01002nas a2200133 4500008004100000245006100041210006000102520056900162100002200731700002100753700001900774700002700793856004800820 2018 en d00aStochastic homogenisation of free-discontinuity problems0 aStochastic homogenisation of freediscontinuity problems3 aIn this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the
existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.1 aCagnetti, Filippo1 aDal Maso, Gianni1 aScardia, Lucia1 aZeppieri, Caterina Ida uhttp://preprints.sissa.it/handle/1963/3530901393nas a2200145 4500008004100000245005300041210005100094260001000145520095500155100002201110700002101132700001901153700002701172856004801199 2017 en d00aGamma-Convergence of Free-discontinuity problems0 aGammaConvergence of Freediscontinuity problems bSISSA3 aWe study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.1 aCagnetti, Filippo1 aDal Maso, Gianni1 aScardia, Lucia1 aZeppieri, Caterina Ida uhttp://preprints.sissa.it/handle/1963/3527600919nas a2200121 4500008004300000245013400043210006900177260004600246520042800292100002200720700001900742856003600761 2011 en_Ud 00aQuasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach0 aQuasistatic crack evolution for a cohesive zone model with diffe bCambridge University Press / EDP Sciences3 aA new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved.1 aCagnetti, Filippo1 aToader, Rodica uhttp://hdl.handle.net/1963/235501111nas a2200121 4500008004300000245007200043210006900115520069700184100002200881700002500903700002500928856003600953 2008 en_Ud 00aA second order minimality condition for the Mumford-Shah functional0 asecond order minimality condition for the MumfordShah functional3 aA new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\\\Gamma)$, $\\\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.1 aCagnetti, Filippo1 aMora, Maria Giovanna1 aMorini, Massimiliano uhttp://hdl.handle.net/1963/1955