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Adriana Garroni

Publications and source records attributed to Adriana Garroni.

At least 19 recordsLinked to original sources
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.

math.FA↗
arXiv2024

Clearing-out of dipoles for minimisers of 2-dimensional discrete energies with topological singularities

A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for controlling the number of defects in the interior. We present a general combinatorial dipole-removal argument for a large class of discrete models including XY systems and screw dislocation models, allowing to prove sharp conditions under which controlled flux and boundary energy guarantee to have minimizers with zero or one charges in the interior. The argument uses the max-flow min-cut theorem in combination with an ad-hoc duality for planar graphs, and is robust with respect to changes of the function defining the interaction energies.

math.AP↗
arXiv2024

A discrete crystal model in three dimensions: the line-tension limit for dislocations

We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of $Γ$-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semi-discrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector.

math.AP↗
arXiv2024

Stacking faults in the limit of a discrete model for partial edge dislocations

In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the $Γ$-limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.

math.AP↗
arXiv2023

On the Read-Shockley energy for grain boundaries in poly-crystals

In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearised elasticity and an ansazt on the distribution of incompatibilities of the lattice at the interface. In this paper we derive a sharp interface limiting functional starting from a nonlinear semidiscrete model for dislocations proposed by Lauteri--Luckhaus. Building upon their analysis we obtain, via $Γ$-convergence, an interfacial energy depending on the rotations of the grains and the relative orientation of the interface which agrees for small angle grain boundaries with the Read and Shockley logarithmic scaling.

math.AP↗
arXiv2023

Homogenization of line tension energies

We prove an homogenization result, in terms of $Γ$-convergence, for energies concentrated on rectifiable lines in $\R^3$ without boundary. The main application of our result is in the context of dislocation lines in dimension $3$. The result presented here shows that the line tension energy of unions of single line defects converge to the energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the $Γ$-Limit, we obtain an alternative proof of the density of rectifiable $1$-currents without boundary in the space of divergence free fields.

math.AP↗
arXiv2022

Line-tension limits for line singularities and application to the mixed-growth case

We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.

math.AP↗
arXiv2021

Density of polyhedral partitions

We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, we consider a decomposition $u$ of a bounded Lipschitz set $Ω\subset\mathbb R^n$ into finitely many subsets of finite perimeter, which can be identified with a function in $SBV_{\rm loc}(Ω;{\cal Z})$ with ${\cal Z}\subset \mathbb R^N$ a finite set of parameters. For all $\varepsilon>0$ we prove that such a $u$ is $\varepsilon$-close to a small deformation of a polyhedral decomposition $v_\varepsilon$, in the sense that there is a $C^1$ diffeomorphism $f_\varepsilon:\mathbb R^n\to\mathbb R^n$ which is $\varepsilon$-close to the identity and such that $u\circ f_\varepsilon-v_\varepsilon$ is $\varepsilon$-small in the strong $BV$ norm. This implies that the energy of $u$ is close to that of $v_\varepsilon$ for a large class of energies defined on partitions. Such type of approximations are very useful in order to simplify computations in the estimates of $Γ$-limits.

math.AP↗
arXiv2020

Homogenization of energies defined on $1$-rectifiable currents

In this paper we study the homogenization of a class of energies concentrated on lines. In dimension $2$ (i.e., in codimension $1$) the problem reduces to the homogenization of partition energies studied by \cite{AB}. There, the key tool is the representation of partitions in terms of $BV$ functions with values in a discrete set. In our general case the key ingredient is the representation of closed loops with discrete multiplicity either as divergence-free matrix-valued measures supported on curves or with $1$-currents with multiplicity in a lattice. In the $3$ dimensional case the main motivation for the analysis of this class of energies is the study of line defects in crystals, the so called dislocations.

math.AP↗
arXiv2020

Damage dynamics, $G$-Convergence, Homogenization in dynamics, Threshold Conditions

In this paper we construct, by means of a variational formulation, the solutions of a problem of elastodynamics which includes the effect of damage for the elastic material. The result is a wave equation with time dependent operators which represents the elastic coefficients of the material undergoing damage. The dynamics that we construct also satisfies a threshold condition with the same threshold value that characterizes the quasi-static evolution of damage (see \cite{GL}).

math.AP↗
arXiv2020

Derivation of strain-gradient plasticity from a generalized Peierls-Nabarro model

We derive strain-gradient plasticity from a nonlocal phase-field model of dislocations in a plane. Both a continuous energy with linear growth depending on a measure which characterizes the macroscopic dislocation density and a nonlocal effective energy representing the far-field interaction between dislocations arise naturally as scaling limits of the nonlocal elastic interaction. Relaxation and formation of microstructures at intermediate scales are automatically incorporated in the limiting procedure based on $Γ$-convergence.

math.AP↗
arXiv2020

Duality arguments for linear elasticity problems with incompatible deformation fields

We prove existence and uniqueness for solutions to equilibrium problems for free-standing, traction-free, non homogeneous crystals in the presence of plastic slips. Moreover we prove that this class of problems is closed under G-convergence of the operators. In particular the homogenization procedure, valid for elliptic systems in linear elasticity, depicts the macroscopic features of a composite material in the presence of plastic deformation.

math.AP↗
arXiv2020

Nonlinear three-dimensional derivation of line tension for dislocations: quadratic growth

In this paper we derive a line tension model for dislocations in 3d starting from a geometrically nonlinear elastic energy with quadratic growth. In the asymptotic analysis, as the amplitude of the Burgers vectors (proportional to the lattice spacing) tends to zero, we show that the elastic energy linearises and the line tension energy density, up to an overall constant rotation, is identified by the linearised cell problem formula given in [17].

math.AP↗
arXiv2018

Convergence and non-convergence of many-particle evolutions with multiple signs

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, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale $δ_n>0$, which converges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on the theory of $λ$-convex gradient flows, a priori estimates for the Groma-Balogh equations and Orlicz spaces. The proof require $δ_n$ to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if $δ_n$ converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations. These results show how the validity of the Groma-Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

math.AP↗
arXiv2017

Minimising movements for the motion of discrete screw dislocations along glide directions

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $Γ$-convergence techniques.

math.AP↗
arXiv2016

Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

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 non-local energy $E_γ$ 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 approximation of the energy $E_γ$ in powers of $1/γ$ by $Γ$-convergence, in the limit $γ\to\infty$. While the zero-order term in the expansion, the $Γ$-limit of $E_γ$, captures the `bulk' profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer' energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

math.AP↗
arXiv2015

Dislocation microstructures and strain-gradient plasticity with one active slip plane

We study dislocation networks in the plane using the vectorial phase-field model introduced by Ortiz and coworkers, in the limit of small lattice spacing. We show that, in a scaling regime where the total length of the dislocations is large, the phase field model reduces to a simpler model of the strain-gradient type. The limiting model contains a term describing the three-dimensional elastic energy and a strain-gradient term describing the energy of the geometrically necessary dislocations, characterized by the tangential gradient of the slip. The energy density appearing in the strain-gradient term is determined by the solution of a cell problem, which depends on the line tension energy of dislocations. In the case of cubic crystals with isotropic elasticity our model shows that complex microstructures may form, in which dislocations with different Burgers vector and orientation react with each other to reduce the total self energy.

math.AP↗