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Item A closure theorem for gAMMA-convergence and H-convergence with applications to non-periodic homogenization(2024-02-29) Braides, Andrea; Dal Maso, Gianni; Le Bris, ClaudeShow more In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coe cients perturbed by a term vanishing at in nity, which has been recently examined in the framework of elliptic PDE.We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by gamma-convergence within a large class of integral functionals. From such result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the G-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried on for non-symmetric operators using the localization properties of H-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.Show more Item Attainment results for nematic elastomers(SISSA, 2013-10-10) Agostiniani, Virginia; Dal Maso, Gianni; DeSimone, Antonio; MathematicsShow more We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by Mueller and Sverak in the context of crystalline solids.Show more Item Compactness for a class of integral functionals with interacting local and non-local terms(2022-12-20) Braides, Andrea; Dal Maso, Gianni; mathematicsShow more We prove a compactness result with respect to -convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the -limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.Show more Item Continuity of some non-local functionals with respect to a convergence of the underlying measures(2022-04-04) Braides, Andrea; Dal Maso, Gianni; mathematicsShow more We study some non-local functionals on the Sobolev space W1,p0(Ω) involving a double integral on Ω × Ω with respect to a measure µ. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of Γ-convergence of the corresponding functionals.Show more Item Existence and uniqueness of dynamic evolutions for a peeling test in dimension one(SISSA, 2016) Dal Maso, Gianni; Lazzaroni, Giuliano; Nardini, Lorenzo; MathematicsShow more In this paper we present a one-dimensional model of a dynamic peeling test for a thin film, where the wave equation is coupled with a Griffith criterion for the propagation of the debonding front. Our main results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data.Show more Item Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition(2015-11-18) Dal Maso, Gianni; Larsen, Christopher J.; Toader, Rodica; MathematicsShow more There are very few existence results for fracture evolution, outside of globally minimizing quasi-static evolutions. Dynamic evolutions are particularly problematic, due to the difficulty of showing energy balance, as well as of showing that solutions obey a maximal dissipation condition, or some similar condition that prevents stationary cracks from always being solutions. Here we introduce a new weak maximal dissipation condition and show that it is compatible with cracks constrained to grow smoothly on a smooth curve. In particular, we show existence of dynamic fracture evolutions satisfying this maximal dissipation condition, subject to the above smoothness constraints, and exhibit explicit examples to show that this maximal dissipation principle can indeed rule out stationary cracks as solutions.Show more Item Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition(2018-03) Dal Maso, Gianni; Larsen, Christopher J.; Toader, Rodica; MathematicsShow more We consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks.Show more Item Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case(SISSA, 2015-04) Dal Maso, Gianni; Orlando, Gianluca; Toader, Rodica; MathematicsShow more We study the asymptotic behavior of a variational model for damaged elastoplastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter " , which forces damage concentration on regions of codimension one. We determine the -limit as " tends to zero and show that it contains an energy term involving the crack opening.Show more Item Gamma-Convergence of Free-discontinuity problems(SISSA, 2017-03-20) Cagnetti, Filippo; Dal Maso, Gianni; Scardia, Lucia; Zeppieri, Caterina Ida; MathematicsShow more We 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.Show more Item Gamma-convergence of quadratic functionals perturbed by bounded linear functionals(2022-12-14) Dal Maso, Gianni; Donati, Davide; mathematicsShow more We study the asymptotic behavior of solutions to elliptic equations of the form (div(Akruk) = fk in ;uk = wk on @; where Rn is a bounded open set, wk is weakly converging in H1(), fk is weakly converging in H1(), and Ak is a sequence square matrices satisfying some uniform ellipticity and boundedness conditions, and H-converging in . In particular, we characterize the weak limits of the solutions uk and of their momenta Akruk . When Ak is symmetric and wk = w = 0, we characterize the limits of the energies for the solutions.Show more Item A global method for deterministic and stochastic homogenisation in BV(SISSA, 2021-01-19) Cagnetti, Filippo; Dal Maso, Gianni; Scardia, Lucia; Zeppieri, Caterina Ida; mathematicsShow more In this paper we study the deterministic and stochastic homogenisation of free discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic 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. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.Show more Item Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms(2023-07-11) Dal Maso, Gianni; Toader, Rodica; mathematicsShow more We study stochastic homogenisation problems for free discontinuity func- tionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimum problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.Show more Item Linearisation of multiwell energies(2017-06) Alicandro, Roberto; Dal Maso, Gianni; Lazzaroni, Giuliano; Palombaro, Mariapia; MathematicsShow more Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.Show more Item Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation(SISSA, 2015-09-24) Dal Maso, Gianni; Orlando, Gianluca; Toader, Rodica; MathematicsShow more We study the lower semicontinuity of some free discontinuity functionals, whose volume term depends on the Euclidean norm of the symmetrized gradient.Show more Item A lower semicontinuity result for a free discontinuity functional with a boundary term(2015-12-15) Almi, Stefano; Dal Maso, Gianni; Toader, Rodica; MathematicsShow more We study the lower semicontinuity in $GSBV^{p}(\Om;\R^{m})$ of a free discontinuity functional~$\F(u)$ that can be written as the sum of a crack term, depending only on the jump set~$S_{u}$, and of a boundary term, depending on the trace of~$u$ on~$\partial\Om$. We give sufficient conditions on the integrands for the lower semicontinuity of~$\F$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of~$\F$ can be represented by the sum of two integrals on~$S_{u}$ and~$\partial\Om$, respectively.Show more Item A minimization approach to the wave equation on time-dependent domains(SISSA, 2018-06) Dal Maso, Gianni; De Luca, Lucia; MathematicsShow more We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.Show more Item A model for the quasistatic growth of cracks with fractional dimension(2016) Dal Maso, Gianni; Morandotti, Marco; MathematicsShow more We study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated.Show more Item A new space of generalised functions with bounded variation motivated by fracture mechanics(2022-01-11) Dal Maso, Gianni; Toader, Rodica; mathematicsShow more We introduce a new space of generalised functions with bound ed variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elasto plastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result.Show more Item A Numerical study of the Jerky crack growth in elastoplastic materials with localized plasticity(SISSA, 2020) Dal Maso, Gianni; Heltai, LucaShow more We present a numerical implementation of a model of quasi-static crack growth in linearly elastic-perfectly plastic materials. We assume that the displacement is antiplane, and that the cracks and the plastic slips are localized on a prescribed path. We provide numerical evidence of the fact that the crack growth is intermittent, with jump characteristics that depend on the material properties.Show more Item On the Cauchy problem for the wave equation on time-dependent domains(SISSA, 2018-04) Dal Maso, Gianni; Toader, Rodica; MathematicsShow more We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous de nition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.Show more