Ricerc@Sapienza

We prove that any function in GSBD p (Ω) , with Ω a n-dimensional open bounded set with finite perimeter, is approximated by functions u k ∈ SBV(Ω ; R n ) ∩ L ∞ (Ω ; R n ) whose jump is a finite union of C 1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies ∫ Ω W(e(u)) dx + H n-1 (J u ) , e(u) and J u being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1.

In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation (GSBD p GSBDp) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration.

We study the Γ-limit of Ambrosio-Tortorelli-type functionals D ϵ (u, v) {D-{arepsilon}(u,v)}, whose dependence on the symmetrised gradient e (u) {e(u)} is different in u {mathbb{A}u} and in e (u)- u {e(u)-mathbb{A}u}, for a â., {mathbb{C}}-elliptic symmetric operator {mathbb{A}}, in terms of the prefactor depending on the phase-field variable v.