Quasistatic evolution of a brittle thin film

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a $\Gamma$-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in $L^\infty$.Comment: 43 page

Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination

This work is devoted so show the appearance of different cracking modes in linearly elastic thin film systems by means of an asymptotic analysis as the thickness tends to zero. By superposing two thin plates, and upon suitable scaling law assumptions on the elasticity and fracture parameters, it is proven that either debonding or transverse cracks can emerge in the limit. A model coupling debonding, transverse cracks and delamination is also discussed

Homogenization of variational problems in manifold valued Sobolev spaces

Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Maly and Trivisa \cite{DFMT}. For energies with superlinear or linear growth, a $\Gamma$-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of \cite{BM}.Comment: 22 page

3D-2D analysis of a thin film with periodic microstructure

The purpose of this article is to study the behavior of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a 3D-2D dimensional reduction through a $\Gamma$-convergence analysis, techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variable.Comment: 19 page

Multiscale nonconvex relaxation and application to thin films

$\Gamma$-convergence techniques are used to give a characterization of the behavior of a family of heterogeneous multiple scale integral functionals. Periodicity, standard growth conditions and nonconvexity are assumed whereas a stronger uniform continuity with respect to the macroscopic variable, normally required in the existing literature, is avoided. An application to dimension reduction problems in reiterated homogenization of thin films is presented.Comment: 40 pages, 5 figure

Hyperbolic structure for a simplified model of dynamical perfect plasticity

This paper is devoted to confront two different approaches to the problem of dynam-ical perfect plasticity. Interpreting this model as a constrained boundary value Friedrichs' system enables one to derive admissible hyperbolic boundary conditions. Using variational methods, we show the well-posedness of this problem in a suitable weak measure theoretic setting. Thanks to the property of finite speed propagation, we establish a new regularity result for the solution in short time. Finally, we prove that this variational solution is actually a solution of the hyperbolic formulation in a suitable dissipative/entropic sense, and that a partial converse statement holds under an additional time regularity assumption for the dissipative solutions

Homogenization of variational problems in manifold valued BV-spaces

This paper extends the result of \cite{BM} on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\Gamma$-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for $BV$-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in \cite{BM}, while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold.Comment: 32 page

A note on the derivation of rigid-plastic models

This note is devoted to a rigorous derivation of rigid-plasticity as the limit of elasto-plasticity when the elasticity tends to infinity