Marcello Ponsiglione

Gradient theory for plasticity via homogenization of discrete dislocations

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region.We show that the G-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form E = fO (W(se) + f (Curl se)) dx, where e represents the elastic part of the macroscopic strain, and Curl se represents the geometrically necessary dislocation density. The plastic energy density f is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.

Dielectric breakdown: optimal bounds

We study dielectric breakdown for composites made of two isotropic phases. We show that Sachss bound is optimal. This simple example is used to illustrate a variational principle which departs from the traditional one. We also derive the usual variational principle by elementary means without appealing to the technology of convex duality.

Elastic Energy Stored in a Crystal Induced by Screw Dislocations: From Discrete to Continuous

This paper deals with the passage from discrete to continuous in modeling the static elastic properties of vertical screw dislocations in a cylindrical crystal, in the setting of antiplanar linear ...

Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials

We prove that the Ciarlet-Necas non-interpenetration of matter condition [9] can be extended to the case of SBV -deformations of hyperelastic brittle materials, and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV -map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden’s materials.

Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity

We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed by Dal Maso, Francfort and Toader,14 which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we also prove a convergence result for the deformations.

Fracture mechanics in perforated domains: A variational model for brittle porous media

This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity. Given the perforated domain Ωe ⊂ ℝN (e being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type Here u is in SBV(Ωe) (the space of special functions of bounded variation), Su is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ωe, and stands for the (N - 1)-Hausdorff measure of the crack Su. We study the asymptotic behavior of the functionals in terms of Γ-convergence as e → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.

A VARIATIONAL MODEL FOR INFINITE PERIMETER SEGMENTATIONS BASED ON LIPSCHITZ LEVEL SET FUNCTIONS: DENOISING WHILE KEEPING FINELY OSCILLATORY BOUNDARIES

We propose a new model for segmenting piecewise constant images with irregular object boundaries: a variant of the Chan–Vese model [T. F. Chan and L. A. Vese, IEEE Trans. Image Process., 10 (2000), pp. 266–277], where the length penalization of the boundaries is replaced by the area of their neighborhood of thickness

Nonlocal Curvature Flows

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A discontinuous finite element approximation of quasi-static growth of brittle fractures

. Our aim is to keep fine details and irregularities of the boundaries while denoising additive Gaussian noise. For the numerical computation we revisit the classical

A Nonlocal Mean Curvature Flow and Its Semi-implicit Time-Discrete Approximation

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