57 research outputs found
Stability of quasi-static crack evolution through dimensional reduction
International audienceThis paper deals with quasi-static crack growth in thin films. We show that, when the thickness of the film tends to zero, any three-dimensional quasi-static crack evolution converges to a two-dimensional one, in a sense related to the Gamma-convergence of the associated total energy. We extend the prior analysis of [2] by adding conservative body and surface forces which allow us to remove the boundedness assumption on the deformation fiel
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
Relaxation approximation of Friedrich's systems under convex constraints
This paper is devoted to present an approximation of a Cauchy problem for
Friedrichs' systems under convex constraints. It is proved the strong
convergence in L^2\_{loc} of a parabolic-relaxed approximation towards the
unique constrained solution
Energy release rate for non smooth cracks in planar elasticity
This paper is devoted to the characterization of the energy release rate of a
crack which is merely closed, connected, and with density at the tip.
First, the blow-up limit of the displacement is analyzed, and the convergence
to the corresponding positively -homogenous function in the cracked plane
is established. Then, the energy release rate is obtained as the derivative of
the elastic energy with respect to an infinitesimal additional crack increment
On the Convergence of critical points of the Ambrosio-Tortorelli functional
This work is devoted to study the asymptotic behavior of critical points
of the Ambrosio-Tortorelli
functional. Under a uniform energy bound assumption, the usual
-convergence theory ensures that
converges in the -sense to some as , where
is a special function of bounded variation. Assuming further the
Ambrosio-Tortorelli energy of to converge to
the Mumford-Shah energy of , the later is shown to be a critical point
with respect to inner variations of the Mumford-Shah functional. As a
byproduct, the second inner variation is also shown to pass to the limit. To
establish these convergence results, interior () regularity
and boundary regularity for Dirichlet boundary conditions are first obtained
for a fixed parameter . The asymptotic analysis is then
performed by means of varifold theory in the spirit of scalar phase transition
problems. Lastly, a complete one-dimensional study allows one to exhibit
non-minimizing critical points of the Ambrosio-Tortorelli functional that do
satisfy our energy convergence assumption
Quasistatic evolution in non-associative plasticity - The cap model
International audienceNon-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled
A variational approach to the local character of G-closure: the convex case
This article is devoted to characterize all possible effective behaviors of
composite materials by means of periodic homogenization. This is known as a
-closure problem. Under convexity and -growth conditions (), it is
proved that all such possible effective energy densities obtained by a
-convergence analysis, can be locally recovered by the pointwise limit
of a sequence of periodic homogenized energy densities with prescribed volume
fractions. A weaker locality result is also provided without any kind of
convexity assumption and the zero level set of effective energy densities is
characterized in terms of Young measures. A similar result is given for cell
integrands which enables to propose new counter-examples to the validity of the
cell formula in the nonconvex case and to the continuity of the determinant
with respect to the two-scale convergence.Comment: 24 pages, 1 figur
Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements
Motivated by models of fracture mechanics, this paper is devoted to the
analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional,
where unilaterality comes from an irreversibility constraint on the fracture
density. In the spirit of gradient flows in metric spaces, such evolutions are
defined in terms of curves of maximal unilateral slope, and are constructed by
means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah
regime is also carried out. It shows the convergence towards a generalized heat
equation outside a time increasing crack set.Comment: accepted in Ann. Inst. H. Poincar\'e, Anal. Nonli
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