This is page xi
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List of Figures
P.1
P.2
An optimal conducting composite structure . . . . . . . . xvi
Anisotropy of an optimal elastic structure . . . . . . . . . xvii
1.1
1.2
1.3
1.4
1.5
Oscillating minimizing sequence . . . . . . . . . . . .
Definition of convexity . . . . . . . . . . . . . . . . .
Weierstrass variation . . . . . . . . . . . . . . . . . .
Oscillating minimizing sequence . . . . . . . . . . . .
Convexification of the Lagrangian and the minimizer
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5
8
13
15
25
2.1
2.2
2.3
The refraction of the current and the field . . . . . . . . .
The fields and the currents in a laminate . . . . . . . . . .
The field in coated circles . . . . . . . . . . . . . . . . . . .
40
52
55
3.1
3.2
3.3
3.4
3.5
Various limits of materials layouts. . . .
Conservation property of G-closure . . .
G-closure in two dimensions . . . . . . .
Domain of attainability, two dimensions
Domain of attainability, three dimensions
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65
71
73
75
76
4.1
4.2
4.3
4.4
4.5
4.6
Problem of the best conductivity of a domain . .
Variation in a strip . . . . . . . . . . . . . . . . .
Forbidden interval . . . . . . . . . . . . . . . . . .
The constitutive relations in the optimal medium
Convex nonsmooth Lagrangian . . . . . . . . . .
The constitutive equation for the optimal medium
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. 82
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xii
List of Figures
4.7
4.8
4.9
4.10
The optimal annular conductors . . . . . . .
Bar of the maximal torsion stiffness . . . . .
Multiwell Lagrangians . . . . . . . . . . . .
Dependence of the volume M2 on the cost γ2
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108
111
112
114
5.1
5.2
5.3
5.4
5.5
5.6
The local problem . . . . . . . . . . .
An optimally conducting cylinder . .
Draft of the fields in the thermolens .
Draft of the optimal thermolens . . .
The fields in the optimal domain . .
Optimal project, conducting domain
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128
135
137
137
138
138
6.1
6.2
A strong local perturbation of a potential . . . . . . . . . . 152
To the Weierstrass variation . . . . . . . . . . . . . . . . . 153
7.1
7.2
7.3
7.4
7.5
A second rank laminate . . . . . . . . .
Constructing an infinite-rank laminate
Matrix laminates . . . . . . . . . . . .
T-structure . . . . . . . . . . . . . . . .
Multicoated matrix laminates . . . . .
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175
186
190
197
198
9.1
9.2
9.3
9.4
Optimality conditions: A family of trial inclusions .
Permitted regions of fields in an optimal composite
The fields in coated circles . . . . . . . . . . . . . .
The fields in matrix laminates . . . . . . . . . . . .
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245
249
254
255
10.1
10.2
10.3
The scheme of bounding of a Gm -closure . . . . . . . . . . 264
Construction of L-closure . . . . . . . . . . . . . . . . . . . 272
The L-closure of conducting materials, 2D . . . . . . . . . 276
11.1
11.2
11.3
11.4
11.5
11.6
The Gm -closure in two dimensions . . . . . . . . . . . .
The Gm -closure in three dimensions . . . . . . . . . . .
G-closure in three dimensions . . . . . . . . . . . . . .
Coupling: The isotropic component of the Gm -closure.
Coupling: Range of the anisotropic conductivities . . .
Geometry of the “secured spheres” . . . . . . . . . . .
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283
288
291
304
306
307
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
Range of three-material composites . . . . . . . . . . .
An extremal anisotropic three-component structure . .
Isotropic optimal multicomponent structures . . . . . .
Optimal three-component structures. Case I . . . . . .
Optimal three-component structures. Case II . . . . . .
Compatible composites from four materials. The fields
Compatible composites from four materials. Geometry
Permitted regions, based on single variations . . . . . .
Scheme of a composite variation . . . . . . . . . . . . .
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311
312
318
319
321
323
324
327
328
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List of Figures
xiii
12.10
12.11
12.12
12.13
12.14
12.15
Permitted region V2 , based on composite variations
Permitted region V1 based on composite variations
The permitted regions, γ = γ1 . . . . . . . . . . . .
The permitted regions, γ ∈ (γ1 , γ2 ) . . . . . . . .
The permitted regions, γ = γ2 . . . . . . . . . . . .
The permitted regions, γ 6∈ [γ1 , γ2 ] . . . . . . . . .
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332
333
336
336
337
337
14.1
14.2
14.3
14.4
Stresses, applied to a unit square . . . . . . . .
Orthogonal matrix laminate of the second rank
Representation of tensors t ⊗ t . . . . . . . . .
Herringbone structure . . . . . . . . . . . . . . .
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361
383
388
390
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
Bounds on the stored energy . . . . . . . . . . . . .
Structures of extremal stiffness . . . . . . . . . . . .
The upper WL and lower Wp bounds of the energy
The quasiconvex envelope . . . . . . . . . . . . . .
Optimal periodic structures . . . . . . . . . . . . .
Optimal cavities: Elongated periodicity cells . . . .
Why does the optimal cavity have corners? . . . . .
Energy outside of the optimal cavity . . . . . . . .
The more cavities, the better . . . . . . . . . . . . .
Optimal cavities, dependence on the loading . . . .
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398
400
401
403
415
415
417
418
419
419
16.1
16.3
16.4
16.5
16.6
Hashin–Shtrikman bounds, Walpole points, and translation
bounds: Well-ordered materials . . . . . . . . . . . . . . .
Hashin–Shtrikman points, Walpole bounds, and translation
bounds: Badly ordered materials . . . . . . . . . . . . . . .
Hexagonal structures by Sigmund . . . . . . . . . . . . . .
An optimal polycrystal, differential scheme . . . . . . . . .
Convergence of the family of bounds . . . . . . . . . . . .
Optimal fractal structure . . . . . . . . . . . . . . . . . . .
428
430
454
456
458
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
A third-rank laminate . . . . . . . . . . . . . . .
Optimal console, piece-wise constant properties
An optimal cylindrical shell . . . . . . . . . . .
Example of an unstable design . . . . . . . . . .
The optimal beam under the “worst” loading .
Optimal structure under multiple loading . . . .
Nonsmooth minimum . . . . . . . . . . . . . . .
Optimal structure of a wheel . . . . . . . . . . .
467
473
476
481
485
487
491
494
16.2
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428