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Vito Crismale

Publications and source records attributed to Vito Crismale.

At least 19 recordsLinked to original sources
arXiv2026

Quasistatic evolution of cohesive-type fracture

We prove the existence of globally stable quasistatic evolutions for a cohesive fracture model with unprescribed crack path and without any topological restriction, in arbitrary dimension. The surface energy density is assumed to be concave and to exhibit an activation threshold, modeling depinning effects and fracture process zones in quasi-brittle materials. We devise a new notion of convergence for memory variables supported on evolving crack sets, inspired by $σ$-convergence in brittle fracture, guaranteeing compactness and lower semicontinuity properties. In contrast to the brittle case, global stability is not preserved under passage to the limit because of oscillation and branching phenomena in the approximating cracks. To overcome this difficulty, we deviate from the classical scheme for proving energetic solutions by first proving the energy balance and convergence of the surface energies, and only afterwards recovering the global stability condition.

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arXiv2026

Phase-field approximation of sharp-interface energies accounting for lattice symmetry

We present a phase-field approximation of sharp-interface energies defined on partitions, designed for modeling grain boundaries in polycrystals. The independent variable takes values in the orthogonal group $\mathrm{O}(d)$ modulo a lattice point group $\mathcal{G}$, reflecting the crystallographic symmetries of the underlying lattice. In the sharp-interface limit, the surface energy exhibits a Read-Shockley-type behavior for small misorientation angles, scaling as $θ|\logθ|$. The regularized functionals are applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.

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arXiv2025

On the approximation of $SBD$ functions and some applications

Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$, and $SBD^p$, whose functions are in $SBD^p_\infty$ and the jump set has finite $\mathcal{H}^{n-1}$-measure. This generalises on the one hand the density result by [Chambolle, 2004-2005] and, on the other hand, extends in some sense the three approximation theorems in by [De Philippis, Fusco, Pratelli, 2017] for $SBV$, $SBV^p_\infty$, $SBV^p$ spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As application, the sharp version of two $Γ$-convergence results for energies defined on $SBD^2$ is derived.

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arXiv2025

Adaptive finite element approximation for quasi-static crack growth

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 triangulation is a part of the unknown of the problem and adaptive in each minimization step. The limit passage is conducted simultaneously in the vanishing mesh size and discretized time step, and results in an evolution for the continuum Griffith model of brittle fracture with isotropic surface energy [FriedrichSolombrino16] which is characterized by an irreversibility condition, a global stability, and an energy balance. Our result corresponds to an evolutionary counterpart of the static Gamma-convergence result in [BabadjianBonhomme23] for which, as a byproduct, we provide an alternative proof.

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arXiv2024

On the visco-plastic approximation of a rate-independent coupled elastoplastic damage model

In this paper we study a rate-independent system for the propagation of damage and plasticity. To construct solutions we resort to approximation in terms of viscous evolutions, where viscosity affects both damage and plasticity with the same rate. The main difficulty arises from the fact that the available estimates do not provide sufficient regularity on the limiting evolutions to guarantee that forces and velocities are in a duality pairing, hence we cannot use a chain rule for the driving energy. Nonetheless, via careful techniques we can characterize the limiting rate-independent evolution by means of an energy-dissipation balance, which encodes the onset of viscous effects in the behavior of the system at jumps.

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arXiv2024

Approximation of topological singularities through free discontinuity functionals: the critical and super-critical regimes

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \cite{DLSVG}. We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes $|\log \varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone near the defects. Further, we remove the \emph{a priori} restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions, approximated through functions with essentially closed jump set, in the strong $BV$ norm.

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arXiv2023

A rate-independent model for geomaterials under compression coupling strain gradient plasticity and damage

We study a strain gradient-enhanced version of a model for geomaterials under compression by Marigo and Kazymyrenko (2019) coupling damage and small-strain associative plasticity. We prove that the jumps in time of the plastic variable may happen only along jumps of the damage variable. Moreover, we perform a vanishing-viscosity analysis showing existence of Balanced Viscosity quasistatic solutions à la Mielke-Rossi-Savaré.

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arXiv2022

A general compactness theorem in $G(S)BD$

We give a new, simpler proof of a compactness result in $GSBD^p$, $p>1$, by the same authors, which is also valid in $GBD$ (the case $p=1$), and shows that bounded sequences converge a.e., after removal of a suitable sequence of piecewise infinitesimal rigid motions, subject to a fixed partition.

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arXiv2022

Singular limits of a coupled elasto-plastic damage system as viscosity and hardening vanish

The paper studies the asymptotic analysis of a model coupling elastoplasticity and damage depending on three parameters -- governing viscosity, plastic hardening, and convergence rate of plastic strain and displacement to equilibrium -- as they vanish in different orders. The notion of limit evolution obtained is proven to coincide in any case with a notion introduced by Crismale and Rossi in 2019; moreover, such solutions are closely related to those obtained in the vanishing-viscosity limit by Crismale and Lazzaroni in 2016, for the analogous model where only the viscosity parameter was present.

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arXiv2021

Energetic solutions for the coupling of associative plasticity with damage in geomaterials

We prove existence of globally stable quasistatic evolutions, referred to as energetic solutions, for a model proposed by Marigo and Kazymyrenko in 2019. The behaviour of geomaterials under compression is studied through the coupling of Drucker-Prager plasticity model with a damage term tuning kinematical hardening. This provides a new approach to the modelling of geomaterials, for which non associative plasticity is usually employed. % The present coupling is such that The kinematical hardening is null where the damage is complete, so there the behaviour is perfectly plastic. We analyse the model combining tools from the theory of capacity and from the treatment of linearly elastic materials with cracks.

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arXiv2021

The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^+$ and $s\to 1^-$

This paper deals with the limit cases for $s$-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to 0^+$ and $s\to 1^-$\,. To this purpose, we describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the $L^2$ metric. First, we provide an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of $Γ$-converging uniformly $λ$-convex energy functionals. Then, we provide the $Γ$-convergence analysis of the $s$-Gagliardo seminorms as $s\to 0^+$ and $s\to 1^-$\,, and apply the general stability result to such specific cases. As a consequence, we prove that $s$-fractional heat flows (suitably scaled in time) converge to the standard heat flow as $s\to 1^-$, and to a degenerate ODE type flow as $s\to 0^+$\,. Moreover, looking at the next order term in the asymptotic expansion of the $s$-fractional Gagliardo seminorm, we show that suitably forced $s$-fractional heat flows converge, as $s\to 0^+$\,, to the parabolic flow of an energy functional that can be seen as a sort of renormalized $0$-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.

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arXiv2020

Integral representation for energies in linear elasticity with surface discontinuities

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 ($GSBD^p$) 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 Bouchittè et al. '98 and a recent Korn-type inequality in $GSBD^p$ (Cagnetti-Chambolle-Scardia '20). Our general strategy also allows to generalize integral representation results in $SBD^p$, obtained in dimension two (Conti-Focardi-Iurlano '16), to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ($GSBV^p$).

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arXiv2020

Fatigue effects in elastic materials with variational damage models: A vanishing viscosity approach

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depend on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time-step $τ$ of the time-discretisation and later the viscosity parameter $ε$ go to zero. As $τ\to 0$, we find $ε$-approximate viscous evolutions; then, as $ε\to 0$, we find a rescaled approximate evolution satisfying an energy-dissipation balance.

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arXiv2020

Existence of strong solutions to the Dirichlet problem for the Griffith energy

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo in 1998. In a recent paper, we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti, Focardi, Iurlano, and of Chambolle, Conti, Iurlano.

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arXiv2020

A derivation of Griffith functionals from discrete finite-difference models

We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $δ$ is smaller than the ellipticity parameter $\varepsilon$, we show the $Γ$-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $L^p$ fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

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arXiv2020

Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity

We extend the results about existence of minimizers, relaxation, and approximation proven by Chambolle et al. in 2002 and 2007 for an energy related to epitaxially strained crystalline films, and by Braides, Chambolle, and Solci in 2007 for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any 'a priori' assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for $(d{-}1)$-rectifiable sets that are jumps of $GSBD^p$ functions, called $σ^p_{\mathrm{sym}}$-convergence.

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