## Abstracts of the minicourses

**Amit Acharya**(April 13-17)

*Continuum mechanical modeling of the dynamics of dislocations*

This mini-course will comprise five 2-hour lectures on the development of a model for the dynamics of dislocations, viewed as a theory of continuum mechanics applicable to atomic and tectonic scale phenomena. Selected topics from classical dislocation theory will be reviewed. Modern developments resulting in a time-dependent pde based model of dislocation dynamics will be described. The model presents open questions for mathematical analysis of new types of pattern-forming equations. Conjectures and open questions related to the development of a physically and mathematically rigorous macroscopic theory of time-dependent plasticity based on the dislocation model will also be laid out.

Outline:

I. Physical basis, small deformation compatibility - H vs. E (displacement 'gradient', vs. strain).

II. Static stress of dislocation fields - Willis equations vs. Kroner Stress function method; comparison.

III. Eshelby's analysis of force on a defect, the Peierls model.

IV. Field Dislocation Mechanics - small deformation theory and results.

V. Field Dislocation Mechanics - finite deformation theory, links to macroscopic plasticity through averaging.

**Roberto Alicandro**(April 27-30)

*Metastability and dynamics of discrete topological singularities*

In this course I will describe a variational approach to the study of a large class of discrete energies accounting for defects, including the elastic energy in crystals with screw dislocations and the energy of XY spin systems with vortices. Based on this analysis, I will describe the phenomenon of pinning of discrete topological singularities, and I will propose a variational approach to describe their depinning and dynamics. More precisely, I will show the existence of many local minimizers that pin the dynamics and I will introduce a discrete in time variational dynamics, which allows to overcome the energy barriers and leads to a rigorous limit evolution.

**Andrea Braides**(April 20-24)

*Topics in the passage discrete-to-continuum for lattice systems*

I will illustrate some results in the passage from energy-driven lattice systems to continuum problems,

mainly using tools from homogenization theory and Gamma-convergence. Time allowing, I will

illustrate the following topics:

- representation results for bulk limit energies. Homogenization formulas and the Cauchy-Born rule.
- optimal design problems for networks
- theories of discrete thin films
- homogenization of defects
- interfacial energies on lattices
- limits of Lennard-Jones systems
- minimizing movements for discrete systems. Pinning and depinning.

**Marco Cicalese**(April 20-24)

*Variational analysis of classical spin systems with discrete and continuous symmetries*

I will give an introduction to the variational analysis of the atomistic-to-continuum limit of lattice spin systems.

The lectures will be mainly focused on the following topics:

- scalar and vectorial Ginzburg-Landau functionals;
- the xy spin system and its equivalence to the screw dislocation model at different scalings;
- frustrated spin systems with S^1 and S^2 symmetry: renormalization and chirality transitions.

**Sergio Conti**(March 9-13)

*Relaxation of variational models in plasticity and nonlinear elasticity*

Crystal plasticity can be studied, under suitable monotonicity assumptions, by variational models which have a similar form as in the theory of elasticity. The presence of several slip systems leads to the spontaneous formation of fine-scale oscillations, related to the physically known phenomena of shear banding and geometric softening. Mathematically, this corresponds to the lack of lower semicontinuity of the functionals.

These lectures shall discuss the theory of relaxation and its relation with the computation of quasiconvex envelopes of the energy density.

I shall start with the classical theory, discuss applications to concrete physical problems and recent analytical developments which permit to account for constraints on the determinant. Then I will address relaxation of functionals defined on one-rectifiable measures, which are appropriate for describing dislocations in crystals.

List of references.

**Gilles Francfort**(May 4-8)

*A few questions in small strain elasto-plasticity: heterogeneity, conditions for the existence of plastic slips, and examples of uniqueness and non-uniqueness.*

After briefly reviewing the basic mathematical theory of elasto-plasticity, I will address the issue of interfaces, that is the interplay between heterogeneities and plastic dissipation.

I will briefly evoke the problem of periodic homogenization in elasto-plasticity without going through the entire homogenization process.

Then, I will investigate the special setting of Von Mises elasto-plasticity. In such a restricted setting, the existence of slips can be analyzed much more precisely and specific conditions that ban plastic slips can be formulated.

Similarly, uniqueness results can be obtained, albeit for specific geometries.

**Adriana Garroni**(February 23-27)

*Multi-scale analysis for dislocations: the phase field approximation of the line tension*

Dislocations are line defects in crystals, that are considered the main mechanism for plasticity. They are interesting at different scales, the microscopic scale at which it is relevant to formulate manageable discrete models, the mesoscopic scale at which these lines carry an energy and interact, and the macroscopic scale at which one should be able to encode the presence of many dislocations in effective models for plasticity. The main goal in the mathematical study of models for dislocations is bridging these scales.

I will give an introduction to dislocations, with an overview of the classical variational model considered in the literature. I will particularly focus on a phase field model, a generalization of the so-called Peierls-Nabarro model, and on the analysis of its sharp interface limit, which corresponds to the line tension energy in the mesoscopic scale.

**Alessandro Giacomini**(May 4-8)

*SBV approach to shape optimization problems with Robin boundary conditions*

I will present some results obtained in collaboration with Dorin Bucur (Université de la Savoie) concerning a variational approach to the existence of optimal solutions for shape optimization problems under Robin conditions at the boundary.

More precisely, a large family of domains is proposed on which such problems are well posed in a way that the extended problem can be considered a relaxed version of the corresponding one on regular domains. The key point of the approach is the study of the regularity properties of minimizers of free discontinuity problems with surface energy of unusual type.

**Cyril Imbert**(March 9-13)

*H*

*omogenization results for some models of dislocation dynamics*

These lectures are focused on the homogenization of some models of dislocation dynamics. Mainly two models are considered: a system of "particles" for the propagation of parallel dislocation lines in a slip plane in a periodic potential, and a fully overdamped Frenkel-Kontorova model. The goal is to describe the dynamics of the density of dislocation lines, that is to say the dynamics at a "macroscopic" scale. From a mathematical point of view, this amounts to solve a singular perturbation problem.

Tentative plan of the course:

Lecture I: presentation of models

Lecture II: viscosity solutions and comparison principles

Lecture III: homogenization of Hamilton-Jacobi equations

Lecture IV: proofs of convergence

Lecture V: constructions of correctors

**Giovanni Leoni**(May 11-15)

*Continuum models for dislocations*

In this mini-course I will give a brief introduction to the physics of dislocations, describe Volterra's dislocations, and study them from a variational point of view in the setting of finite elasticity. I will address both the static and dynamic case.

**Alexander Mielke**(March 16-20)

*Mathematical approaches to finite-strain elastoplasticity*

We will discuss the mathematical modeling and the analysis of finite-strain elastoplasticity based on the multiplicative decomposition of the deformation gradient in an elastic part and a plastic part. We will use the energetic description in terms of energy-storage functional and a dissipation potential, which clears up many confusions concerning the relevant stress for the plastic flow rule.

In the rate-independent case we introduce the dissipation distance on the Lie group of plastic tensors and will be led to the concept of energetic solution as limit of incremental minimization.

In the middle part of the lecture series we discuss the general construction of energetic solutions (also known as quasistatic evolutions) in the general topological setting without Banach space structure.

Finally, we apply the abstract theory to obtain existence results for finite-strain elastoplasticity including a regularization via the gradient of the plastic tensor. As fundamental tools we use polyconvexity and multiplicative bounds for suitable stresses in terms of the energy.

If time permits, we discuss more recent results (with Riccarda Rossi and Giuseppe Savare) on finite-strain viscoplasticity.

References

A. Mainik and A. Mielke. Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci., 19(3): 221-248, 2009.

A. Mielke. Formulation of thermoelastic dissipative material behavior using GENERIC. Contin. Mech. Thermodyn., 23(3): 233-256, 2011.

A. Mielke. Differential, energetic, and metric formulations for rate-independent processes. Chapter 3 in "S. Bianchini, E.A. Carlen, A. Mielke, C. Villani: Nonlinear PDE's and Application, LNM 2028, Springer 2011", pages 87-170.

**Marcello Ponsiglione**(March 16-20)

*Gamma-convergence analysis for dislocations and vortices within the core radius approach*

In this course I will describe the core radius approach to topological singularities in two dimensions. This, together with the Ginzburg-Landau approach, represents a canonical formalism to cut off the infinite logarithmic tail of the energy, induced by topological singularities.

First, I will discuss the scalar cases of vortices and screw dislocations in crystals, drawing a parallel between the core radius approach and the Ginzburg-Landau model. Then, I will describe the vectorial case of edge dislocations.

The main goal is to develop a Gamma-convergence analysis of these models as the length-scale parameter (representing the lattice spacing, in the dislocations' case) tends to zero.

**Tomas Roubicek**(March 2-6)

*Plasticity at small strains combined with damage*

Coupling of plasticity with damage allows for modelling many complex processes occurring in solid continuum mechanics and physics, in contrast to mere plasticity or mere damage.

First, a quasistatic model of linearized plasticity with hardening at small strains combined with gradient damage will be presented in its basic scenario with unidirectional damage and in the fully rate-independent setting. Various concepts of weak solutions will be discussed, ranging from the concept of energetic (i.e., in particular, energy conserving) solutions to stress-driven local solutions, and also the maximal-dissipation principle will be discussed.

Further, some variants of this basic model amenable to rigorous analysis will be exposed. In particular a rate-dependent damage allowing possibly also healing, and plasticity possibly without hardening and with damageable yield stress. This variant needs the concept of 2nd-grade non-simple materials and allows e.g. for modelling of thin shearbands surrounded by a wider damage zone. An "opposite" variant is rate-dependent plasticity but damage again rate independent and unidirectional, which allows for energy conservation and in particular for extension towards anisothermal processes, and possibly also combination with some other rate-dependent processes like various phase transformations or diffusion of some "fluidic" medium with wide applications covering e.g. heat/moisture transport in concrete or rocks, or a metal/hybrid transformation under diffusion of hydrogen, etc.

A combination of these options leads to rate-dependent plasticity and damage which, in the anisothermal situation, allows us to imitate a popular rate-and-state dependent friction model devised in geophysics by Dieterich and Ruina for tectonic earthquakes modelling or (when augmented by porosity evolution flow rule) may cover some used models for poroelastic rocks undergoing damage.

Eventually, an interfacial variants of the above bulk damage/plasticity models allows for modelling various adhesive contacts, in particular delamination processes sensitive to mode of delamination (exhibiting more dissipation in shear mode than in pure opening mode).

**Lucia Scardia**(May 11-15)

*Multiscale problems in dislocation theory*

One of the hard open problems in mechanical engineering is the upscaling of large numbers of dislocations.

In this course I will discuss the derivation of continuum dislocation density models and strain-gradient plasticity models from semi-discrete dislocations theories. The derivation is done via Gamma-convergence. An important tool is a generalisation of the rigidity estimate of Friesecke, James & Mueller to fields that are not gradients.

I will also discuss progress and open problems in modelling dislocation interactions.

This is based on work in collaboration with Marc Geers, Maria Giovanna Mora, Stefan Mueller, Ron Peerlings, Mark Peletier and Caterina Zeppieri.

**Ulisse Stefanelli**(March 23-27)

*Gamma-convergence for rate-independent systems and linearization in finite plasticity.*

This series of lectures will be focusing on evolutive Gamma-convergence for rate-independent systems. I shall present the general abstract theory from [1] and apply it to the rigorous discussion on the small-deformation limit in plasticity [2]. Precisely, we will prove that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

If time permits, I will present an existence and linearization theory for an isotropic elasto-plastic model based on the positive definite plastic metric tensor [3].

[1] A. Mielke, T. Roubícek, U. Stefanelli. Gamma-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 3:387-416

[2] A. Mielke, U. Stefanelli. Linearized plasticity is the evolutionary Gamma-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15 (2013) 3:923-948.

[3] D. Grandi, U. Stefanelli. Finite plasticity based on plastic metric tensor: global existence and linearization. In preparation, 2015.

## Abstracts of the contributed talks

**Marco Artina**(March 20)

*A robust and efficient anisotropic mesh adaptation strategy to numerically simulate quasi-static crack propagation in brittle materials*.

The numerical simulation of quasi-static fracturing of brittle material, where no pre-defined crack path is imposed, is a challenging problem. In particular, we deal with the Francfort-Marigo model which requires the minimization of the well-known nonconvex and nonsmooth Mumford-Shah functional. To deal with a smoother functional which eases the minimization process of the energy, we consider the Γ-approximation of the Mumford-Shah functional via the Ambrosio-Tortorelli functional. Moreover, from a modeling viewpoint, we in- duce the quasi-static crack evolution via an applied displacement slowly chang- ing in time.

It is a well-established fact that the numerical simulation is often biased by an improper numerical discretization, with potential non-physical results. In particular, a non-optimal or a fixed computational mesh can affect the fracture evolution, driving it along non-realistic directions, like the edges of the triangulation. This issue can be successfully tackled by resorting to anisotropic adapted meshes, which have been successfully employed to model phenomena exhibiting strongly directional features.

To drive mesh adaptation, we elaborated an a-posteriori anisotropic error estimator for the proper discretization of the energy functional. Through anisotropic meshes we are able to locate, size and orient the mesh elements in order to follow the intrinsic directionalities of the solution at hand. Thanks to this grid adaptation process, we obtained solutions reliable from a physical viewpoint and with a relatively small computational cost.

In this talk, we introduce the Francfort-Marigo model and its Γ-approximation in case of plane and antiplane displacements. Then, we focus on the anisotropic mesh adaptation strategy. We finally assess the reliability, robustness, and efficiency of the proposed approach on some benchmark tests.

Joint work with Massimo Fornasier, Stefano Micheletti, and Simona Perotto.

**Jean-François**

**Babadjian**(April 15)

*Elasto-plasticity models in soil mechanics.*

This talk will present elasto-plasticity models arising in soil mechanics. Contrary to the typical models mainly used for metals, it is required here to take into account plastic dilatancy due to the sensitivity of granular materials to hydrostatic pressure. The yield criterion thus depends on the mean stress, and the elasticity domain is unbounded and not invariant in the direction of hydrostatic matrices. In the mechanical literature, so-called cap models have been introduced, where the elasticity domain is cut in the direction of hydrostatic stresses by means of a strain-hardening yield surface, called a cap. Well-posedness results of such models in dynamical and quasi-static regimes will be presented, as well as an asymptotic analysis as the cap is moved to infinity. It enables one to recover solutions to the uncapped model of perfect elasto-plasticity.

Finally, a generalization to non-associative flow rules, where plastic flow directions are not necessarily normal to the elasticity set, will be presented.

**Mattia Bongini**(March 19)

*Mean-field Pontryagin maximum principle.*

The mean-field approach is a powerful tool for giving sense to the notion that a discrete dynamical system converges to a continuous one as the number of agents increases. This technique has been recently used in connection with Gamma-convergence to show that a two-populations discrete optimal control problem converges to an ODE-PDE constrainted optimal control problem: in this model, a discrete population of agents (called "leaders") interacts with a continuous one (the mass of "followers"), and its target is to steer the entire population towards configurations minimizing a given cost. In this paper we address the problem of deriving optimality conditions for this optimal control problem. We show that these optimality conditions can be seen as the mean-field limit

*N—>∞*of the Pontryagin Maximum Principle applied to the two-population discrete system with

*N*followers. This, in turn, enables us a constructive method to derive solutions as limits of solutions of the discrete optimality conditions, which in the end let us establish existence results for these kinds of Hamiltonian systems. Finally, we prove that the resulting optimality conditions are indeed Hamiltonian flows in the Wasserstein space of probability measures.

**Manuel Friedrich**(March 24)

*A quantitative geometric rigidity result in SBD and the derivation of linearized models from nonlinear Griffith energies.*

We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame-indifferent energies in brittle fracture via Gamma-convergence. The convergence is given in terms of rescaled configurations measuring the displacement of the deformations from piecewise rigid motions which are constant on each connected component of the cracked body. The key ingredient to establish a compactness result is a quantitative geometric rigidity result for special functions of bounded deformation. This estimate generalizes the result of Friesecke, James, Mueller in nonlinear elasticity theory and the piecewise rigidity result of Chambolle, Giacomini, Ponsiglione for SBV functions which do not store elastic energy.

**Gurgen (Greg) Hayrapetyan**(March 4)

*Stability and Evolution of Bilayer Interfaces in Amphiphilic Systems.*

Functionalized energies, such as the Functionalized Cahn-Hilliard, model phase separation in amphiphilic systems, in which interface production is limited only by competition for surfactant phase, which wets the interface. Consequently single-layer interfaces, connecting phase A to a phase B, are typically not global minimizers, and typically bifurcate to either bilayer structures in which a thin, codimension one morphology of phase B interpenetrates phase A or lead to higher co-dimensional, pore and micelle like morphologies. This is in contrast to classical phase-separating energies, such as the Cahn-Hilliard, in which interfacial area is energetically penalized.

We discuss the stability and evolution of bilayer interfaces in the Functionalized Cahn-Hilliard equation, including the onset of pearling bifurcations which lead to development of pore dominated network morphologies.

**Peter Hornung**(April 23)

*Some remarks about intrinsically strained plates in nonlinear elasticity*.

In this talk, we make some remarks about stationary points of generalised nonlinear bending theories for plates, which model intrinsically strained plates (such as plant leaves) and shells.

**Flaviana Iurlano**(May 13)

*An integral representation result in dimension*2

*for functionals defined on*SBD^p

*.*

The set

*SBD^p(*

*Ω*

*)*is defined as the space of special functions of bounded deformation whose strain is

*p*-th power integrable and whose jump set has finite measure. In dimension

*2*we show that lower semicontinuous functionals defined on

*SBD^p(*

*Ω*

*)*admit an integral representation with Caratheodory integrands, under some growth and continuity conditions.

**Martin Jesenko**(March 26)

*Closure and commutability results for Γ-limits.*

Under a suitable notion of equivalence of integral densities we prove a Γ-closure theorem for integral functionals with standard p-growth: The limit of a sequence of Γ-convergent families is again a Γ-convergent family. Its Γ-limit can be recovered from Γ-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute. We then also address the case

*p=1*with its difficulties.

This is a joint work with Bernd Schmidt (Augsburg University).

**Carolin Kreisbeck**(May 14)

*Homogenization of layered materials with rigid components in single-slip finite plasticity.*

This talk reports on first progress toward a better quantitative understanding of the effective behavior of polycrystalline solids in the framework of geometrically nonlinear plasticity. More precisely, we study a variational model for plastic material composed of fine parallel layers of two types. While one component is completely rigid in the sense that it does not admit any plastic deformation but only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, explicit homogenization formulas are determined by means of Gamma-convergence. Owing to the anisotropic nature of the problem, the findings depend critically on the orientation of the layers relative to the slip direction, leading to three qualitatively different regimes. In particular, one observes macroscopic shearing and blocking effects. The technical difficulties are rooted in the intrinsic rigidity resulting from the layered geometry, which calls for new rigidity estimates as well as a careful analysis of the admissible microstructures restricted by a differential inclusion.

This is joint work with Fabian Christowiak (Universität Regensburg).

**Matthias Ruf**(March 11)

*Interfacial energies on stochastic lattices.*

We present some results on the variational limit of discrete interfacial energies defined on scaled disordered lattices. Under suitable geometric constraints, every Gamma-limit (as the lattice spacing tends to zero) can be represented as a surface integral. If the lattice is given by a random stationary point process we prove a stochastic homogenization result.

The results are joint work with Roberto Alicandro and Marco Cicalese.

**Riccardo Scala**(February 26)

*A closeness result for the class of deformations in the presence of dislocations.*

We consider the energy of a single crystal with dislocations, and want to minimize it among a suitably large class of deformations and dislocations satisfying some boundary condition. Assuming the energy depending on the crystal deformation via a polyconvex function and on the dislocations density with some coerciveness conditions, we prove a closeness and compactness result for a big class of competitors, by studying the graphs of the deformations seen as 3-currents in

*ΩxR^3*.

**Igor Velcic**(May 5)

*Non-periodic homogenization and dimensional reduction in non-linear elasticity in small strain regimes on the example of bending rod.*

We will discuss the derivation of the rod model in the bending regime by simultaneous homogenization and dimensional reduction without periodicity assumption. We show kind of stability result for the equations, i.e., in the limit we always obtain quadratic energy density in the standard strain for this regime. This kind of stability is not valid for the plate equations in the bending regime, while it is valid for the plate equations in more linear regime (von Karman). The approach requires slight variation of standard Gamma-convergence techniques as well as the standard dimension reduction techniques based on the theorem on geometric rigidity.