Special Session 124: Mathematical methods for heterogeneous media

Riesz fractional gradient functionals defined on partitions: Nonlocal-to-local variational limits

Stefano Almi University of Naples Federico II Italy Co-Author(s):    Maicol Caponi, Manuel Friedrich, Francesco Solombrino

Abstract: We address the asymptotics of functionals with linear growth depending on the Riesz s-fractional gradient on piecewise constant functions. We consider a general class of varying energy densities and we characterize their local limiting functionals in the sense of Gamma-convergence.

The effect of almost-periodic microstructures in phase separation

Riccardo Cristoferi Radboud University Netherlands Co-Author(s):    Lorenza D`Elia

Abstract: Many of the new media needed by, and sought for, in the applications are heterogeneous materials. Being able to describe and predict stable equilibrium configurations of phases is thus fundamental in order to take full advantages of these materials. In this talk we will work within the gradient theory of phase separations, namely using a van der Waals-Cahn-Hilliard type of energy. We consider the case where the material has a microstructures that can be described by an almost-periodic function. This is intended to model quasi-crystalline materials, whose properties are in between periodic and random materials. We will discuss the effect of these types of microstructures in the sharp interface limit of the energy. This work is in collaboration with Lorenza D`Elia (TU Wien).

Homogenization in magnetoelasticity under small elastic response

Lorenza D`Elia TU Wien Austria Co-Author(s):

Abstract: In this talk, we perform a simultaneous homogenization and linearization analysis for a magnetoelastic energy functional featuring a mixed Eulerian-Lagrangian structure. Neglecting Zeeman and anisotropic contributions, we characterize the asymptotic behaviour in the sense of $\Gamma$-convergence for the sum of a nonlinear magnetoelastic energy, a symmetric exchange term defined on the actual configuration, and for the associated magnetostatic self-energy. This is a joint work with M. Cherdantsec (Cardiff University), E. Davoli (TU Wien) and S. Ricc\`{o} (TU Wien).

Effective transmission through an interface with evolving microstructure

Lucas M Fix University of Augsburg Germany Co-Author(s):    Gianna Goetzmann, Malte A. Peter, Jan-F. Pietschmann

Abstract: We study the asymptotic behaviour of a system of nonlinear reaction--diffusion--advection equations in a domain consisting of two bulk regions connected via microscopic channels distributed within a thin membrane. Both the width of the channels and the thickness of the membrane are of order $\varepsilon \ll 1$, and the geometry evolves in time in an a priori known way. We consider nonlinear flux boundary conditions at the lateral boundaries of the channels and critical scaling of the diffusion inside the layer. Extending the method of homogenisation in domains with evolving microstructure to thin layers, we employ two-scale convergence and unfolding techniques to derive an effective model in the limit $\varepsilon \to 0$, in which the membrane is reduced to a lower-dimensional interface. We obtain jump conditions for the solution and the total fluxes, which involve the solutions of local, space--time-dependent cell problems in the reference channel.

Variational models for material voids

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

Abstract: In this talk, I present variational models for materials with voids in both geometrically linear and nonlinear elasticity. Specifically, we consider functionals featuring elastic bulk terms and a surface contribution proportional to the surface area of the voids within the material. We first discuss a relaxation result that leads to an additional surface term related to the phenomenon of collapsing voids. We then present an application to the derivation of effective homogenization limits for heterogeneous materials.

Relaxation of variational models for multiphase elastoplastic materials

Carolin Kreisbeck KU Eichst\"att-Ingolstadt Germany Co-Author(s):    Elisa Davoli, Samuele Ricco

Abstract: The coupling of phase transitions in shape memory alloys with plastic deformation induces intricate microstructure formation and gives rise to highly nonconvex energy functionals. In this talk, we study a class of variational models arising in single-slip finite crystal plasticity. In contrast to classical settings with a single elastic well, the elastic energy features two wells, leading to interactions between the slip system and the elastic phases under different compatibility constraints. To better understand the effective deformation behavior of these models, we analyze their relaxation, with focus on the quasiconvexification of the underlying extended-valued energy densities. After identifying the quasiconvex hull of the admissible deformation gradients, which shows that the effective states can vary within a range of volumetric responses, we derive bounds on the quasiconvex envelope from above and below. The upper bounds on the rank-one convex envelopes require careful constructions of suitable rank-one lines, while the established polyconvex lower bounds are generally smaller, but are shown to be optimal in a special case.

Variational analysis for the nonlinear elastic energy induced by edge dislocations: the dilute regime

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

Abstract: We introduce nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. We analyze the asymptotic behavior of the nonlinear elastic energy, as the lattice spacing (in the purely discrete model) and the core-radius (in the semi-discrete framework) vanish. We work within the dilute regime, corresponding to a finite number of effective singularities. Joint work with Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro, Marcello Ponsiglione.

Membranes in nonlocal hyperelasticity

Anastasia Molchanova TU Wien Austria Co-Author(s):    Dominik Engl, Hidde Sch\onberger

Abstract: Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, initially isotropic interaction ranges naturally become anisotropic, which leads us to develop a theory of anisotropic nonlocal gradients with direction-dependent interaction ranges. In contrast to existing nonlocal derivatives with finite horizon, which are typically defined via interaction kernels supported on balls of positive radius, our formulation employs ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework interpolating between fully nonlocal, partially nonlocal, and purely local models. Using these tools, we establish a $\Gamma$-convergence analysis for the associated nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered exactly when all interaction radii vanish. This talk is based on joint work with Dominik Engl and Hidde Sch\onberger.

Variational formulation of planar linearized elasticity with incompatible kinematic

Marco Morandotti Politecnico di Torino Italy Co-Author(s):    Pierluigi Cesana and Edoardo Fabbrini

Abstract: We address mechanical equilibrium in the planar strain regime for systems with incompatible kinematic from a variational standpoint. We show that, for non-simply connected domains, the equilibrium problem for a non-liftable strain-stress pair can be reformulated as a well-posed minimization problem for the Airy potential. We characterize the additional compatibility conditions on the internal boundaries of the domain as Volterra dislocations or disclination. Finally, we establish that the minimization problem for the Airy potential can be reduced to a finite-dimensional optimization involving cell formulas.

Dynamics of screened particles towards equispaced ground states and applications to misfit dislocations

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

Abstract: We analyse the dynamics - driven by the gradient flow of negative fractional seminorms - of empirical measures towards equispaced ground states. In particular, we prove the optimality of equispaced misfit dislocations at semi-coherent interfaces

Derivative-free descriptions of smoothness in variable exponent Sobolev spaces and generalized Orlicz-Sobolev spaces

Ana Margarida Ribeiro NOVA School of Science and Technology (NOVA FCT) Portugal Co-Author(s):    R. Ferreira, P. Hasto, C. Kreisbeck, E. Zappale

Abstract: Function spaces with spatially heterogeneous behaviour have become an important tool in modern analysis. In this talk, I will discuss derivative-free descriptions of smoothness in the framework of variable exponent and generalized Orlicz spaces. The approach considered has its origins in the work of Bourgain, Brezis and Mironescu [2002, J. Anal. Math.] where difference quotients were used to study classical Sobolev spaces. In the setting of variable exponent and generalized Orlicz spaces, difference quotients are no longer adequate. It will be shown that the averaging operator introduced by Diening and Hasto [2007, Studia Math.] to address the trace space appropriately captures the desired smoothness of functions in a variable exponent Sobolev space as well as in a generalized Orlicz-Sobolev space.

Sharp-interface limit for non-isothermal and nonlocal Modica-Mortola functionals

Emanuele Tasso TU Wien Austria Co-Author(s):    Elisa Davoli

Abstract: In this talk, we analyze a non-isothermal and nonlocal variant of the Modica-Mortola diffuse model for phase transitions. Here the classical gradient penalization is replaced by a nonlocal singular perturbation and the double-well potential is space-dependent. Our main result is the identification of the sharp-interface limit as the width of the transition layers converges to zero. This is joint work with Elisa Davoli.

Homogenization of elasto-plastic plate equations

Igor Velcic University of Zagreb, Faculty of Electrical Engineering and Computing Croatia Co-Author(s):    Marin Buzancic, Elisa Davoli, Josip Zubrinic

Abstract: Starting from $3d$ quasistatic evolution of perfect plasticity we discuss the model of $2d$ quasistatic evolution of perfect plasticity by doing simultaneous homogenization and dimension reduction. Depending on the ratio between the thickness of the plate and period of the oscillations of the material we obtain different models in the limit and discuss the effective plastic dissipation potential at the interface of the different phases of the material.