Special Session 126: Defects, Microstructures, and Failure: Multiscale Variational Models

Nonlocal aproximation of a Griffith-type energy

Stefano Almi University of Naples Federico II Italy Co-Author(s):    Elisa Davoli, Anna Kubin, Emanuele Tasso

Abstract: We prove that De Giorgi's conjecture for nonlocal approximation of free discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. For a suitable class of continuous finite difference functionals, we show the compactness of deformations with equibounded energies, as well as their Gamma convergence. The compactness (and closure) analysis builds on a Fr\`echet-Kolmogorov approach and a novel characterization of GSBD.

Homogenization of the stochastic double-porosity model.

Elise Bonhomme Universite de Bretagne Occidentale France Co-Author(s):    Mitia Duerinckx, Antoine Gloria

Abstract: In this talk, I will present joint work with Mitia Duerinckx and Antoine Gloria, in which we prove the homogenization of the so-called double-porosity model in a random setting where the resonant inclusions are not uniformly bounded nor separated. This mesoscopic model (used to describe flows in fractured porous media) is obtained as the limit of a diffusion process in a highly heterogeneous material made of two pure phases: a connected healthy phase (with conductivity of order 1) randomly perforated by a dense network of small inclusions of a second, nearly soft phase whose conductivity scales like the square of their size and tends to zero. In this specific regime, so-called resonance phenomena occur, in the sense that the homogenized model keeps memory of nontrivial interactions between the micro and macroscopic scales of the material.

Curvature-dependent cavitation energies

Marco Bresciani Johannes Kepler University Linz Austria Co-Author(s):    Carlos Mora-Corral, Fabian Rupp

Abstract: The modeling of cavitation in nonlinear elastostatics typically rests upon the minimization of an energy functional comprising a bulk term, accounting for strain concentration, and a surface term proportional to the area of the cavity boundaries. Motivated by studies on void stability and the search for a regularizing effect, we propose a variational model featuring a surface term that depends on the curvature of the cavity boundaries. The rigorous formulation employs oriented varifolds on the deformed configuration and is based on a suitable coupling between these varifolds and the underlying deformation. Our main findings ensure the well-posedness of the resulting free-discontinuity problem and identify first-order necessary conditions for minimality.

Energy concentration in a two-dimensional magnetic skyrmion model

Luca Briani Technical University of Munich Germany Co-Author(s):    Marco Cicalese, Leonard Kreutz

Abstract: In this talk, I present the formation of singularities in a chiral Skyrme-type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In the presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, I show how to establish a weak compactness result for the topological charge density and prove that it converges to an atomic measure with quantized weights. I characterize the $\Gamma$-limit of the energies as the total variation of this measure. Then, I consider the case of lattice-type energies and prove a corresponding compactness and $\Gamma$-convergence result. To this end, I will first carefully define a notion of discrete topological charge for $\mathbb{S}^2$ -valued maps. This is a joint work with Marco Cicalese and Leonard Kreutz.

A fractional approach to strain-gradient theory for plasticity: beyond core-radius of discrete dislocations

Maicol Caponi University of L`Aquila Italy Co-Author(s):

Abstract: In this talk we derive a strain-gradient theory for plasticity as the $\Gamma$-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido, Cueto, and Mora-Corral, we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order $1-\alpha$. As $\alpha$ goes to $0$, we show that suitably rescaled energies $\Gamma$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione. This is a joint work with Stefano Almi (University of Naples Federico II), Manuel Friedrich (Johannes Kepler University Linz), and Francesco Solombrino (University of Salento).

Variational phase-field models of cohesive fracture

Sergio Conti University of Bonn Germany Co-Author(s):

Abstract: Cohesive-zone models are free-discontinuity problems with a surface energy that converges to zero for small values of the jump, and to a positive, finite value for large values of the jump. Their approximation with a phase- field model requires an interaction between the damage variable and the strain that depends on the regularization parameter. I will discuss recent progress on the development of these approximations and on the study of the limiting model. The talk is based on joint work with Flaviana Iurlano and Matteo Focardi.

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

Vito Crismale Sapienza Universita di Roma Italy Co-Author(s):

Abstract: The talk concerns a phase field approximation for sharp interface energies, defined on partitions, as appropriate for modeling grain boundaries in polycrystals. The label takes value in $O(d)/G$, where $G$ is the point group of a lattice. The limiting surface energy behaves for small angles as $s|\log s|$, according to the Read and Shockley law. These functionals can be used for image reconstruction of grain boundaries. Joint work with S. Conti (HCM Bonn), A. Garroni, A. Malusa (Sapienza).

Gamma-convergence of the square sticky disk to the octagonal crystalline perimeter

Giacomo Del Nin Max Planck Institute Leipzig Germany Co-Author(s):

Abstract: We consider a variant of the sticky disk model for N interacting particles in the plane, where distances are evaluated by means of the supremum norm instead of the Euclidean norm. We show crystallization for minima of such an energy (for fixed N) and we prove Gamma-convergence (in the limit as N goes to infinity) of suitably rescaled energies to the anisotropic perimeter with octagonal Wulff shape. The key result to establish this is an energy decomposition for graphs in the plane that hinges upon the notion of angular defect, and that is potentially adaptable to other energies. The talk is based on joint work with Lucia De Luca (IAC-CNR).

Homogenisation of free discontinuity problems with cohesive type surface terms

Davide Donati SISSA, Trieste Italy Co-Author(s):    Gianni Dal Maso

Abstract: We study homogenisation problems for free discontinuity functionals depending on vector-valued functions under new hypotheses on the surface energies. We prove necessary and sufficient conditions for an homogenisation phenomenon to occur and show that these conditions are always satisfied under the usual hypotheses of stochastic homogenisation.

Quasi-static crack growth in cohesive-type fracture

Manuel Friedrich JKU Linz Austria Co-Author(s):

Abstract: In this talk, I present an existence result for a variational model of quasi-static crack growth for a class of cohesive fracture energies with an activation threshold. The evolution is characterized by an irreversibility condition in terms of a maximal opening variable, a global stability, and an energy balance.

Variational problems with nonlocal gradients: Heterogeneous horizons and local boundary conditions

Carolin Kreisbeck KU Eichst\"att-Ingolstadt Germany Co-Author(s):    Hidde Sch\onberger

Abstract: Motivated by peridynamic models capturing discontinuities and singularities in material behavior, and building on recent advances in nonlocal hyperelasticity, this talk studies a class of variational problems involving integral functionals with nonlocal gradients. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values. We will discuss properties of the associated Sobolev spaces, focusing in particular on the analysis of a trace operator and the proof of a Poincar\`e inequality. A central ingredient of our approach is to exploit connections with pseudo-differential operator theory. As an application, we show the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet-, Neumann- or mixed-type boundary conditions.

On quantitative properties of Griffith quasi-minimizers

Camille Labourie University of Lorraine France Co-Author(s):    Antoine Lemenant, Lorenzo Lamberti, Yana Teplitskaya

Abstract: Griffith quasiminimizers (in the sense of David and Semmes) are meant to represent minimizers of Griffith-type energies with possibly highly irregular coefficients. I will present recent regularity results such as uniform rectifiability, existence of espace curves, finiteness of traces and explain why no stronger regularity property can hold for all such quasiminimizers.

A variational approach to topological singularities through Mumford-Shah type functionals

Lucia De Luca IAC-CNR Roma Italy Co-Author(s):

Abstract: We will present variational approaches to the analysis of topological singularities in the plane, starting from the - nowadays - classical Ginzburg-Landau (GL) model and core-radius (CR) approach. We will introduce a third approach inspired by the Mumford-Shah functional used in the context of image segmentation. Within our framework, the order parameter is an $SBV$ map taking values in the unit sphere of the plane; the bulk energy is the squared $L^2$ norm of the approximate gradient whereas the penalization term is given by the length of the jump set, scaled by a small parameter. After providing a notion of Jacobian determinant for $SBV$ maps, we show that at any logarithmic scale our functional is ``variationally equivalent'' to the ``standard'' (CR) and (GL) models. Joint work with Vito Crismale, Nicolas Van Goethem and Riccardo Scala.

Lower energy scaling bounds of singular perturbation models for higher order laminates via Fourier-based localization methods

Lennart Machill Rheinische Friedrich-Wilhelms-Universit\\"{a}t Bonn Germany Co-Author(s):    Angkana R\uland, Antonio Tribuzio, Timo Hofmann

Abstract: Fourier-based localization methods can be used to obtain lower scaling bounds for singular perturbation models. To illustrate this approach, we begin with a three-well problem within the theory of geometrically linearized elasticity. We then consider settings with more wells and discuss the singular perturbed Tartar square, revisiting the results by R\uland and Tribuzio (2022). We also briefly compare the technique with the results by Chan and Conti (2015), which rely on real-space localization methods, and discuss the extent to which the Fourier-based approach can be applied in the geometrically nonlinear case. The talk is based on joint works with Angkana R\uland, Antonio Tribuzio, and Timo Hofmann.

Gamma-convergence for the plane-to-wrinkles transition problem

Roberta Marziani University of Siena Italy Co-Author(s):    Peter Bella

Abstract: Motivated by physical experiments, we consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet. We start with a description of the model and of some related results due to Bella and Kohn. We then perform an asymptotic analysis, via $\Gamma$-convergence, as the sheet thickness goes to zero. The limiting problem is scalar and convex, but constrained and posed for measures. We complement the analysis by proving existence of a minimizer for the limiting functional as well as equipartition of the energy.

Tuning chirality and the emergence of antiskyrmions

Christof Melcher RWTH Aachen University Germany Co-Author(s):

Abstract: We investigate the role of anisotropic chirality in chiral field theories and its impact on the stabilization and emergence of antiskyrmions. Extending classical models of chiral magnetism, we incorporate direction-dependent Dzyaloshinskii-Moriya interactions to capture anisotropic symmetry breaking. This framework reveals that anisotropy fundamentally alters the topological energy landscape, enabling the formation of antiskyrmionic textures alongside conventional skyrmions. We analyze the conditions under which antiskyrmions are energetically favorable, emphasizing the interplay between crystalline symmetry, chirality, and external fields. Our results provide a unified theoretical perspective consistent with recent experimental observations and prior work on micromagnetic energy minimization, highlighting anisotropic chirality as a key mechanism for engineering topological spin structures in magnetic materials.

Analysis of a Nonlocal Variational Model for Pattern Formation in Biomembranes

Anastasija Pesic WIAS Berlin Germany Co-Author(s):    Sara Daneri, Eris Runa

Abstract: In this talk, we present a variational model motivated by lipid raft formation in cellular membranes. The energy combines a short-range attractive interaction, modeled by a Modica--Mortola term, with a long-range repulsive interaction given by a Yukawa kernel. We study minimizers of this functional in a certain parameter regime, in which the competition between these two effects leads to pattern formation. We will present a new $\Gamma$-convergence result. This talk is based on a joint work with Sara Daneri and Eris Runa.

A notion of s-fractional mass for 1-currents in higher codimension

Marcello Ponsiglione Sapienza, University of Roma Italy Co-Author(s):

Abstract: We will introduce and discuss a notion of s-fractional mass for 1-currents, generalizing the s-fractional perimeter in the plane to higher codimension singularities. We will present basic compactness and density properties for such a notion, and discuss possible future directions of investigation

From discrete to continuum in the helical XY-model: emergence of chirality transitions.

Francesco SOLOMBRINO Universit\`a del Salento Italy Co-Author(s):    Marco Cicalese, Dario Reggiani, and Matthias Ruf

Abstract: We study the energy per particle of a ferromagnetic-anti-ferromagnetic frustrated spin chain with nearest and next-to-nearest interactions close to the Landau-Lifschitz point (where the helimagnetic-ferromagnetic transition occurs) , as the number of particles diverges. We rigorously prove the emergence of chiral ground states and we compute, by performing the $\Gamma$-limits of proper renormalizations and scalings, the energy for a chirality transition, if spins take value in the unit sphere $S^1$. Such a result does not hold if spins are $S^2$ valued, as in this case , as it is well established that in this case chirality transitions may emerge with vanishing energy. Inspired by recent work on the N-clock model, we consider a spin model where spins are constrained to a diverging number $k_n$ of copies of $S^1$ covering $S^2$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n$, below which the $\Gamma$-limit of the discrete energies capture chirality transitions while retaining an $S^2$-valued energy description in the continuum limit.

Energy of semi-coherent interfaces

Emanuele Spadaro University of Rome La Sapienza Italy Co-Author(s):    Valerio Corica

Abstract: In this talk I will discuss a variational model for the energy of inter-crystalline boundaries that arise from differences in atomic spacing. The functional is a variant of the well-known Lauteri-Luckhaus energy for the proof of the Read-Shockley formula for tilt grain boundaries. In the case of semi-coherent interfaces (i.e., with small misfit between the crystal lattices), we show that the interfacial energy scales super-linearly with the misfit parameter, thus justifying in a more general setting the results of van der Merwe.

Linearization of Quasistatic Fracture Evolution in Brittle Materials

Pascal Steinke Institute for Applied Mathematics, University of Bonn Germany Co-Author(s):    S. Conti, M. Friedrich, K. Stinson

Abstract: In this talk, we discuss a linearization result for quasistatic fracture evolution in nonlinear elasticity. As the stiffness of the material tends to infinity, we show that rescaled displacement fields and their associated crack sets converge to a solution of quasistatic crack growth in linear elasticity without any a priori assumptions on the geometry of the crack set. This result corresponds to the evolutionary counterpart of the static linearization result by M. Friedrich, where a Griffith model for nonsimple brittle materials has been considered featuring an elastic energy which also depends suitably on the second gradient of the deformations. The proof relies on a careful study of unilateral global minimality, as determined by the nonlinear evolutionary problem, and its linearization together with a variant of the jump transfer lemma in GSBD proven by M. Friedrich and F. Solombrino. This is a joint project with M. Friedrich (Linz, Austria) and K. Stinson (Salt Lake City, Utah).

Concentration-Compactness in Fracture Mechanics

Kerrek Stinson University of Utah USA Co-Author(s):    W. M. Feldman

Abstract: Motivated by variational models for fracture, we discuss a new proof of compactness for $GSBV^p$ functions without a priori bounds on the function itself. Our proof is based on the classical idea of concentration-compactness, making it quite intuitive. Further, so far as we are aware, this is the first time the connection to concentration-compactness has been made explicit for problems in fracture mechanics.

Multiscale analysis and homogenization of nonlocal thin films

Antonio Tribuzio University of Bonn Germany Co-Author(s):

Abstract: In this talk, we will introduce a nonlocal, variational model for thin films. We consider family of functionals of convolution-type defined on a thin domain of thickness $\gamma$ with the size of the most effective interactions between points being of order $\varepsilon$. After discussing the correct rescaling, we study the $\Gamma$-convergence of these energies as both parameters go to zero to a local integral functional defined on a lower dimensional domain. In the case of periodic homogenization, we can show that a separation of scales takes place. This is a work in collaboration with Nadia Ansini.

Measure-valued solutions for non-associative finite plasticity

Andreas Vikelis University of Vienna Austria Co-Author(s):    Ulisse Stefanelli

Abstract: The variational treatment of evolutionary nonassociative elasto-plasticity at finite strains remains unexplored. In this direction, following the concept of energetic solutions, we present an existence result for measure-valued so- lutions of the quasistatic evolution problem which are stable and balance the energy. In particular, we apply a modification of the standard time- discretization scheme, considering Young measures generated by piecewise constant interpolants of time-discrete solutions of a properly defined mini- mization problem. A key point in our analysis is the limit passage in the dissipation. The later calls for time-continuity properties of the stresses which are not expected in the quasistatic framework. To overcome this ob- stacle we introduce a regularization of the generalized stress in the definition of our energetic solutions.

Dimension reduction in dynamical continuum mechanics

Riccardo Voso UTIA, Czech Academy of Sciences Czech Rep Co-Author(s):    Martin Kruzik

Abstract: In this talk, we discuss the derivation of a membrane model for hyper viscoelastic materials with inertia, starting from the full three-dimensional theory. At first, the hyperbolic minimizing movements scheme is introduced. This is a two time scale variational approximation which accounts for the reformulation of the problem in terms of incremental minimization of a energy-dissipation functional, where inertia is discretized in time. Within this framework, the asymptotic analysis as the thickness of the body vanishes is carried out via a $\Gamma$ convergence argument at the level of the functional and, subsequently, the continuous in time dynamics is recovered.

The random fractional obstacle problem

Caterina Ida Zeppieri University of Muenster Germany Co-Author(s):    Francesco Deangelis, Matteo Focardi

Abstract: Nonlocal energies, such as fractional Sobolev seminorms, arise naturally in mathematical models involving long-range interactions. In this talk, we study minimizers of such energies that vanish on a collection of small balls with random centers and radii, leading to a bilateral (fractional) obstacle problem. We will present a homogenization result that holds under minimal assumptions on the distribution and size of the obstacles, which are generated by a stationary marked point process. In particular, the obstacles may overlap and form clusters with positive probability, giving rise to a complex microstructure. Our analysis identifies the limiting energy and shows how it reflects the underlying probability distribution of the obstacles.