Die Funktionsweise vieler Komponenten in moderneren Geräten beruht auf spezifischen Eigenschaften so genannter multifunktionaler Materialien. Diese Materialien zeichnen sich dadurch aus, dass sie wenigstens zwei der folgenden Eigenschaften besitzen: elastische Verformbarkeit, thermische Ausdehnbarkeit, Magnetisierbarkeit oder Polarisierbarkeit. Wichtig ist, dass diese Eigenschaften auf nichttriviale Weise miteinander wechselwirken. Das heißt, eine Anregung der einen Eigenschaft (z.B. mechanische Spannungen, äußeres Magnetfeld) ruft eine Reaktion der anderen Eigenschaften hervor. Dies ist zum Beispiel in PiezoKristallen der Fall, wo Deformationen elektrische Ströme erzeugen.
Ein typisches Merkmal sowohl multifunktionaler Materialien als auch elastoplastischer Materialien ist das Auftreten von Hysterese. Hysterese bedeutet, dass die Reaktion mit einer gewissen Verzögerung erfolgt, ganz gleich wie langsam sich die Anregung ändert. Dies kann am besten anhand zyklischer Belastungen dargestellt werden, siehe die folgende Graphik. Die umschlossenen Bereiche in den SpannungsDehnungsdiagrammen entsprechen der Energie, die in einem Zyklus dissipiert wird.
Hysteresekurven. Links: Superelastische Hysterese in Formgedächtnismaterialien; Rechts: Plastische Hysterese.
Elastoplastische Materialien und viele multifunktionale Materialien können als Materialien mit inneren Variablen modelliert werden. Diese Variablen charakterisieren die innere Struktur des Materials und umfassen beispielsweise die Magnetisierung, plastische Verzerrungen oder Informationen über Phasenanteile bei verschiedenen Phasen. Des Weiteren kann die Mikrostruktur, die in einem makroskopischen Punkt auftritt, durch innere Variablen erfasst werden. Auch Hybridsysteme sind hierbei von Interesse, bei denen gewisse Effekte nur durch die Wechselwirkung von Volumen und Grenzflächenvariablen entstehen. Solche Modelle finden sich z.B. in geophysikalischen Prozessen, die durch Kopplung von elastischen Verformungen und Reibungseffekten auf verschiedensten Orts und Zeitskalen entstehen.
Die Teilprozesse, bei denen Hysterese auftritt, zeigen häufig, zumindest näherungsweise, ein ratenunabhängiges EingangsAusgangsVerhalten. Dies bedeutet, dass eine Umskalierung der Eingangsgröße des Teilprozesses bezüglich der Zeit zu einer gleichartigen Umskalierung der Ausgangsgröße des Prozesses führt.
Am WIAS werden derartige (Teil)Prozesse mit zwei Ansätzen modelliert:
 Zum einen versucht man, unter Verwendung der Eingangsgröße, eine ratenunabhängige quasistatische Evolutionsgleichung zu formulieren, so dass die Ausgangsgröße aus der zugehörigen Lösung bestimmen werden kann. Wird diese Strategie zur Beschreibung komplexer hysteresischer Zusammenhänge verwendet, dann müssen eventuell zusätzliche neue innere Variablen zur Beschreibung des momentanen Zustandes eingeführt werden.
 Zum anderen können in vielen Fällen HystereseOperatoren zur Beschreibung des EingangsAusgangsVerhaltens des Teilprozesses gefunden werden, die nur die ursprünglichen Variablen verwenden. Diese Strategie vermeidet die zusätzlichen inneren Variablen in der Formulierung des Lösungsbegriffs, führt aber zu einem Gedächtnis, das heißt, dass der Zustand zu einer Zeit von der gesamten Vorgeschichte abhängt.
Modellierung mit ratenunabhängigen quasistatischen Evolutionsgleichungen
Die Entwicklungen im Material werden durch eine Evolutionsgleichung mit einem Energiefunktional und einem Dissipationspotential beschrieben. Dabei können auch Situationen mit mehreren möglichen Entwicklungen behandelt werden. Das Energiefunktional ist typischerweise ein Integral über eine Energiedichte, die die im Körper gespeicherte Energie erfasst, plus ein weiterer Term, der die Arbeit der zeitabhängigen äußeren Kräfte enthält. Das Dissipationspotential hängt von den inneren Variablen und ihren Zeitableitungen ab. Da Hysterese vor allem im Fall sich langsam ändernder Kräfte eine besondere Rolle spielt, ist es sinnvoll, die Modellierung auf den ratenunabhängigen Limes einzuschränken. In diesem Fall ist das Dissipationspotential positiv homogen vom Grad eins bezüglich der Zeitableitungen der inneren Variablen, und die Evolutionsgleichung ist ein so genanntes ratenunabhängiges System. Da die Lösungen in diesem Fall zeitlich unstetig sein können, wurde in den letzten Jahren die so genannte energetische Formulierung für ratenunabhängige Prozesse entwickelt. Der Vorteil des energetischen Zugangs ist, dass keinerlei Glattheit für die Daten, die Energie, die Dissipation und die Lösungen vorausgesetzt werden muss. Des Weiteren braucht der Zustandsraum keine lineare Struktur zu besitzen. Dies ist insbesondere wichtig bei der Modellierung von Mikrostrukturevolution.
Modellierung mit HystereseOperatoren
Zur Modellierung von ratenunabhängigen Zusammenhängen werden alternativ die so genannten HystereseOperatoren verwendet. Diese bilden zeitabhängige InputFunktionen (also die Eingangsgröße) auf zeitabhängige OutputFunktionen (also die Ausgangsgröße) ab. Hierbei gilt:
 Der Operator ist ratenunabhängig: Die OutputFunktion, die aus einer zeitlich transformierten InputFunktion folgt, ist identisch mit der entsprechend zeitlich transformierten OutputFunktion zur ursprünglichen InputFunktion.
 Der Operator ist kausal: Für einen beliebig gegebenen Zeitpunkt hängt der Wert der OutputFunktion nur von den Werten der InputFunktion bis zu diesem Zeitpunkt ab.
Bei der Modellierung elektromagnetomechanischer Komponenten werden HystereseOperatoren verwendet. Für diese sind dann die Parameter aus Messungen herzuleiten, und daher mit gewissen Unsicherheiten behaftet. Daher werden jetzt die Methoden der Uncertainty Quantification auf derartige Modelle angewendet, siehe z.B. Klein 2016.
Höhepunkte
Modellierung mit ratenunabhängigen quasistatischen Evolutionsgleichungen
In den letzten Jahren wurden im Bereich der Materialmodellierung elastoplastische Systeme mit inneren Variablen untersucht, und für die rateunabhängigen Systeme eine neue Existenztheorie auf Basis der abstrakten ratenunabhängigen Systeme entwickelt. Die sich bei der Berücksichtigung allgemeiner Materialmodelle ergebenden gekoppelten Systeme ratenunabhängiger und ratenabhängiger Gleichungen wurden mittels variationeller Methoden (inkrementelle Minimierung) studiert. Dabei zeigte es sich, dass die für ratenunabhängige Systeme im Zusammenhang mit den energetischen Lösungen entwickelten Methoden in vielen Fällen flexibel auf allgemeinere Situationen angewendet werden können.
Insbesondere konnten erstmals Existenz und numerische Approximationssätze für mehrdimensionale Systeme mit großen Deformationen für die Elastoplastizität und für Formgedächtnislegierungen hergeleitet werden, siehe die Forschungsmonographie Mielke, Roubíček 2015. Für Modelle bei kleinen Deformationen konnten diverse Schädigungsprozesse und in gewissen Fällen deren Kopplung mit thermischen Effekten behandelt werden, siehe z.B. Lazzaroni, Rossi, Thomas, Toader 2016.
Modellierung mit HystereseOperatoren
Zur Anwendung von HystereseOperatoren auf Materialmodelle wurden in den letzten Jahren die folgenden Resultate erzielt:
 Im Zusammenhang mit der Modellierung von Vorgängen in HerzMuskeln wurde eine partielle Differentialgleichung mit einem HystereseOperator hergeleitet. Für diese konnte gezeigt werden, dass eine periodische Anregung zu einer nichttrivialen periodischen Lösung führt, siehe Seidman, Klein 2013.
 Es wurde für alle Erweiterungen des sogenannten sweeping process auf InputFunktionen mit beschränkter Variation bewiesen, dass die Erweiterung nicht stetig ist, wenn man den Urbildraum mit der strikten Topologie betrachtet und den Bildraum mit der L^{1}Topologie, siehe Klein, Recupero 2016.
Ein weiteres anwendungsorientiertes Forschungsthema mit HystereseEffekten
 HystereseEffekte in Batterien und anderen Energiespeichersystemen werden im anwendungsorientierten Forschungsthema Phasenübergänge und Hysterese im Zusammenhang mit Speicherproblemen untersucht.
Publikationen
Monografien

O. Marquardt, V.M. Kaganer, P. Corfdir, Chapter 12: Nanowires, in: Vol. 1 of Handbook of Optoelectronic Device Modeling and Simulations: Fundamentals, Materials, Nanostructures, LEDs, and Amplifiers, J. Piprek, ed., Series in Optics and Optoelectronics, CRC Press, Taylor & Francis Group, Boca Raton, 2017, pp. 395415, (Chapter Published).

H.Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

M. Dimian, P. Gurevich, O. Klein, D. Knees, D. Rachinskii, S. Tikhomirov, eds., MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, 252 pages, (Collection Published).

A. Mielke, T. Roubíček, Rateindependent Systems. Theory and Application, 193 of Applied Mathematical Sciences, Springer International Publishing, New York, 2015, vii+660 pages, (Monograph Published).
Abstract
This monograph provides both an introduction to and a thorough exposition of the theory of rateindependent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rateindependent systems, first on an abstract level either with or even without a linear structure, discussing various concepts of solutions with full mathematical rigor. Then, usefulness of the abstract concepts is demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesivetype contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to ratedependent thermodynamic models are considered as well. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. In this book, the mathematical framework for a rigorous mathematical treatment of "rateindependent systems" is presented in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well written book useful. 
A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125155, (Chapter Published).
Abstract
In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gammaconvergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rateindependent system. On the other hand we show how the concept of BalancedViscosity solution arise as in the vanishingviscosity limit.
As applications we discuss, first, the evolution of laminate microstructures in finitestrain elastoplasticity and, second, a twophase model for shapememory materials, where Hmeasures are used to construct the mutual recovery sequences needed in the existence theory. 
G. Dal Maso, A. Mielke, U. Stefanelli, eds., Rateindependent Evolutions, 6 (No. 1) of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2013, 275 pages, (Collection Published).

R. Helmig, A. Mielke, B. Wohlmuth, eds., Multifield Problems in Solid and Fluid Mechanics, 28 of Lecture Notes in Applied and Computational Mechanics, Springer, Heidelberg, 2006, xi+571 pages, (Collection Published).
Artikel in Referierten Journalen

D. Peschka, A. Zafferi, L. Heltai, M. Thomas, Variational approach to fluidstructure interaction via GENERIC, Journal of NonEquilibrium Thermodynamics, (2022), published online on 11.02.2022, DOI 10.1515/jnet20210081 .
Abstract
We present a framework to systematically derive variational formulations for fluidstructure interaction problems based on thermodynamical driving functionals and geometric structures in different coordinate systems by suitable transformations within this formulation. Our approach provides a promising basis to construct structurepreserving discretization strategies. 
M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambdaconvex gradient flows, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 427453, DOI 10.3934/dcdss.2020328 .
Abstract
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λconvex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are AllenCahn type equations and evolutionary equations driven by the pLaplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic twoscale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the wellestablished notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ)convex functionals. 
G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ricerche di Matematica. A Journal of Pure and Applied Mathematics, published online on 01.07.2021, DOI 10.1007/s11587021006103 .
Abstract
We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (2627): 45854608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case. 
M. Thomas, S. Tornquist, Discrete approximation of dynamic phasefield fracture in viscoelastic materials, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 38653924, DOI 10.3934/dcdss.2021067 .
Abstract
This contribution deals with the analysis of models for phasefield fracture in viscoelastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rateindependent law with a positively 1homogeneous dissipation potential. Both evolution laws encode a nonsmooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional. 
R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 89119 (published online in March 2020), DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phasefield fracture model at finite strains based on modified invariants, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 100 (2020), pp. e201900288/1e201900288/51, DOI 10.1002/zamm.201900288 .
Abstract
Phasefield models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phasefield fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phasefield model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right CauchyGreen strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the timediscrete solutions converge in a weak sense to a solution of the timecontinuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results. 
O. Klein, D. Davino, C. Visone, On forward and inverse uncertainty quantification for models involving hysteresis operators, Mathematical Modelling of Natural Phenomena, 15 (2020), pp. 53/153/19, DOI https://doi.org/10.1051/mmnp/2020009 .
Abstract
Parameters within hysteresis operators modeling real world objects have to be identified from measurements and are therefore subject to corresponding errors. To investigate the influence of these errors, the methods of Uncertainty Quantification (UQ) are applied. 
A. Mielke, T. Roubíček, Thermoviscoelasticity in KelvinVoigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), pp. 145, DOI 10.1007/s0020502001537z .
Abstract
The frameindifferent thermodynamicallyconsistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the secondgrade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under timedependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization. 
M. Heida, B. Schweizer, Stochastic homogenization of plasticity equations, ESAIM. Control, Optimisation and Calculus of Variations, 24 (2018), pp. 153176.
Abstract
In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flowrule function are given through a dynamical system on a probability space. A parameter ε > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ε → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution [0,T]∋ t ↦ξ(t) ∈ ℝ_{s}^{dxd} induces a stress evolution [0,T]∋ t ↦Σ (ξ) (t)∈ℝ_{s}^{dxd}. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by ∇ ⋅ Σ(∇^{s}u)=f. 
G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rateindependent damage in thermoviscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 13111364, DOI 10.1007/s108840189666y .
Abstract
We present a model for rateindependent, unidirectional, partial damage in viscoelastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rateindependent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the AmbrosioTortorelli phasefield model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled timediscrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rateindependent limit model for displacements and damage, which is independent of temperature. 
M. Thomas, C. Bilgen, K. Weinberg, Phasefield fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMMMitteilungen, 40 (2018), pp. 207237, DOI 10.1002/gamm.201730004 .
Abstract
Phasefield models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phasefield fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phasefield model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right CauchyGreen strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions %Second the mathematical background of the approach is examined and and we show that the timediscrete solutions converge in a weak sense to a solution of the timecontinuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results. 
R. Rossi, M. Thomas, From adhesive to brittle delamination in viscoelastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 14891546, DOI 10.1142/S0218202517500257 .
Abstract
In this paper we analyze a system for brittle delamination between two viscoelastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rateindependent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rateindependent systems to the present mixed ratedependent/rateindependent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit. 
R. Rossi, M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 14191494.
Abstract
We address the analysis of an abstract system coupling a rateindependet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised timediscretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rateindependent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rateindependent processes in viscoelastic solids with inertia, and to a recently proposed model for damage with plasticity. 
M. Thomas, Ch. Zanini, Cohesive zonetype delamination in viscoelasticity, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 14871517, DOI 10.3934/dcdss.2017077 .
Abstract
We study a model for the rateindependent evolution of cohesive zone delamination in a viscoelastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.
Due to the presence of multivalued and unbounded operators featuring nonpenetration and the `memory'constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIASPreprint2113]. 
P. Gussmann, A. Mielke, Linearized elasticity as Moscolimit of finite elasticity in the presence of cracks, Advances in Calculus of Variations, 13 (2020), pp. 3352 (published online on 17.10.2017), DOI 10.1515/acv20170010 .
Abstract
The smalldeformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Gamma converge to the linearized elastic energy with a local constraint of noninterpenetrability along the crack. 
M. Hintermüller, C.N. Rautenberg, S. Rösel, Density of convex intersections and applications, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 473 (2017), pp. 20160919/120160919/28, DOI 10.1098/rspa.2016.0919 .
Abstract
In this paper we address density properties of intersections of convex sets in several function spaces. Using the concept of Gammaconvergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite element discretizations of sets associated to convex constraints. Finally, two applications are provided, which include elastoplasticity and image restoration problems. 
M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasivariational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 135.
Abstract
A class of abstract nonlinear evolution quasivariational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semidiscrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradienttype. 
A. Mielke, R. Rossi, G. Savaré, Global existence results for viscoplasticity at finite strain, Archive for Rational Mechanics and Analysis, 227 (2018), pp. 423475 (published online on 20.09.2017), DOI 10.1007/s0020501711646 .
Abstract
We study a model for ratedependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of globalintime solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finitestrain elasticity as well as the multiplicative decomposition of finitestrain plasticity. Moreover, the dissipation potential depends on the leftinvariant plastic rate and thus, depends on the plastic state variable.
The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energydissipationbalance (EDB) and energydissipationinequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory. 
S. Heinz, A. Mielke, Existence, numerical convergence, and evolutionary relaxation for a rateindependent phasetransformation model, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 374 (2016), pp. 20150171/120150171/23, DOI 10.1098/rsta.2015.0171 .
Abstract
We revisit the twowell model for phase transformation in a linearly elastic body introduced and studied in A. Mielke, F. Theil, and V.I. Levita “A variational formulation of rateindependent phase transformations using an extremum principle", Arch. Rational Mech. Anal., 162, 137177, 2002 ([MTL02]). This energetic rateindependent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable numerical approximations via spacetime discretization, and (iii) to perform the evolutionary relaxation from the purestate model to the relaxed mixture model. All these results rely on weak converge and involve the Hmeasure as an essential tool. 
M. Hintermüller, S. Rösel, A dualitybased pathfollowing semismooth Newton method for elastoplastic contact problems, Journal of Computational and Applied Mathematics, 292 (2016), pp. 150173.

A. Mielke, T. Roubíček, Rateindependent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 22032236.
Abstract
Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rateindependent evolution. The energy functional with a frameindifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonselfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.
In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations. 
R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermoviscoelasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015), pp. 159.
Abstract
We address the analysis of a model for brittle delamination of two viscoelastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques. 
T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stressdriven localsolution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 645663.
Abstract
A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rateindependent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semiimplicit scheme and regularized by a BVtype gradient term. An analytical zerodimensional example motivates the model and a specific localsolution concept. Twodimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli. 
CH. Heinemann, Ch. Kraus, A degenerating CahnHilliard system coupled with complete damage processes, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 388403.
Abstract
Complete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a timedepending domain for phase separation and complete damage processes under timevarying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasistatic balance equation for the displacement field. All these equations are strongly nonlinearly coupled. Because of the doubly degenerating character and the doubly nonlinear differential inclusion we are confronted with introducing a suitable notion of weak solutions. We choose a notion of weak solutions which consists of weak formulations of the diffusion equation and the momentum balance, a onesided variational inequality for the damage function and an energy estimate. For the introduced degenerating system, we prove existence of weak solutions in an $SBV$framework. The existence result is based on an approximation system, where the elliptic degeneracy of the displacement field and the parabolic degeneracy of the concentration are eliminated. In the framework of phase separation and damage, this means that the approximation system allows only for partial damage and a nondegenerating mobility tensor. For the approximation system, existence results are established. Then, a passage to the limit shows existence of weak solutions of the degenerating system. 
CH. Heinemann, Ch. Kraus, Complete damage in linear elastic materials  Modeling, weak formulation and existence results, Calculus of Variations and Partial Differential Equations, 54 (2015), pp. 217250.
Abstract
We introduce a complete damage model with a timedepending domain for linearelastically stressed solids under timevarying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasistatic balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove localintime existence and globalintime existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a timedepending domain. In this context, two major challenges arise: Firstly, the timedependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $Gamma$convergence techniques. Secondly, the timedepending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolicparabolic phase field system with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 20442073.
Abstract
The aim of this paper is to prove existence of weak solutions of hyperbolicparabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing timediscretization, $H^2$regularization and variational techniques. 
CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 45874612.
Abstract
In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damagedependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging. 
M. Radszuweit, H. Engel, M. Bär, An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum, PLOS ONE, 9 (2014), pp. e99220/1e99220/15.
Abstract
Motivated by recent experimental studies, we derive and analyze a twodimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic twophase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the twophase medium. The reactiondiffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling.
A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets. 
P. Hornung, S. Neukamm, I. Velcic, Derivation of a homogenized nonlinear plate theory from 3D elasticity, Calculus of Variations and Partial Differential Equations, 51 (2014), pp. 677699.

A. Mielke, Ch. Ortner, Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM Journal on Mathematical Analysis, 46 (2014), pp. 13171347.
Abstract
We formulate quasistatic nonlinear finitestrain viscoelasticity of ratetype as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure timedependent frameindifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the timeincremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the onedimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the timecontinuous in some case in a specific case. 
A. Mielke, S. Zelik, On the vanishingviscosity limit in parabolic systems with rateindependent dissipation terms, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XIII (2014), pp. 67135.
Abstract
We consider quasilinear parabolic systems with a nonsmooth rateindependent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishingviscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishingviscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rateindependent systems. 
S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014), DOI 10.1007/s0052601407652 .
Abstract
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gammaconvergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using twoscale convergence. This is a nontrivial task, since one has to treat twoscale convergence in connection with a nonlinear differential constraint. 
CH. Heinemann, Ch. Kraus, Existence results for diffuse interface models describing phase separation and damage, European Journal of Applied Mathematics, 24 (2013), pp. 179211.
Abstract
In this paper we analytically investigate CahnHilliard and AllenCahn systems which are coupled with elasticity and unidirectional damage processes. We are interested in the case where the free energy contains logarithmic terms of the chemical concentration variable and quadratic terms of the gradient of the damage variable. For elastic CahnHilliard and AllenCahn systems coupled with unidirectional damage processes, an appropriate notion of weak solutions is presented as well as an existence result based on certain regularization methods and an higher integrability result for the strain. 
S. Neukamm, I. Velcic, Derivation of a homogenized vonKármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 27012748.
Abstract
We rigorously derive a homogenized vonKármán plate theory as a Gammalimit from nonlinear threedimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, threedimensional plate with spatially periodic material properties. The functional features two small length scales: the period $epsilon$ of the elastic composite material, and the thickness h of the slender plate. We study the behavior as $epsilon$ and h simultaneously converge to zero in the vonKármán scaling regime. The obtained limit is a homogenized vonKármán plate model. Its effective material properties are determined by a relaxation formula that exposes a nontrivial coupling of the behavior of the outofplane displacement with the oscillatory behavior in the inplane directions. In particular, the homogenized coefficients depend on the relative scaling between h and $epsilon$, and different values arise for h<<$epsilon$, $epsilon$ h and $epsilon$ << h. 
A. Fiaschi, D. Knees, S. Reichelt, Global higher integrability of minimizers of variational problems with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 401 (2013), pp. 269288.
Abstract
We consider integral functionals with densities of pgrowth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of pgrowth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys. 
TH.I. Seidman, O. Klein, Periodic solutions of isotone hybrid systems, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 18 (2013), pp. 483493.
Abstract
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work (Krasnosel'skiĭPokrovskiĭ 1974) of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells. 
M. Thomas, Quasistatic damage evolution with spatial BVregularization, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 235255.
Abstract
An existence result for energetic solutions of rateindependent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W^{1,r}(Ω) with r∈(1,∞) for Ω⊂R^{d}. We now cover the case r=1. The lack of compactness in W^{1,1}(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γlimit of functionals of ModicaMortola type. 
A. Mielke, R. Rossi, G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calculus of Variations and Partial Differential Equations, 46 (2013), pp. 253310.
Abstract
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a timediscretization scheme with variational techniques. Finally, we discuss an application to a material model in finitestrain elasticity. 
A. Mielke, U. Stefanelli, Linearized plasticity is the evolutionary Gamma limit of finite plasticity, Journal of the European Mathematical Society (JEMS), 15 (2013), pp. 923948.
Abstract
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the smalldeformations limit, we prove via Gammaconvergence for rateindependent processes that energetic solutions of the quasistatic finitestrain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity 
S. Bartels, A. Mielke, T. Roubíček, Quasistatic smallstrain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM Journal on Numerical Analysis, 50 (2012), pp. 951976.
Abstract
The quasistatic rateindependent evolution of the PragerZieglertype model of linearized plasticity with hardening is shown to converge to the rateindependent evolution of the PrandtlReuss elastic/perfectly plastic model. Based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of Gammaconvergence for rateindependent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A twosided energy inequality is a posteriori verified to document experimental convergence rates. 
K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finitestrain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012), pp. 888909.
Abstract
We study the time evolution in elastoplasticity within the rateindependent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of timeincremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented. 
A. Mielke, R. Rossi, G. Savaré, BV solutions and viscosity approximations of rateindependent systems, ESAIM. Control, Optimisation and Calculus of Variations, 18 (2012), pp. 3680.
Abstract
In the nonconvex case solutions of rateindependent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finitedimensional case we study the vanishingviscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rateindependent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rateindependent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the timecontinuous and for the timediscretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rateindependent systems. 
A. Mielke, T. Roubíček, M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Journal of Elasticity. The Physical and Mathematical Science of Solids, 109 (2012), pp. 235273.
Abstract
Brittle Griffithtype delamination of compounds is deduced by means of Gammaconvergence from partial, isotropic damage of threespecimensandwichstructures by flattening the middle component to the thickness 0. The models used here allow for nonlinearly elastic materials at small strains and consider the processes to be unidirectional and rateindependent. The limit passage is performed via a double limit: first, we gain a delamination model involving the gradient of the delamination variable, which is essential to overcome the lack of a uniform coercivity arising from the passage from partial damage to delamination. Second, the delaminationgradient is supressed. Noninterpenetration and transmissionconditions along the interface are obtained. 
A. Mielke, L. Truskinovsky, From discrete viscoelasticity to continuum rateindependent plasticity: Rigorous results, Archive for Rational Mechanics and Analysis, 203 (2012), pp. 577619.
Abstract
We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a viscoelastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishingviscosity limit and the discrete to continuum limit. In the course of these limits a nonconvex elastic problem transforms into a convex elastic problem while the quadratic ratedependent dissipation of viscoelastic solid transforms into a singular rateindependent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shapememory alloys. The approach, however, is sufficiently general and can be used for similar reductions in the cases of more general plasticity and damage models. 
A. Mielke, Emergence of rateindependent dissipation from viscous systems with wiggly energies, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 591606.
Abstract
We consider the passage from viscous system to rateindependent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R^{*}) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization. 
A. Mielke, Generalized PrandtlIshlinskii operators arising from homogenization and dimension reduction, Phys. B, 407 (2012), pp. 13301335.
Abstract
We consider rateindependent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a generalized PrandtlIshlinskii operator 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for CahnHilliard systems coupled with elasticity and damage, Advances in Mathematical Sciences and Applications, 21 (2011), pp. 321359.
Abstract
The CahnHilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the socalled CahnLarché system by combining it with elasticity to capture nonneglecting deformation phenomena, which occurs during phase separation in the material. Evolving microstructures such as phase separation and coarsening processes have a strong influence on damage initiation and propagation in alloys. We develop the existing framework of CahnHilliard and CahnLarché systems by coupling these systems with a unidirectional evolution inclusion for an internal variable, describing damage processes. After establishing a weak notion of the corresponding evolutionary system, we prove existence of weak solutions for ratedependent damage processes under certain growth conditions of the energy functional. 
M. Liero, A. Mielke, An evolutionary elastoplastic plate model derived via $Gamma$convergence, Mathematical Models & Methods in Applied Sciences, 21 (2011), pp. 19611986.
Abstract
This paper is devoted to dimension reduction for linearized elastoplasticity in the rateindependent case. The reference configuration of the threedimensional elastoplastic body has a twodimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear KirchhoffLove plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gammaconvergence theory for rateindependent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energystorage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energystorage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Moscoconvergent quadratic energies. 
H. Hanke, Homogenization in gradient plasticity, Mathematical Models & Methods in Applied Sciences, 21 (2011), pp. 16511684.
Abstract
This paper yields a twoscale homogenization result for a rateindependent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor ε^{γ} (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the twoscale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part. 
H. Hanke, Homogenization in gradient plasticity, GAMMMitteilungen, 34 (2011), pp. 102106.
Abstract
This paper yields a twoscale homogenization result for a rateindependent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor ε^{γ} (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the twoscale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part. 
D. Knees, Global spatial regularity for a regularized elastoplastic model, GAMMMitteilungen, 34 (2011), pp. 2127.
Abstract
In this note the spatial regularity of weak solutions for a class of elastoviscoplastic evolution models is studied for nonsmooth domains. The considered class comprises models which are obtained through a Yosida regularization from classical, rateindependent elastoplastic models. The corresponding evolution model consists of an elliptic PDE for the (generalized) displacements which is coupled with an ordinary differential equation with a Lipschitz continuous nonlinearity describing the evolution of the internal variable. It is shown that the global spatial regularity of the displacements and the inner variables is exactly determined through the mapping properties of the underlying elliptic operator. 
CH. Kraus, The degenerate and nondegenerate Stefan problem with inhomogeneous and anisotropic GibbsThomson law, European Journal of Applied Mathematics, 22 (2011), pp. 393422.
Abstract
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic GibbsThomson condition at the phase boundary. We show the longtime existence of weak solutions for the nondegenerate Stefan problem with a spatially inhomogeneous and anisotropic GibbsThomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic GibbsThomson law in a weak generalized BVformulation. 
A. Mielke, Completedamage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems  Series S, 4 (2011), pp. 423439.
Abstract
The rateindependent damage model recently developed in Bouchitté, Mielke, Roubíček “A completedamage problem at small strains" allows for complete damage, such that the deformation is no longer welldefined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but nonquadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions. 
A. Mielke, Formulation of thermoelastic dissipative material behavior using GENERIC, Continuum Mechanics and Thermodynamics, 23 (2011), pp. 233256.
Abstract
The theory of GENERIC (general equations for nonequilibrium reversible irreversibel coupling) is presented in a mathematical form. It is applied first to simple mechanical systems and then generalized to standard generalized materials. It is shown that nonisothermal viscoplasticity can be cast into the form of GENERIC, if the dissipative structure is generalized from linear functionals to the more general subdifferential of convex potentials. 
A. Mielke, On thermodynamically consistent models and gradient structures for thermoplasticity, GAMMMitteilungen, 34 (2011), pp. 5158.
Abstract
Dedicated to HansDieter Alber on the occasion of his sixtieth birthday
It is investigated in what sense thermoplasticity can be written as a generalized gradient system with respect to the total entropy and the entropyproduction potential. The difficulty is that the quasistatic equilibrium equation for the elastic forces is obtained by minimizing the total energy and that this condition must be eliminated suitably. The subtle interplay between energy and entropy is treated via the formalism of GENERIC. 
M. Thomas, A. Mielke, Damage of nonlinearly elastic materials at small strain  Existence and regularity results, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 88112.
Abstract
In this paper an existence result for energetic solutions of rateindependent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [MielkeRoubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W^1,r (Omega) with r>d for Omega ⊂ R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz and Höldercontinuity of solutions with respect to time. 
H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified GibbsThomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511545.
Abstract
We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified GibbsThomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the EulerLagrange equation of the diffuse phase field energy. 
P. Gruber, D. Knees, S. Nesenenko, M. Thomas, Analytical and numerical aspects of timedependent models with internal variables, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 861902.
Abstract
In this paper some analytical and numerical aspects of timedependent models with internal variables are discussed. The focus lies on elasto/viscoplastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elastoplasticity with hardening and viscous models of the NortonHoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rateindependent processes is explained and temporal regularity results based on different convexity assumptions are presented. 
D. Knees, Ch. Zanini, A. Mielke, Crack growth in polyconvex materials, Physica D. Nonlinear Phenomena, 239 (2010), pp. 14701484.
Abstract
We discuss a model for crack propagation in an elastic body, where the crack path is described apriori. In particular, we develop in the framework of finitestrain elasticity a rateindependent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energyrelease rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed. 
D. Knees, On global spatial regularity and convergence rates for time dependent elastoplasticity, Mathematical Models & Methods in Applied Sciences, 20 (2010), pp. 18231858.

A. Mielke, T. Roubíček, J. Zeman, Complete damage in elastic and viscoelastic media and its energetics, Computer Methods in Applied Mechanics and Engineering, 199 (2010), pp. 12421253.
Abstract
A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The “flow rule” for the damage parameter is rateindependent. The stored energy involves the gradient of the damage variable, which determines an internal lengthscale. Quasistatic fully rateindependent evolution is considered as well as ratedependent evolution including viscous/inertial effects. Illustrative 2dimensional computer simulations are presented, too. 
A. Mielke, L. Paoli, A. Petrov, U. Stefanelli, Error estimates for spacetime discretizations of a rateindependent variational inequality, SIAM Journal on Numerical Analysis, 48 (2010), pp. 16251646.
Abstract
This paper deals with error estimates for spacetime discretizations in the context of evolutionary variational inequalities of rateindependent type. After introducing a general abstract evolution problem, we address a fullydiscrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit spacetime convergence rates for the isothermal SouzaAuricchio model for shapememory alloys. 
A. Petrov, M. Schatzman, Mathematical results on existence for viscoelastodynamic problems with unilateral constraints, SIAM Journal on Mathematical Analysis, 40 (2009), pp. 18821904.
Abstract
We study a damped wave equation and the evolution of a KelvinVoigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution. 
G. Bouchitté, A. Mielke, T. Roubíček, A completedamage problem at small strains, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 205236.
Abstract
The complete damage of a linearlyresponding material that can thus completely disintegrate is addressed at small strains under timevarying Dirichlet boundary conditions as a rateindependent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a onedimensional example. 
K. Kuhnen, P. Krejčí, Compensation of complex hysteresis and creep effects in piezoelectrically actuated systems  A new Preisach modeling approach, IEEE Transactions on Automatic Control. Institute of Electrical and Electronics Engineers, Inc., New York, NY (US). Control Systems Society., 54 (2009), pp. 537550.

A. Mainik, A. Mielke, Global existence for rateindependent gradient plasticity at finite strain, Journal of Nonlinear Science, 19 (2009), pp. 221248.
Abstract
We provide a global existence result for the timecontinuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables. 
D. Knees, On global spatial regularity in elastoplasticity with linear hardening, Calculus of Variations and Partial Differential Equations, 36 (2009), pp. 611625.
Abstract
We study the global spatial regularity of solutions of elastoplastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L^{∞}((0,T);H^{3/2δ}(Ω)) for the displacements and z in L^{∞}((0,T);H^{1/2δ}(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument. 
A. Mielke, F. Rindler, Reverse approximation of energetic solutions to rateindependent processes, NoDEA. Nonlinear Differential Equations and Applications, 16 (2009), pp. 1740.
Abstract
Energetic solutions to rateindependent processes are usually constructed via timeincremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit. 
A. Mielke, T. Roubíček, Numerical approaches to rateindependent processes and applications in inelasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), pp. 399429.
Abstract
A general abstract approximation scheme for rateindependent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shapememory alloys. 
A. Mielke, L. Paoli, A. Petrov, On existence and approximation for a 3D model of thermallyinduced phase transformations in shapememory alloys, SIAM Journal on Mathematical Analysis, 41 (2009), pp. 13881414.
Abstract
This paper deals with a threedimensional model for thermal stressinduced transformations in shapememory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rateindependent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also spacetime discretizations and establish convergence of these approximations. 
J.A. Griepentrog, W. Höppner, H.Chr. Kaiser, J. Rehberg, A biLipschitz continuous, volume preserving map from the unit ball onto a cube, Note di Matematica, 28 (2008), pp. 185201.
Abstract
We construct two biLipschitz, volume preserving maps from Euclidean space onto itself which map the unit ball onto a cylinder and onto a cube, respectively. Moreover, we characterize invariant sets of these mappings. 
S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), pp. 795801.

F. Auricchio, A. Mielke, U. Stefanelli, A rateindependent model for the isothermal quasistatic evolution of shapememory materials, Mathematical Methods in the Applied Sciences, 18 (2008), pp. 125164.
Abstract
This note addresses a threedimensional model for isothermal stressinduced transformation in shapememory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rateindependent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasistatic evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics. 
R. HallerDintelmann, H.Chr. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées, 89 (2008), pp. 2548.

J.A.C. Martins, M.D.P. Monteiro Marques, A. Petrov, On the stability of elasticplastic systems with hardening, Journal of Mathematical Analysis and Applications, 343 (2008), pp. 10071021.
Abstract
This paper discusses the stability of quasistatic paths for a continuous elasticplastic system with hardening in a onedimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasistatic problems involving elasticplastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasistatic paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasistatic ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasistatic paths of these elasticplastic systems is the main result proved in the paper. 
D. Knees, P. Neff, Regularity up to the boundary for nonlinear elliptic systems arising in timeincremental infinitesimal elastoplasticity, SIAM Journal on Mathematical Analysis, 40 (2008), pp. 2143.
Abstract
In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the timediscretization of models from infinitesimal elastoplasticity. Our main focus lies on an elastoplastic Cosserat model. More specifically we show that the time discretization renders $H^2$regularity of the displacement and $H^1$regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and PrandtlReuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elastoplasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (LegendreHadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the meshwidth $h$ in error estimates. 
D. Knees, A. Mielke, Energy release rate for cracks in finitestrain elasticity, Mathematical Methods in the Applied Sciences, 31 (2008), pp. 501528.
Abstract
Griffith's fracture criterion describes in a quasistatic setting whether or not a preexisting crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the wellknown Griffith formula and the $J$integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. 
D. Knees, A. Mielke, Ch. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models & Methods in Applied Sciences, 18 (2008), pp. 15291569.
Abstract
We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions. 
D. Knees, A. Mielke, On the energy release rate in finitestrain elasticity, Mechanics of Advanced Materials and Structures, 15 (2008), pp. 421427.

D. Knees, Global stress regularity of convex and some nonconvex variational problems, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 187 (2008), pp. 157184.

A. Mielke, A. Petrov, J.A.C. Martins, Convergence of solutions of kinetic variational inequalities in the rateindependent quasistatic limit, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 10121020.
Abstract
This paper discusses the convergence of kinetic variational inequalities to rateindependent quasistatic variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rateindependent quasistatic problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rateindependent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to threedimensional elasticplastic systems with hardening is given. 
J.A.C. Martins, M.D.P. Monteiro, A. Petrov, On the stability of quasistatic paths for finite dimensional elasticplastic systems with hardening, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 303313.

J. Elschner, H.Chr. Kaiser, J. Rehberg, G. Schmidt, $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models & Methods in Applied Sciences, 17 (2007), pp. 593615.

P. Krejčí, J. Sprekels, Elasticideally plastic beams and PrandtlIshlinskii hysteresis operators, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 23712393.

A. Mielke, A. Petrov, Thermally driven phase transformation in shapememory alloys, Advances in Mathematical Sciences and Applications, 17 (2007), pp. 667685.
Abstract
This paper analyzes a model for phase transformation in shapememory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rateindependent processes. Existence and uniqueness results are proved. 
A. Mielke, R. Rossi, Existence and uniqueness results for a class of rateindependent hysteresis problems, Mathematical Models & Methods in Applied Sciences, 17 (2007), pp. 81123.

A. Mielke, A. Timofte, Twoscale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007), pp. 642668.
Abstract
This paper is devoted to the twoscale homogenization for a class of rateindependent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energystorage functional and a dissipation functional. Using the recently developed method of weak and strong twoscale convergence via periodic unfolding, we show that these two functionals have a suitable twoscale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using socalled joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals. 
A. Mielke, A model for temperatureinduced phase transformations in finitestrain elasticity, IMA Journal of Applied Mathematics, 72 (2007), pp. 644658.

F. Schmid, A. Mielke, Existence results for a contact problem with varying friction coefficient and nonlinear forces, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 616631.
Abstract
We consider the rateindependent problem of a particle moving in a three  dimensional half space subject to a timedependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rateindependent systems as developed by Mielke and coworkers. We generalize the timeincremental minimization procedure of Mielke and Rossi for the present situation of a nonassociative flow rule. 
M.A. Efendiev, A. Mielke, On the rateindependent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), pp. 151167.

M. Kočvara, A. Mielke, T. Roubíček, A rateindependent approach to the delamination problem, Mathematics and Mechanics of Solids, 11 (2006), pp. 423447.

H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA. Nonlinear Differential Equations and Applications, 13 (2006), pp. 287310.

D. Knees, Global regularity of the elastic fields of a powerlow model on Lipschitz domains, Mathematical Methods in the Applied Sciences, 29 (2006), pp. 13631391.

D. Knees, Griffithformula and Jintegral for a crack in a powerlaw hardening material, Mathematical Models & Methods in Applied Sciences, 16 (2006), pp. 17231749.

A. Mielke, G. Francfort, Existence results for a class of rateindependent material models with nonconvex elastic energies, Journal fur die Reine und Angewandte Mathematik, 595 (2006), pp. 5591.

A. Mielke, S. Müller, Lower semicontinuity and existence of minimizers in incremental finitestrain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 3 (2006), pp. 233250.

A. Mielke, T. Roubíček, Rateindependent damage processes in nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 16 (2006), pp. 177209.

M. Kružík, A. Mielke, T. Roubíček, Modelling of microstructure and its evolution in shapememoryalloy singlecrystals, in particular in CuAINi, Meccanica. International Journal of the Italian Association of Theoretical and Applied Mechanics, 40 (2005), pp. 389418.

A. Mielke, Necessary and sufficient conditions for polyconvexity of isotropic functions, Journal of Convex Analysis, 12 (2005), pp. 291314.

O. Klein, Asymptotic behaviour for a phasefield model with hysteresis in onedimensional thermoviscoplasticity, Applications of Mathematics, 49 (2004), pp. 309341.

P. Krejčí, J. Sprekels, U. Stefanelli, Onedimensional thermoviscoplastic processes with hysteresis and phase transitions, Advances in Mathematical Sciences and Applications, 13 (2003), pp. 695712.

O. Klein, P. Krejčí, Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 4 (2003), pp. 755785.

P. Krejčí, J. Sprekels, U. Stefanelli, Phasefield models with hysteresis in onedimensional thermoviscoplasticity, SIAM Journal on Mathematical Analysis, 34 (2002), pp. 409434.
Beiträge zu Sammelwerken

R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 89119, DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
O. Klein, On uncertainty quantification for models involving hysteresis operators, in: Extended Abstracts Spring 2018  Singularly Perturbed Systems, Multiscale Phenomena and Hysteresis: Theory and Applications, A. Korobeinikov, M. Caubergh, T. Lázaro, J. Sardanyés, eds., 11 of Research Perspectives CRM Barcelona, Birkhäuser, Cham, 2019, pp. 271275, DOI 10.1007/9783030252618 .

R. Rossi, M. Thomas, From nonlinear to linear elasticity in a coupled ratedependent/independent system for brittle delamination, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 127157, DOI 10.1007/9783319759401_7 .
Abstract
We revisit the weak, energetictype existence results obtained in [Rossi/ThomasESAIMCOCV21(1):159,2015] for a system for rateindependent, brittle delamination between two viscoelastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of viscoelastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Moscoconvergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the timecontinuous level, and secondly, to pass from a timediscrete to a timecontinuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, superquadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature. 
S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$regularization, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179203, DOI 10.1007/9783319759401_9 .
Abstract
We address a model for rateindependent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BVregularization. Discrete solutions are obtained using an alternate timediscrete scheme and the VariableADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rateindependent system. Moreover, we present our numerical results for two benchmark problems. 
M. Thomas, A comparison of delamination models: Modeling, properties, and applications, in: Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, P. VAN Meurs, M. Kimura, H. Notsu, eds., 30 of Mathematics for Industry, Springer Nature, Singapore, 2018, pp. 2738, DOI 10.1007/9789811062834_3 .
Abstract
This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed. 
A. Mielke, Three examples concerning the interaction of dry friction and oscillations, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 159177, DOI 10.1007/9783319759401_8 .
Abstract
We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with statedependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved. 
G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rateindependent damage in thermoviscoelastodynamics, in: MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1012009/20.
Abstract
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the ratedependent heat equation, and with the momentum balance featuring inertia and viscosity according to KelvinVoigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a timedependent Dirichlet loading. 
O. Klein, V. Recupero, Hausdorff metric BV discontinuity of sweeping processes, in: MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes & HYSteresis (MURPHYS) & the 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, 2016, pp. 012006/1012006/12, DOI 10.1088/17426596/727/1/012006 .
Abstract
Sweeping processes are a class of evolution differential inclusions arising in elastoplasticity and were introduced by J.J. Moreau in the early seventies. The solution operator of the sweeping processes represents a relevant example of emphrate independent operator containing as a particular case the so called emphplay operator which is widely used in hysteresis. The continuity properties of these operators were studied in several works. In this note we address the continuity with respect to the strict metric in the space of functions of bounded variation with values in the metric space of closed convex subsets of a Hilbert space. We provide a counterexample showing that the solution operator of the sweeping process is not continuous when its domain is endowed with the strict topology of $BV$ and its codomain is endowed with the $L^1$topology. This is at variance with the case of the play operator which instead is continuous in this sense. 
A. Mielke, Deriving effective models for multiscale systems via evolutionary $Gamma$convergence, in: Control of SelfOrganizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235251.

A. Mielke, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for MultiField and MultiScale Problems. In Honor of Michael Ortiz's 60th Birthday, K. Weinberg, A. Pandolfi, eds., 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 135160.
Abstract
In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the nonisothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.
Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving timeincremental minimization procedures, where the entropyproduction potential minus the total entropy is minimized with respect to the internal variables and the temperature.
We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundaryvalue problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation. 
A. Mielke, Relaxation of a rateindependent phase transformation model for the evolution of microstructure, in: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure, Workshop, March 1418, 2016, R. Kienzler, D.L. Mcdowell, S. Müller, E.A. Werner, eds., 13 of Oberwolfach Reports, European Mathematical Society, 2016, pp. 840842.

A. Mielke, Evolutionary relaxation of a twophase model, in: Scales in Plasticity, MiniWorkshop, November 814, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 30273030.

D. Knees, R. Kornhuber, Ch. Kraus, A. Mielke, J. Sprekels, C3  Phase transformation and separation in solids, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189203.

A. Mielke, L. Paoli, A. Petrov, U. Stefanelli, Error bounds for spacetime discretization of a 3D model for shapememory materials, in: IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, K. Hackl, ed., 21 of IUTAM Bookseries, Springer, Heidelberg, 2010, pp. 185197.

A. Mielke, Existence theory for finitestrain crystal plasticity with gradient regularization, in: IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, K. Hackl, ed., 21 of IUTAM Bookseries, Springer, Heidelberg, 2010, pp. 171183.

A. Mielke, Lie groups in plasticity at finite strains, in: Applied Dynamics and Geometric Mechanics, Workshop, July 2125, 2008, 5 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2008, pp. 18901893.

A. Mielke, Numerical approximation techniques for rateindependent inelasticity, in: Proceedings of the IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, B.D. Reddy, ed., 11 of IUTAM Bookseries, Springer, 2008, pp. 5363.

A. Petrov, J.A.C. Martins, M.D.P. Monteiro Marques, Mathematical results on the stability of quasistatic paths of elasticplastic systems with hardening, in: Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific, Singapore, 2007, pp. 167182.

A. Petrov, Thermally driven phase transformation in shapememory alloys, in: Analysis and Numerics of RateIndependent Processes, Workshop, February 26  March 2, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 605607.

D. Knees, Energy release rate for cracks in finitestrain elasticity, in: Analysis and Numerics of RateIndependent Processes, Workshop, February 26  March 2, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 627630.

E. Gürses, A. Mainik, Ch. Miehe, A. Mielke, Analytical and numerical methods for finitestrain elastoplasticity, in: Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., 28 of Lect. Notes Appl. Comput. Mech., Springer, Heidelberg, 2006, pp. 443481.

A. Mielke, A mathematical framework for standard generalized materials in the rateindependent case, in: Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., 28 of Lect. Notes Appl. Comput. Mech., Springer, Heidelberg, 2006, pp. 351379.

A. Mielke, Modeling and analysis of rateindependent processes in continuum mechanics, in: Trends in Applications of Mathematics to Mechanics. Proceedings of the XIVth International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM'2004), Seeheim, Germany, 2228 August 2004, K. Hutter, Y. Wang, eds., Berichte aus der Mathematik, Shaker, Aachen, 2005, pp. 299310.

A. Mielke, Some existence results in finitestrain plasticity, in: Analysis and computation of microstructures in finite plasticity, MiniWorkshop, November 1319, 2005, 52 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2005, pp. 1316.

O. Klein, P. Krejčí, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, in: Proceedings of the Fourth International Symposium on Hysteresis and Micromagnetic Modeling, Salamanca, Spain, 2830 May 2003, L. LopezDias, L. Torres, O. Alejos, eds., 343 of Physica B: Condensed Matter, Elsevier B.V., 2004, pp. 5358.
Preprints, Reports, Technical Reports

L. Schmeller, D. Peschka, Gradient flows for coupling order parameters and mechanics, Preprint no. 2909, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2909 .
Abstract, PDF (7667 kByte)
We construct a formal gradient flow structure for phasefield evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally very brief, the focus of this paper is on an extensible functional analytical framework and a discretization approach that preserves an appropriate variational structure as much as possible. As examples, we first present phase separation and swelling of gels and then the approach of stationary states of multiphase systems with surface tension and show the robustness of the general approach. 
A. Mielke, R. Rossi, BalancedViscosity solutions to infinitedimensional multirate systems, Preprint no. 2902, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2902 .
Abstract, PDF (778 kByte)
We consider generalized gradient systems with rateindependent and ratedependent dissipation potentials. We provide a general framework for performing a vanishingviscosity limit leading to the notion of parametrized and true BalancedViscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $eps^alpha$ and an internal variable $z$ with relaxation time $eps$ we obtain different limits for the three cases $alpha in (0,1)$, $alpha=1$ and $alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics. 
M. Heida, M. Thomas, GENERIC for dissipative solids with bulkinterface interaction, Preprint no. 2872, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2872 .
Abstract, PDF (322 kByte)
The modeling framework of GENERIC was originally introduced by Grmela and Öttinger for thermodynamically closed systems. It is phrased with the aid of the energy and entropy as driving functionals for reversible and dissipative processes and suitable geometric structures. Based on the definition functional derivatives we propose a GENERIC framework for systems with bulkinterface interaction and apply it to discuss the GENERIC structure of models for delamination processes. 
M. Heida, Stochastic homogenization on perforated domains I: Extension operators, Preprint no. 2849, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2849 .
Abstract, PDF (750 kByte)
This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W^{1,p}functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W^{1,p} to W^{1,r} with the strict inequality r<p. In particular, we estimate the L^{r}norm of the extended gradient in terms of the L^{p}norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝ^{d}valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local (δ,M)regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process, for which we can explicitly estimate the connectivity terms. 
A. Zafferi, D. Peschka, M. Thomas, GENERIC framework for reactive fluid flows, Preprint no. 2841, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2841 .
Abstract, PDF (805 kByte)
We describe reactive fluid flows in terms of the formalism General Equation for NonEquilibrium ReversibleIrreversible Coupling also known as GENERIC. Together with the formalism, we present the thermodynamical and mechanical foundations for the treatment of fluid flows using continuous fields and present a clear relation and transformation between a Lagrangian and an Eulerian formulation of the corresponding systems of partial differential equations. We bring the abstract framework to life by providing many physically relevant examples for reactive compressive fluid flows. 
M. Heida, Stochastic homogenization on randomly perforated domains, Preprint no. 2742, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2742 .
Abstract, PDF (1175 kByte)
We study the existence of uniformly bounded extension and trace operators for W^{1,p}functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local (δ, M)regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ”mesoscopic” connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on R^{d} and Ω in order to construct a stochastic twoscale convergence method and apply the resulting theory to the homogenization of a pLaplace problem on a randomly perforated domain. 
S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rateindependent damage models with gradient regularization, Preprint no. 2707, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2707 .
Abstract, PDF (3444 kByte)
This work provides a convergence analysis of a timediscrete scheme coupled with a finiteelement approximation in space for a model for partial, rateindependent damage featuring a gradient regularization as well as a nonsmooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rateindependent gradient damage is based on a Variable ADMMmethod to approximate the nonsmooth contribution. Spacediscretization is based on P1 finite elements and the algorithm directly couples the timestep size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BVfunctions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.
Vorträge, Poster

M.H. Farshbaf Shaker, D. Peschka , M. Thomas, B. Wagner, Variational methods for viscoelastic flows and gelation, MATH+ Day 2021 (Online Event), November 5, 2021.

M. Heida, Stochastic homogenization on random geometries (online talk), Applied Analysis Seminar, Universität Heidelberg, June 17, 2021.

M. Heida, Stochastic homogenization on randomly perforated domains (online talk), SIAM Conference on Mathematical Aspects of Materials Science (Online Event), May 17  28, 2021, May 24, 2021.

A. Zafferi, Coupling of thermoviscoelastic solids and reactive flows via GENERIC (online talk), CRC 1114 Conference 2021 (Online Event), March 1  3, 2021, Freie Universität Berlin, March 2, 2021.

A. Zafferi, Dynamics of rock dehydration on multiple scales, CRC 1114 Conference 2021 (Online Event), March 1  3, 2021.

A. Zafferi, Thermodynamics of reactiondiffusioninduced rock dehydration processes (online talk), 16thJoint European Thermodynamics Conference (Hybrid Event), June 14  18, 2021, Charles University Prague, Czech Republic, June 16, 2021.

M. Thomas, Convergence analysis for fully discretized damage and phasefield fracture models (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13  17, 2021, WIAS Berlin, September 14, 2021.

M. Thomas, GENERIC structures with bulkinterface interaction (online talk), 16th Joint European Thermodynamics Conference (Hybrid Event), June 14  18, 2021, Charles University Prague, Czech Republic, June 17, 2021.

A. Mielke, Thermoviscoelasticity at finite strain (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 19, 2021.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Thomas, Thermodynamical modelling via energy and entropy functionals (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin.

M. Liero, A. Mielke, Analysis for thermomechanical models with internal variables, Presentation of project proposals in DFG SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

A. Mielke, Finitestrain viscoelasticity with temperature coupling, Calculus of Variations and Applications, January 27  February 1, 2020, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, January 28, 2020.

A. Mielke, Variational structures for the analysis of PDE systems (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin, October 13, 2020.

M. Heida, A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, Zeuthen, May 20  22, 2019.

G. Nika, Homogenization for a multiscale model of magnetorheological suspension, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Minisymposium MS ME13 1 ``Emerging Problems in the Homogenization of Partial Differential Equations'', July 15  19, 2019, Valencia, Spain, July 15, 2019.

D. Peschka, Gradient structures for flows of concentrated suspensions  jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

S. Tornquist, Variational problems involving Caccioppoli partitions, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis", February 18  22, 2019, Technische Universität Wien, Austria, February 19, 2019.

A. Zafferi, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, Zeuthen, May 20  22, 2019.

M. Thomas, Analytical and numerical aspects for the approximation of gradientregularized damage models, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS A3226 ``PhaseField Models in Simulation and Optimization'', July 15  19, 2019, Valencia, Spain, July 17, 2019.

M. Thomas, Analytical and numerical aspects of rateindependent gradientregularized damage models, Conference ``Dynamics, Equations and Applications (DEA 2019)'', Session D444 ``Topics in the Mathematical Modelling of Solids'', September 16  20, 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems, MURPHYSHSFS 2019 Summer School on MultiRate Processes, SlowFast Systems and Hysteresis, June 17  19, 2019, Politecnico di Torino, Turin, Italy.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Mathematics for Mechanics, October 29  November 1, 2019, Czech Academy of Sciences, Prague, Czech Republic, October 31, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.

M. Thomas, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, May 20  22, 2019, Freie Universität Berlin, Zeuthen, May 21, 2019.

M. Thomas, GENERIC structures with bulkinterface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

M. Thomas, Rateindependent evolution of sets and application to fracture processes, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

A. Mielke, An existence result for thermoviscoelasticity at finite strains, Mathematics for Mechanics, October 29  November 1, 2019, Czech Academy of Sciences, Institute for Information Theory and Automation, Prague, Czech Republic, November 1, 2019.

A. Mielke, On initialboundary value problems for materials with internal variables or temperature dependence, Workshop on Mathematical Methods in Continuum Physics and Engineering: Theory, Models, Simulation, November 6  7, 2019, Technische Universität Darmstadt, November 6, 2019.

A. Mielke, Variational methods in timedependent material models with finitestrain deformations, Hausdorff School on Modeling and Analysis of Evolutionary Problems in Materials Science, September 23  27, 2019, Hausdorff Center for Mathematics, Universität Bonn.

A. Mielke, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, May 20  22, 2019, SFB 1114, Freie Universität Berlin, Zeuthen, May 21, 2019.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section DFG Priority Programmes PP1748 ``Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretization Methods, Mechanical and Mathematical Analysis'', March 19  23, 2018, Technische Universität München, March 20, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.

M. Thomas, Analytical and numerical approach to a class of damage models, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 75 ``Mathematics and Materials: Models and Applications'', July 5  9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 6, 2018.

O. Klein, On uncertainty quantification for models involving hysteresis operators, MURPHYSHSFS2018: Interdisciplinary Workshop on Multiple Scale Systems, Systems with Hysteresis and Trends in Dynamical Systems, May 28  June 1, 2018, Centre de Recerca Matemàtica (CRM), Barcelona, Spain, May 31, 2018.

A. Mielke, EDPconvergence: Gammaconvergence for gradient systems in the sense of the energydissipation principle, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section S14 ``Applied Analysis'', March 19  23, 2018, Technische Universität München, March 20, 2018.

A. Mielke, Global existence for finitestrain viscoplasticity, Workshop ``Variational Methods for the Modelling of Inelastic Solids'', February 5  9, 2018, Mathematisches Forschungsinstitut Oberwolfach, February 6, 2018.

M. Thomas, Rateindependent delamination processes in viscoelasticity, Miniworkshop on Dislocations, Plasticity, and Fracture, February 13  16, 2017, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, February 15, 2017.

O. Klein, Uncertainty quantification for models involving hysteresis operators, Summer School on MultiRate Processes, SlowFast Systems and Hysteresis MURPHYSHSFS2017, June 19  23, 2017, DISMA Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy.

A. Mielke, Global existence results for viscoplasticity, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 10, 2017.

A. Mielke, On selfinduced oscillations for friction reduction with applications to walking, Conference ``Dynamical Systems and Geometric Mechanics'', June 12  14, 2017, Technische Universität München, Zentrum für Mathematik, June 13, 2017.

TH. Frenzel, Evolutionary Gammaconvergence for amplitude equations and for wiggly energy models, Winter School 2016: Calculus of Variations in Physics and Materials Science, Würzburg, February 15  19, 2016.

M. Heida, Stochastic homogenization of 1homogeneous functionals, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18  22, 2016, Technische Universität Berlin, July 22, 2016.

M. Heida, Stochastic homogenization of rateindependent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

M. Thomas, Coupling rateindependent and ratedependent processes: Delamination models in viscoelastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, 7th European Congress of Mathematics (ECM), minisymposium ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18  22, 2016, Technische Universität Berlin, Berlin, July 22, 2016.

M. Thomas, Energetic concepts for coupled rateindependent and ratedependent processes: Damage & delamination in viscoelastodynamics, International Conference ``Mathematical Analysis of Continuum Mechanics and Industrial Applications II'' (CoMFoS16), October 22  24, 2016, Kyushu University, Fukuoka, Japan.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, special session ``Ratedependent and Rateindependent Evolution Problems in Continuum Mechanics: Analytical and Numerical Aspects'', July 1  5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 4, 2016.

M. Thomas, Nonsmooth PDEs in material failure: Towards dynamic fracture, Joint Annual Meeting of DMV and GAMM, Section 14 ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, March 10, 2016.

M. Thomas, Rateindependent evolution of sets, INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5  8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 6, 2016.

M. Thomas, Rateindependent evolution of sets & application to fracture processes, Seminar on Analysis, Kanazawa University, Institute of Science and Engineering, Kanazawa, Japan, October 28, 2016.

O. Klein, On uncertainty quantification for hysteresis operators, Silesian University, Mathematical Institute, Opava, Czech Republic, December 7, 2016.

O. Klein, Uncertainty quantification for hysteresis operators, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18  22, 2016, Technische Universität Berlin, July 22, 2016.

O. Klein, Uncertainty quantification for hysteresis operators and models for magnetomechanical hysteresis, Conference ``Advances in Magnetics'' (AIM) 2016, March 14  16, 2016, Bormio, Italy, March 14, 2016.

A. Mielke, Evolutionary relaxation for a rateindependent phasetransformation model, Workshop ``Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure'', March 14  18, 2016, Mathematisches Forschungszentrum Oberwolfach, March 14, 2016.

A. Mielke, Global existence for finitestrain viscoplasticity via the energydissipation principle, Seminar ``Analysis & Mathematical Physics'', Institute of Science and Technology Austria (IST Austria), Vienna, Austria, July 7, 2016.

A. Mielke, Microstructure evolution via relaxation for a rateindependent elastic twophase model, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

A. Mielke, Mutual recovery sequences and evolutionary relaxation of a twophase problem, 2nd Workshop on CENTRAL Trends in Analysis and Numerics for PDEs, May 26  28, 2016, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, May 27, 2016.

A. Mielke, On a model for the evolution of microstructures in solids  Evolutionary relaxation, KTGUIMU Mathematics Colloquia, March 30  31, 2016, Kyoto University, Department of Mathematics, Japan, March 31, 2016.

A. Mielke, Rateindependent microstructure evolution via relaxation of a twophase model, Workshop ``Advances in the Mathematical Analysis of Material Defects in Elastic Solids'', June 6  10, 2016, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, June 10, 2016.

M. Heida, Stochastic homogenization of PrandtlReuss plasticity, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 1, 2015.

CH. Heinemann, Solvability of differential inclusions describing damage processes and applications to optimal control problems, Universität EssenDuisburg, Fakultät für Mathematik, Essen, December 3, 2015.

M. Landstorfer, Theory, structure and experimental justification of the metal/electrolyte interface, Minisymposium `` Recent Developments on Electrochemical Interface Modeling'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 11, 2015.

M. Thomas, Analysis of nonsmooth PDE systems with application to material failuretowards dynamic fracture, Minisymposium ``Analysis of Nonsmooth PDE Systems with Application to Material Failure'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Pavia, Italy, March 5, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Evolutionary Gammaconvergence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Session on Applied Analysis, March 23  27, 2015, Università del Salento, Lecce, Italy, March 26, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence and evolutionary Gamma convergence, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 19, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), GAMM Juniors Poster Session, Lecce, Italy, March 23  27, 2015.

M. Thomas, Evolutionary Gamma convergence with application to damage and delamination, Seminar DICATAM, Università di Brescia, Dipartimento di Matematica, Brescia, Italy, June 3, 2015.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 2, 2015.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, Workshop on CENTRAL Trends in PDEs, November 12  13, 2015, University of Vienna, Faculty of Mathematics, Vienna, Austria, November 13, 2015.

M. Thomas, Rateindependent damage models with spatial BVregularization  Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

O. Klein, A representation result for rateindependent systems, 10th International Symposium on Hysteresis Modeling and Micromagnetics (HMM), May 18  20, 2015, Iasi, Romania, May 19, 2015.

A. Mielke, A mathematical approach to finitestrain viscoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 20, 2015.

A. Mielke, Abstract approach to energetic solutions for rateindependent solutions, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 18, 2015.

A. Mielke, Evolutionary relaxation of a twophase model, MiniWorkshop ``Scales in Plasticity'', November 8  14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

A. Mielke, Existence results in finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 19, 2015.

A. Mielke, Homogenizing the PenroseFife system via evolutionary $Gamma$convergence, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  20, 2015, Rome, Italy, May 19, 2015.

A. Mielke, Mathematical modeling for finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 16, 2015.

A. Mielke, The multiplicative strain decomposition in finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 17, 2015.

A. Mielke, Variational approaches and methods for dissipative material models with multiple scales, Workshop ``Analysis and Computation of Microstructure in Finite Plasticity'', May 4  5, 2015, Hausdorff Center for Mathematics, Bonn, May 4, 2015.

S. Heinz, Analysis and numerics of a phasetransformation model, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

M. Thomas, A stressdriven localsolution approach to quasistatic brittle delamination, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 91: Variational Methods for Evolution Equations, July 7  11, 2014, Madrid, Spain, July 7, 2014.

M. Thomas, Existence & stability results for rateindependent processes in viscoelastic materials, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, March 18, 2014.

M. Thomas, Existence and stability results for rateindependent processes in viscoelastic materials, Women in Partial Differential Equations & Calculus of Variations Workshop, March 6  8, 2014, University of Oxford, Mathematical Institute, UK, March 6, 2014.

M. Thomas, GENERIC for solids with dissipative interface processes, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), GAMM Juniors' Poster Session, FriedrichAlexander Universität ErlangenNürnberg, March 10  14, 2014.

M. Thomas, Rateindependent systems with viscosity and inertia: Existence and evolutionary Gammaconvergence, Workshop ``Variational Methods for Evolution'', December 14  20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, International Workshop ``Variational Modeling in Solid Mechanics'', September 22  24, 2014, University of Udine, Department of Mathematics and Informatics, Italy, September 23, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

M. Thomas, Stressdriven localsolution approach to quasistatic brittle delamination, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 11, 2014.

M. Thomas, Thermomechanical modeling of dissipative processes in elastic media via energy and entropy, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7  11, 2014, Madrid, Spain, July 8, 2014.

A. Mielke, Modeling jumps in rateindependent systems using balancedviscosity solutions, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

P. Gussmann, Linearized elasticity as $Gamma$limit of finite elasticity in the case of cracks, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Section ``Applied Analysis'', March 18  22, 2013, University of Novi Sad, Serbia, March 20, 2013.

D. Knees, Recent results in nonlinear elasticity and fracture mechanics, Universität der Bundeswehr, Institut für Mathematik und Bauinformatik, München, August 13, 2013.

CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013  Diffuse Interface Models, September 10  13, 2013, Levico Terme, Italy, September 11, 2013.

CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

M. Thomas, A stressdriven local solution approach to quasistatic brittle delamination, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 29, 2013.

M. Thomas, A stressdriven local solution approach to quasistatic brittle delamination, Seminar on Functional Analysis and Applications, International School of Advanced Studies (SISSA), Trieste, Italy, November 12, 2013.

M. Thomas, Existence & fine properties of solutions for rateindependent brittle damage models, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1  2, 2013.

A. Mielke, Rateindependent plasticity as vanishingviscosity limit for wiggly energy landscape, Workshop on Evolution Problems for Material Defects: Dislocations, Plasticity, and Fracture, September 30  October 4, 2013, International School of Advanced Studies (SISSA), Trieste, Italy, September 30, 2013.

J. Sprekels, PrandtlIshlinskii operators and elastoplasticity, Spring School on ``Rateindependent Evolutions and Hysteresis Modelling'', May 27  31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

P. Gussmann, Linearisierte Elastizität als Grenzwert finiter Elastizität im Falle von Schlitzgebieten, Jahrestagung der Deutsche MathematikerVereinigung (DMV), Studierendenkonferenz, September 17  20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 2012.

S. Heinz, Regularization and relaxation of timecontinuous problems in plasticity, 11th GAMM Seminar on Microstructures, Universität DuisburgEssen, January 20  21, 2012.

S. Heinz, Rigorous derivation of a dissipation for laminate microstructures, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Material Modelling in Solid Mechanics, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

D. Knees, Modeling and mathematical analysis of elastoplastic phenomena, Winter School on Modeling Complex Physical Systems with Nonlinear (S)PDE (DoM$^2$oS), February 27  March 2, 2012, Technische Universität Dortmund, Fakultät für Mathematik.

CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

CH. Kraus, CahnLarché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11  15, 2012, Frauenchiemsee, June 12, 2012.

CH. Kraus, Phasenfeldsysteme für Entmischungs und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

CH. Kraus, The Stefan problem with inhomogeneous and anisotropic GibbsThomson law, 6th European Congress of Mathematics, July 2  6, 2012, Cracow, Poland, July 5, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19  21, 2012, University of Udine, Department of Mathematics, Italy, September 21, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17  21, 2012, Cortona, Italy, September 17, 2012.

M. Thomas, Analytical aspects of rateindependent damage models with spatial BVregularization, Seminar, SISSA  International School for Advanced Studies, Functional Analysis and Applications, Trieste, Italy, November 28, 2012.

M. Thomas, Coupling of reactiondiffusion processes with thermomechanics using GENERIC, Winter School ``Calculus of Variations in Physics and Materials Science'', Würzburg, January 8  13, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11  15, 2012, Universität Regensburg, Frauenchiemsee, June 11, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, Seminar on Applied Mathematics, Università di Brescia, Dipartimento di Matematica, Italy, March 14, 2012.

M. Thomas, Mathematical methods in continuum mechanics of solids, COMMAS (Computational Mechanics of Materials and Structures) Summer School, October 8  12, 2012, Universität Stuttgart, Institut für Mechanik (Bauwesen).

M. Thomas, Modellierung und Analysis von Delaminationsprozessen, Sitzung des Wissenschaftlichen Beirats des WIAS, Berlin, October 5, 2012.

M. Thomas, Thermomechanical modeling via energy and entropy, Seminar on Applied Mathematics, University of Pavia, Department of Mathematics, Italy, February 14, 2012.

M. Thomas, Thermomechanical modeling via energy and entropy using GENERIC, Workshop ``Mechanics of Materials'', March 19  23, 2012, Mathematisches Forschungsinstitut Oberwolfach, March 22, 2012.

A. Mielke, Finitestrain viscoelasticity as a gradient flow, Analysis and Applications of PDEs: An 80th Birthday Meeting for Robin Knops, December 10  11, 2012, International Center for Mathematical Sciences, Edinburgh, UK, December 11, 2012.

A. Mielke, From smallstrain to finitestrain elastoplasticity via evolutionary Gamma convergence, Variational Models and Methods for Evolution, September 10  12, 2012, Centro Internazionale per la Ricerca Matematica (CIRM) and Istituto di Matematica Applicata e Tecnologie Informatiche/Consiglio Nazionale delle Ricerche (IMATICNR), Levico, Italy, September 11, 2012.

A. Mielke, Gamma convergence and evolution, International Conference ``Trends in Mathematical Analysis'', March 1  3, 2012, Politecnico di Milano, Dipartimento di Matematica ``F. Brioschi'', Italy, March 1, 2012.

A. Mielke, Linearized elastoplasticity is the evolutionary Gamma limit of finite elastoplasticity, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

A. Mielke, Smallstrain elastoplasticity is the evolutionary Gamma limit of finitestrain elastoplasticity, International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2012), September 3  6, 2012, Israel Institute of Technology (Technion), Faculty of Aerospace Engineering, Haifa, September 4, 2012.

S. Heinz, A model for the texture evolution in polycrystalline materials, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Solid Mechanics, April 18  21, 2011, Technische Universität Graz, Austria, April 19, 2011.

A. Petrov, Sur la modélisation mathématique de matériaux à mémoire de forme, Université de Lyon, Institut Camille Jordan, France, May 13, 2011.

S. Heinz, Regularizations and relaxations of timecontinuous problems in plasticity, Workshop der Forschergruppe 797 ``Analysis and Computation of Microstructure in Finite Plasticity'', Universität Bonn, Mathematisches Institut, November 14, 2011.

H. Hanke, Derivation of an effective damage model based on microstructure evolution, Workshop on Phase Separation, Damage and Fracture, September 21  23, 2011, WIAS, September 22, 2011.

CH. Heinemann, Existence results for CahnHilliard equations coupled with elasticity and damage, Workshop on Phase Separation, Damage and Fracture, September 21  23, 2011, WIAS, September 23, 2011.

CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18  22, 2011, Vancouver, Canada, July 18, 2011.

A. Mielke, An evolutionary elastoplastic plate model obtained via Mosco convergence, 10th GAMM Seminar on Microstructures, January 20  22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

A. Mielke, Complex hysteresis operators arising from homogenization and dimension reduction, 8th International Symposium on Hysteresis Modelling and Micromagnetics (HMM2011), Session ``Mathematics of Hysteresis II'', May 9  11, 2011, Università degli Studi di Trento, Centro Internazionale per la Ricerca Matematica, Levico Terme, Italy, May 10, 2011.

A. Mielke, Evolution for dissipative materials at finite strains, From Polymer Physics to Rubber Elasticity, January 17  19, 2011, Institut National de Recherche en Informatique et en Automatique (INRIA), Paris, France, January 19, 2011.

R. Müller, Finite element simulation of damage and phase separation, Workshop on Phase Separation, Damage and Fracture, September 21  23, 2011, WIAS, September 23, 2011.

M. Thomas, Delamination in viscoelastic materials with thermal effects, Workshop on Phase Separation, Damage and Fracture, September 21  23, 2011, WIAS, September 21, 2011.

M. Thomas, Delamination in viscoelastic materials with thermal effects, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, November 24, 2011.

A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7  12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

M. Liero, Rateindependent Kurzweil processes, Workshop ``Rateindependent Systems: Modeling, Analysis, and Computations'', August 30  September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 3, 2010.

A. Petrov, On a 3D model for shapememory alloys, Workshop ``Rateindependent Systems: Modeling, Analysis, and Computations'', August 30  September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 2, 2010.

M. Thomas, From damage to delamination in nonlinearly elastic materials, 6th Singular Days on Asymptotic Methods for PDEs, April 29  May 1, 2010, WIAS, May 1, 2010.

M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Workshop ``Microstructures in Solids: From Quantum Models to Continua'', March 14  20, 2010, Mathematisches Forschungsinstitut Oberwolfach, March 18, 2010.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified GibbsThomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22  26, 2010, Karlsruhe, March 23, 2010.

CH. Kraus, Inhomogeneous and anisotropic phasefield quantities in the sharp interface limit, 6th Singular Days 2010, April 29  May 1, 2010, WIAS, Berlin, April 30, 2010.

A. Mielke, A mathematical model for the evolution of microstructures in elastoplasticity, Fifth International Conference on Multiscale Materials Modeling, Symposium on Mathematical Methods, October 4  8, 2010, Fraunhofer Institut für Werkstoffmechanik (IWM), Freiburg, October 4, 2010.

A. Mielke, Approaches to finitestrain elastoplasticity, SIAM Conference on Mathematical Aspects of Materials Science (MS10), May 23  26, 2010, Philadelphia, USA, May 23, 2010.

A. Mielke, Geometrische Nichtlinearitäten und LieGruppen in der Elastoplastizität, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, January 27, 2010.

A. Mielke, Rateindependent plasticity as Gamma limit of a slow viscous gradient flow for wiggly discrete energy, ZweiStädteKolloquium zur Analysis, FriedrichAlexanderUniversität ErlangenNürnberg, Fachbereich Mathematik, November 26, 2010.

A. Mielke, Rateindependent plasticity as Gamma limit of a slow viscous gradient flow for wiggly energies, Partial Differential Equations Seminar, University of Oxford & Queen's College, Centre for Nonlinear PDE, UK, February 22, 2010.

A. Mielke, Rateindependent plasticity as Gamma limit of a slow viscous gradient flow for wiggly energies, Nečas Seminar on Continuum Mechanics, Jindrich Nečas Center for Mathematical Modeling, Prague, Czech Republic, November 8, 2010.

M. Thomas, Rateindependent damage and delamination processes, Workshop ``Rateindependent Systems: Modeling, Analysis, and Computations'', August 30  September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, August 31, 2010.

S. Heinz, A model for the evolution of laminates, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Young Researchers Minisymposium ``Mathematics and Mechanics of Microstructure Evolution in Finite Plasticity'', February 9  13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

S. Heinz, The evolution of laminates, 8th GAMM Seminar on Microstructures, January 15  17, 2009, Universität Regensburg, NWFI Mathematik, January 17, 2009.

A. Petrov, On existence for viscoelastodymanic problems with unilateral boundary conditions, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Session ``Applied analysis'', February 9  13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

A. Petrov, On the error estimates for spacetime discretizations of rateindependent processes, 8th GAMM Seminar on Microstructures, January 15  17, 2009, Universität Regensburg, NWFI Mathematik, January 17, 2009.

A. Petrov, On the existence and error bounds for spacetime discretizations of a 3D model for shapememory alloys, Lisbon University, Center for Mathematics and Fundamental Applications, Portugal, September 17, 2009.

A. Petrov, On the numerical approximation of a viscoelastic problem with unilateral constrains, 7th EUROMECH Solid Mechanics Conference (ESMC2009), Minisymposium on Contact Mechanics, September 7  11, 2009, Instituto Superior Técnico, Lisbon, Portugal, September 8, 2009.

H. Hanke, Gammalimits in rateindependent evolutionary problems and homogenization in gradient plasticity, DKSeminar ``Numerical Simulations in Technical Sciences'', Graz University of Technology, Faculty of Technical Mathematics and Technical Physics, Austria, November 18, 2009.

H. Hanke, Homogenization in gradient plasticity, ISIMM Workshop on Mathematical Problems of Solid Mechanics, October 8  9, 2009, Technische Universität Darmstadt, Fachbereich Mathematik, October 9, 2009.

A. Mielke, Vanishingviscosity approximation of rateindependent systems, Workshop ``Material Theories'', December 14  18, 2009, Mathematisches Forschungsinstitut Oberwolfach, December 15, 2009.

J. Polzehl, Sequential multiscale procedures for adaptive estimation, The 1st Institute of Mathematical Statistics Asia Pacific Rim Meeting, June 28  July 1, 2009, Seoul National University, Institute of Mathematical Statistics, Korea (Republic of), July 1, 2009.

A. Petrov, Error estimates for spacetime discretizations of a 3D model for shapememory materials, IUTAM Symposium ``Variational Concepts with Applications to the Mechanics of Materials'', September 22  26, 2008, RuhrUniversität Bochum, Lehrstuhl für allgemeine Mechanik, September 24, 2008.

A. Petrov, Existence and approximation for 3D model of thermally induced phase transformation in shapememory alloys, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), Session ``Material models in solids'', March 31  April 4, 2008, Universität Bremen, April 1, 2008.

A. Petrov, Some mathematical results for a model of thermallyinduced phase transformations in shapememory materials, sc MatheonICM Workshop on Free Boundaries and Materials Modeling, March 17  18, 2008, WIAS, March 18, 2008.

A. Petrov, On the convergence for kinetic variational inequality to quasistatic variational inequality with application to elasticplastic systems with hardening, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 17, 2007.

A. Petrov, Thermally driven phase transformation in shapememory alloys, Workshop ``Analysis and Numerics of RateIndependent Processes'', February 26  March 2, 2007, Mathematisches Forschungsinstitut Oberwolfach, February 27, 2007.

J. Sprekels, Models of phase transitions and hysteresis operators, Joint International Meeting UMIDMV 2007, Minisymposium ``Phase Transitions and Hysteresis in Free Boundary Problems'', June 18  22, 2007, Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, Italy, June 21, 2007.

A. Petrov, Mathematical result on the stability of elasticplastic systems with hardening, European Conference on Smart Systems, October 26  28, 2006, Researching Training Network "New Materials, Adaptive Systems and their Nonlinearities: Modelling, Control and Numerical Simulation" within the European Commission's 5th Framework Programme, Rome, Italy, October 27, 2006.

O. Klein, Asymptotic behavior for a phasefield model for thermoviscoplasticity involving outwards pointing hysteresis operators, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 26, 2006.

J. Sprekels, PrandtlIshlinskii hysteresis operators and 1D elastoplasticity, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 26, 2006.

O. Klein, Asymptotic behaviour for a phasefield model with hysteresis in thermoviscoplasticity, INdAM Workshop ``Dissipative Models in Phase Transitions'', September 5  11, 2004, Cortona, Italy, September 9, 2004.

O. Klein, Longtime behaviour of solutions to equations involving outwards pointing hysteresis operators, International Workshop on Hysteresis & MultiScale Asymptotics (HAMSA 2004), March 17  21, 2004, University College Cork, Ireland, March 19, 2004.

O. Klein, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, 4th International Symposium on Hysteresis and Micromagnetic Modeling (HMM2003), May 28  30, 2003, Universidad de Salamanca, Departamento de Física Aplicada, Spain, May 30, 2003.
Preprints im Fremdverlag

D. Knees, Griffithformula and Jintegral for a crack in a powerlaw hardening material, Preprint no. 2005/12, Universität Stuttgart, SFB 404, 2005.
Ansprechpartner
Mathematischer Kontext
 Analysis partieller Differentialgleichungen und Evolutionsgleichungen
 Funktionalanalysis und Operatortheorie
 HystereseOperatoren und ratenunabhängige Systeme
 Mehrskalenmodellierung und Hybridmodelle
 Modellierung, Analysis und Numerik von Phasenfeldmodellen
 Systeme partieller Differentialgleichungen: Modellierung, numerische Analysis und Simulation
 Variationsrechnung
Projekte/Drittmittel
 Analysis für thermomechanische Modelle mit internen Variablen
 Effektive Modelle für mikroskopisch strukturierte Materialien und Grenzflächen
 Elektromechanische Kopplung für Halbleiterbauelemente
 Nichtlinearer dynamischer Bruch: Modellierung, Analyse, Approximation und Anwendungen
 Störungszonennetzwerke und Skaleneigenschaften von Deformationsakkumulation