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The main results in this note concern the characterization of the length of continua 1 (Theorems 2.5) and the parametrization of continua with finite length (Theorem 4.4). Using these results we give two independent and relatively…
Using a generalized cut vertex expansion we introduce the concept of an extended fracture function for the description of semi-inclusive deep inelastic processes in the target fragmentation region. Extended fracture functions are shown to…
We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed in [13] by G. Dal Maso, G.A. Francfort, and R. Toader, which takes into account body forces and surface loads.…
We provide an adaptive finite element approximation for a model of quasi-static crack growth in dimension two. The discrete setting consists of integral functionals that are defined on continuous, piecewise affine functions, where the…
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 Griffith's criterion. We restrict our…
In this paper we study the asymptotic behaviour via Gamma-convergence of some integral functionals which model some multi-dimensional structures and depend explicitly on the linearized strain tensor. The functionals are defined in…
We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and…
We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower…
We investigate a quasicontinuum method by means of analytical tools. More precisely, we compare a discrete-to-continuum analysis of an atomistic one-dimensional model problem with a corresponding quasicontinuum model. We consider next and…
We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type $$ %\int_\O {1/2}({\mathbb C} \E u, \E u)dx + \int_{J_{u}} \Theta(u^+, u^-, \nu_{u})d \H^{N-1} \enspace, \enspace…
This paper introduces the proper notion of variational quasiconvexity associated to a group of diffeomorphisms. We prove a lower semicontinuity theorem connected to this notion. In the second part of the paper we apply this result to a…
We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show…
This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a…
The method of iterated conformal maps is developed for quasi-static fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to mode I and II. The latter require…
We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed…
Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes.…
We deal with a family of functionals depending on curvatures and we prove for them compactness and semicontinuity properties in the class of closed and bounded sets which satisfy a uniform exterior and interior sphere condition. We apply…
Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals…
The quasistatic, Prandtl-Reuss perfect plasticity at small strains is combined with a gradient, reversible (i.e. admitting healing) damage which influences both the elastic moduli and the yield stress. Existence of weak solutions of the…
We introduce and study two new relations between function spaces over measure spaces of infinite measure, motivated by the question of establishing compactness. The first relation captures the uniform decay of function (quasi-)norms ``at…