Overview
The variety of models and numerical methods within the theory of damage processes reflects the variety of the involved physical properties of the material. Each model has, depending on the mechanism that dominates the evolution, its benefits and drawbacks. For complex materials where phase separation and heat transport play an important role beside its elastic properties, continuum models have proven to be successful. These can be derived from variational (in)equalities and are in compliance with the laws of thermodynamics. Within this theory, the degree of damage is described by inner variables that represents the local structural integrity. The dynamics of this order parameter has to obey the Griffith criterion.Different types of damages models were studied in our research group. This included rateindependent models that were developed for brittle materials but also ratedependent models. The latter are of particular use in fatigue failure analysis. Furthermore, different types of partial and complete damage were considered, as well as crack models. The partial damage models are, for instance, obtained by regularization of a complete damage model. This is important for numerical simulations. For the associated system of partial differential equations, we were engaged with existence, uniqueness, smoothness properties of solutions, with derivations of effective models via homogenization, and with their simulation on computers via numerical methods. In the following, two types of damage and crack models that are considered at the WIAS are explained in more detail.
Phase Field Models
Within the framework of phase field theory damage is represented by an order parameter which reflects the local degree of structural integrity. In the cases we study this is a scalarvalued function that has values inside the interval [0,1]. A value of 1 is interpreted is a completely intact material, whereas 0 gives the completely disintegrated state. In contrast to other damage models that have to keep track of the crack manifold, in the field approach even complex crack patterns are described easily. It can be employed to approximate sharpinterface crack models, where the interfaces are smeared out on a length scale not relevant to the considered physical processes. Alternatively, a damagephase field model can be obtained as a result of a mathematical homogenization process (see WIAS preprints 1749 and 1880 ). In this case, values between 0 and 1 for the order parameter are interpreted as the presence of microcracks and microvoids. consequently, the material possesses partial stability. By making the assumption that spontaneous healing of the material is not allowed, the evolution of the damageorder parameter is unidirectional and bounded on the interval [0,1]. These constraints result in nonsmooth evolution equations and make it necessary to develop novel notions of weak solutions. In the rateindependent case, the evolution can be characterized by a stability criterion and an energy balance equation. We refer to WIAS preprints 1633 and 1867 for further details. By considering viscous terms of the damage variable, the evolution becomes ratedependent. A weak notion for the partial damage case has been developed in WIAS preprint 1520, which consists of a variational inequality together with an energy inequality. Existence of such a weak solution has been proven. The complete damage case has also been investigated by performing a degenerate limit procedure as done in WIAS preprint 1722. Concerning the coupling of damage processes with further physical mechanisms like phase separation, we refer to 1569, 1759, and 1841. The nonlinear and nonsmooth properties properties of the partial differential equations for a realistic damage model makes the numerical simulations challenging. An aggravating point is the multiscale nature of the processes, both in time and space. In order to resolve all the relevant structures in a single simulation, an adaptive Finite Element code was developed, which deals with the nonsmoothness by introducing inequality constraints. The methodology is described in WIAS preprint 2299.Adaptive Finite Element simulation of a stretched bimaterial drop. Left: Initial Instability of spinodal decomposition. Right: After a coarsening process the drop is strained vertically, and as a consequence cracks evolve.
Crack Models
The propagation of cracks along specific crack paths is studied in fracture mechanics. In this case, the socalled Griffith criterion is commonly applied to model the evolution of cracks in brittle elastic materials. The central quantity entering the Griffith criterion is the energy release rate (ERR), which is defined as the negative derivative of the elastic energy with respect to the crack length. According to the Griffith criterion a crack does not propagate if the ERR is strictly less than the fracture toughness, which is a material dependent constant characterizing the energy that is needed to create the new crack surface. Assuming that the crack path is known in advance, a local, rateindependent model was developed for crack propagation in geometrically nonlinear elastic materials (finite strain setting). Due to the nonuniqueness of minimizing deformations the reduced elastic energy is not differentiable with respect to the crack length. Thus, the fracture evolution has to be formulated using a general subdifferential instead of the classical formulation based on the energy release rate. The existence of solutions is proved via a vanishing viscosity method (see WIAS preprints 1100 and 1351 ).Left: Domain with surface loading h(t,x)=t h_{1}(x); Right: Global energetic solution based on global energy minimization principles (green), local solution obtained as vanishing viscosity limit (red) and viscous solutions for different viscosities (blue curves).
Phase Separation in Modern Solder Materials for Microelectronic Devices
Solder materials consist of eutectic mixtures with usually two or three chemical components. That special composition leads to a fine morphology, that guarantees high stiffness. The micro structure of classical solder materials consists of two kinds of solid solutions. SnPb, which nowadays is exclusively used in some rare but important applications, consists of a tinrich phase with a small amount of dissolved lead, and a leadrich phase with a small amount of dissolved tin. The classical solder CuAg shows similar behavior. In contrast, modern leadfree solder materials, like the binary SnAg or the ternary SnAgCu do not consist of solid solutions, rather they form fine structured Sn polycrystals that enclose in their interior or between grain boundaries intermetallic phases.Different physical processes are involved in the evolution of microstructures in solder alloys. In the first instance, phase separation occurs when the solder material is cooled below a critical temperature. Finegrained structures of different chemical phases emerge in this situation. On a longer timescale, these structures coarsen such that the bulk and the surface energy will be minimized. Due to different physical properties of the phases (e.g. thermal expansion coefficients and stressfree strains), high stress concentrations may appear especially at phase boundaries and damage behavior is initiated. It has been observed that the microvoids and microcracks aggregate along these phase boundaries and finally form macroscopic cracks. The appearance of such a socalled cold solder joint mostly causes a malfunction of the whole electronic circuit.
Within research group Thermodynamic Modeling and Analysis of Phase Transitions and junior group Modeling of Damage Processes various aspects of solder materials were investigated:
 Modeling and analysis of unifying PDE systems coupling phase separation, damage processes and elasticity
 Establishment of a SharpInterfaceModel to explain special shapes of intermetallic inclusions within the Sn matrix. Various effects of anisotropy play also here a crucial role
 Investigation of degenerate limits such that complete damage and degenerating diffusion mobilities are allowed
 Modeling of the coarsening by various different phase field models that take elastic properties and anisotropy of the various phases, as well as initially different orientations of grains into account
 Development of fast and robust numerical simulation tools
Publications
Monographs

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).
Abstract
HAGs von Christoph bestätigen lassen 
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).
Articles in Refereed Journals

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rateindependent damage in thermoviscoelastic materials with inertia, Journal of Dynamics and Differential Equations, (2018), published online on 10.05.2018, 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. Hintermüller, M. Holler, K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 064002/1064002/39, DOI 10.1088/13616420/aab586 .
Abstract
In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semicontinuous envelope (relaxation) of a suitable total variation type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted total variation for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddlepoint problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson loglikelihood data discrepancy terms. Finally, we provide proofofconcept numerical examples where we solve the saddlepoint problem for weighted TV denoising as well as for MR guided PET image reconstruction. 
M. Liero, S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary Gammaconvergence, NoDEA. Nonlinear Differential Equations and Applications, (2018), published online on 29.01.2018, DOI 10.1007/s0003001804959 .
Abstract
In this paper we discuss two approaches to evolutionary Γconvergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γconvergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the timedependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energydissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for CahnHilliardtype equations. Using the method of weak and strong twoscale convergence via periodic unfolding, we show that the energy and dissipation functionals Γconverge. In conclusion, we will give specific examples for the applicability of each of the two approaches. 
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. 
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. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 25182546.
Abstract
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by HawkinsDaruud et al. in citeHZO. The model consists of a CahnHilliard equation for the tumor cell fraction $vp$ coupled to a reactiondiffusion equation for a function $s$ representing the nutrientrich extracellular water volume fraction. The distributed control $u$ monitors as a righthand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the controltostate operator is Fréchet differentiable between appropriate Banach spaces and derive the firstorder necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a CahnHilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 3754.
Abstract
This paper is concerned with a phase field system of CahnHilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of wellposedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates 
CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperaturedependent phasefield model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177247.
Abstract
In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for CahnHilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461490] in the framework of FourierNavierStokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 13451369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 25192586] for the study of PDE systems for phase transition and damage. Our globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme. 
A. Roggensack, Ch. Kraus, Existence of weak solutions for the CahnHilliard reaction model including elastic effects and damage, Journal of Partial Differential Equations, 30 (2017), pp. 111145, DOI 10.4208/jpde.v30.n2.2 .
Abstract
In this paper, we introduce and study analytically a vectorial CahnHilliard reaction model coupled with ratedependent damage processes. The recently proposed CahnHilliard reaction model can e.g. be used to describe the behavior of electrodes of lithiumion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithiumion batteries. Mathematically, this is realized by a CahnLarché system with a nonlinear Newton boundary condition for the chemical potential and a doubly nonlinear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the nonlinear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system. 
CH. Kraus, M. Radszuweit, Modeling and simulation of nonisothermal ratedependent damage processes in inhomogeneous materials using the phasefield approach, Computational Mechanics, 60 (2017), pp. 163179, DOI 10.1007/s0046601713934 .
Abstract
We present a continuum model that incorporates ratedependent damage and fracture, a material order parameter field and temperature. Different material characteristics throughout the medium yield a strong inhomogeneity and affect the way fracture propagates. The phasefield approach is employed to describe degradation. For the material order parameter we assume a Cahn Larchétype dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation during fracture, thermalexpansionin duced failure and elasticinhomogeneity effects. We furthermore present an adaptive Finite Element code in two space dimensions, that is capable of solving such a highly nonlinear and nonconvex system of partial differential equations. With the help of this tool we conduct numerical experiments of different complexity in order to investigate the possibilities and limitations of the presented model. A main feature of our model is that we can describe the process of microcrack nucleation in regions of partial damage to form macrocracks in a unifying approach. 
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, pp. published online on 17.10.2017, urlhttps://doi.org/10.1515/acv20170010, 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. 
S.P. Frigeri, Global existence of weak solutions for a nonlocal model for twophase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 19571993.
Abstract
We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a NavierStokes type system coupled with a convective nonlocal CahnHilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular doublewell potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model. 
M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016), pp. 189210.
Abstract
We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible NavierStokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 . 
P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous CahnHilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016), pp. 195225, DOI 10.1007/s002450159299z .
Abstract
A boundary control problem for the viscous CahnHilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved. 
E. Rocca, R. Rossi, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM Journal on Mathematical Analysis, 74 (2015), pp. 25192586.
Abstract
In this paper we analyze a PDE system modelling (nonisothermal) phase transitions and dam age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the righthand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as “entropic”, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain globalintime existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its “entropic” formulation), and of the a priori estimates performed on it. Our timediscrete analysis could be useful towards the numerical study of this model. 
E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective CahnHilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015), pp. 16541680.
Abstract
In this paper we study a distributed optimal control problem for a nonlocal convective CahnHilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the wellposedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate firstorder necessary optimality conditions for the optimal control problem. 
S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for twophase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), pp. 12571293.
Abstract
We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the NavierStokes system coupled with a convective nonlocal CahnHilliard equation with nonconstant mobility. We first prove the existence of a global weak solution in the case of nondegenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal CahnHilliard equation with degenerate mobility and singular potential in dimension three. 
E. Bonetti, Ch. Heinemann, Ch. Kraus, A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, Journal of Partial Differential Equations, 258 (2015), pp. 39283959.
Abstract
In this work, we analytically investigate a multicomponent system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques. 
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. 
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the BallMajumdar free energy, 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., 194 (2015), pp. 12691299.
Abstract
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landaude Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible NavierStokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Qtensor $QQ$, namely a tensorvalued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Qtensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blowup rate, is carried out. 
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. 
H. Hanke, D. Knees, Homogenization of elliptic systems with nonperiodic, state dependent coefficients, Asymptotic Analysis, 92 (2015), pp. 203234.
Abstract
In this paper, a homogenization problem for an elliptic system with nonperiodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the nonperiodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to timedependent damage models (discussed in a future paper), we provide a Γconvergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided. 
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 PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 25652590.
Abstract
In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolicparabolic system. To this end, we first generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, timediscretization and different variational techniques. 
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. 
E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 12651341.

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. Colli, G. Gilardi, J. Sprekels, On the CahnHilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014), pp. 972994.
Abstract
The CahnHilliard and viscous CahnHilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some wellposedness and regularity results are proved. 
S. Melchionna, E. Rocca, On a nonlocal CahnHilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014), pp. 461497.
Abstract
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u). 
A. Miranville, E. Rocca, G. Schimperna, A. Segatti, The PenroseFife phasefield model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 42594290.

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. 
M. Liero, Passing from bulk to bulk/surface evolution in the AllenCahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013), pp. 919942.
Abstract
In this paper we formulate a boundary layer approximation for an AllenCahntype equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2gradient flow with the corresponding AllenCahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions. 
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. 
D. Knees, R. Rossi, Ch. Zanini, A vanishing viscosity approach to a rateindependent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 565616.
Abstract
We analyze a rateindependent 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 bynow classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rateindependent damage models as limits of systems driven by viscous, ratedependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparameterization. In this way, in the limit we obtain a novel formulation for the rateindependent damage model, which highlights the interplay of viscous and rateindependent 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 quasistatic crack propagation, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 6399.
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 rateindependent 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 selfcontact 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 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. Fiaschi, D. Knees, U. Stefanelli, Young measure quasistatic damage evolution, Archive for Rational Mechanics and Analysis, 203 (2012), pp. 415453.

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. 
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. 
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. 
J.A. Griepentrog, L. Recke, Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems, Journal of Evolution Equations, 10 (2010), pp. 341375.
Abstract
A general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in SobolevMorrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data. 
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. 
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. 
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. 
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.

TH. Böhme, W. Dreyer, W.H. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Continuum Mechanics and Thermodynamics, 18 (2007), pp. 411441.

H. Gajewski, J.A. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete and Continuous Dynamical Systems, 15 (2006), pp. 505528.

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

W. Dreyer, B. Wagner, Sharpinterface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005), pp. 199227.

J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004), pp. 153164.

E. Bonetti, W. Dreyer, G. Schimperna, Global solution to a generalized CahnHilliard equation with viscosity, Advances in Differential Equations, 8 (2003), pp. 231256.

W. Dreyer, W.H. Müller, C.M. Brown, The convergence of a DFT algorithm for solution of stress strain problems in composite materials, Advances in Differential Equations, 125 (2003), pp. 2737.

E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna, J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Physica D. Nonlinear Phenomena, 165 (2002), pp. 4865.

W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000), pp. 38413871.

W. Dreyer, W.H. Müller, Computer modeling of micromorphological change by phase field models: Applications to metals and ceramics, Journal of the Australasian Ceramic Society, 36 (2000), pp. 8394.
Contributions to Collected Editions

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, 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. 
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. 
D. Knees, R. Rossi, C. Zanini, A vanishing viscosity approach in damage mechanics, in: Variational Methods for Evolution, Workshop, December 410, 2011, A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 31533155.

D. Knees, A survey on energy release rates, in: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Miniworkshop, April 2429, 2011, 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 12161219.

M. Thomas, Modeling and analysis of rateindependent damage and delamination processes, in: Proceedings of the 19th International Conference on Computer Methods in Mechanics (online only), 2011, pp. 16.

M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, in: Microstructures in Solids: From Quantum Models to Continua, Workshop, March 1420, 2010, 7 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2010, pp. 783785.

P. Philip, A quasistatic crack propagation model allowing for cohesive forces and crack reversibility, in: Proceedings of APCOM'07  EPMESC XI (CDROM), December 36, 2007, Kyoyo, Japan, 2007, pp. 10 pages.
Preprints, Reports, Technical Reports

P. Colli, G. Gilardi, J. Sprekels, Optimal distributed control of a generalized fractional CahnHilliard system, Preprint no. 2519, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2519 .
Abstract, PDF (323 kByte)
In the recent paper “Wellposedness and regularity for a generalized fractional CahnHilliard system” by the same authors, general wellposedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous CahnHilliard system, in which nonlinearities of doublewell type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B, having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated controltostate mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and firstorder necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system. 
P. Colli, G. Gilardi, J. Sprekels, Wellposedness and regularity for a generalized fractional CahnHilliard system, Preprint no. 2509, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2509 .
Abstract, PDF (309 kByte)
In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous CahnHilliard system in which nonlinearities of doublewell type occur. Standard cases like regular or logarithmic potentials, as well as nondifferentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators A and B, where the latter are densely defined, unbounded, selfadjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard secondorder elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourthorder systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general wellposedness and regularity results that extend corresponding results which are known for either the nonfractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. These results are entirely new if at least one of the operators A and B differs from the Laplacian. It turns out that the first eigenvalue λ_{1} of A plays an important und not entirely obvious role: if λ_{1} is positive, then the operators A and B may be completely unrelated; if, however, λ_{1} equals zero, then it must be simple and the corresponding onedimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of B. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system. 
M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phasefield fracture model at finite strains based on modified invariants, Preprint no. 2456, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2456 .
Abstract, PDF (1542 kByte)
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.
Talks, Poster

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, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 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, Analysis for the discrete approximation of damage and fracture, Applied Analysis Day, June 28  29, 2018, Technische Universität Dresden, Chair of Partial Differntial Equations, Germany, June 29, 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.

M. Thomas, Phasefield fracture at finite strains based on modified invariants, Special Materials and Complex Systems (SMACS 2018), June 17  22, 2018, University of Milan, Department of Mathematics, Gargnano, Italy, June 18, 2018.

M. Thomas, Rateindependent evolution of sets & applications to damage and delamination, PDEs Friends, June 21  22, 2018, Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy, June 22, 2018.

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, June 7, 2017.

M. Hintermüller, Optimal control of multiphase fluids based on non smooth models, 14th International Conference on Free Boundary Problems: Theory and Applications, Theme Session 8 ``Optimization and Control of Interfaces'', July 9  14, 2017, Shanghai Jiao Tong University, China, July 10, 2017.

M. Hintermüller, Optimal control of nonsmooth phasefield models, DFGAIMS Workshop on ``Shape Optimization, Homogenization and Control'', March 13  16, 2017, Mbour, Senegal, March 14, 2017.

TH. Frenzel, Evolutionary Gammaconvergence for a delamination model, Workshop on Industrial and Applied Mathematics 2016, 5th Symposium of German SIAM Student Chapters, August 31  September 2, 2016, University of Hamburg, Department of Mathematics, Hamburg, September 1, 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, From adhesive contact to brittle delamination in viscoelastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22  26, 2016, WIAS Berlin, Berlin, February 26, 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 & application to fracture processes, Seminar on Analysis, Kanazawa University, Institute of Science and Engineering, Kanazawa, Japan, October 28, 2016.

S.P. Frigeri, On a diffuse interface model of tumor growth, 9th European Conference on Elliptic and Parabolic Problems, May 23  27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, 66th Workshop ``Advances in Convex Analysis and Optimization'', July 5  10, 2016, International Centre for Scientific Culture ``E. Majorana'', School of Mathematics ``G. Stampacchia'', Erice, Italy, July 9, 2016.

M. Hintermüller, Optimal control of multiphase fluids and droplets, WIASPGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10  12, 2016, WIAS Berlin, May 11, 2016.

M. Hintermüller, Optimal control of multiphase fluids and droplets, The Fifth International Conference on Continuous Optimization, Session: ``Recent Developments in PDEconstrained Optimization I'', August 6  11, 2016, Tokyo, Japan, August 10, 2016.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, June 9, 2016.

M. Hintermüller, Towards sharp stationarity conditions for classes of optimal control problems for variational inequalities of the second kind, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20  24, 2016, Cortona, Italy, June 20, 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.

E. Rocca, Optimal control of a nonlocal convective CahnHilliard equation by the velocity, Numerical Analysis Seminars, Durham University, UK, March 13, 2015.

S.P. Frigeri, On a diffuse interface model of tumor growth, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 22, 2015.

S.P. Frigeri, On a nonlocal diffuse interface model for binary incompressible fluids with different densities, Mathematical Thermodynamics of Complex Fluids, June 28  July 3, 2015, Fondazione CIME ``Roberto Conti'' (International Mathematical Summer Center), Cetraro, Italy, July 2, 2015.

S.P. Frigeri, Recent results on optimal control for CahnHilliard/NavierStokes systems with nonlocal interactions, Control Theory and Related Topics, April 13  14, 2015, Politecnico di Milano, Italy, April 13, 2015.

CH. Heinemann, On elastic CahnHilliard systems coupled with evolution inclusions for damage processes, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Young Researchers' Minisymposium 2, March 23  27, 2015, Lecce, Italy, March 23, 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.

J. Sprekels, Optimal boundary control problems for CahnHilliard systems with dynamic boundary conditions, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 21, 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.

CH. Heinemann, Wellposedness of strong solutions for a damage model in 2D, Universitá di Brescia, Department DICATAM  Section of Mathematics, Italy, March 13, 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.

E. Rocca, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, Symposium on Trends in Application of Mathematics to Mechanics (STAMM), September 8  11, 2014, International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

D. Knees, A quasilinear differential inclusion for viscous and rateindependent damage systems in nonsmooth domains, Analysis & Stochastics Seminar, Technische Universität Dresden, Institut für Analysis, January 16, 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.

A. Mielke, Homogenization of parabolic gradient systems via evolutionary $Gamma$convergence, Second Workshop of the GAMM Activity Group on ``Analysis of Partial Differential Equations'', September 29  October 1, 2014, Universität Stuttgart, Institut für Analysis, Dynamik und Modellierung, September 30, 2014.

A. Mielke, On the microscopic origin of generalized gradient structures for reactiondiffusion systems, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8  11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 11, 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.

CH. Heinemann, Analysis of degenerating CahnHilliard systems coupled with complete damage processes, 2013 CNA Summer School, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, USA, May 30  June 7, 2013.

CH. Heinemann, Degenerating CahnHilliard systems coupled with complete damage processes, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 2013.

CH. Heinemann, Degenerating CahnHilliard systems coupled with mechanical effects and complete damage processes, Equadiff13, MS27  Recent Results in Continuum and Fracture Mechanics, August 26  30, 2013, Prague, Czech Republic, August 27, 2013.

CH. Heinemann, On a PDE system describing damage processes and phase separation, Oberseminar Analysis, Universität Augsburg, July 11, 2013.

H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the NavierStokesKorteweg model, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 2013.

D. Knees, A vanishing viscosity approach to a rateindependent 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.

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

D. Knees, Weak solutions for rateindependent systems illustrated at an example for crack propagation, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 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, Damage and delamination processes in thermoviscoelastic materials, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Young Reserchers' Minisymposium ``Analytical and Engineering Aspects in the Material Modeling of Solids'', March 18  22, 2013, University of Novi Sad, Serbia, March 19, 2013.

M. Thomas, Existence & fine properties of solutions for rateindependent brittle damage models, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM Juniors Poster Exhibition, Novi Sad, Serbia, March 18  22, 2013.

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

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

M. Thomas, Rateindependent damage models with spatial BVregularization  Existence & fine properties of solutions, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 24, 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.

H. Hanke, Derivation of an effective damage model with evolving microstructure, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 29, 2013.

H. Hanke, Derivation of an effective damage model with nonperiodic evolving microstructure, 12th GAMM Seminar on Microstructures, February 8  9, 2013, HumboldtUniversität zu Berlin, Institut für Mathematik, February 9, 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.

CH. Heinemann, Complete damage in linear elastic materials, Variational Models and Methods for Evolution, Levico, Italy, September 10  12, 2012.

CH. Heinemann, Damage processes coupled with phase separation in elastically stressed alloys, GAMM Jahrestagung 2012 (83rd Annual Meeting), March 26  30, 2012, Technische Universität Darmstadt, March 27, 2012.

CH. Heinemann, Existence of weak solutions for ratedependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, LeibnizDoktorandenForum der Sektion D, Berlin, June 7  8, 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.

D. Knees, A vanishing viscosity approach in fracture mechanics, Oberseminar zur Analysis, Universität DuisburgEssen, Fakultät für Mathematik, November 13, 2012.

D. Knees, A vanishing viscosity approach to a rateindependent damage model, Oberseminar zur Analysis, Universität DuisburgEssen, Fakultät für Mathematik, November 20, 2012.

D. Knees, Global spatial regularity for elastic fields with cracks and contract, 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.

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.

D. Knees, On a vanishing viscosity approach in damage mechanics, Kolloquium der AG Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, Institut für Mathematik, August 21, 2012.

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, 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, Rateindependent evolution of sets, Variational Models and Methods for Evolution, Levico, Italy, September 10  12, 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.

H. Hanke, Derivation of an effective damage evolution model, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

H. Hanke, Derivation of an effective damage evolution model using twoscale convergence techniques, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Damage Processes and Contact Problems, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 29, 2012.

H. Hanke, Derivation of an effective damage evolution model using twoscale convergence techniques, 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 19, 2012.

J.A. Griepentrog, On nonlocal phase separation processes in multicomponent systems, 10th GAMM Seminar on Microstructures, January 20  22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

J.A. Griepentrog, The role of nonsmooth regularity theory in the analysis of phase separation processes, Ehrenkolloquium anlässlich des 60. Geburtstages von PD Dr. habil. Lutz Recke, HumboldtUniversität zu Berlin, Institut für Mathematik, November 21, 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.

D. Knees, A survey on energy release rates, MiniWorkshop ``Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture'', April 24  29, 2011, Mathematisches Forschungsinstitut Oberwolfach, April 26, 2011.

D. Knees, A vanishing viscosity approach in damage mechanics, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), June 20  24, 2011, Gargnano (Brescia), Italy, June 22, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, Workshop ``Perspectives in Continuum Mechanics'' in Honor of Gianfranco Capriz's 85th Birthday, University of Florence, Department of Mathematics, Italy, January 28, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 7th International Congress on Industrial and Applied Mathematics, Session ``Materials Science II'', July 18  22, 2011, Society for Industrial and Applied Mathematics, Vancouver, Canada, July 19, 2011.

D. Knees, On a vanishing viscosity approach for a model in damage mechanics, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Applied Analysis, April 18  21, 2011, Technische Universität Graz, Austria, April 20, 2011.

D. Knees, A vanishing viscosity approach in fracture mechanics, Seminar on Partial Differential Equations, Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague, March 1, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 10th GAMM Seminar on Microstructures, January 20  22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 21, 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.

M. Thomas, Modeling and analysis of rateindependent 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.

D. Knees, A vanishing viscosity approach in damage mechanics, Workshop ``Variational Methods for Evolution'', December 5  10, 2011, Mathematisches Forschungsinstitut Oberwolfach, December 5, 2011.

D. Knees, Analysis und Numerik für quasistatische Rissausbreitung, Lectures in Continuum Mechanics, Universität Kassel, Fachbereich für Mathematik und Naturwissenschaften, December 12, 2011.

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.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 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.

D. Knees, On crackpropagation in polyconvex materials, 81th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2010), Session on Damage and Fracture Mechanics, March 22  26, 2010, Universität Karlsruhe, March 24, 2010.

D. Knees, On the vanishing viscosity method in fracture mechanics, BerlinLeipzig Seminar on Analysis and Probability Theory, Technische Universität Berlin, Institut für Mathematik, July 9, 2010.

D. Knees, On the vanishing viscosity method in fracture mechanics, Seminar Technomathematik, Technische Universität Graz, Institut für Numerische Mathematik, Austria, February 25, 2010.

D. Knees, Quasistatic crack propagation in polyconvex materials, 9th GAMM Seminar on Microstructures, January 21  23, 2010, Universität Stuttgart, Institut für Mechanik (Bauwesen), January 22, 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.

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.

J.A. Griepentrog, Maximal regularity for nonsmooth parabolic boundary value problems in SobolevMorrey spaces, International Conference on Elliptic and Parabolic Equations, November 30  December 4, 2009, WIAS, December 1, 2009.

D. Knees, Crack propagation in polyconvex materials, Colloquium Dedicated to A.M. Sändig's 65th Birthday, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, July 17, 2009.

D. Knees, On quasistatic crack propagation in finitestrain elasticity, University of Pavia, Institute of Applied Mathematics and Information Technology (IMATI), Italy, September 8, 2009.

D. Knees, On the inviscid limit of a model for crack propagation, Multiscale Asymptotics and Computational Approximation for Surface Defects and Applications in Mechanics (MACADAM), August 31  September 1, 2009, Ecole Normale Supérieure de Cachan, France, August 31, 2009.

CH. Kraus, A phasefield model with anisotropic surface tension in the sharp interface limit, Second GAMMSeminar on Multiscale Material Modelling, July 10  12, 2008, Universität Stuttgart, Institut für Mechanik (Bauwesen), July 12, 2008.

CH. Kraus, Ein Phasenfeldmodell vom CahnHilliardTyp im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

CH. Kraus, Phase field models and corresponding GibbsThomson laws. Part II, SIMTECH Seminar Multiscale Modelling in Fluid Mechanics, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, November 5, 2008.

CH. Kraus, On jump conditions at phase interfaces, Oberseminar über Angewandte Mathematik, December 10  15, 2007, Universität Freiburg, Abteilung für Angewandte Mathematik, December 11, 2007.

CH. Kraus, Equilibria conditions in the sharp interface limit of the van der WaalsCahnHilliard phase model, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14  16, 2006, Levico, Italy, September 14, 2006.

CH. Kraus, The sharp interface limit of the van der WaalsCahnHilliard model, PolishGerman Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

CH. Kraus, On the sharp limit of the Van der WaalsCahnHilliard model, WIAS Workshop ``Dynamic of Phase Transitions'', November 30  December 3, 2005, Berlin, December 2, 2005.

CH. Kraus, On the sharp limit of the Van der WaalsCahnHilliard model, Workshop ``MicroMacro Modeling and Simulation of LiquidVapor Flows'', November 16  18, 2005, Universität Freiburg, Mathematisches Institut, Kirchzarten, November 17, 2005.

CH. Kraus, Maximale Konvergenz in höheren Dimensionen, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, May 24, 2005.
External Preprints

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

A. Mielke, On an evolutionary model for complete damage based on energies and stresses, Preprint no. 29PV10/27/0, pp. 2332, in: Rateindependent evolutions and material modeling, T. Roubíček, U. Stefanelli (eds.), Istituto di Matematica Applicata e Technologie Informatiche, 2010.
Contact
Mathematical Context
 Analysis of Partial Differential Equations and Evolutionary Equations
 Free boundary problems for partial differential equations
 Modeling, analysis and numerics of phase field models
 Multi scale modeling and hybrid models
 Nonlinear kinetic equations
 Systems of partial differential equations: modeling, numerical analysis and simulation
 Variational methods