Maria Giovanna Mora

Derivation of nonlinear bending theory for shells from three dimensional nonlinear elasticity by Gamma-convergence

Abstract We show that the nonlinear bending theory of shells arises as a Γ -limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Globally stable quasistatic evolution in plasticity with softening

We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

Non-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 mollication of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.

Local calibrations for minimizers of the mumford-shah functional with a regular discontinuity set

Abstract Using a calibration method, we prove that, if w is a function which satisfies all Euler conditions for the Mumford–Shah functional on a two-dimensional open set Ω , and the discontinuity set Sw of w is a regular curve connecting two boundary points, then there exists a uniform neighbourhood U of Sw such that w is a minimizer of the Mumford–Shah functional on U with respect to its own boundary conditions on ∂U. We show that Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w on ∂Ω and whose extended graph is contained in a neighbourhood of the extended graph of w, and we give a sufficient condition in terms of the geometrical properties of Ω and Sw under which this kind of minimality holds.

Convergence of equilibria of three-dimensional thin elastic beams

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter h of the cross-section goes to zero. More precisely, we show that stationary points of the nonlinear elastic functional E , whose energies (per unit cross-section) are bounded by Ch , converge to stationary points of the Γ-limit of E/h . This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James, and Muller [4] and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

Time-dependent systems of generalized Young measures

In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows us to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.

Finite-difference approximation of free-discontinuity problems

We approximate functionals depending on the gradient of

Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets

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Large Time Existence for Thin Vibrating Plates

and on the behaviour of

Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement

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