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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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9 g7 b) M, ~- `+ g目录
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7 ?! p! Z8 _9 N# B8 o" B. qContents
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( u6 B, j! _6 ]! Y& Z; @1 ^. \Preface page xvii
5 G3 y4 {- Y8 ]& V1 N: F1 Introduction: Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1' _" F9 P1 d7 x
1.1 Viscoelastic Phenomena 1
7 M+ ~/ x$ p, r' ^7 L- B5 \1.2 Motivations for Studying Viscoelasticity 3
C5 C7 s" q% p- e0 v1.3 Transient Properties: Creep and Relaxation 3* }& e5 H1 R7 w8 y( n
1.3.1 Viscoelastic Functions J (t), E(t) 3) Y5 b7 G5 S' x
1.3.2 Solids and Liquids 7$ e+ q# u3 x0 D/ I
1.4 Dynamic Response to Sinusoidal Load: E∗, tanδ 8
7 F2 u" @; \; t% C2 a, e+ S0 J+ Z+ d1.5 Demonstration of Viscoelastic Behavior 10
- g3 D( l0 K, U; n; e4 V: K1.6 Historical Aspects 10
0 y0 O3 t% f: Z, A: p. K1.7 Summary 11! ^1 S2 h1 E8 \# J
1.8 Examples 11$ q9 X( u/ T. s3 [ a- U
1.9 Problems 121 w. V5 s+ `5 t, O2 J
Bibliography 12
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4 ?; L8 k: H6 x, o2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14- B( h9 A$ c5 q' H4 h
2.1 Introduction 14( J1 F# O8 h: i5 l6 V7 D
2.2 Prediction of the Response of Linearly Viscoelastic Materials 14' b" }9 |6 L$ \5 Y
2.2.1 Prediction of Recovery from Relaxation E(t) 14
% }5 D7 U: X6 w* x3 d% w# Y# o. {2.2.2 Prediction of Response to Arbitrary Strain History 15
% {8 x- ]0 B6 G5 \0 a F8 O w2.3 Restrictions on the Viscoelastic Functions 17
! m4 D# p" F& r) V3 T2.3.1 Roles of Energy and Passivity 17/ X }" @0 J6 [' V, f
2.3.2 Fading Memory 18
3 q' m) i% @. d7 v7 ~; e2.4 Relation between Creep and Relaxation 19; \; L, s! B' T5 Q$ l: F9 i, l* D
2.4.1 Analysis by Laplace Transforms: J (t) ↔ E(t) 19
2 Z$ R4 F+ J( k2 M7 r9 s- P2.4.2 Analysis by Direct Construction: J (t) ↔ E(t) 20
. K) ~, o* x. u; p1 t2 c2.5 Stress versus Strain for Constant Strain Rate 20+ B# R# ? C# K- z: h& N/ a& z6 b) O; o' x
2.6 Particular Creep and Relaxation Functions 21
7 l% o1 W3 m+ l2.6.1 Exponentials and Mechanical Models 21
6 ]6 C% s3 s& P( R5 j) p8 ~4 v2.6.2 Exponentials and Internal Causal Variables 265 m7 U5 Z- i! j/ ^/ E& P
2.6.3 Fractional Derivatives 27% _6 [8 x5 W9 e5 ^2 r' D! ?
2.6.4 Power-Law Behavior 28. {, e V2 y4 U% z1 |7 v7 b
2.6.5 Stretched Exponential 29
# j2 j4 X! f' K- i! V/ e+ I2.6.6 Logarithmic Creep; Kuhn Model 29
( i; i4 {& z+ L+ J2.6.7 Distinguishing among Viscoelastic Functions 30
+ C/ z4 D6 p$ d# z6 }1 o0 k6 Y2.7 Effect of Temperature 30# i. j3 `, _; r- x% c
2.8 Three-Dimensional Linear Constitutive Equation 33
7 T" H' z" o: d4 S2.9 Aging Materials 35& O5 s* R$ g" P7 I5 ^8 \! y
2.10 Dielectric and Other Forms of Relaxation 35" J3 u( ^: e! U
2.11 Adaptive and “Smart” Materials 36
2 [- _0 \1 E; H" X2.12 Effect of Nonlinearity 37 t% R& p ?) U$ w% b
2.12.1 Constitutive Equations 372 m( J7 |9 C3 o2 {! \6 \7 r& o
2.12.2 Creep–Relaxation Interrelation: Nonlinear 40) v$ s7 k9 f2 N" v7 }$ d5 h% b* ~0 _
2.13 Summary 43 d2 m' }' x- y' C
2.14 Examples 43+ Z9 H! a4 g( H/ T3 U% ~$ ^2 \
2.15 Problems 510 d$ g$ C' H7 y
Bibliography 52
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3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 M8 r3 r/ R/ {& A' _
3.1 Introduction and Rationale 55
6 p! h9 _- @0 y4 k/ z3.2 The Linear Dynamic Response Functions E∗, tanδ 56+ K g0 w0 x% B2 T- [) l, I
3.2.1 Response to Sinusoidal Input 57' k4 `+ }; V0 ^9 ~
3.2.2 Dynamic Stress–Strain Relation 598 `' G, m8 R! W# M( B( I( D N; Q
3.2.3 Standard Linear Solid 62, W' F' N4 J3 I! R6 Q" m6 o, @
3.3 Kramers–Kronig Relations 63
: q3 y2 o0 t5 h1 s3.4 Energy Storage and Dissipation 65
/ u4 t7 I, t$ ]7 o3 H' i ^# u& z3.5 Resonance of Structural Members 67
( Q3 v3 F# w# {# Z& t3.5.1 Resonance, Lumped System 67
, [ ?+ t# h7 e& T3.5.2 Resonance, Distributed System 716 m6 }- Y' P2 O9 m
3.6 Decay of Resonant Vibration 74: W. X! P8 e$ v- I, w! t, U; U
3.7 Wave Propagation and Attenuation 779 A, w- a. J! [2 |1 f: D5 K2 }
3.8 Measures of Damping 79
9 E. \( Z" Y3 T, O1 s8 J: @3.9 Nonlinear Materials 79
; Z- Z3 d1 {4 e: X1 L5 C+ G4 ]4 t& R% `3.10 Summary 81
* L! p$ i' H9 \) p. f# |3.11 Examples 818 f( P/ b# J8 y% G
3.12 Problems 88 M# k8 ~! B( W2 ?, c6 ]- G
Bibliography 898 u$ I8 e2 n# i* S. y. Y: m6 }2 B2 s1 p
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4 Conceptual Structure of Linear Viscoelasticity . . . . . . . . . . . . . . . 91
$ }) i2 U q# t4 G3 H4.1 Introduction 91
! k+ O C1 U+ i+ X0 N" b- Q( c4.2 Spectra in Linear Viscoelasticity 929 R) C# N: L! R) E0 y
4.2.1 Definitions H(τ ), L(τ ) and Exact Interrelations 92) @. a) V" Y2 b8 N! ~
4.2.2 Particular Spectra 93- L& {0 Z2 S, T7 _& I! l1 G
4.3 Approximate Interrelations of Viscoelastic Functions 95) j3 W3 z. ? u a
4.3.1 Interrelations Involving the Spectra 95
- T- E7 h$ v3 W X5 }4.3.2 Interrelations Involving Measurable Functions 98
* b7 u2 Z& D5 d8 S* h7 H4.3.3 Summary, Approximate Relations 101
% w# y$ Y1 y' D+ m5 _" [; v4 h4.4 Conceptual Organization of the Viscoelastic Functions 101
3 G$ c+ c% l( ?+ J4 h% Y4.5 Summary 104- f, H' H& F0 ?9 A
4.6 Examples 104
0 u8 a/ ~% k0 C1 h8 s4.7 Problems 109 \0 v6 ^; r) B% W% A& y2 o! K {
Bibliography 109
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5 Viscoelastic Stress and Deformation Analysis . . . . . . . . . . . . . . . 111
# Y( I/ R/ R% k, J. R* q, u( I5.1 Introduction 111$ w* j# G, p o1 y/ H1 M$ R2 t
5.2 Three-Dimensional Constitutive Equation 111
. z; P `) O& p7 E4 L& O" I5.3 Pure Bending by Direct Construction 112 s! R# F( H- h" @+ w2 a. y; Z q
5.4 Correspondence Principle 114
3 C3 q1 d8 |! }5 n" |- O! @6 O5.5 Pure Bending by Correspondence 116
y! b- n6 a, W- O, y5.6 Correspondence Principle in Three Dimensions 1162 ]! x4 K4 H$ z: ]
5.6.1 Constitutive Equations 116$ W: O# U0 _ ~/ ]- k
5.6.2 Rigid Indenter on a Semi-Infinite Solid 117
' j" I" n q1 l- u9 l/ M4 A# S5.6.3 Viscoelastic Rod Held at Constant Extension 119
( k5 w: [+ ?+ ^4 B5.6.4 Stress Concentration 119
6 F9 K4 p: s" x6 t/ L; h3 c5.6.5 Saint Venant’s Principle 120$ o- P/ h& e8 I, M0 l. `
5.7 Poisson’s Ratio ν(t) 1212 n: H$ E. C. E
5.7.1 Relaxation in Tension 121
9 Y5 O5 X! k% o" x* |) g( `5.7.2 Creep in Tension 123* G0 r! b! ]# {2 l: G- V! p4 b$ G
5.8 Dynamic Problems: Effects of Inertia 124- M0 ~2 O7 X) q! j. Q
5.8.1 Longitudinal Vibration and Waves in a Rod 124
! m- k' d1 ~% B/ G1 Q/ e5.8.2 Torsional Waves and Vibration in a Rod 125
, m {/ @; H0 n' w* Z5.8.3 Bending Waves and Vibration 1280 k- a" p6 X3 I' t; l& D4 f) t
5.8.4 Waves in Three Dimensions 129" X4 i% g' U; N2 f% Q
5.9 Noncorrespondence Problems 131# M" C6 t' m6 j m4 y0 j$ q, ~3 e
5.9.1 Solution by Direct Construction: Example 131
0 e" ?0 I* y1 R5.9.2 A Generalized Correspondence Principle 132
2 w9 g8 L. S7 O \$ N1 \5.9.3 Contact Problems 132
8 ^+ s M$ J) i. y5.10 Bending in Nonlinear Viscoelasticity 133
) O1 H* x! Y' ~2 k* u5.11 Summary 134& G: ^1 @+ i0 [; _
5.12 Examples 134
, W( t; Q) }+ S1 L" w: S& O5.13 Problems 1421 F0 C: t+ X9 C4 C O
Bibliography 142
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6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145$ a, E- m. ^- P% k: V5 v
6.1 Introduction and General Requirements 145/ i5 L" V" N! a
6.2 Creep 146
+ n5 p4 Z9 A- f5 I) h8 Y7 ^% ~6.2.1 Creep: Simple Methods to Obtain J (t) 146 ]/ Q0 V9 t( ^5 \8 I
6.2.2 Effect of Risetime in Transient Tests 146# l! F7 N( J G# i! v
6.2.3 Creep in Anisotropic Media 148: v* }( {6 n! x: Y1 j4 _
6.2.4 Creep in Nonlinear Media 148: m2 j2 |9 t1 [) K. a
6.3 Inference of Moduli 1503 O* A' i- P4 W
6.3.1 Use of Analytical Solutions 150
8 q7 \7 E( h- d$ j6 e L0 Z0 R4 G6.3.2 Compression of a Block 151$ t5 [' j$ k( K, d" {
6.4 Displacement and Strain Measurement 152
" g0 {! T& W6 ~) _1 s6.5 Force Measurement 156
. X A" I. |, Y2 ?2 E; Y: J+ w6.6 Load Application 1571 z& f$ e/ @# G6 t; T" W$ _8 h# J
6.7 Environmental Control 157
8 W) i" A3 C+ b6.8 Subresonant Dynamic Methods 158
% e5 ~: \( j* s8 a6 w9 h6.8.1 Phase Determination 158
! Z' E! n' O2 ^7 H4 t. k) @5 ]6.8.2 Nonlinear Materials 160
! Y" |/ W* T; B' n" S( P. S6.8.3 Rebound Test 161$ v! r# M! ^, g$ r3 z, K2 T" R3 q
6.9 Resonance Methods 161" {+ E/ U+ ~0 l o
6.9.1 General Principles 161
$ u, f+ T7 D8 M4 Q' n6 B/ g% [6.9.2 Particular Resonance Methods 163! I; l& `! [. |! g1 A/ G
6.9.3 Methods for Low-Loss or High-Loss Materials 166. p% w) B4 ^$ L* p
6.9.4 Resonant Ultrasound Spectroscopy 168$ @4 J; ^) O, {( Z* ?! h" V
6.10 Achieving a Wide Range of Time or Frequency 171
) s3 H3 N2 x. F# @8 N6.10.1 Rationale 171, C4 w3 \% B$ j' r2 z% Q
6.10.2 Multiple Instruments and Long Creep 172( d. c! m! o( `! ]; k
6.10.3 Time–Temperature Superposition 172
7 }2 Q% \* I, T( Q% S* Z6.11 Test Instruments for Viscoelasticity 173 ~$ f" V. ` c+ g
6.11.1 Servohydraulic Test Machines 173" S! g- e: Y- z' p; U% q% D
6.11.2A Relaxation Instrument 174
6 ]2 }, Q, W9 Q# P6.11.3 Driven Torsion Pendulum Devices 174
9 e/ o: C. w2 V) C: v6.11.4 Commercial Viscoelastic Instrumentation 178
$ F9 p2 \( \% I6.11.5 Instruments for a Wide Range of Time and Frequency 1790 X' H. ^7 Q1 m! e4 i
6.11.6 Fluctuation–Dissipation Relation 182# f/ x2 }3 F. \
6.11.7 Mapping Properties by Indentation 183
: U7 r$ o% }' U: G4 X0 v- m6.12 Wave Methods 184
\9 H: ~( z- @& Q7 o" ~4 m" |6.13 Summary 1885 R' W4 S. V( C: G
6.14 Examples 188% r* {3 O+ W) `
6.15 Problems 2004 i4 E; j+ z( b2 j& E" R, M
Bibliography 201. p c7 d. f2 _# u; X
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) J7 N A7 S3 j7 Viscoelastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . 207, Y4 k1 ]/ w! ~ }7 N) y) @
7.1 Introduction 207
7 l0 ^# I" g6 d/ b0 O; K4 @7.1.1 Rationale 207; D: C" [$ c r( [" _ i# W
7.1.2 Overview: Some Common Materials 207% T% Q, S0 V- y0 J" \2 ]
7.2 Polymers 2081 p3 t: P. U/ M" ]% q7 B& o; V
7.2.1 Shear and Extension in Amorphous Polymers 208
/ w, m6 N& F) d7.2.2 Bulk Relaxation in Amorphous Polymers 212
: W& U5 {2 t$ t" l+ T6 H6 h7.2.3 Crystalline Polymers 213
% T4 D- e, v! M2 M+ F2 S7.2.4 Aging and other Relaxations 214
& Q+ {5 i) w$ F4 D4 t4 R b5 m6 q: u7.2.5 Piezoelectric Polymers 214. j; k" o9 |/ C( F% n9 x
7.2.6 Asphalt 214
% _3 t- a* l( A7 _- q7 n; [6 a: G7.3 Metals 215
! w) u* u& a4 _# L5 p* l& b7.3.1 Linear Regime of Metals 215: m# e. B2 @, Z' }; r# h
7.3.2 Nonlinear Regime of Metals 217
+ s' x7 _" |' U! z5 M! W6 R% L7.3.3 High-Damping Metals and Alloys 219
g e. @6 @5 ]* p& X4 Q3 A7.3.4 Creep-Resistant Alloys 224& y: A0 m# N1 P& z
7.3.5 Semiconductors and Amorphous Elements 225
4 M+ o2 ?# x8 M* C* z" ?7.3.6 Semiconductors and Acoustic Amplification 226
. O+ R0 ?/ J) X0 A1 R+ U7.3.7 Nanoscale Properties 226
: f! h8 E+ j& ^6 A$ ? E7.4 Ceramics 227
. g4 L" I* p3 c. @. L7.4.1 Rocks 227
Y2 [6 G1 I% x$ b7 W* z7.4.2 Concrete 229
9 ~2 q( @; p+ H7.4.3 Inorganic Glassy Materials 231
0 d9 z! E$ g+ L" X% G* e' i7.4.4 Ice 231+ P% z- l0 }( z2 ?
7.4.5 Piezoelectric Ceramics 232
' `; ?6 E8 E# U! _$ g$ g8 X7.5 Biological Composite Materials 2332 T9 R) F9 K' b8 ^- f
7.5.1 Constitutive Equations 234
& S. \& d/ i+ i) G: [ `7.5.2 Hard Tissue: Bone 234
! B4 W# f# h" c: T$ c' [* D5 s7.5.3 Collagen, Elastin, Proteoglycans 236& {6 P2 [/ R8 k+ A" `% V( _
7.5.4 Ligament and Tendon 2376 A& `* P$ S% e/ Q
7.5.5 Muscle 240% k4 c* m7 @( n# N2 B
7.5.6 Fat 243% x: O& Q% F; L1 ], K5 H( H
7.5.7 Brain 243
2 L8 U- \/ {& W5 A7.5.8 Vocal Folds 244
' o% B6 |% E8 D3 P% n* x7.5.9 Cartilage and Joints 2444 _1 A6 ^3 E! u$ }
7.5.10 Kidney and Liver 246# ^. F, n3 q* H1 j+ T' _6 h$ {8 g
7.5.11 Uterus and Cervix 246
0 P6 y& @9 p, [1 {* b. ~* i6 U7.5.12 Arteries 247& |5 C! g% L% H! A. M
7.5.13 Lung 248
& Z# O3 j r, s$ w2 ~' @- K. K7.5.14 The Ear 248
) I* t% |& e% |: e$ h9 F) {7.5.15 The Eye 249
# q5 ^& ]' j, ~7.5.16 Tissue Comparison 251$ w8 c% w; g3 m: ]2 ^) [
7.5.17 Plant Seeds 252 M: b* }! S* S2 Y. _0 Z1 R
7.5.18 Wood 252. e4 S4 V) Q9 y6 K# p+ U
7.5.19 Soft Plant Tissue: Apple, Potato 253
7 O" L5 A3 w4 }' u7.6 Common Aspects 253: q7 `) ~" l1 l3 i( r0 b
7.6.1 Temperature Dependence 253: s! l' z# K' X& h- f! Q0 s5 ~" G
7.6.2 High-Temperature Background 254
) |5 C1 ?' w: z- i8 }! s7.6.3 Negative Damping and Acoustic Emission 2559 t1 ?' k- I# n" G% n" \
7.7 Summary 255
' [# a" P, }# G1 x% D& `7.8 Examples 255/ j2 ~: ~. @" I( i
7.9 Problems 256
5 T4 f4 l2 r$ C2 P" EBibliography 257: V- X4 T# s. p2 F% e% S
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4 X2 [, X" u, T" a1 h, _, w8 Causal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271$ N# n' A, w6 ?& \
8.1 Introduction 2711 a$ v+ P5 A' a# m) l. r
8.1.1 Rationale 2715 n% Z$ \+ r$ {! `+ u, i
8.1.2 Survey of Viscoelastic Mechanisms 271
* `# K, Z. u) r$ V. S. k8.1.3 Coupled Fields 273
4 A# O7 B9 |8 H7 k, E- @4 I, K8.2 Thermoelastic Relaxation 274
1 g/ `# j) ]6 O0 ?1 f4 T8.2.1 Thermoelasticity in One Dimension 274
6 A" b: i/ o/ a7 L. D. P+ H6 h3 Z/ o8.2.2 Thermoelasticity in Three Dimensions 275% n* v5 Q, H( s+ Q! g. s5 {
8.2.3 Thermoelastic Relaxation Kinetics 2765 B6 A. s7 U) E8 V3 h8 U
8.2.4 Heterogeneity and Thermoelastic Damping 278
$ I% Z$ E/ c; ^$ k V8.2.5 Material Properties and Thermoelastic Damping 280
& V6 \: u) e9 P; h2 x. y8.3 Relaxation by Stress-Induced Fluid Motion 280
1 w$ X h! e3 [% |+ ]# X6 S" A8.3.1 Fluid Motion in One Dimension 2804 n5 k$ R9 F6 e9 N
8.3.2 Biot Theory: Fluid Motion in Three Dimensions 281* P3 f* v; h3 z n) y
8.4 Relaxation by Molecular Rearrangement 286
+ r0 O% U( H* D. I8.4.1 Glassy Region 286" E2 _9 O) `0 I9 [3 _
8.4.2 Transition Region 2879 l' D' ?+ E9 [1 j0 Z( E7 J( ]
8.4.3 Rubbery Behavior 2895 v- H' l: ~. q
8.4.4 Crystalline Polymers 291
* v0 m. N7 X/ D. X- j% L2 G5 `; _8.4.5 Biological Macromolecules 292$ z+ Q4 c( P+ S
8.4.6 Polymers and Metals 292
& S& t# b5 P6 ~+ h) A$ f. u8.5 Relaxation by Interface Motion 2921 \3 [' T/ e w4 z( j8 p
8.5.1 Grain Boundary Slip in Metals 292! E( t w; o: Q0 s8 W* b7 k7 q5 H
8.5.2 Interface Motion in Composites 2948 p8 J! V6 K+ F
8.5.3 Structural Interface Motion 294: i. L8 E( D$ U- p3 k* ]
8.6 Relaxation Processes in Crystalline Materials 294/ A) _4 U: t1 k5 C
8.6.1 Snoek Relaxation: Interstitial Atoms 294
1 P6 H5 R2 G" q0 P: z8.6.2 Zener Relaxation in Alloys: Pairs of Atoms 298: ~. M& [, h! m
8.6.3 Gorsky Relaxation 299- T2 a. P9 N, n+ [3 K
8.6.4 Granato–L ¨ ucke Relaxation: Dislocations 300. y6 m$ B" M! ]5 ?$ _1 g: u- ~$ c
8.6.5 Bordoni Relaxation: Dislocation Kinks 303
2 _4 @' f* e' p. X/ n4 v8.6.6 Relaxation Due to Phase Transformations 305
2 ]3 _# ^$ J, H5 y6 T, L8.6.7 High-Temperature Background 314
; b: _7 @, O9 N8 F8 B8.6.8 Nonremovable Relaxations 315
M) S# ?0 n* H ?9 C8.6.9 Damping Due to Wave Scattering 316
# O- k* O8 Q% A( g' M7 H4 a8.7 Magnetic and Piezoelectric Materials 3161 n M4 z2 p( ? Z
8.7.1 Relaxation in Magnetic Media 316$ b( k) a5 G" D6 W& s6 Q
8.7.2 Relaxation in Piezoelectric Materials 318
1 U6 \& m% m. a( V" ~8.8 Nonexponential Relaxation 322
* X2 ] }0 T4 Y: v- q8.9 Concepts for Material Design 323
/ f- O; V" D1 T. ]' J/ p8.9.1 Multiple Causes: Deformation Mechanism Maps 3232 [4 Z6 _& N2 |3 N
8.9.2 Damping Mechanisms in High-Loss Alloys 326
: y8 p/ A+ B& |/ m7 ?" s s8.9.3 Creep Mechanisms in Creep-Resistant Alloys 326
2 c9 h% X5 d( g5 I3 _ G: u: W8.10 Relaxation at Very Long Times 3276 @$ ^. t5 [$ m2 V4 k0 A
8.11 Summary 327+ c& b8 z; Y; T
8.12 Examples 328
; g" y" e! T! x& _) e. l8.13 Problems and Questions 332
1 O2 P( j+ P( ^$ }0 q6 xBibliography 332+ j e: a9 s# s
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& e/ v9 I! r, g3 O- h9 Viscoelastic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 341
$ Z3 c6 h1 Y( [: A6 X9.1 Introduction 341 b1 m A4 |8 t a& o
9.2 Composite Structures and Properties 341
$ V0 J! W3 r; _8 }& k9.2.1 Ideal Structures 3413 S3 i# n {. _% X3 R8 K
9.2.2 Anisotropy due to Structure 342! H/ n0 P7 j. o' D# X
9.3 Prediction of Elastic and Viscoelastic Properties 344$ Q3 U1 ]# Q0 I6 o1 Y$ q1 @
9.3.1 Basic Structures: Correspondence Solutions 344: F u8 [: K5 O! M( a
9.3.2 Voigt Composite 345) a" T' e, i8 X' p0 P' x1 V0 i
9.3.3 Reuss Composite 345: A7 [" ?) ?* y
9.3.4 Hashin–Shtrikman Composite 346
$ l% F+ O' a4 E: ]9.3.5 Spherical Particulate Inclusions 347
2 C D1 v1 j/ l4 W [9.3.6 Fiber Inclusions 349
9 Z9 G6 }/ h& R0 C5 M# m3 S9.3.7 Platelet Inclusions 349/ Y4 N* S& D' v9 q5 [/ z
9.3.8 Stiffness-Loss Maps 350
6 u6 G, F' O- p* A, ~2 M9.4 Bounds on the Viscoelastic Properties 353
9 E, Y$ w R+ F4 k) m9.5 Extremal Composites 3542 C2 e. Y+ M) L+ }. ?. v" a3 C- i6 B
9.6 Biological Composite Materials 356
+ _( {! `; J" G1 t9.7 Poisson’s Ratio of Viscoelastic Composites 357
& U& q6 W" Z( d& `& R6 g- `9.8 Particulate and Fibrous Composite Materials 358* O# D* X. D% R# E& Q
9.8.1 Structure 358
9 i0 s! e- S1 j3 o9.8.2 Particulate Polymer Matrix Composites 359; }8 ]" W- x- t) x% U+ S
9.8.3 Fibrous Polymer Matrix Composites 3617 ]' x6 j1 b# O. w
9.8.4 Metal–Matrix Composites 362
. | T* L: ^+ p+ s+ i, G9.9 Cellular Solids 363
- X. y! F' {: v* X N2 K/ P( G9.10 Piezoelectric Composites 366
1 `# Z% ~# O1 e9.11 Dispersion of Waves in Composites 3662 R! C4 v& V. D: o4 n- f
9.12 Summary 367+ Q$ v) f$ t5 x4 _
9.13 Examples 367
5 B' g0 ?/ w% t0 e! j; \9.14 Problems 3703 p# ~+ @8 v/ r# [: l; t
Bibliography 3704 B- C. v6 K+ Q6 {8 f
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2 P6 P0 p3 }9 y5 S( S i
& {) m `# b( n6 k% I. [8 j! ?10 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 377
! R' ^. T3 L7 i, i4 w10.1 Introduction 377
$ r" X" c8 m' |% r$ x0 b% m1 l+ }10.2 A Viscoelastic Earplug: Use of Recovery 377- y/ j# t4 _" i$ j. f# F* s: U d
10.3 Creep and Relaxation of Materials and Structures 3787 ]$ q O& ~9 K5 r& I& S! q: a2 d% b
10.3.1 Concrete 3782 O0 I# A! u( E: C$ e% p
10.3.2 Wood 378
3 L5 i* L$ C' R% b1 A! }10.3.3 Power Lines 379
* b% y: C/ |$ F- U10.3.4 Glass Sag: Flowing Window Panes 380
" G3 c4 j% H/ O& v# C10.3.5 Indentation: Road Rutting 380
# a( v; N; {) H' K$ Z, n10.3.6 Leather 381
0 G) \: j, U2 c10.3.7 Creep-Resistant Alloys and Turbine Blades 3815 x1 ^. `3 U" {4 v0 f" n
10.3.8 Loosening of Bolts and Screws 382
- G( ~/ [1 A9 X' B- c10.3.9 Computer Disk Drive: Case Study of Relaxation 384* p0 \/ K ]. B& |9 n$ p2 g/ A. w
10.3.10 Earth, Rock, and Ice 385
$ W8 g! g5 F5 u10.3.11 Solder 386/ i; Y6 b/ r2 B {8 P
10.3.12 Filamentsi nL ight Bulbs and Other Devices 387
6 p H/ U8 J# k8 Y) @# M9 A10.3.13Tires: Flat-Spotting and Swelling 388
, F/ ~8 e6 t/ N: `, P% ?& ` h4 E10.3.14Cushionsfor Seats and Wheelchairs 3889 p4 @ Z g; F/ G" u/ V
10.3.15 Artificial Joints 389
: M: `8 f4 m; ]2 b: u10.3.16 Dental Fillings 389 e( r5 A1 X* K$ T8 W1 Z7 g$ B
10.3.17 Food Products 389! ~' @% Q+ @+ X2 l5 H* B9 u
10.3.18 Seals and Gaskets 390
8 _- D9 Q0 \. ~; T0 b10.3.19 Relaxationi nM usical Instrument Strings 390
9 O/ y5 q2 H6 i# {10.3.20 Winding of Tape 391
4 _0 w4 g# ~/ [3 q$ N6 n" b; l! {10.4 Creep and Recovery in Human Tissue 3911 g7 L t" o2 u
10.4.1 Spinal Discs: Height Change 391& h* I4 l$ q2 j' o$ O
10.4.2 The Nose 392* ^& J, v1 C: d; A( ?0 C
10.4.3 Skin 3927 ]8 I9 l& |$ u/ U9 _
10.4.4 The Head 393
% e9 p9 e7 `1 X10.5 Creep Damage and Creep Rupture 394- z# F: M7 Y X. r
10.5.1 Vajont Slide 394* {4 F x2 e0 p
10.5.2 Collapse of a Tunnel Segment 394
3 G4 ~% D" c- ?* t8 L" G5 c10.6 Vibration Control and Waves 394# _+ L" x% H4 _
10.6.1 Analysis of Vibration Transmission 3943 S" |8 j! Q/ k: B
10.6.2 Resonant (Tuned) Damping 3974 k9 C/ w5 s' B$ B( y9 R
10.6.3 Rotating Equipment Vibration 397' t8 g4 A6 M7 R0 a6 `. @3 i
10.6.4 Large Structure Vibration: Bridges and Buildings 398
; ~* L2 e2 n+ I% l: R5 z10.6.5 Damping Layers for Plate and Beam Vibration 399
! O! ~6 W4 f4 u5 [" h10.6.6 Structural Damping Materials 400, Y. l, ~, j7 m! h
10.6.7 Piezoelectric Transducers 402
! @4 ~# _) \; Y10.6.8 Aircraft Noise and Vibration 402/ M& c O' g0 ]6 J( c. {2 T
10.6.9 Solid Fuel Rocket Vibration 404
$ a+ K$ M5 F) C' h0 J- A4 u9 P10.6.10 Sports Equipment Vibration 404- b5 U2 d! [5 ? _8 n- R8 Z
10.6.11 Seat Cushions and Automobiles: Protection of People 404
' @, l( c/ z |1 l10.6.12 Vibrationi n ScientificI nstruments 406
" W$ P8 w- n5 F; Z10.6.13 Waves 406' ~ I: [) m; Q2 I' A8 q. N
10.7 “Smart” Materials and Structures 407
5 y* D+ |3 {, C" c: Q$ I3 h' W1 u10.7.1 “Smart” Materials 407# D9 y# O; Z0 T8 ^+ u
10.7.2 Shape Memory Materials 408
' u! e, g- K4 \( c10.7.3 Self-Healing Materials 409! y; t2 e: H# j) y) d3 F a7 @) ]
10.7.4 Piezoelectric Solid Damping 409
0 C& ~$ H4 {4 e) A& u* F2 d5 R8 d10.7.5 Active Vibration Control: “Smart” Structures 409
+ G/ J" f. w) l. _5 ` Y10.8 Rolling Friction 409
' K& I7 J- n" x9 O4 r1 i; `1 Y9 l10.8.1 Rolling Analysis 4107 V: E, w# Y/ ^
10.8.2 Rolling of Tires 4117 R. w6 Q! i6 U# }* H* o. V
10.9 Uses of Low-Loss Materials 412( V1 `( D% P# L- ~& W
10.9.1 Timepieces 4122 v5 P' s# k9 M9 Y. z$ c, v7 w
10.9.2 Frequency Stabilization and Control 4138 z( R" F: [6 Y' [9 k( |: _- n
10.9.3 Gravitational Measurements 413+ Z0 L5 k- Z! N, W, |) c
10.9.4 Nanoscale Resonators 4142 r: ]' v5 [3 t5 x3 D7 [
10.10 Impulses, Rebound, and Impact Absorption 414) ?/ T3 b0 E1 p1 ?2 G5 g
10.10.1 Rationale 414
$ ^" ]" u; ~# [7 K10.10.2 Analysis 4157 X+ P2 H+ I# d* p+ k% c
10.10.3 Bumpers and Pads 418/ S2 F! `% ?+ d
10.10.4 Shoe Insoles, Athletic Tracks, and Glove Liners 419
$ V' Z4 t8 N. W10.10.5 Toughness of Materials 419
% }; H3 |: m g. w1 r, D) c+ U10.10.6 Tissue Viscoelasticity in Medical Diagnosis 420' E4 O/ |2 M; B: P. n
10.11Rebound of a Ball 421/ n( a$ q1 D* @, z! H! {
10.11.1 Analysis 4216 P- h& q) ^3 r+ j
10.11.2 Applications in Sports 422" Y- C$ R2 p% s0 n$ g
10.12 Applications of Soft Materials 424
; ] S; Y X- K- @+ P$ c10.12.1 Viscoelastic Gels in Surgery 424, l9 t; y# H% b E3 I5 B4 O
10.12.2 Hand Strength Exerciser 424. v/ Q! Q2 ~& R2 Q
10.12.3 Viscoelastic Toys 424
9 ]: P/ x. L8 z5 }10.12.4 No-Slip Flooring, Mats, and Shoe Soles 425/ {1 c/ s/ o* E# W0 S
10.13 Applications Involving Thermoviscoelasticity 425$ W0 p3 j( r( @8 O
10.14 Satellite Dynamics and Stability 426
" F) S, [7 x. ]9 s; ~10.15 Summary 428
( a7 `7 P, ~2 w10.16 Examples 429
% V! Y) ]% ?, B8 ~* w10.17 Problems 431
9 [7 t( S i) aBibliography 4313 w9 ?; O# \! m& w% J F" `7 y
; F2 V# p" J/ n4 Z2 g9 ?6 D2 {
0 h5 W- G. Z( ^+ l2 F( y/ w, e" [* p/ d S$ b
A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441% p7 ^- N2 o# F( U
A.1 Mathematical Preliminaries 4410 h: q6 {& G# m+ Q
A.1.1 Introduction 441
% q- E) \ r3 ?; ^3 Q( {A.1.2 Functionals and Distributions 441
. q( t2 H( A- T1 J$ v0 r1 H; @+ f* W* RA.1.3 Heaviside Unit Step Function 442
0 _- \6 h% s! i, J1 UA.1.4 Dirac Delta 442
7 O% z# j& J# X6 RA.1.5 Doublet 443
9 s D$ O4 M" R: h% r4 N6 lA.1.6 Gamma Function 4452 E* T" `* p9 D9 @
A.1.7 Liebnitz Rule 445! O6 _# a) D2 I' W+ ]: l3 D5 {$ I+ o, s
A.2 Transforms 445$ S( G5 P. a( l K! Y' Q9 J+ Y* ]
A.2.1 Laplace Transform 446
) F7 A, F6 s" ~! B. Z# sA.2.2 Fourier Transform 446% ^0 d5 K: z6 r- w2 Z' K2 i8 Y
A.2.3 Hartley Transform 447" U( q3 R$ j" [# d5 k0 \+ G5 b
A.2.4 Hilbert Transform 447$ b, G' [4 t- `4 ?
A.3 Laplace Transform Properties 448, B3 f j+ I# p3 C b5 W
A.4 Convolutions 449: @; M: K# A* S2 d8 t x
A.5 Interrelations in Elasticity Theory 451% z/ O3 F( k8 r! Y& t
A.6 Other Works on Viscoelasticity 451
% g& H$ ]: C& Z p# MBibliography 4529 [+ Y* x3 A' U5 y
$ U9 ]* q8 Y5 c* G6 `# C; q
$ g, B6 t) Y* O& d9 [" |
B: Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
4 \! _* E- y& q9 p3 zB.1 Principal Symbols 455
3 F: [* D: Q& V- L' U5 R9 n' C- ^Index 457
4 z0 S" N) T2 ?$ ]
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