This is page v
Printer: Opaque this
Contents
List of Figures
xiii
Preface
I
xv
Preliminaries
1 Relaxation of One-Dimensional Variational Problems
1.1 An Optimal Design by Means of Composites . . . . . .
1.2 Stability of Minimizers and the Weierstrass Test . . . .
1.2.1 Necessary and Sufficient Conditions . . . . . . .
1.2.2 Variational Methods: Weierstrass Test . . . . . .
1.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Nonconvex Variational Problems . . . . . . . . .
1.3.2 Convex Envelope . . . . . . . . . . . . . . . . . .
1.3.3 Minimal Extension and Minimizing Sequences . .
1.3.4 Examples: Solutions to Nonconvex Problems . .
1.3.5 Null-Lagrangians and Convexity . . . . . . . . .
1.3.6 Duality . . . . . . . . . . . . . . . . . . . . . . .
1.4 Conclusion and Problems . . . . . . . . . . . . . . . . .
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
7
7
10
14
14
16
19
24
27
29
32
2 Conducting Composites
2.1 Conductivity of Inhomogeneous Media . . . . . . . . . . .
2.1.1 Equations for Conductivity . . . . . . . . . . . . .
2.1.2 Continuity Conditions in Inhomogeneous Materials
2.1.3 Energy, Variational Principles . . . . . . . . . . . .
2.2 Composites . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Homogenization and Effective Tensor . . . . . . . .
.
.
.
.
.
.
35
35
35
39
42
45
46
vi
Contents
2.3
2.2.2 Effective Properties of Laminates . . . . . . . . . . .
2.2.3 Effective Medium Theory: Coated Circles . . . . . .
Conclusion and Problems . . . . . . . . . . . . . . . . . . .
3 Bounds and G-Closures
3.1 Effective Tensors: Variational Approach . . . . . . . .
3.1.1 Calculation of Effective Tensors . . . . . . . . .
3.1.2 Wiener Bounds . . . . . . . . . . . . . . . . . .
3.2 G-Closure Problem . . . . . . . . . . . . . . . . . . . .
3.2.1 G-convergence . . . . . . . . . . . . . . . . . .
3.2.2 G-Closure: Definition and Properties . . . . . .
3.2.3 Example: The G-Closure of Isotropic Materials
3.2.4 Weak G-Closure (Range of Attainability) . . .
3.3 Conclusion and Problems . . . . . . . . . . . . . . . .
II
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Optimization of Conducting Composites
4 Domains of Extremal Conductivity
4.1 Statement of the Problem . . . . . . . . . . . . . . . .
4.2 Relaxation Based on the G-Closure . . . . . . . . . . .
4.2.1 Relaxation . . . . . . . . . . . . . . . . . . . .
4.2.2 Sufficient Conditions . . . . . . . . . . . . . . .
4.2.3 A Dual Problem . . . . . . . . . . . . . . . . .
4.2.4 Convex Envelope and Compatibility Conditions
4.3 Weierstrass Test . . . . . . . . . . . . . . . . . . . . .
4.3.1 Variation in a Strip . . . . . . . . . . . . . . .
4.3.2 The Minimal Extension . . . . . . . . . . . . .
4.3.3 Summary . . . . . . . . . . . . . . . . . . . . .
4.4 Dual Problem with Nonsmooth Lagrangian . . . . . .
4.5 Example: The Annulus of Extremal Conductivity . . .
4.6 Optimal Multiphase Composites . . . . . . . . . . . .
4.6.1 An Elastic Bar of Extremal Torsion Stiffness .
4.6.2 Multimaterial Design . . . . . . . . . . . . . .
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . .
51
55
57
59
59
59
61
63
63
67
73
75
76
79
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
81
82
83
83
85
89
90
92
92
99
101
103
108
110
110
111
115
5 Optimal Conducting Structures
5.1 Relaxation and G-Convergence . . . . . . . . . . . . . . .
5.1.1 Weak Continuity and Weak Lower Semicontinuity
5.1.2 Relaxation of Constrained Problems by G-Closure
5.2 Solution to an Optimal Design Problem . . . . . . . . . .
5.2.1 Augmented Functional . . . . . . . . . . . . . . . .
5.2.2 The Local Problem . . . . . . . . . . . . . . . . . .
5.2.3 Solution in the Large Scale . . . . . . . . . . . . .
5.3 Reducing to a Minimum Variational Problem . . . . . . .
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
117
117
117
121
123
123
126
129
130
134
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Contents
5.5
III
vii
Conclusion and Problems . . . . . . . . . . . . . . . . . . . 139
Quasiconvexity and Relaxation
6 Quasiconvexity
6.1 Structural Optimization Problems . . . . . . . . . .
6.1.1 Statements of Problems of Optimal Design .
6.1.2 Fields and Differential Constraints . . . . . .
6.2 Convexity of Lagrangians and Stability of Solutions .
6.2.1 Necessary Conditions: Weierstrass Test . . .
6.2.2 Attainability of the Convex Envelope . . . . .
6.3 Quasiconvexity . . . . . . . . . . . . . . . . . . . . .
6.3.1 Definition of Quasiconvexity . . . . . . . . . .
6.3.2 Quasiconvex Envelope . . . . . . . . . . . . .
6.3.3 Bounds . . . . . . . . . . . . . . . . . . . . .
6.4 Piecewise Quadratic Lagrangians . . . . . . . . . . .
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . .
143
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
145
145
145
148
151
151
155
158
158
163
165
167
170
7 Optimal Structures and Laminates
7.1 Laminate Bounds . . . . . . . . . . . . . . . . . . . . .
7.1.1 The Laminate Bound . . . . . . . . . . . . . .
7.1.2 Bounds of High Rank . . . . . . . . . . . . . .
7.2 Effective Properties of Simple Laminates . . . . . . . .
7.2.1 Laminates from Two Materials . . . . . . . . .
7.2.2 Laminate from a Family of Materials . . . . . .
7.3 Laminates of Higher Rank . . . . . . . . . . . . . . . .
7.3.1 Differential Scheme . . . . . . . . . . . . . . . .
7.3.2 Matrix Laminates . . . . . . . . . . . . . . . .
7.3.3 Y -Transform . . . . . . . . . . . . . . . . . . .
7.3.4 Calculation of the Fields Inside the Laminates
7.4 Properties of Complicated Structures . . . . . . . . . .
7.4.1 Multicoated and Self-Repeating Structures . .
7.4.2 Structures of Contrast Properties . . . . . . . .
7.5 Optimization in the Class of Matrix Composites . . .
7.6 Discussion and Problems . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
171
171
172
174
176
177
180
182
183
189
193
195
198
198
201
206
211
8 Lower Bound: Translation Method
8.1 Translation Bound . . . . . . . . . . . . . . . . . . . . .
8.2 Quadratic Translators . . . . . . . . . . . . . . . . . . .
8.2.1 Compensated Compactness . . . . . . . . . . . .
8.2.2 Determination of Quadratic Translators . . . . .
8.3 Translation Bounds for Two-Well Lagrangians . . . . . .
8.3.1 Basic Formulas . . . . . . . . . . . . . . . . . . .
8.3.2 Extremal Translations . . . . . . . . . . . . . . .
8.3.3 Example: Lower Bound for the Sum of Energies .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
213
213
220
220
224
228
228
229
232
viii
Contents
8.4
8.3.4 Translation Bounds and Laminate Structures . . . . 235
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9 Necessary Conditions and Minimal Extensions
9.1 Variational Methods for Nonquasiconvex Lagrangians
9.2 Variations . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Variation of Properties . . . . . . . . . . . . . .
9.2.2 Increment . . . . . . . . . . . . . . . . . . . . .
9.2.3 Minimal Extension . . . . . . . . . . . . . . . .
9.3 Necessary Conditions for Two-Phase Composites . . .
9.3.1 Regions of Stable Solutions . . . . . . . . . . .
9.3.2 Minimal Extension . . . . . . . . . . . . . . . .
9.3.3 Necessary Conditions and Compatibility . . . .
9.3.4 Necessary Conditions and Optimal Structures .
9.4 Discussion and Problems . . . . . . . . . . . . . . . . .
IV
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
G-Closures
10 Obtaining G-Closures
10.1 Variational Formulation . . . . . . . . . . .
10.1.1 Variational Problem for Gm -Closure
10.1.2 G-Closures . . . . . . . . . . . . . .
10.2 The Bounds from Inside by Laminations . .
10.2.1 The L-Closure in Two Dimensions .
239
239
241
241
242
246
248
248
249
251
253
257
259
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
261
261
262
269
270
274
11 Examples of G-Closures
11.1 The Gm -Closure of Two Conducting Materials
11.1.1 The Variational Problem . . . . . . . .
11.1.2 The Gm -Closure in Two Dimensions . .
11.1.3 Three-Dimensional Problem . . . . . . .
11.2 G-Closures . . . . . . . . . . . . . . . . . . . .
11.2.1 Two Isotropic Materials . . . . . . . . .
11.2.2 Polycrystals . . . . . . . . . . . . . . . .
11.2.3 Two-Dimensional Polycrystal . . . . . .
11.2.4 Three-Dimensional Isotropic Polycrystal
11.3 Coupled Bounds . . . . . . . . . . . . . . . . .
11.3.1 Statement of the Problem . . . . . . . .
11.3.2 Translation Bounds of Gm -Closure . . .
11.3.3 The Use of Coupled Bounds . . . . . . .
11.4 Problems . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
279
279
279
280
284
289
289
291
292
293
296
296
299
305
308
12 Multimaterial Composites
12.1 Special Features of Multicomponent Composites . . . . . .
12.1.1 Attainability of the Wiener Bound . . . . . . . . . .
12.1.2 Attainability of the Translation Bounds . . . . . . .
309
311
311
316
.
.
.
.
.
Contents
12.1.3 The Compatibility of Incompatible Phases . . .
12.2 Necessary Conditions . . . . . . . . . . . . . . . . . . .
12.2.1 Single Variations . . . . . . . . . . . . . . . . .
12.2.2 Composite Variations . . . . . . . . . . . . . .
12.3 Optimal Structures for Three-Component Composites
12.3.1 Range of Values of the Lagrange Multiplier . .
12.3.2 Examples of Optimal Microstructures . . . . .
12.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
13 Supplement: Variational Principles for
Dissipative Media
13.1 Equations of Complex Conductivity .
13.1.1 The Constitutive Relations . .
13.1.2 Real Second-Order Equations .
13.2 Variational Principles . . . . . . . . .
13.2.1 Minimax Variational Principles
13.2.2 Minimal Variational Principles
13.3 Legendre Transform . . . . . . . . . .
13.4 Application to G-Closure . . . . . . .
V
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ix
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
321
325
326
328
334
334
338
341
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
343
344
344
347
348
349
351
352
353
Optimization of Elastic Structures
357
14 Elasticity of Inhomogeneous Media
14.1 The Plane Problem . . . . . . . . . . . . . . . . . . . . .
14.1.1 Basic Equations . . . . . . . . . . . . . . . . . .
14.1.2 Rotation of Fourth-Rank Tensors . . . . . . . . .
14.1.3 Classes of Equivalency of Elasticity Tensors . . .
14.2 Three-Dimensional Elasticity . . . . . . . . . . . . . . .
14.2.1 Equations . . . . . . . . . . . . . . . . . . . . . .
14.2.2 Inhomogeneous Medium. Continuity Conditions .
14.2.3 Energy, Variational Principles . . . . . . . . . . .
14.3 Elastic Structures . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Elastic Composites . . . . . . . . . . . . . . . . .
14.3.2 Effective Properties of Elastic Laminates . . . . .
14.3.3 Matrix Laminates, Plane Problem . . . . . . . .
14.3.4 Three-Dimensional Matrix Laminates . . . . . .
14.3.5 Ideal Rigid-Soft Structures . . . . . . . . . . . .
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
359
359
359
363
371
373
373
377
378
379
379
380
382
385
387
391
15 Elastic Composites of Extremal Energy
15.1 Composites of Minimal Compliance . . .
15.1.1 The Problem . . . . . . . . . . .
15.1.2 Translation Bounds . . . . . . .
15.1.3 Structures . . . . . . . . . . . . .
15.1.4 The Quasiconvex Envelope . . .
.
.
.
.
.
.
.
.
.
.
393
393
393
395
398
402
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
x
Contents
15.1.5 Three-Dimensional Problem . . . . . . . .
15.2 Composites of Minimal Stiffness . . . . . . . . . .
15.2.1 Translation Bounds . . . . . . . . . . . .
15.2.2 The Attainability of the Convex Envelope
15.3 Optimal Structures Different from Laminates . .
15.3.1 Optimal Structures by Vigdergauz . . . .
15.3.2 Optimal Shapes under Shear Loading . .
15.4 Problems . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
405
407
408
409
412
412
416
420
16 Bounds on Effective Properties
16.1 Gm -Closures of Special Sets of Materials . . . . .
16.2 Coupled Bounds for Isotropic Moduli . . . . . . .
16.2.1 The Hashin–Shtrikman Bounds . . . . . .
16.2.2 The Translation Bounds . . . . . . . . . .
16.2.3 Functionals . . . . . . . . . . . . . . . . .
16.2.4 Translators . . . . . . . . . . . . . . . . .
16.2.5 Modification of the Translation Method .
16.2.6 Appendix: Calculation of the Bounds . . .
16.3 Isotropic Planar Polycrystals . . . . . . . . . . .
16.3.1 Bounds . . . . . . . . . . . . . . . . . . .
16.3.2 Extremal Structures: Differential Scheme
16.3.3 Extremal Structures: Fixed-Point Scheme
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
421
421
424
425
427
431
433
435
438
449
450
452
456
17 Some Problems of Structural Optimization
17.1 Properties of Optimal Layouts . . . . . . . . . . . .
17.1.1 Necessary Conditions . . . . . . . . . . . . .
17.1.2 Remarks on Instabilities . . . . . . . . . . . .
17.2 Optimization of the Sum of Elastic Energies . . . . .
17.2.1 Minimization of the Sum of Elastic Energies .
17.2.2 Optimal Design of Periodic Structures . . . .
17.3 Arbitrary Goal Functionals . . . . . . . . . . . . . .
17.3.1 Statement . . . . . . . . . . . . . . . . . . . .
17.3.2 Local Problem . . . . . . . . . . . . . . . . .
17.3.3 Asymptotics . . . . . . . . . . . . . . . . . .
17.4 Optimization under Uncertain Loading . . . . . . . .
17.4.1 The Formulation . . . . . . . . . . . . . . . .
17.4.2 Eigenvalue Problem . . . . . . . . . . . . . .
17.4.3 Multiple Eigenvalues . . . . . . . . . . . . . .
17.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
461
461
462
465
467
468
471
474
474
475
477
479
480
483
486
494
References
497
Author/Editor Index
529
Subject Index
537