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We consider a model for dislocations in crystals introduced by Koslowski, Cuitiño and Ortiz, which includes elastic interactions via a singular kernel behaving as the H 1/2 norm of the slip. We obtain a sharp-interface limit of the model within the framework of Γ-convergence. From an analytical point of view, our functional is a vector-valued generalization(More)
We present an algorithm for shape-optimization under stochastic loading, and representative numerical results. Our strategy builds upon a combination of techniques from two-stage stochastic programming and level-set-based shape optimization. In particular, usage of linear elasticity and quadratic objective functions permits to obtain a computational cost(More)
We consider rate-independent crystal plasticity with constrained elasticity, and state the variational formulation of the incremental problem. For generic boundary data, even the first time increment does not admit a smooth solution, and fine structures are formed. By using the tools of quasiconvexity, we obtain an explicit relaxation of the first(More)
We consider the nonlinear elastic energy of a thin membrane whose boundary is kept fixed, and assume that the energy per unit volume scales as h β , with h the film thickness and β ∈ (0, 4). We derive, by means of Γ convergence, a limiting theory for the scaled displacements , which takes a form similar to the one proposed by Föppl in 1907. The difference(More)
The intermediate state of a type-I superconductor is a classical example of energy-driven pattern-formation, first studied by Landau in 1937. Three of us recently derived five different rigorous upper bounds for the ground state energy, corresponding to different microstructural patterns, but only one of them was complemented by a lower bound This paper(More)
The generalization to gradient vector elds of the classical double-well, singularly perturbed func-tionals, I" (u;) := Z 1 " W(ru) + "jr 2 uj 2 dx; where W() = 0 if and only if = A or = B, and A ? B is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that I" ?-converge to +1 otherwise, where K is the(More)
We study coarsening of a binary mixture within the Mullins-Sekerka evolution in the regime where one phase has small volume fraction φ 1. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as φt −1/3 as t → ∞. We prove rigorously a corresponding weak lower bound. Moreover, we establish a(More)