Surface energies emerging in a microscopic, two-dimensional two-well problem

Kitavtsev, Georgy; Luckhaus, Stephan; Rüland, Angkana

doi:10.1017/s0308210516000433

Cited by 13 publications

(20 citation statements)

References 23 publications

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“…In dimension two, the analysis was extended to the frame-indifferent linearized setting by S. Conti and B. Schweizer in [20] who also accomplished the characterization of the fully nonlinear framework (1.3) for d = 2 in the two subsequent papers [19,21]. Recently, some related microscopic models for twodimensional martensitic transformations as well as their discrete-to-continuum limits have been analyzed in [38,39].…”

Section: Introductionmentioning

confidence: 99%

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

Davoli

Friedrich

2020

Calc. Var.

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We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of Γ-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.

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Section: Introductionmentioning

confidence: 99%

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

Davoli

Friedrich

2020

Calc. Var.

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“…In [15] it is shown that the Γ-limit of (0.6) is an interfacial energy concentrated on the jump set of ∇u. A microscopic derivation of such result has been recently obtained in [18] in the context of squareto-rectangular martensitic phase transitions. We point out that in [15,18] the two wells of K consist of matrices with positive determinant, while this is not the case in the present context.…”

Section: Introductionmentioning

confidence: 91%

On the effect of interactions beyond nearest neighbours on non-convex lattice systems

2017

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“…Remark 7. Then there exists a constant C < ∞ and a sequence ( (34). Set E := lim sup ε↓0 E ε (ω; u ε , A).…”

Section: Outline Of the Proof Of Theorem 31mentioning

confidence: 99%

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

Neukamm¹,

Schäffner²,

Schlömerkemper³

2017

SIAM J. Math. Anal.

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Recently, there has been considerable effort to understand periodic and stochastic homogenization of elliptic equations and integral functionals with degenerate growth, as well as related questions on the effective behavior of conductance models in degenerate, random environments. In the present paper we prove stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions. In particular, we study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions d ≥ 2. We consider energy functionals with random (stationary and ergodic) pair interactions; thus our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a uniform p-growth condition, the homogenization problem is well-understood. In this paper, we are interested in a degenerate situation, when the interactions neither satisfy a uniform growth condition from above nor from below. We consider interaction potentials that obey a p-growth condition with a random growth weight λ. We show that if λ satisfies the moment condition E[λ α + λ −β ] < ∞ for suitable values of α and β, then the discrete energy Γ-converges to an integral functional with a non-degenerate energy density. In the scalar case, it suffices to assume that α ≥ 1 and β ≥ 1 p−1 (which ensures the non-degeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that α > 1 and 1 α + 1 β ≤ p d .

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Surface energies emerging in a microscopic, two-dimensional two-well problem

Cited by 13 publications

References 23 publications

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

On the effect of interactions beyond nearest neighbours on non-convex lattice systems

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

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