Homogenization of nonlinear inextensible pantographic structures by Γ-convergence

Homogenization of Cahn – Hilliard - type equations via evolutionary Γ - convergence

In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage o...

Homogenization of Random Fractional Obstacle Problems via Γ-convergence

Γ-convergence methods are used to prove homogenization results for fractional obstacle problems in periodically perforated domains. The obstacles have random sizes and shapes and their capacity scales according to a stationary ergodic process. We use a trace-like representation of fractional Sobolev norms in terms of weighted Sobolev energies established in [8], a weighted ergodic theorem and a...

Two-Scale Convergence and Homogenization of Periodic Structures

Asymptotic expansions by Γ-convergence

We propose a new framework aimed at constructing approximations of different order to static mechanical models with variational structure. Our starting point is a parameterized family of functionals (a ‘theory’) and we are interested in approximating the global minimuma of the energy when a secondary small parameter goes to zero. The goal is to develop a set of increasingly accurate asymptotic ...

Γ-convergence approximation of fracture and cavitation in nonlinear elasticity

Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The exp...