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- ItemGlobal spatial regularity for a regularized elasto-plastic model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Bumb, Andreas; Knees, DorotheeIn this note the spatial regularity of weak solutions for a class of elasto-viscoplastic evolution models is studied for nonsmooth domains. The considered class comprises e.g. models which are obtained through a Yosida regularization from classical, rate-independent models. The corresponding evolution model consists of an elliptic PDE for the (generalized) displacements which is coupled with an ordinary differential equation with a Lipschitz continuous nonlinearity describing the evolution of the internal variable. It is shown that the global spatial regularity of the displacements and the inner variables is exactly determined through the mapping properties of the underlying elliptic operator.
- ItemGlobal spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Knees, Dorothee; Schröder, AndreasA global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B 3/2 2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.
- ItemGlobal higher integrability of minimizers of variational problems with mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Fiaschi, Alice; Knees, Dorothee; Reichelt, SinaWe consider integral functionals with densities of p-growth, 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 p-growth 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+e for a uniform e >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.
- ItemEnergy release rate for cracks in finite-strain elasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Knees, Dorothee; Mielke, AlexanderGriffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing 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 well-known 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.
- ItemGlobal spatial regularity for time dependent elasto-plasticity and related problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Knees, DorotheeWe study the global spatial regularity of solutions of generalized elasto-plastic models with linear hardening on smooth domains. Under natural smoothness assumptions on the data and the boundary we obtain that the displacements belong to L^8((0,T);H^(3/2-d)(O)) whereas the internal variables belong to L^8((0,T);H^(1/2-d)(O)). The key step in the proof is a reflection argument which gives the regularity result in directions normal to the boundary on the basis of tangential regularity results
- ItemAnalytical and numerical aspects of time-dependent models with internal variables(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Gruber, Peter; Knees, Dorothee; Nesenenko, Sergiy; Thomas, MaritaIn this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented
- ItemDerivation of an effective damage model with evolving micro-structure(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hanke, Hauke; Knees, DorotheeIn this paper rate-independent damage models for elastic materials are considered. The aim is the derivation of an effective damage model by investigating the limit process of damage models with evolving micro-defects. In all presented models the damage is modeled via a unidirectional change of the material tensor. With progressing time this tensor is only allowed to decrease in the sense of quadratic forms. The magnitude of the damage is given by comparing the actual material tensor with two reference configurations, denoting completely undamaged material and maximally damaged material (no complete damage). The starting point is a microscopic model, where the underlying micro-defects, describing the distribution of either undamaged material or maximally damaged material (but nothing in between), are of a given shape but of different time-dependent sizes. Scaling the micro-structure of this microscopic model by a parameter " > 0 the limit passage " ! 0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guarantee enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z : [0, T ] ! W1,p( ) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x 2 is possible. Moreover, this damage variable is connected to the material tensor via an explicit formula, namely, a unit cell formula known from classical homogenization results
- ItemHomogenization of elliptic systems with non-periodic, state dependent coefficients(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hanke, Hauke; Knees, DorotheeIn this paper, a homogenization problem for an elliptic system with non-periodic, 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 non-periodic 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 time-dependent damage models (discussed in a future paper), we provide a Γ-convergence 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.
- ItemA vanishing viscosity approach to a rate-independent damage model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Knees, Dorothee; Rossi, Riccarda; Zanini, ChiaraWe analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.
- ItemCrack growth in polyconvex materials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Knees, Dorothee; Zanini, Chiara; Mielke, AlexanderWe discuss a model for crack propagation in an elastic body, where the crack path is described a-priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release 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.