Searcharxiv⌕ Search

arXiv subjects

Flaviana Iurlano

Publications and source records attributed to Flaviana Iurlano.

At least 19 recordsLinked to original sources
arXiv2026

Shape optimization of light structures with general Hooke's laws

For the minimum-compliance shape optimization problem it was established recently [\textit{Duke Math.\ J.} 172 (2023), 43--103] that a sequence of approximately optimal shapes of exact volume \(\e\) converges to a limiting generalized shape, which is represented by a possibly diffuse probability measure, as \(\e \todown 0\). This generalized shape minimizes the so-called light-structures compliance functional, which --- as conjectured by Bouchitté --- is expressed in terms of a relaxed integrand, reflecting that optimal shapes must only involve ``lower-dimensional'' structures, like the Michell-type trusses that are well known in the engineering domain. However, the aforementioned result only applies for the simplest isotropic quadratic energy density, that is, the squared Frobenius norm, in dimensions two and three. The present work extends the relaxation result to the entire class of (isotropic or anisotropic) elasticity tensors and to any dimension, thus solving Bouchitté's conjecture in full generality. Since the previous work relied on explicit formulas for the relaxed integrands (due to Allaire, Kohn, and Strang), which are not available in the general case, several new arguments and substantial modifications of the original approach are required. We also significantly simplify and streamline other parts of the original argument, in addition to clarifying some related statements in the literature.

math.AP↗
arXiv2026

Superlinear free-discontinuity models: relaxation and phase-field approximation

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase-field models.

math.AP↗
arXiv2026

Existence Results for the Time-incremental Elastic Contact Problem with Coulomb Friction in 2D

In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.

math.AP↗
arXiv2025

Transition from Continuous to Jumping Solutions in 2D Quasi-static Elastic Contact Problems with Coulomb Friction: the Mathematics Underlying the Onset of Brake Squeal

We formulate the quasi-static elastic contact problem with Coulomb friction in a very general setting, with possible jumps in time for both the load and the solution. Exploiting ideas originating in our recent paper [4], we exhibit an optimal condition on the magnitude of the friction coefficient under which we prove the existence of an absolutely continuous solution for arbitrary absolutely continuous loads in the case of the most general 2D problem. We provide examples showing that, when the condition is violated, spontaneous jumps in time of the solution may occur, even when the load varies absolutely continuously in time. We argue that these spontaneous jumps in time of the solution in the quasi-static problem reveal a transition of the process from a quasi-static nature to a dynamic nature, interpreted as the mathematical signature of the onset of friction-induced vibrations in the elastodynamic contact problem with dry friction.

math.AP↗
arXiv2024

Convergence of critical points for a phase-field approximation of 1D cohesive fracture energies

Variational models for cohesive fracture are based on the idea that the fracture energy is released gradually as the crack opening grows. Recently, [Conti, Focardi, and Iurlano, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2016] proposed a variational approximation via $Γ$-convergence of a class of cohesive fracture energies by phase-field energies of Ambrosio-Tortorelli type, which may be also used as regularization for numerical simulations. In this paper we address the question of the asymptotic behaviour of critical points of the phase-field energies in the one-dimensional setting: we show that they converge to a selected class of critical points of the limit functional. Conversely, each critical point in this class can be approximated by a family of critical points of the phase-field functionals.

math.AP↗
arXiv2023

Piecewise rank-one approximation of vector fields with measure derivatives

This work adresses the question of density of piecewise constant (resp. rigid) functions in the space of vector valued functions with bounded variation (resp. deformation) with respect to the strict convergence. Such an approximation property cannot hold when considering the usual total variation in the space of measures associated to the standard Frobenius norm in the space of matrices. It turns out that oscillation and concentration phenomena interact in such a way that the Frobenius norm has to be homogenized into a (resp. symmetric) Schatten-1 norm which coincides with the Euclidean norm on rank-one (resp. symmetric) matrices. By means of explicit constructions consisting of the superposition of sequential laminates, the validity of an optimal approximation property is established at the expense of endowing the space of measures with a total variation associated with the homogenized norm in the space of matrices.

math.AP↗
arXiv2023

Approximation of $SBV$ functions with possibly infinite jump set

We prove an approximation result for functions $u\in SBV(Ω;\mathbb R^m)$ such that $\nabla u$ is $p$-integrable, $1\leq p<\infty$, and $g_0(|[u]|)$ is integrable over the jump set (whose $\mathcal H^{n-1}$ measure is possibly infinite), for some continuous, nondecreasing, subadditive function $g_0$, with $g_0^{-1}(0)=\{0\}$. The approximating functions $u_j$ are piecewise affine with piecewise affine jump set; the convergence is that of $L^1$ for $u_j$ and the convergence in energy for $|\nabla u_j|^p$ and $g([u_j],ν_{u_j})$ for suitable functions $g$. In particular, $u_j$ converges to $u$ $BV$-strictly, area-strictly, and strongly in $BV$ after composition with a bilipschitz map. If in addition $\mathcal H^{n-1}(J_u)<\infty$, we also have convergence of $\mathcal H^{n-1}(J_{u_j})$ to $\mathcal H^{n-1}(J_u)$.

math.AP↗
arXiv2022

Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $Γ$-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.

math.AP↗
arXiv2022

Homogenization of friction in a 2D linearly elastic contact problem

Contact problems with Coulomb friction in linear elasticity are notoriously difficult and their mathematical analysis is still largely incomplete. In this paper, a model problem with heterogeneous friction coefficient is considered in two-dimensional elasticity. For this model problem, an existence and uniqueness result is proved, relying heavily on harmonic analysis. A complete and rigorous homogenization analysis can be performed in the case of a highly oscillating friction coefficient, being the first result in that direction. The Coulomb law is found to hold in the limit, and an explicit formula is provided to calculate the effective friction coefficient. This effective friction coefficient is found to differ from the spatial average, showing an influence of the coupling between friction and elasticity on the homogenized limit.

math.AP↗
arXiv2022

Shape optimization of light structures and the vanishing mass conjecture

This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size $\varepsilon \downarrow 0$ to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitté in 2001 and predicted heuristically before in works of Allaire & Kohn and Kohn & Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, settling this open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) $\mathrm{div}$-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang integrand, and the characterization of limit minimizers due to Bouchitté & Buttazzo.

math.AP↗
arXiv2020

Cohesive fracture in 1D: quasi-static evolution and derivation from static phase-field models

In this paper we propose a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. We investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model arises naturally via Gamma-convergence from a phase-field model of the generalized Ambrosio-Tortorelli type, which may be used as regularization for numerical simulations.

math.AP↗
arXiv2019

Concentration versus oscillation effects in brittle damage

This work is concerned with an asymptotic analysis, in the sense of $Γ$-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order $\varepsilon$, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at $\varepsilon$ fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as $\varepsilon\to 0$, concentration and saturation of damage are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at $\varepsilon$ fixed) to being of linear-growth type (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. In particular, the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far. We explicitly identify the $Γ$-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky plasticity type. We further consider the regime where the divergence remains square-integrable in the limit, which leads to a Tresca-type model.

math.AP↗
arXiv2019

Partial regularity for the crack set minimizing the two-dimensional Griffith energy

In this paper we prove a $\mathcal C^{1,α}$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a $\mathcal C^{1,α}$ curve outside a set of zero Hausdorff measure.

math.AP↗
arXiv2018

A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

We prove partial regularity for minimizers to elasticity type energies in the nonlinear framework {with $p$-growth, $p>1$,} in dimension $n\geq 3$. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than {$n-(p^*\wedge 2)$, and actually $n-2$ in the autonomous case} (full regularity is well-known in dimension $2$). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions $2$ and $3$.

math.AP↗
arXiv2018

Approximation of fracture energies with $p$-growth via piecewise affine finite elements

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSBD^p(Ω)$, $p\in(1,\infty)$, their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of $GSBD^p$ functions, for $p\in(1,\infty)$, with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of $C^1$ hypersurfaces. The strains of the approximating functions converge strongly in $L^p$ to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudhental partition of a cubic grid.

math.AP↗
arXiv2017

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation. This is based on replacing the coarea formula by a method to approximate $SBD^p$ functions with small jump set by Sobolev functions and is restricted to two dimensions. The other two ingredients are contained in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSBD^p$ functions by $SBV^p$ ones.

math.AP↗
arXiv2016

Existence of minimizers for the $2$d stationary Griffith fracture model

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $SBD^2$ and for which existence is well-known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford-Shah problem.

math.AP↗