Lower semicontinuity for non autonomous surface integrals

Some lower semicontinuity results are established for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. The proof of the semicontinuity results is based on some suitable approximations from below with appropriate functionals

Lower semicontinuity in GSBD for nonautonomous surface integrals

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of \emph{nonautonomous symmetric joint convexity}, which extends the analogous definition devised for autonomous integrands in \cite{FPS} where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in \cite{ACDD}, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

We study the $\Gamma$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$

Quasistatic evolution of a brittle thin film

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a $\Gamma$-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in $L^\infty$.Comment: 43 page

k-quasiconvexity reduces to quasiconvexity

The relation between quasi-convexity and k-quasiconvexity (k greater than or equal to 2) is investigated. It is shown that every smooth strictly k-quasi-convex integrand with p-growth at infinity, p > 1, is the restriction to kth-order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for kth-order variational problems are deduced as corollaries of well-known first-order theorems. This generalizes a previous work by Dal Maso et al., in which the case where k = 2 was treated