Overview: rate-independent processes

Time-dependent processes in physics, biology, and economics often exhibit a rate-independent input-output behavior. In those processes a rescaling of the input function with respect to time leads to the same rescaling of the response function.

Quite often, such processes are accompanied by the occurrence of hysteresis phenomena induced by inherent memory effects. For example, an elasto-plastic body that is deformed by some given load, may stay deformed if the load is removed. If periodic load cycles are considered, one may observe hysteresis-loops:

Figure 1: Load-deformation curve with elasto-plastic behavior
m6OK-hkurve-en.png

There are two methods to describe such processes considered at WIAS:
  • In order to describe the current state, one can add further internal variables to the ordinal variables, so that the modeling of the memory is not necessary. In this case, however, one must describe the solution concept using these internal variables explicitly.
  • For many cases one can find hysteresis operators directly describing the input-output-behavior, using only the observable variables. These operators avoid the direct use of internal variables in the formulation, but lead to a memory, i.e., they depend on the history of the considered process.
Figure 2: Input-output behavior for two processes with internal memory.

Modeling with rate-independent systems

Rate-independent systems are rate-independent, quasistatic evolution equations, formulated with an energy functional and a dissipation potential, where the dissipation potential is positively homogeneous of degree one in the rate of the internal variables. The theory typically describes the state of those systems by internal variables, so that an evolution theory in the state space can be developed, that does not need any memory. Typical evolution equations can be formulated as differential inclusions, e.g., a force balance between the dissipative forces as functions of the change rate of the state variables and the restoring forces. Quite often, these evolution equation have multiple solutions.

The analysis of rate-independent system may rely on the general theory of dissipative evolution systems. There are some additional difficulties since in general the solutions are no longer continuous. Especially for non-convex energy functionals the typical solution have jumps that must be additionally modeled. This can be done by the vanishing-viscosity method, where the introduction of physical (or artificial) viscosity generates continuous solutions, whose limits define a new class of solutions. A further strategy relies on so-called energetic solutions, where the dissipative forces are described by a dissipation potential. The energetic solutions are described by a stationary global stability condition and an energy balance.

Modeling with hysteresis operators

Hysteresis operators map time-dependent functions, the so-called input-functions, to time-dependent functions, the so-called output-functions, such that the following holds:

  1. The operator is rate-independent: For every input-function and all admissible time-transformations there holds: The output-function that is obtained from a time-transformed input-function is exactly the time-transformed version of the output to the original input-function.
  2. The operator is causal: For every input-function and all times there holds: The value of the output-function at a given time only depends on the values of the input-function at previous times.

If data in models with hysteresis operators, like, e.g., models for electric-magnetic-mechanic components, are determined from measurements, these data are afflicted with uncertainness. Therefore, the methods of Uncertainty Quantification are applied to such models, see, e.g., Klein 2016.


Highlights

Modeling with rate-independent systems

The existence of energetic solutions can be shown for a broad range of applications, for example in elasto-plastic materials (including situations with large deformations), in shape memory alloys, damage models, and for fracture growth, cf. Mielke,Roubíček 2015.

Moreover, for a large abstract class of rate-independent systems, is was proved that the vanishing-viscosity limit can be identified by considering the balanced viscosity solutions, cf. Mielke, Rossi, Savaré 2016.

Modeling with hysteresis operators

The representation result in Brokate-Sprekels 1996, for hysteresis operators acting on scalar-valued input functions, was generalized to hysteresis operators acting on special vector-valued input functions, and also further extended, cf. Klein 2014.

It was proved that for every extension of the sweeping processes to input functions of bounded variation it holds that the extension is not continuous when its domain is endowed with the strict topology of and its codomain is endowed with the L1-topology, cf. Klein, Recupero 2016.

Publications

  Monographs

  • M. Dimian, P. Gurevich, O. Klein, D. Knees, D. Rachinskii, S. Tikhomirov, eds., MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, 252 pages, (Collection Published).

  • A. Mielke, T. Roubíček, Rate-independent 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 rate-independent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rate-independent 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 adhesive-type contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to rate-dependent 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 "rate-independent 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.

  • G. Dal Maso, A. Mielke, U. Stefanelli, eds., Rate-independent Evolutions, 6 (No. 1) of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Sciences, Springfield, 2013, 275 pages, (Collection Published).

  • P. Colli, A. Damlamian, N. Kenmochi, M. Mimura, J. Sprekels, eds., Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation, 29 of Gakuto International Series Mathematical Sciences and Applications, Gakkōtosho, Tokyo, 2008, 475 pages, (Collection Published).

  Articles in Refereed Journals

  • M. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017) pp. 1303--1327.
    Abstract
    We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

  • R. Rossi, M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017) pp. 1419--1494.
    Abstract
    We address the analysis of an abstract system coupling a rate-independet 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 time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

  • R. Rossi, M. Thomas, From adhesive to brittle delamination in visco-elastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017) pp. 1489--1546, DOI 10.1142/S0218202517500257 .
    Abstract
    In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent 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 rate-independent systems to the present mixed rate-dependent/rate-independent 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.

  • M. Thomas, Ch. Zanini, Cohesive zone-type delamination in visco-elasticity, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017) pp. 1487--1517, DOI 10.20347/WIAS.PREPRINT.2350 .
    Abstract
    We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic 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 non-penetration 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&Thomas15WIAS-Preprint2113].

  • O. Klein, A representation result for rate-independent systems, Phys. B, 486 (2016) pp. 81--83.

  • A. Mielke, R. Rossi, G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, Journal of the European Mathematical Society (JEMS), 18 (2016) pp. 2107--2165.
    Abstract
    Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for in nite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their di erentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on re ned lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

  • R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015) pp. 1--59.
    Abstract
    We address the analysis of a model for brittle delamination of two visco-elastic 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, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015) pp. 645--663.
    Abstract
    A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent 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 semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional 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.

  • O. Klein, On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2591--2614.
    Abstract
    In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e., being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function. In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space. of the set of all strings of pairs of elements of the vector space.

  • O. Klein, A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials, Phys. B, 435 (2014) pp. 113--115.
    Abstract
    In this work, hysteresis operators mapping continuous vector-valued input functions being piecewise monotaffine, i.e. being piecewise the composition of a monotone with an affine function, to vector-valued output functions are considered. It is shown that the operator can be generated by a uniquely defined function on the convexity triple free strings. A formulation of a congruence property for periodic inputs is presented and reformulated as a condition for the generating string function.

  • A. Mielke, S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XIII (2014) pp. 67--135.
    Abstract
    We consider quasilinear parabolic systems with a nonsmooth rate-independent 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 vanishing-viscosity 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 vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

  • 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. 483--493.
    Abstract
    Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work (Krasnosel'skiĭ-Pokrovskiĭ 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.

  • D. Knees, R. Rossi, Ch. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013) pp. 565--616.
    Abstract
    We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

  • D. Knees, A. Schröder, Computational aspects of quasi-static crack propagation, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 63--99.
    Abstract
    The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

  • M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 235--255.
    Abstract
    An existence result for energetic solutions of rate-independent 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∈ W1,r(Ω) with r∈(1,∞) for Ω⊂Rd. We now cover the case r=1. The lack of compactness in W1,1(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 Γ-limit of functionals of Modica-Mortola type.

  • M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$-convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012) pp. 437--457.
    Abstract
    This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. 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 which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

  • A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasi-static damage evolution, Archive for Rational Mechanics and Analysis, 203 (2012) pp. 415--453.

  • S. Bartels, A. Mielke, T. Roubíček, Quasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM Journal on Numerical Analysis, 50 (2012) pp. 951--976.
    Abstract
    The quasistatic rate-independent evolution of the Prager--Ziegler-type model of linearized plasticity with hardening is shown to converge to the rate-independent evolution of the Prandtl--Reuss 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 Gamma-convergence for rate-independent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A two-sided energy inequality is a posteriori verified to document experimental convergence rates.

  • O. Klein , Representation of hysteresis operators for vector-valued inputs by functions on strings, Phys. B, 407 (2012) pp. 1399--1400.

  • O. Klein, Representation of hysteresis operators acting on vector-valued monotaffine functions, Advances in Mathematical Sciences and Applications, 22 (2012) pp. 471--500.
    Abstract
    In Brokate-Sprekels-1996, it was shown that scalar-valued hysteresis operators for scalar-valued continuous piecewise monotone input functions can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers. Using this representation, various properties of these hysteresis operators were investigated. In this work, it is shown that a similar representation result can be derived for hysteresis operators dealing with inputs in a general topological linear vector space. Introducing a new class of functions, the so-called emphmonotaffine functions, which can be considered as a vector generalization of monotone scalar functions, and the convexity triple free strings on a vector space as a generalization of the alternating strings allows to formulate the corresponding representation result. As an example for the application of the representation result, a vectorial formulation of the second and third Madelung rule are discussed.

  • A. Mielke, R. Rossi, G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM. Control, Optimisation and Calculus of Variations, 18 (2012) pp. 36--80.
    Abstract
    In the nonconvex case solutions of rate-independent 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 finite-dimensional case we study the vanishing-viscosity 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 rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent 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 time-continuous and for the time-discretized 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 rate-independent systems.

  • A. Mielke, R. Rossi, G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan Journal of Mathematics, 80 (2012) pp. 381--410.
    Abstract
    We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.

  • 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. 235--273.
    Abstract
    Brittle Griffith-type delamination of compounds is deduced by means of Gamma-convergence from partial, isotropic damage of three-specimen-sandwich-structures 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 rate-independent. 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 delamination-gradient is supressed. Noninterpenetration- and transmission-conditions along the interface are obtained.

  • A. Mielke, L. Truskinovsky, From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Archive for Rational Mechanics and Analysis, 203 (2012) pp. 577--619.
    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 visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete to continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic solid transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shape-memory 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 rate-independent dissipation from viscous systems with wiggly energies, Continuum Mechanics and Thermodynamics, 24 (2012) pp. 591--606.
    Abstract
    We consider the passage from viscous system to rate-independent 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 Prandtl--Ishlinskii operators arising from homogenization and dimension reduction, Phys. B, 407 (2012) pp. 1330--1335.
    Abstract
    We consider rate-independent 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 Prandtl-Ishlinskii operator

  • M. Liero, A. Mielke, An evolutionary elastoplastic plate model derived via $Gamma$-convergence, Mathematical Models & Methods in Applied Sciences, 21 (2011) pp. 1961--1986.
    Abstract
    This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional 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 Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage 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 energy-storage 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 Mosco-convergent quadratic energies.

  • A. Mielke, Formulation of thermo-elastic dissipative material behavior using GENERIC, Continuum Mechanics and Thermodynamics, 23 (2011) pp. 233--256.
    Abstract
    The theory of GENERIC (general equations for non-equilibrium 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.

  • 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. 88--112.
    Abstract
    In this paper an existence result for energetic solutions of rate-independent 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 [Mielke-Roubicek 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 Hölder-continuity of solutions with respect to time.

  • P. Gruber, D. Knees, S. Nesenenko, M. Thomas, Analytical and numerical aspects of time-dependent models with internal variables, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010) pp. 861--902.
    Abstract
    In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff 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 rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.

  • M. Eleuteri, J. Kopfová, P. Krejčí, Magnetohydrodynamic flow with hysteresis, SIAM Journal on Mathematical Analysis, 41 (2009) pp. 435--464.
    Abstract
    We consider a model system describing the 2D flow of a conducting fluid surrounded by a ferromagnetic solid under the influence of the hysteretic response of the surrounding medium. We assume that this influence can be represented by the Preisach hysteresis operator. Existence and uniqueness of solutions for the resulting system of PDEs with hysteresis nonlinearities is established in the convexity domain of the Preisach operator.

  • 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. 537--550.

  • P. Krejčí, M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009) pp. 117--145.
    Abstract
    The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

  • M. Brokate, M. Eleuteri, P. Krejčí, On a model for electromagnetic processes inside and outside a ferromagnetic body, Mathematical Methods in the Applied Sciences, 31 (2008) pp. 1545-1567.

  • M. Eleuteri, J. Kopfová, P. Krejčí, On a model with hysteresis arising in magnetohydrodynamics, Phys. B, 403 (2008) pp. 448-450.

  • M. Eleuteri, O. Klein, P. Krejčí, Outward pointing inverse Preisach operators, Phys. B, 403 (2008) pp. 254-256.

  • P. Krejčí, K. Kuhnen, Existence, uniqueness and $L_infty$-stability of the Prandtl--Ishlinskii hysteresis and creep compensator, European Journal of Control, 14 (2008) pp. 409--417.

  • P. Krejčí, J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl--Ishlinskii hysteresis operators, Discrete and Continuous Dynamical Systems -- Series S, 1 (2008) pp. 283--292.

  • P. Krejčí, A higher order energy bound in a singular Preisach circuit, Phys. B, 403 (2008) pp. 297-300.

  • M. Eleuteri, P. Krejčí, An asymptotic convergence result for a system of partial differential equations with hysteresis, Communications on Pure and Applied Analysis, 6 (2007) pp. 1131-1143.

  • M. Eleuteri, Well-posedness results for a class of partial differential equations with hysteresis arising in electromagnetism, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 8 (2007) pp. 1494-1511.

  • P. Krejčí, J. Sprekels, Elastic-ideally plastic beams and Prandtl--Ishlinskii hysteresis operators, Mathematical Methods in the Applied Sciences, 30 (2007) pp. 2371--2393.

  • P. Krejčí, J. Sprekels, Long time behaviour of a singular phase transition model, Discrete and Continuous Dynamical Systems, 15 (2006) pp. 1119-1135.

  • O. Klein, Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity, Applications of Mathematics, 49 (2004) pp. 309--341.

  • P. Krejčí, J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Advances in Mathematical Sciences and Applications, 14 (2004) pp. 593--612.

  • P. Krejčí, J. Sprekels, U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Advances in Mathematical Sciences and Applications, 13 (2003) pp. 695-712.

  • 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. 755--785.

  • J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, Journal of Mathematical Analysis and Applications, 279 (2003) pp. 97-110.

  • N. Kenmochi, J. Sprekels, Phase-field systems with vectorial order parameters including diffusional hysteresis effects, Communications on Pure and Applied Analysis, 1 (2002) pp. 495-511.

  • P. Krejčí, J. Sprekels, U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermovisco-plasticity, SIAM Journal on Mathematical Analysis, 34 (2002) pp. 409-434.

  • P. Krejčí, J. Sprekels, Parabolic regularization of differential inclusions and the stop operator, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 4 (2002) pp. 423-435.

  • J. Sprekels, P. Krejčí, Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics, Mathematical Methods in the Applied Sciences, 25 (2002) pp. 309-325.

  • P. Krejčí, J. Sprekels, S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis, Journal of Differential Equations, 175 (2001) pp. 88--107.

  • P. Krejčí, J. Sprekels, On a class of multi-dimensional Prandtl-Ishlinskii operators, Phys. B, 306 (2001) pp. 185-190.

  • P. Krejčí, J. Sprekels, A hysteresis approach to phase-field models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 39 (2000) pp. 569--586.

  • P. Krejčí, J. Sprekels, Phase-field models with hysteresis, Journal of Mathematical Analysis and Applications, 252 (2000) pp. 198--219.

  • G. Gilardi, P. Krejčí, J. Sprekels, Hysteresis in phase-field models with thermal memory, Mathematical Methods in the Applied Sciences, 23 (2000) pp. 909--922.

  Contributions to Collected Editions

  • G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast 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. 012009/1--012009/20.
    Abstract
    This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt 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 time-dependent Dirichlet loading.

  • O. Klein, V. Recupero, Hausdorff metric BV discontinuity of sweeping processes, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes & HYSteresis (MURPHYS) & the 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, 2016, pp. 012006/1--012006/12, DOI 10.1088/1742-6596/727/1/012006 .
    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, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for Multi-Field and Multi-Scale 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. 135--160.
    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 non-isothermal 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 time-incremental minimization procedures, where the entropy-production 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 boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 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.

  • M. Thomas, Modeling and analysis of rate-independent damage and delamination processes, in: Proceedings of the 19th International Conference on Computer Methods in Mechanics (online only), 2011, pp. 1--6.

  • A. Mielke, Geometry and thermodynamics for the coupling of quantum mechanics and dissipative systems, in: Applied Dynamics and Geometric Mechanics, Workshop, August 14--20, 2011, 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 2260--2263.

  • M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, in: Microstructures in Solids: From Quantum Models to Continua, Workshop, March 14--20, 2010, 7 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2010, pp. 783--785.

  • O. Klein, Outward pointing properties for vectorial hysteresis operators and some applications, in: International Workshop on Multi-Rate Processes and Hysteresis'', March 31 -- April 5, 2008, Cork, Ireland, M.P. Mortell, R.E. O'Malley, A. Pokrovskii, D. Rachinskii, V.A. Sobolev, eds., 138 of J. Phys.: Conf. Ser., Inst. Phys., 2008, pp. 012009/1--012009/14.

  • P. Krejčí, Hysteresis rarefaction in the Riemann problem, in: International Workshop on Multi-Rate Processes and Hysteresis, March 31 -- April 5, 2008, Cork, Ireland, M.P. Mortell, R.E. O'Malley, A. Pokrovskii, D. Rachinskii, V.A. Sobolev, eds., 138 of J. Phys.: Conf. Ser., Inst. Phys., 2008, pp. 012010/1--012010/10.

  • M. Eleuteri, An existence result for a P.D.E. with hysteresis, convection and a nonlinear boundary condition, in: Dynamical Systems, Differential Equations and Applications. Dedicated to the 6th AIMS Conference, Poitiers, France, B. Belinsky, K. Lan, X. Lu, A. Miranville, R. Shivaji, eds., American Institute of Mathematical Sciences, Springfield, 2007, pp. 344-353.

  • M. Eleuteri, Some P.D.E.s with hysteresis, in: Free Boundary Problems. Theory and Applications, I.N. Figueiredo, J.F. Rodrigues, L. Santos, eds., 154 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 2007, pp. 159-168.

  • P. Krejčí, Asymptotic hysteresis patterns in a phase separation problem, in: Free Boundary Problems. Theory and Applications, I.N. Figueiredo, J.F. Rodrigues, L. Santos, eds., 154 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 2007, pp. 283-290.

  • M. Brokate, O. Klein, P. Krejčí, Outward pointing properties for Preisach operators, in: Proceedings of the Fifth International Symposium on Hysteresis and Micromagnetic Modeling, Budapest, Hungary, 30 May--01 June 2005, A. Ivanyi, G. Kádár, eds., 372 of Phys. B, Elsevier, Amsterdam, 2006, pp. 5--8.

  • J. Sprekels, Asymptotic behavior of the stop hysteresis operator, in: Applications of Asymptotic Analysis, Workshop, June 18--24, 2006, 3 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2006, pp. 1696-1697.

  • 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, 28--30 May 2003, L. Lopez-Dias, L. Torres, O. Alejos, eds., 343 of Physica B: Condensed Matter, Elsevier B.V., 2004, pp. 53-58.

  • J. Sprekels, D. Tiba, Optimization of differential systems with hysteresis, in: Analysis and Optimization of Differential Systems, IFIP TC7/WG7.2 International Working Conference on Analysis and Optimization of Differential Systems, September 10--14, 2002, Constanta, Romania, V. Barbu, I. Lasiecka, D. Tiba, C. Varsan, eds., Kluwer Academic Publishers, Boston, 2003, pp. 387--398.

  • P. Krejčí, J. Sprekels, Phase-field systems and vector hysteresis operators, in: Free Boundary Problems: Theory and Applications II, N. Kenmochi, ed., 14 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, Tokyo, 2000, pp. 295--310.

  Preprints, Reports, Technical Reports

  • R. Rossi, M. Thomas, From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination, Preprint no. 2409, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2409 .
    Abstract, PDF (291 kByte)
    We revisit the weak, energetic-type existence results obtained in [Rossi/Thomas-ESAIM-COCV-21(1):1-59,2015] for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence 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 time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.

  • A. Mielke, Three examples concerning the interaction of dry friction and oscillations, Preprint no. 2405, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2405 .
    Abstract, PDF (3492 kByte)
    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 state-dependent 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.

  • M. Thomas, A comparison of delamination models: Modeling, properties, and applications, Preprint no. 2393, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2393 .
    Abstract, PDF (140 kByte)
    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.

  • S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$-regularization, Preprint no. 2388, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2388 .
    Abstract, PDF (566 kByte)
    We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM 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 rate-independent system. Moreover, we present our numerical results for two benchmark problems.

  • O. Klein, Uncertainty quantification for hysteresis operators and a model for magneto-mechanical hysteresis, Preprint no. 2246, WIAS, Berlin, 2016.
    Abstract, PDF (708 kByte)
    Many models for magneto-mechanical components involve hysteresis operators. The parameter within these operators have to be identified from measurements and are therefore subject to uncertainties. To quantify the influence of these uncertainties, the parameter in the hysteresis operator are considered as functions of random variables. Combining this with the hysteresis operator, we get new random variables and we can compute stochastic properties of the output of the model. For two hysteresis operators corresponding numerical results are presented in this paper. Moreover, the influence of the variation of the parameters in a model for a magneto-mechanical component is investigated.

  • B. Albers, P. Krejčí, E. Rocca, Solvability of an unsaturated porous media flow problem with thermomechanical interaction, Preprint no. 2211, WIAS, Berlin, 2016.
    Abstract, PDF (259 kByte)
    A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.

  Talks, Poster

  • M. Thomas, Rate-independent delamination processes in visco-elasticity, Miniworkshop on Dislocations, Plasticity, and Fracture, February 13 - 16, 2017, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, February 15, 2017.

  • A. Mielke, On self-induced 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.

  • M. Heida, Stochastic homogenization of 1-homogeneous 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 rate-independent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

  • M. Heida, Stochastic homogenization of rate-independent systems, Joint Annual Meeting of DMV and GAMM, Session ``Multiscales and Homogenization'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Delamination models in visco-elastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

  • M. Thomas, Coupling rate-independent and rate-dependent 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 rate-independent and rate-dependent processes: Damage & delamination in visco-elastodynamics, 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 visco-elastodynamics, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, special session ``Rate-dependent and Rate-independent 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, From adhesive contact to brittle delamination in visco-elastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • M. Thomas, Non-smooth 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, Rate-independent evolution of sets, INdAM-ISIMM 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, Rate-independent 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 magneto-mechanical hysteresis, Conference ``Advances in Magnetics'' (AIM) 2016, March 14 - 16, 2016, Bormio, Italy, March 14, 2016.

  • 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 failure---towards 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 rate-independent and rate-dependent processes: Existence results, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Pavia, Italy, March 5, 2015.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Evolutionary Gamma-convergence 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 rate-independent and rate-dependent 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 rate-independent and rate-dependent 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 visco-elastodynamics, 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, Rate-independent damage models with spatial BV-regularization --- Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

  • O. Klein, A representation result for rate-independent 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 finite-strain 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 rate-independent 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, Existence results in finite-strain 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, Mathematical modeling for finite-strain 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 finite-strain 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.

  • M. Thomas, A stress-driven local-solution 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 rate-independent processes in viscoelastic materials, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, March 18, 2014.

  • M. Thomas, Existence and stability results for rate-independent 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, Friedrich-Alexander Universität Erlangen-Nürnberg, March 10 - 14, 2014.

  • M. Thomas, Rate-independent systems with viscosity and inertia: Existence and evolutionary Gamma-convergence, Workshop ``Variational Methods for Evolution'', December 14 - 20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

  • M. Thomas, Rate-independent, partial damage in thermo-viscoelastic materials, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 8, 2014.

  • M. Thomas, Rate-independent, partial damage in thermo-viscoelastic 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, Rate-independent, partial damage in thermo-viscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

  • M. Thomas, Stress-driven local-solution 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, Friedrich-Alexander Universität Erlangen-Nürnberg, March 11, 2014.

  • O. Klein, Classification of hysteresis operators for vector-valued inputs by using their representation as functions on strings, RIPE60 -- Rate Independent Processes and Evolution Workshop, June 24 - 26, 2014, Prague, Czech Republic, June 24, 2014.

  • O. Klein, String-representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions, MURPHYS-HSFS-2014, The 7th Multi-Rate Processes and Hysteresis Workshop, in conjunction with the Second International Workshop on Hysteresis and Slow-Fast Systems, April 7 - 11, 2014, Weierstrass Institute, Berlin, April 11, 2014.

  • A. Mielke, Modeling jumps in rate-independent systems using balanced-viscosity solutions, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 8, 2014.

  • S. Heinz, On a way to control oscillations for a special evolution equation, Conference ``Nonlinearities'', June 10 - 14, 2013, University of Warsaw, Institute of Mathematics, Male Ciche, Poland, June 11, 2013.

  • D. Knees, A vanishing viscosity approach to a rate-independent damage model, Seminar ``Wissenschaftliches Rechnen'', Technische Universität Dortmund, Fachbereich Mathematik, January 31, 2013.

  • D. Knees, Crack evolution models based on the Griffith criterion, Workshop on Mathematical Aspects of Continuum Mechanics, October 12 - 14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 13, 2013.

  • D. Knees, Global spatial regularity results for crack with contact and application to a fracture evolution model, Oberseminar Nichtlineare Analysis, Universität Köln, Mathematisches Institut, October 28, 2013.

  • D. Knees, Modeling and analysis of crack evolution based on the Griffith criterion, Nonlinear Analysis Seminar, Keio University of Science, Yokohama, Japan, October 9, 2013.

  • D. Knees, On energy release rates for nonlinearly elastic materials, Workshop on Mathematical Aspects of Continuum Mechanics, October 12 - 14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 12, 2013.

  • M. Thomas, Fine properties of solutions for rate-independent brittle damage models, XXIII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, Italy, February 3 - 8, 2013.

  • M. Thomas, Local versus energetic solutions in rate-independent brittle delamination, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 13, 2013.

  • M. Thomas, Rate-independent damage models with spatial BV-regularization -- Existence & fine properties of solutions, Oberseminar zur Analysis, Universität Duisburg-Essen, Fachbereich Mathematik, Essen, January 24, 2013.

  • M. Thomas, A stress-driven 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 stress-driven 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 rate-independent brittle damage models, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1 - 2, 2013.

  • O. Klein, A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials, 9th International Symposium on Hysteresis Modelling and Micromagnetics (HMM 2013), May 13 - 15, 2013, Taormina, Italy, May 13, 2013.

  • O. Klein, Representation of hysteresis operators for vector-valued monotaffine inputs by functions on strings, Politecnico di Torino, DISMA Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy, April 23, 2013.

  • A. Mielke, Emergence of rate independence in gradient flows with wiggly energies, SIAM Conference on Mathematical Aspects of Materials Science (MS13), Minisymposium ``The Origins of Hysteresis in Materials'' (MS56), June 9 - 12, 2013, Philadelphia, USA, June 12, 2013.

  • J. Sprekels, Prandtl--Ishlinskii operators and elastoplasticity, Spring School on ``Rate-independent Evolutions and Hysteresis Modelling'', May 27 - 31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

  • G. Witterstein, Compressible phase change flows and the existence of transition profiles, PDEs for multiphase advanced materials, ADMAT2012, September 17 - 21, 2012, Cortona, Italy, September 18, 2012.

  • P.-É. Druet, Some problems associated with the second order optimal shape of a crystallisation interface, PDEs for multiphase advanced materials, ADMAT2012, September 17 - 21, 2012, Cortona, Italy, September 20, 2012.

  • D. Knees, A vanishing viscosity approach in fracture mechanics, Nonlocal Models and Peridynamics, November 5 - 7, 2012, Technische Universität Berlin, Institut für Mathematik, November 5, 2012.

  • M. Thomas, Analytical aspects of rate-independent damage models with spatial BV-regularization, Seminar, SISSA -- International School for Advanced Studies, Functional Analysis and Applications, Trieste, Italy, November 28, 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).

  • W. Dreyer, Sharp limits of diffuse interface models in the context of energy storage problems, PDEs for Multiphase Advanced Materials (ADMAT2012), September 17 - 21, 2012, Cortona, Italy, September 18, 2012.

  • O. Klein, A representation result for hysteresis operators acting on vector-valued continuous, piecewise monotaffine input functions, 6th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2012), May 21 - 24, 2012, Stefan cel Mare University, Suceava, Romania, May 22, 2012.

  • O. Klein, Representation of hysteresis operators for vector-valued inputs by string functions, PDEs for multiphase advanced materials, ADMAT2012, September 17 - 21, 2012, Cortona, Italy, September 17, 2012.

  • A. Mielke, From small-strain to finite-strain 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 (IMATI-CNR), 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, Small-strain elastoplasticity is the evolutionary Gamma limit of finite-strain 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.

  • J. Sprekels, Mathematical modeling of hysteresis, Seminar ``Thermodynamische Modellierung und Analyse von Phasenübergängen'', WIAS, Berlin, November 13, 2012.

  • J. Sprekels, Mathematical modeling of hysteresis - Part II, Seminar ``Thermodynamische Modellierung und Analyse von Phasenübergängen'', WIAS, Berlin, December 11, 2012.

  • O. Klein, Representation of hysteresis operators for vector-valued inputs by functions on strings, International Symposium on Hysteresis Modelling and Micromagnetics (HMM 2011), Levico (Trento), Italy, May 9 - 11, 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.

  • A. Mielke, Multiscale problems in systems driven by functionals, ISAM-TopMath Summer School 2011 on Variational Methods, September 12 - 16, 2011, Technische Universität München, Fakultät für Mathematik.

  • M. Thomas, Modeling and analysis of rate-independent damage and delamination processes, 19th International Conference on Computer Methods in Mechanics, Minisymposium ``Growth Phenomena and Evolution of Microstructures. Applications in Solids'', May 9 - 12, 2011, Warsaw University of Technology, Poland, May 11, 2011.

  • O. Klein, Hysteresis operators for vector-valued inputs and their representation by functions on strings, International Workshop on Hysteresis and Slow-Fast Systems (HSFS-2011), December 12 - 14, 2011, Lutherstadt Wittenberg, December 14, 2011.

  • J. Sprekels, Phase field models and hysteresis operators, Trends in Thermodynamics and Materials Theory 2011, December 15 - 17, 2011, Technische Universität Berlin, December 16, 2011.

  • M. Liero, Rate-independent Kurzweil processes, Workshop ``Rate-independent 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 shape-memory alloys, Workshop ``Rate-independent 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.

  • 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, Rate-independent plasticity as Gamma limit of a slow viscous gradient flow for wiggly discrete energy, Zwei-Städte-Kolloquium zur Analysis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Fachbereich Mathematik, November 26, 2010.

  • A. Mielke, Rate-independent 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, Rate-independent 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, Rate-independent damage and delamination processes, Workshop ``Rate-independent Systems: Modeling, Analysis, and Computations'', August 30 - September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, August 31, 2010.

  • O. Klein, Outward pointing properties for vectorial hysteresis operators and some applications, International Workshop on Multi-Rate Processes & Hysteresis, March 31 - April 5, 2008, University College Cork, Ireland, April 4, 2008.

  • J. Sprekels, Oscillating elastoplastic bodies: Dimensional reduction, hysteresis operators, existence results, Direct, Inverse and Control Problems for PDE's (DICOP 08), September 22 - 26, 2008, Cortona, Italy, September 22, 2008.

  • J. Sprekels, Oscillating thin elastoplastic bodies: Dimensional reduction, hysteresis operators, existence results, Seminar Partial Differential Equations: Models and Applications, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, May 20, 2008.

  • J. Sprekels, Models of phase transitions and hysteresis operators, Joint International Meeting UMI-DMV 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.

  • O. Klein, Asymptotic behavior for a phase-field model for thermo-visco-plasticity 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.

  • O. Klein, Outwards pointing properties for Preisach operators, International Workshop on Multi-Rate Processes & Hysteresis (MURPHYS 2006), April 3 - 7, 2006, University College Cork, Ireland, April 4, 2006.

  • J. Sprekels, Asymptotics of the stop hysteresis operator, Workshop ``Applications of Asymptotic Analysis'', June 18 - 22, 2006, Mathematisches Forschungsinstitut Oberwolfach, June 21, 2006.

  • J. Sprekels, Prandtl--Ishlinskii 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, Outward pointing properties for Preisach operators, 5th International Symposium on Hysteresis and Micromagnetic Modeling (HMM 2005), May 30 - June 1, 2005, Budapest, Hungary, May 30, 2005.

  • O. Klein, Asymptotic behaviour for a phase-field model with hysteresis in thermo-visco-plasticity, INdAM Workshop ``Dissipative Models in Phase Transitions'', September 5 - 11, 2004, Cortona, Italy, September 9, 2004.

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

  • J. Sprekels, On nonlocal phase-field models, Workshop ``Thermodynamische Materialtheorien'', December 12 - 15, 2004, Mathematisches Forschungsinstitut Oberwolfach, December 14, 2004.

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

  • J. Sprekels, Mathematical modelling of hysteresis phenomena, February 18 - 20, 2003, Chiba University, Department of Mathematics, Japan.

  • J. Sprekels, Hysteresis operators in phase-field modelling, Workshop ``Phasenübergänge'', April 29 - May 4, 2001, Mathematisches Forschungsinstitut Oberwolfach, May 1, 2001.

  • J. Sprekels, Hysteresis operators in phase-field systems, University of Warsaw, Interdisciplinary Centre for Mathematics and Computer Modelling, Poland, January 5, 2001.

  • J. Sprekels, On nonlocal phase transition models for non-conserved order parameters, Università di Pavia, Dipartimento di Matematica, Italy, September 19, 2001.

  • J. Sprekels, Phase-field systems and vector hysteresis operators, Workshop "`Phase Transitions and Interfaces in Evolution Equations: Analysis, Control and Approximation"', February 8 - 12, 2000, Santa Margherita Ligure, Italy, February 11, 2000.

  • J. Sprekels, Phase-field systems with hysteresis, Konferenz "`Evolution Equations 2000: Applications to Physics, Industry, Life Sciences and Economics"', October 30 - 31, 2000, Levico, Italy, October 30, 2000.

  External Preprints

  • A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasi-static damage evolution, Preprint no. 28PV10/26/0, Istituto di Matematica Applicata e Technologie Informatiche, 2010.