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Wednesday, 6 April:
- 9:30 - 10:00: Registration
- 10:00 - 11:00: Patrick Dondl
We will consider a non-convex model of strain-gradient, single-crystal elasto-plasticity, where the non-convexity arises through the imposition of a hard ``single-plane" side condition on the plastic deformation. This well-posed variational model arises as the relaxation of a more fundamental ``single-slip" model in which at most one slip system can be activated at each spatial point. The relaxation procedure is motivated by the desire for efficient, oscillation-free simulation of single-crystal plasticity, but it is not immediately obvious how to implement the strict side-condition (that at most one slip-plane may be activated at each point) numerically. Our approach to this problem is to regularize the side-condition by introducing a large, but finite, cross-hardening penalty into the plastic energy. The regularized model is then amenable to implementation with finite-element methods, and, with the aid of div-curl arguments, one can show that it Gamma-converges to the single-plane model for large penalization. Finally we show some microstructures arising in the numerical implementation of this model.
- 11:00 - 11:30: break
- 11:30 - 12:30: Hanuš Seiner
The talk will summarize the recent understanding of the patter-formation mechanism in irreversibly formed NiTi shape memory alloys. It will be shown that the massively observed ‘plastic twinning’ can be interpreted as energy-minimizing coupling between the martensitic microstructure and the plastic slip. As a result, the whole pattern formation can be captured by a (relatively simple) two-parameter model in the framework of non-linear elasticity continuum theory. The model suggests that the plastic twins are in fact combinations of plastic kinks and reversible martensitic twins, which leads us to introducing a new term kwinks for them. It will be discussed why the kwinks tend to get oriented along specific crystallographic planes – most probably this comes from an interplay between bulk and surface energy, both being involved in the model.
Joint work with: Petr Sedlák, Petr Šittner, Orsloya Molnárová and Miroslav Frost (Prague).
- 12:30 - 14:00: lunch
- 14:00 - 15:00: Dorothee Knees
Macroscopic damage and failure phenomena of solids typically are the result of the accumulation of small cracks or defects on a microscopic scale. In engineering literature, various multi-scale or homogenized models are proposed in order to describe time-dependent damage phenomena with microscopic origins like the growth of micro-cracks or micro-voids. In this lecture, we discuss these approaches in the framework of homogenization and evolutionary Γ-convergence, allowing for micro-defects that may grow individually with respect to the time-dependent loadings. The lecture relies on joint work with Hauke Hanke (formerly WIAS Berlin).
- 15:00 - 15:30: break
- 15:30 - 16:30: John Ball
For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it seems necessary to impose a nondegeneracy condition on the constitutive equation for the stress. The talk will explain this, and show how in some cases the nondegeneracy condition can be proved using the monodromy group of a holomorphic map. This is joint work with Inna Capdeboscq and Yasemin Şengül.
- 16:30 - 17:30: Georg Dolzmann
Structures on small scales can be found in many elastic or plastic materials which are described by free energy densities which fail to be convex. In this lecture we present numerical schemes and analytical results for the identification of effective energies. This is joint work with Sergio Conti (Bonn).
- 17:30 - 18:00: poster presentation
Thursday, 7 April:
- 9:00 - 10:00: Tomonari Inamura (online)
Improving the compatibility of phases and hence reducing the introduction of plastic deformations during the phase transition is crucial to suppress the functional fatigue of shape memory alloys. We focus here on a new condition of supercompatibility between martensitic variants, called triplet condition (TC), that allows the formation of compatible three-fold microstructures. We show that in the case of cubic to orthorhombic transformations TC generalizes the cofactor conditions, conditions of supercompatibility between phases that empirically seem to influence reversibility, but much more difficult to achieve in the alloy design. We then analyse the martensite microstructure of a Ti-Ni based alloy, which closely satisfies TC, observing an unusual microstructure comprising 2-fold (ordinary twin), 3-fold (TC), and 4-fold (crossing twin) building blocks connected to each other. Transmission electron microscopy observation revealed that the accumulation of dislocation is drastically suppressed in the alloy. We finally discuss the possibility of the design of durable shape memory alloys based on TC.
- 10:00 - 11:00: Antonio Tribuzio
Motivated by the appearence of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of simplified N-well problems without gauge invariances with surface energy given by a singular higher-order term. We first consider the stress-free states to be given by the Tartar square, a prototypical model allowing for so-called "wild microstructures". We provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of an "infinite order of lamination". We then show how to exploit this bootstrap argument to find lower scaling-bounds related to specific models allowing for laminations of higher-orders. In these cases the upper bound is provided by iterated branching constructions. This is a joint work with Angkana Rüland.
- 11:00 - 11:30: break
- 11:30 - 12:30: Martin Kružík
We investigate variational problems in large-strain magnetoelasticity, both in the static and in the quasistatic setting. The model contemplates a mixed Eulerian-Lagrangian formulation: while deformations are defined on the reference configuration, magnetizations are defined on the deformed set in the actual space. In the static setting, we establish the existence of minimizers. In particular, we provide a compactness result for sequences of admissible states with equi-bounded energies which gives the convergence of the composition of magnetizations with deformations. In the quasistatic setting, we consider a notion of dissipation which is frame-indifferent and we show that the incremental minimization problem is solvable. Then, we propose a regularization of the model in the spirit of gradient polyconvexity and we prove the existence of energetic solutions for the regularized model. This is a joint work with M. Bresciani (Vienna) and E. Davoli (Vienna).
- 12:30 - 14:00: lunch
- 14:00 - 15:00: Benedikt Wirth (online)
It is well-known that point sources with sufficient mutual distance can be reconstructed exactly from finitely many Fourier measurements by solving a convex optimization problem with Tikhonov-regularization (this property is sometimes termed superresolution). In case of noisy measurements one can bound the reconstruction error in unbalanced Wasserstein distances or weak Sobolev-type norms. A natural question is to what other regularizers the phenomenon of superresolution extends. We show that also for anisotropic total variation, exact reconstruction results and L1 error bounds hold under sufficient mutual distance between the horizontal and vertical edges. This is joint work with Martin Holler.
- 15:00 - 15:30: break
- 15:30 - 16:30: Mariya Ptashnyk
Many biological tissues must be structured in such a way as to be able to adapt to two extreme biomechanical scenarios: they have to be strong to resist high pressure and mechanical forces and yet be flexible to allow large expansions and growth. A part of nature's solution to this intriguing problem are the complex microstructures and microscopic (cellular) processes, that modify tissue's elastic properties. To analyse the interplay between the mechanics, microstructure, and the chemistry we derive microscopic models for plant biomechanics, assuming that the elastic properties depend on the chemical processes and chemical reactions depend on the mechanical stresses. The microscopic models constitute strongly coupled systems of reaction-diffusion-convection equations for chemical processes and equations of elasticity or poroelasticity for elastic deformations. Multiplicative decomposition of the deformation gradient into elastic and growth parts is used to model growth of a biological tissue. To analyse the properties and behaviour of plant tissues, the macroscopic models are derived using homogenization techniques. In the multiscale analysis we distinguish between periodic and random distribution of cells in a plant tissue. Numerical solutions for macroscopic models demonstrate the impact of the microstructure on tissue deformations and growth.
- 16:30 - 17:30: poster session
- 19:30: conference dinner
Friday, 8 April:
- 9:00 - 10:00: Dennis Kochmann (online)
Ferroelectric ceramics show rich domain microstructures, in which the different domain variants tend to arrange in complex laminate-type patterns. Upon the application of electric and/or mechanical fields, those microstructural patterns evolve over time, giving rise to the macroscopically observable electro-mechanically coupled material behavior. We will discuss modeling strategies to simulate such processes with the overall objective of (i) predicting realistic microstructures and (ii) capturing the intricate kinetic evolution mechanisms. We will show that classical phase-field models are appropriate for (i), while new approaches are required for (ii).
- 10:00 - 11:00: Adriana Garroni
Inspired by a recent result of Lauteri and Luckhaus, with derive, via Gamma convergence, a surface tension model for polycrystals in dimension two. The starting point is a semi-discrete model accounting for the possibility of having crystal defects. The presence of defects is modelled by incompatible strain fields with quantised curl. In the limit as the lattice spacing tends to zero we obtain an energy for grain boundaries that depends on the relative angle of the orientations of the two neighbouring grains. The energy density is defined through an asymptotic cell problem formula. By means of the bounds obtained by Lauteri and Luckhaus we also show that the energy density exhibits a logarithmic behaviour for small angle grain boundaries in agreement with the classical Shockley Read formula. The talk is based on a paper in preparation in collaboration with Emanuele Spadaro.
- 11:00 - 11:30: break
- 11:30 - 12:30: Manuel Friedrich
In this talk, I present a quasistatic nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. The force balance is formulated in the reference configuration by resorting to the concept of nonsimple materials whereas the heat transfer equation is governed by the Fourier law in the deformed configurations. Weak solutions are obtained by means of a staggered in-time discretization where the deformation and the temperature are updated alternatingly. Our result refines a recent work by Mielke and Roubicek since our approximation does not require any regularization of the viscosity term. Afterwards, we focus on the case of deformations near the identity and small temperatures, and we show by a rigorous linearization procedure that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Based on joint work with Rufat Badal (Erlangen) and Martin Kružík (Prague).
You can find a pdf version of the schedule here.
