 Q! Z6 K k! ~
A.1 Mathematical Preliminaries 441( w4 N, w R* S7 i. J
A.1.1 Introduction 4412 Z( C- A4 r9 O8 _
A.1.2 Functionals and Distributions 441 c7 T7 P: P' f+ x, H$ `* x2 q" k& W7 {
A.1.3 Heaviside Unit Step Function 442
/ k# k% ?* ^* E- {+ x# WA.1.4 Dirac Delta 442
; U- h7 O+ W. q* {0 {A.1.5 Doublet 443
: k! ?* O+ L* y3 g% `6 Z8 ~4 B5 QA.1.6 Gamma Function 445
! M4 g: \; [/ n' a1 W' r3 x4 z, CA.1.7 Liebnitz Rule 445- W0 B& S2 s' J* s+ s: T
A.2 Transforms 445
4 e8 \4 b* J# P7 jA.2.1 Laplace Transform 446
5 m h( t4 {, H# b, lA.2.2 Fourier Transform 446
* k4 a4 V& x9 D! _! y6 f: mA.2.3 Hartley Transform 447
- `" z Y1 S# J0 P& ?3 F* VA.2.4 Hilbert Transform 4470 `6 h( w A0 Y' S% Q7 ]: @
A.3 Laplace Transform Properties 448) B" K1 _" D1 ]3 L# z6 X- Z
A.4 Convolutions 4491 ^/ C. |0 d* }+ I% X/ k
A.5 Interrelations in Elasticity Theory 451" V8 l5 O* D( ~/ u: {( J
A.6 Other Works on Viscoelasticity 451
! Q' z: b" a/ e" _; oBibliography 4521 {7 I: Z/ c4 k" ]1 n& r3 w
! D3 \1 |0 Y* A! j& e# P
- y$ U/ y4 G! G5 P% f0 @B: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4557 P! _8 \; s0 t8 d9 Y2 S
B.1 Principal Symbols 455
2 R. k$ {+ R# C! pIndex 457' @1 x8 F. M& j/ |% Y
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