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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.
- 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
- 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.
- ItemRegularity of elastic fields in composites(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Knees, Dorothee; Sändig, Anna-MargareteIt is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 "Stress concentrations in heterogeneous materials" of the SFB 404 "Multifield Problems in Solid and Fluid Mechanics" it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results.
- ItemOn the inviscid limit of a model for crack propagation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Knees, Dorothee; Mielke, Alexander; Zanini, ChiaraWe 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 rate-independent 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.
- ItemRegularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elastoplasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Knees, Dorothee; Neff, PatrizioIn this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.
- ItemShort note on global spatial regularity in elasto-plasticity with linear hardening(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Knees, DorotheeWe study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L∞((0,T);H3/2-[delta]([omega])) for the displacements and z in L∞((0,T);H1/2-[delta]([omega])) for the internal variables. The proof is based on a difference quotient technique and a reflection argument.