Lucia Scardia

• Professor (Associate) at Heriot-Watt University

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_\alpha$ defined on probability measures in $\R^n$, with $n\geq 3$. The energy $I_\alpha$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $\alpha=0$ and is anisotropic ot...

We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the s...

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under \emph{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. Comb...

In 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 converg...

In this paper we consider a nonlocal energy $I_\alpha$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $\alpha\in \R$. The case $\alpha=0$ corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; $\alpha=1$ corresponds to the energy of...

In this paper we characterise the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic.

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\odds...

In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies E ε {E_{\varepsilon}} defined in a randomly perforated domain, as ε goes to zero. The functionals E ε {E_{\varepsilon}} model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences...

In this paper we study the convergence of integral functionals with q-growth in a randomly perforated domain of R^n , with 1 < q < n. Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged analogue of the n...

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional elli...

In this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ G S B D p , for every $$p>1$$ p > 1 and any dimension $$n\ge 2$$ n ≥ 2 . We show that a displacement in $$GSBD^p$$ G S B D p with a small jump set coincides with a $$W^{1,p}$$ W 1 , p function, up to a small set whose perimeter and volume are controlled by the s...

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 t...

In this paper we show that ellipses persist as energy minimisers under small perturbations, starting from a nonlocal energy inspired by edge dislocations. The case of Coulomb energies has been recently considered by the same authors.

In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. Thi...

In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is , , with . This kernel is anisotropic except for the Coulomb case . We present a short compact proof of the known surprising fact that the unique minimizer of the...

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ n ≥ 3 . The energy $$I_\alpha $$ I α consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potent...

В работе рассматриваются нелокальные функционалы энергии, заданные на множестве вероятностных мер на плоскости как сумма свертки, описывающей взаимодействие, и квадратичного ограничения. Ядро взаимодействия имеет вид $-\log|z|+\alpha x^2/|z|^2$, $z=x+iy$, где $-1<\alpha<1$. Оно анизотропно, если не считать кулоновского случая $\alpha=0$. Дается кор...

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies Iα defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in Iα is of convolution type, and is isotropic for α = 0 and anisotropic otherwise. The cases α = 0 and α = 1 are special: The f...

In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+\alpha\, x^2/|z|^2, \; z=x+iy,$ with $-1 < \alpha< 1.$ This kernel is anisotropic except for the Coulombic case $\alpha=0.$ We present a short compact pro...

In this paper we study the asymptotic behaviour of a family of random free-discontinuity energies $E_\varepsilon$ defined on a randomly perforated domain, as $\varepsilon$ goes to zero. The functionals $E_\varepsilon$ model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences o...

We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign,...

We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ε > 0. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the in...

In 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 converg...

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...

In this paper, we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum nonlocal energy Eγ modelling the interactions — at a typical length-scale of 1/γ — of the walls subjected to a constant shear stress, we derive a first-order appro...

The aim of this paper is to characterise the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semi-circle law, a well-known measure...

These lecture notes are a step-by-step guide to \(\varGamma \)-convergence and its applications to the upscaling of discrete systems. In many cases of interest—atoms, defects in metals, crowds—one has to face the challenging problem of deriving a macroscopic model to describe the collective behaviour of the interacting particles. \(\varGamma \)-con...

This article gives a short description and a slight refinement of recentwork [MSZ15], [SZ12] on the derivation of gradient plasticity models fromdiscrete dislocations models.We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in t...

This paper unravels the problem of an idealised pile-up of n infinite, equi-spaced walls of edge dislocations at equilibrium. We define a dimensionless parameter that depends on the geometric, constitutive and loading parameters of the problem, and we identify five different scaling regimes corresponding to different values of that parameter for la...

We consider systems of $n$ parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit $n\to\infty$. The main aim o...

In this paper we show that a strain-gradient plasticity model arises as the Γ-limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field. A key ingredient in the derivation is the extension of the rigidity estim...

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the on...

We consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain. We are interested in a good approximation of the discrete energy of this system for a large n...

In this paper we rigorously derive a line-tension model for plasticity as the G-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Γ-limit we obtain as the length of the Burgers vector tends to zero has the same form as the G-limit obtained by starting from a linear, semidiscre...

In this work we consider a one-dimensional chain of atoms which interact through nearest and next-to-nearest neighbour interactions of Lennard–Jones type. We impose Dirichlet boundary conditions and in addition prescribe the deformation of the second and last but one atoms of the chain. This corresponds to prescribing the slope at the boundary of t...

The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a noncoercive homogenization problem.

MaTe poster

A homogenization result is given for a material with brittle periodic inclusions, under the requirement that the interpenetration of matter is forbidden. According to the ratio between the softness of the inclusions and the size of the microstucture, three different limit models are deduced via Γ-convergence. In particular it is shown that in the l...

MaTe Poster

We study the problem of the rigorous derivation of one-dimensional mod- els for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particu- lar, we prove that the limit functional corresponding to higher scalings coincides with the one derived by...

We study delamination of two elastic bodies glued together by an adhesive that can undergo a unidirectional inelastic rate-independent process. The quasistatic delamination process is thus activated by time-dependent external loadings, realized through body forces and displacements prescribed on parts of the boundary. The novelty of this work consi...

The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional $\mathcal E^h$, whose energies (per unit thickness) are bounded by $Ch^4$, converge to critical points of the $\Gamma$-lim...

A homogenization result is given for a material having brittle inclusions arranged in a periodic structure. According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Γ-convergence. In particular, damage is obtained as limit of periodically distributed microfractures.

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending-torsion theory for rods are deduced.

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dim...