Multiscale Modeling of Materials — the Role of Analysis

Conti, Sergio; DeSimone, Antonio; Dolzmann, Georg; Müller, Stefan; Otto, Félix

doi:10.1007/978-3-662-05281-5_11

Cited by 13 publications

(15 citation statements)

References 48 publications

(65 reference statements)

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“…Remark 3.3 Strictly speaking, the second-rank laminate is a sequence of structures, the corresponding energies converge to the calculated values when the ratio of laminates scales goes to zero, see for example (Conti, 2003). Rank-one path through given field ranges.…”

Section: Optimal Fields and Optimal Laminatesmentioning

confidence: 76%

Optimal anisotropic three-phase conducting composites: Plane problem

Cherkaev¹,

Zhang²

2011

International Journal of Solids and Structures

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The paper establishes tight lower bound for effective conductivity tensor K * of two-dimensional threephase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin-Shtrikman and Translation bounds to multiphase structures, it is derived using a combination of Translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi (1995) and Cherkaev (2009) for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites with effective conductivity tensor K * . The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of K * , that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by and Cherkaev (2009). The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.Keywords: multimaterial composites, optimal microstructures, bounds for effective properties, structural optimization, multiscale the periodicity cell Ω. The computed value of J allows for computation of the add outer bound of G-closure, as discussed in Sections 2.2 and 2.3. The matching microstructures (minimizing sequences) are found by a different technique that was introduces in (Albin et al, 2007a, Cherkaev, 2009 and is described in Sections 3.3 and 6; by assumption, optimal structures are laminates of some rank. The effective properties of the structures form the inner bound of G-closure. When the outer and inner bounds coincide, they are exact and the G-closure is determined. We show that our bounds are exact in all domains of parameters but one. In the last domain, we estimate the gap between the outer and inner bounds. Remark 1.1 The complementary upper bound can be established by a solution of a dual problem, in which conductivity ki are replaced by resistivity ρi = 1/ki. In the considered problem, one of the component is a superconductor (k3 = ∞) which makes the dual bound trivial -the effective resistivity can be arbitrary large, or K −1 * ≥ 0. The obtained results allows for the upper bound determination for the G-closure of materials with conductivities k1 = 0 < k2 < k3 < ∞.Bounds The problem of exact bounds has a long history. It started with the bounds by Voigt (1928) and Reuss (1929), called also Wiener bounds or the arithmetic and harmonic mean bounds. The bounds are valid for all microstructures and become in a sense exact for laminates: One of the eigenvalues of K * of a laminate is equal to the harmonic mean of the mixed materials' conductivities, and the other oneto the arithmetic mean of them. The pioneering paper by Hashin and Shtrikman (1963) found the bounds and the matchi...

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Section: Optimal Fields and Optimal Laminatesmentioning

confidence: 76%

Optimal anisotropic three-phase conducting composites: Plane problem

Cherkaev¹,

Zhang²

2011

International Journal of Solids and Structures

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“…The original functional proposed by Aviles and Giga [4] to study liquid crystals also has this trace condition. In addition for the micro-magnetic analogue of functional I there is nothing like a pointwise condition on the trace [8,10]. This micro-magnetic functional is given by M (v) = −1…”

Section: Theorem 11 ([18]) Let ω Be a Convex Set With Diameter 2 Cmentioning

confidence: 99%

“…In addition there is a closely related functional modeling thin magnetic films [1,[8][9][10]19]. For function u ∈ W 2,2 0 (Ω) we refer to I (u) as the Aviles Giga energy of u.…”

Section: Introductionmentioning

confidence: 99%

A simple proof of the characterization of functions of low Aviles Giga energy on a ballviaregularity

Lorent

2011

ESAIM: COCV

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“…An existence theorem for the physically-relevant equal-determinant case (in which the rank-one convex hull of K has codimension 1) was proven by Müller and Šverák [23] combining a more refined extension of Gromov's ideas with a subtle construction, which permits to enforce the determinant constraint pointwise. This paper presents the first application to a multi-well problem with a discrete point group in three dimensions and with physical relevance; for related cases with continuous symmetry see [8,12,2].…”

Section: Introductionmentioning

confidence: 99%

Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions

Conti

Dolzmann

Kirchheim

2007

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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The three-well problem consists in looking for minimizers u : Ω ⊂ R 3 → R 3 of a functional I (u) = Ω W (∇u) dx, where the elastic energy W models the tetragonal phase of a phase-transforming material. In particular, W attains its minimum on K = 3 i=1 SO(3)U i , with U i being the three distinct diagonal matrices with eigenvalues (λ, λ,λ), λ,λ > 0 and λ =λ. We show that, for boundary values F in a suitable relatively open subset of M 3×3 ∩ {F : detF = det U 1 }, the differential inclusionhas Lipschitz solutions. RésuméLe problème de type triple puits consiste en la recherche de minimizers u : Ω ⊂ R 3 → R 3 d'une fonctionnelle I (u) = Ω W (∇u) dx, où l'énergie élastique W modèle la phase tétragonale d'un matériel à mémoire de forme. En particulier, W atteint son minimum sur K = 3 i=1 SO(3)U i , avec U i les trois matrices diagonales distinctes avec les valeurs propres (λ, λ,λ), λ,λ > 0 et λ =λ. Nous montrons que, pour des conditions au bord F dans un sous-ensemble bien choisi relativement ouvert dea des solutions u ∈ W 1,∞ (Ω; R 3 ). 954S. Conti et al. / Ann. I. H. Poincaré -AN 24 (2007) [953][954][955][956][957][958][959][960][961][962]

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Multiscale Modeling of Materials — the Role of Analysis

Cited by 13 publications

References 48 publications

Optimal anisotropic three-phase conducting composites: Plane problem

Optimal anisotropic three-phase conducting composites: Plane problem

A simple proof of the characterization of functions of low Aviles Giga energy on a ballviaregularity

Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions

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