The theory of composites

Milton G.W.051104092X, 0521781256

The theory of composite materials is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the past four decades has dramatically increased our understanding of the relationship among the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli that govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results.

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Table of contents :
Half-title……Page 5
Series-title……Page 6
Title……Page 7
Copyright……Page 8
Dedication……Page 9
Contents……Page 11
List of figures……Page 21
Preface……Page 25
References……Page 28
1.1. What are composites, and why study them?……Page 31
1.2. What makes composites useful?……Page 32
1.3. The effective tensors of composites……Page 35
1.4. Homogenization from an intuitive viewpoint……Page 37
1.5. Periodic homogenization……Page 38
1.6. Homogenization in random media……Page 41
1.7. Homogenization in the settings of G -, H -, and ………Page 42
References……Page 44
2.1. The conductivity and related equations……Page 49
2.2. Magnetotransport and convection enhanced diffusion……Page 51
2.3. The elasticity equations……Page 52
2.4. Thermoelectric, piezoelectric, and similar coupled equations……Page 58
2.5. Thermoelasticity and poroelasticity……Page 60
2.6. Pyroelectric equations and their relation to conductivity and magnetotransport equations in fibrous composites……Page 63
2.7. The equivalence between elasticity in fibrous composites and two-dimensional piezoelectricity and thermoelasticity……Page 65
2.8. Numerical methods for finding effective tensors……Page 68
References……Page 70
3.1. Duality transformations for conductivity……Page 77
3.2. Phase interchange identities for two-phase media……Page 79
3.3. The conductivity of two-dimensional polycrystals……Page 80
3.5. Duality transformations for elasticity……Page 81
3.6. Duality transformations for other elastic media……Page 83
3.7. The effective shear modulus of incompressible two-dimensional polycrystals and symmetric materials……Page 85
References……Page 87
4.1. Translations applied to conductivity……Page 89
4.2. A formula for the Hall coefficient in two-dimensional polycrystals……Page 90
4.3. A formula for the Hall coefficient in two-phase, two-dimensional media……Page 91
4.4. Inhomogeneous translations for three-dimensional conductivity……Page 95
4.5. Translations for elasticity……Page 96
4.6. A proof that the Young’s modulus of a metal plate with holes does not depend on the Poisson’s ratio of the metal……Page 97
4.7. The elastic moduli of certain two-dimensional polycrystals and symmetric materials……Page 99
References……Page 100
5.1. The uniform field argument……Page 105
5.2. The bulk modulus of polycrystals with cubic symmetry……Page 106
5.3. The elastic moduli of a composite with a constant shear modulus……Page 107
5.4. The thermal expansion tensor and constant of specific heat in a composite of two isotropic phases……Page 109
5.5. The extension to nonlinear thermal expansion……Page 111
5.6. The thermal expansion tensor and specific heat in composites of two anisotropic phases……Page 112
5.7. Exact thermoelastic relations for polycrystals……Page 113
5.8. The effective poroelastic moduli of two-phase media……Page 114
5.9. The elastic moduli of two-phase fibrous composites……Page 116
5.10. Exact relations for pyroelectric, conductivity, and magnetotransport equations……Page 117
5.11. The bulk modulus of a suspension of elastic particles in a fluid……Page 118
References……Page 119
6.1. The covariance property of the effective tensor……Page 123
6.2. The reduction to uncoupled equations for two-phase composites with isotropic phases……Page 125
6.3. Translations for coupled equations……Page 127
6.4. Elasticity as a special case of coupled field equations……Page 128
6.5. Equivalent coupled field problems in two dimensions……Page 131
6.6. The two-dimensional equations as a system of first-order partial differential equations……Page 133
6.7. The covariance property of the fundamental matrix……Page 134
6.8. Linking special classes of antiplane and planar elasticity problems……Page 135
6.9. Expressing the fields in each phase in terms of analytic functions……Page 136
References……Page 140
7.1. The coated sphere assemblage……Page 143
7.2. Multicoated sphere assemblages……Page 147
7.3. A phase interchange identity and inequality……Page 148
7.4. Assemblages of spheres with varying radial and tangential conductivity……Page 150
7.5. The conductivity of Schulgasser’s sphere assemblage……Page 151
7.6. The conductivity of an assemblage of spheres with an isotropic core and polycrystalline coating……Page 153
7.7. Assemblages of ellipsoids and their associated Ricatti equations……Page 154
7.8. The conductivity of an assemblage of coated ellipsoids……Page 157
7.9. A solution of the elasticity equations in the coated ellipsoid assemblage……Page 160
7.10. Expressions for the depolarization factors……Page 162
7.11. Neutral coated inclusions……Page 164
References……Page 169
8.1. Modifying the material moduli so the field is not disturbed……Page 173
8.2. Assemblages of coated spheres and coated ellipsoids with anisotropic cores……Page 174
8.3. Making an affine coordinate transformation……Page 175
8.4. The conductivity of an assemblage of coated ellipsoids with an anisotropic core and coating……Page 178
8.5. Making a curvilinear coordinate transformation……Page 179
8.6. Quasiconformal mappings……Page 182
8.7. Generating microgeometries from fields……Page 183
References……Page 185
9.2. Elementary lamination formulas……Page 189
9.3. Lamination formulas when the direction of lamination is arbitrary……Page 194
9.4. Tartar’s lamination formula for two-phase simple and coated laminates……Page 195
9.5. Lamination formulas for elasticity, thermoelasticity, thermoelectricity, and piezoelectricity……Page 197
9.6. The lamination formula for a coated laminate with anisotropic coating and anisotropic core……Page 201
9.7. Reference transformations……Page 202
9.8. Explicit formulas for the conductivity and elasticity tensors of a coated laminate……Page 203
9.9. Ordinary differential laminates……Page 205
9.10. Partial differential laminates……Page 207
References……Page 211
10.1. Polarizability of a dielectric inclusion……Page 215
10.2. Dielectric constant of a dilute suspension of inclusions to the first order in the volume fraction……Page 218
10.3. Dielectric constant of a suspension of well-separated spheres to the second order in the volume fraction……Page 219
10.4. The Maxwell approximation formula……Page 222
10.5. The effective medium approximation for the dielectric constant of an aggregate with spherical grains……Page 225
10.6. Average field approximations……Page 228
10.7.The differential scheme for the effective conductivity of a suspension of spheres……Page 231
10.8.The effective medium approximation as the attractor of a differential scheme……Page 233
10.9. Approximation formulas for effective elastic moduli……Page 234
10.10. Asymptotic approximation formulas……Page 237
10.11. Critical exponents and universality……Page 241
References……Page 243
11 Wave propagation in the quasistatic limit……Page 251
11.1. Electromagnetic wave propagation in the quasistatic limit……Page 252
11.2. Electromagnetic signals can propagate faster in a composite than in the constituent phases……Page 258
11.3. Elastic wave propagation in the quasistatic limit……Page 260
11.4. The correspondence principle and the attenuation of sound in a bubbly fluid……Page 263
11.5. Transformation to real equations……Page 264
11.6. Correspondence with thermoelectricity in two dimensions……Page 267
11.7. Resonance and localized resonance in composites……Page 268
References……Page 272
12.1. Resolving a periodic field into its three component fields………Page 275
12.2. A wider class of partial differential equations with associated effective tensors……Page 278
12.3. A related………Page 280
12.4. The equation satisfied by the polarization field……Page 281
12.5. The effective tensor of dilute suspensions of aligned ellipsoids……Page 282
12.6. Expressions for the action of the………Page 287
12.7. A framework for defining effective tensors in a more general context……Page 290
12.8. Various solutions for the fields and effective tensor……Page 291
12.9. The duality principle……Page 292
12.10. The effective tensor of the adjoint equation……Page 293
12.11. Magnetotransport and its equivalence to thermoelectricity in two dimensions……Page 294
References……Page 297
13.1. Classical variational principles and inequalities……Page 301
13.3. Null Lagrangians……Page 304
13.4. Variational principles for problems with a complex or other non-self-adjoint tensor……Page 306
13.5. Hashin-Shtrikman variational principles and inequalities……Page 308
13.6. Relation between the Hashin-Shtrikman and classical variational inequalities……Page 311
13.7. Variational inequalities for nonlinear media……Page 312
References……Page 316
14.1. Expanding the formulas for the effective tensors and fields in power series……Page 321
14.2. The series expansion in a composite to second order……Page 322
14.3. Thermoelastic composites for which the third and higher order terms in the expansion vanish……Page 324
14.4. A large class of exactly solvable materials with complex moduli……Page 325
14.5. Reducing the dimensionality of the problem……Page 327
14.6. Convergence of the expansions and the existence and uniqueness of the fields and effective tensors……Page 328
14.7. Convergence when L is not self-adjoint……Page 330
14.8. Extending the domain of convergence……Page 331
14.9. A series with a faster convergence rate……Page 332
14.10. A related series that converges quickly……Page 334
14.11. Numerical computation of the fields and effective tensor using series expansions……Page 336
References……Page 339
15.1. Expressing the third-order term of the series expansion in terms of correlation functions……Page 343
15.2. The terms in the series expansion for random media……Page 345
15.3. Correlation functions for penetrable spheres……Page 349
15.4. Correlation functions for cell materials……Page 350
15.5. Reduced correlation functions……Page 353
15.6. Expansions for two-phase random composites with geometric isotropy……Page 357
15.7. Series expansions for cell materials with geometric isotropy……Page 363
References……Page 365
16.1. Effect of a small variation in the material moduli……Page 371
16.2. Application to weakly coupled equations of thermoelectricity or piezoelectricity……Page 372
16.4. The variance of the electric field in a two-phase conducting composite……Page 374
16.5. Bounds on the conductivity tensor of a composite of two isotropic phases……Page 376
16.6. The change in the effective tensor due to a shift in the phase boundary……Page 377
16.7. Perturbing the lamination directions in a multiple-rank laminate……Page 381
References……Page 382
17.1. Links between effective tensors as exact relations: The idea of embedding……Page 385
17.2. Necessary conditions for an exact relation……Page 387
17.3. Sufficient conditions for an exact relation……Page 389
17.4. An exact formula for the shear modulus of certain three-dimensional polycrystals……Page 391
17.5. More exact relations for coupled equations……Page 392
17.6. Exact relations with limited statistical information……Page 393
17.7. Additional necessary conditions for an exact relation……Page 395
References……Page 397
18.1. Analyticity of the effective dielectric constant of two-phase media……Page 399
18.2. Analyticity of the effective tensor for problems involving many eigenvalues……Page 400
18.3. Integral representations for the effective tensor for problems involving two eigenvalues……Page 405
18.4. The correspondence between energy functions and microgeometries……Page 411
18.5. The correspondence between effective conductivity functions and microgeometries in two dimensions……Page 413
18.6. Integral representations for problems involving more than two eigenvalues: The trajectory method……Page 417
18.7. The lack of uniqueness in the choice of integral kernel: Constraints on the measure……Page 419
References……Page 421
19.1. The Y -tensor in two-phase composites……Page 427
19.2. The Y -tensor in multiphase composites……Page 429
19.3. A formula for the effective thermoelastic tensor in terms of the elasticity Y -tensor……Page 433
19.4. The Hilbert space setting for the Y -tensor problem……Page 436
19.5. The Y -tensor polarization problem……Page 438
19.6. Variational inequalities and principles for Y -tensors……Page 439
References……Page 441
20.1. The incidence matrix and the fields of potential drops and currents……Page 443
20.2. The subdivision of bonds in an electrical circuit……Page 445
20.3. The Y -tensor of the electrical circuit……Page 447
20.4. The effective tensor of the passive network……Page 448
20.5. The interpretation of the subspace………Page 449
20.6. The relation between the effective tensor and the Y -tensor in an electrical circuit……Page 451
References……Page 453
21.1. Why are bounds useful?……Page 455
21.2. What are bounds?……Page 456
21.3. The role of bounds in structural optimization: A model problem……Page 459
References……Page 463
22.1. Multiphase conducting composites attaining energy bounds……Page 467
22.2. Optimal bounds on the conductivity of isotropic polycrystals……Page 469
22.3. Optimal bounds on the bulk modulus of isotropic polycrystals……Page 471
22.4. The complete characterization of the set ………Page 474
22.5. The G-closure in two dimensions of an arbitrary set of conducting materials……Page 476
22.6. Bounds on complex effective tensors……Page 480
References……Page 482
23.1. Bounds on the effective conductivity of an isotropic composite of n isotropic phases……Page 487
23.2. Optimal bounds on the effective conductivity of an anisotropic composite of two isotropic phases……Page 491
23.3. Bounds for two-phase, well-ordered materials……Page 492
23.4. Bounds on the energy that involve only the volume fractions……Page 495
23.5. Bounds on the effective tensor that involve only the volume fractions……Page 498
23.6. Bounds for two-phase composites with non-well-ordered tensors……Page 504
23.7. Bounding the complex effective moduli of an isotropic composite of two isotropic phases……Page 506
23.8. Using quasiconformal mappings to obtain bounds……Page 510
23.9. Optimal two-dimensional microgeometries: Reduction to a Dirichlet problem……Page 511
23.10. Bounds for cell polycrystals……Page 517
References……Page 520
24.1. The translation bound and comparison bound……Page 529
24.2. Upper bounds on the bulk modulus of two-phase composites and polycrystals in two dimensions……Page 530
24.3. Allowing quasiconvex translations……Page 533
24.4. A lower bound on the effective bulk modulus of a three-dimensional, two-phase composite……Page 534
24.5. Using the idea of embedding to extend the translation method……Page 535
24.6. Bounds on the conductivity tensor of a composite of two isotropic phases……Page 536
24.7. The translation bounds as a corollary of the comparison bounds……Page 539
24.8. Embedding in a higher order tensorial problem: A lower bound on the conductivity tensor of a polycrystal……Page 540
24.9. A geometric characterization of translations……Page 542
24.10. Translation bounds on the Y -tensor……Page 546
24.11. Deriving the trace bounds……Page 548
24.12. Mixed bounds……Page 549
24.13. Volume fraction independent bounds on the conductivity of a mixture of two isotropic phases……Page 550
24.14. Bounds correlating different effective tensors……Page 552
References……Page 555
25.1. Other derivations of the translation bounds and their extension to nonlinear problems……Page 559
25.2. Extremal translations……Page 562
25.3. Attainability criteria for the comparison bounds……Page 565
25.4. Isotropic polycrystals with minimum conductivity constructed from a fully anisotropic crystal……Page 567
25.5. Attainability criteria for the translation bounds……Page 571
25.6. Attainability criteria for the Hashin-Shtrikman-Hill bounds on the conductivity and bulk modulus……Page 572
25.7. A general procedure for finding translations that generate optimal bounds on sums of energies……Page 574
25.8. Translations for three-dimensional elasticity……Page 577
References……Page 580
26.1. A brief history of bounds incorporating correlation functions……Page 583
26.2. Three-point bounds on the conductivity of a two-phase mixture……Page 584
26.3. Three-point bounds on the elastic moduli of a two-phase mixture……Page 587
26.4. Correlation function independent elasticity bounds: Improving the Hashin-Shtrikman-Hill-Walpole bounds……Page 588
26.5. Using the translation method to improve the third-order bounds……Page 590
26.6. Third-order bounds from cross-property bounds……Page 591
26.7. General third-order bounds for a two-phase composite……Page 592
References……Page 594
27.1. A brief history of bounds derived using the analytic method……Page 599
27.2. A topological classification of rational conductivity functions……Page 601
27.3. Bounds that incorporate a sequence of series expansion coefficients……Page 603
27.4. Relation between the bounds and Padé approximants……Page 608
27.5. Bounds incorporating known real or complex values of the function and series expansion coefficients……Page 609
27.6. Numerical computation of the bounds……Page 613
27.7. Bounds for two-dimensional isotropic composites……Page 615
27.8. Bounds for symmetric materials……Page 618
27.9. Reducing the set of independent bounds……Page 619
27.10. Proving elementary bounds using the method of variation of poles and residues……Page 620
27.11. Proving the bounds using the method of variation of poles and zeros……Page 622
References……Page 626
28.1. Eliminating the constraints imposed by known series expansion coefficients……Page 633
28.2. Eliminating the constraints imposed by known real values of the function……Page 637
28.3. An alternative approach that treats the components on a symmetric basis……Page 640
28.4. The extension of the fractional linear transformations to matrix-valued analytic functions……Page 644
References……Page 647
29.1. Associations between operations on analytic functions and operations on subspace collections……Page 649
29.2. Hints of a deeper connection between analytic functions and subspace collections……Page 654
29.3. The field equation recursion method for two-phase composites……Page 656
29.4. Representing the operators as infinite-dimensional matrices……Page 661
29.5. The field equation recursion method for multiphase composites with isotropic components……Page 663
29.6. Bounds on the energy function of a three-phase conducting composite……Page 668
References……Page 671
30.1. An equivalence between G -closure problems with and without prescribed volume fractions……Page 673
30.2. Stability under lamination and the convexity properties of the G-closure……Page 675
30.3. Characterizing the G-closure through minimums of sums of energies and complementary energies……Page 677
30.4. Characterization of the G-closure by single energy minimizations……Page 680
30.5. Extremal families of composites for elasticity: Proving that any positive-definite tensor can be realized as the………Page 682
30.6. An extremal family of unimode materials for two-dimensional elasticity……Page 688
30.7. An extremal family of bimode materials for two-dimensional elasticity……Page 693
30.8. Extremal materials for three-dimensional elasticity……Page 696
References……Page 697
31.1. Quasiconvexification problems in elasticity theory……Page 701
31.2. The independence of the quasiconvexified function on the shape and size of the region………Page 704
31.3. Replacing the affine boundary conditions with periodic boundary conditions……Page 705
31.4. The equivalence of bounding the energy of multiphase linear composites and quasiconvexification……Page 707
31.5. The link between the lamination closure and ………Page 709
31.6. Quasiconvex hulls and rank-1 convex hulls……Page 711
31.7. Laminate fields built from rank-1 incompatible matrices……Page 713
31.8. Example of a rank-1 function that is not quasiconvex……Page 714
31.9. A composite with an elasticity tensor that cannot be mimicked by a multiple-rank laminate material……Page 720
References……Page 725
Author index……Page 729
Subject index……Page 741