Alessandro Giacomini

Quasi-static evolution for a model in strain gradient plasticity

We prove the existence of a quasi-static evolution for a model in strain gradient plasticity proposed by Gurtin and Anand concerning isotropic, plastically irrotational materials under small deformations. This is done by means of the energetic approach to rate-independent evolution problems. Finally we study the asymptotic behavior of the evolution as the strain gradient length scales tend to zero recovering in the limit a quasi-static evolution in perfect plasticity.

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.

Size effects on quasi-static growth of cracks

We perform an analysis of the size effect for quasi-static growth of cracks in isotropic linearly elastic bodies under antiplanar shear. In the framework of the variational model proposed by Francfort and Marigo in [J. Mech. Phys. Solids, 46 (1998), pp. 1319--1342], we prove that if the size of the body tends to infinity, and even if the surface energy is of cohesive form, under suitable boundary displacements the crack propagates following the Griffiths functional.

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

Abstract We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffiths criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation.

On periodic homogenization in perfect elasto-plasticity

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

Whitney property in two dimensional Sobolev spaces

For p > 1, we prove that all the functions of W 2,p loc (R 2 ) satisfy the Whitney property; i.e., if u ∈ W 2,p loc (R 2 ) is such that ∇u = 0 (in the sense of capacity) on a connected set K C R 2 , then u is constant on K.

The taming of plastic slips in von Mises elasto-plasticity

We derive sufficient conditions that prevent the formation of plastic slips in three-dimensional small strain Prandtl-Reuss elasto-plasticity when the yield criterion is of the Von Mises type.

The most dangerous flaw orientation in brittle materials and structures

In the present paper a sheet of material is considered. It is loaded by uniaxial tensile stress and contains a random distribution of flaw orientations, with the flaws thought of as flat pre-cracks of comparable length, and with all crack planes being oriented perpendicular to the faces of the sheet. Intuition suggests that the most likely flaw to initiate fracture, which will be termed the “most dangerous defect”, lies orthogonally to the major load axis. The purpose of the present paper is to show that such an assumption is incorrect. Neither the most dangerous defect nor the first increments of crack growth will be oriented perpendicularly to the stress direction (nor will they be co-planar with the orientation of the most critical flaw).

A GENERALIZATION OF {\RM GO{\L}\c{A}B'S} THEOREM AND APPLICATIONS TO FRACTURE MECHANICS

We study the lower semicontinuity for functionals of the form K → ∫K φ (x, ν)dℋ1 defined on compact sets in ℝ2 with a finite number of connected components and finite ℋ1 measure and apply the result to the study of quasi-static growth of brittle fractures in linearly elastic inhomogeneous and anisotropic bodies.