We propose a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition, a thermodynamically admissible flow rule for Fp is described involving as plastic gradient Curl Fp. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on the plastic metric [Formula: see text]. The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: [Formula: see text] on the microscopically hard boundary ΓD ⊂ ∂Ω and [ Curl p] · τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. A weak reformulation of the infinitesimal model allows for a global in-time solution of the rate-independent initial boundary value problem. The method is based on a mixed variational inequality with symmetric and coercive bilinear form. We use a new Hilbert-space suitable for dislocation density dependent plasticity.

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Typesetting: Camera-ready TEX output by the author SPIN: 10649759 46/3143-5432 IO -Printed on acid-free paper Dedicated to Rolf Leis and to Jutta, Anne and Ina PrefaceContinuum mechanics offers a rising number of fascinating problems to mathematical analysis, and the interest of mathematicians in these problems has been revived and is steadily growing.The reason behind this development is the computer. It makes more and more mathematical problems accessible to numerical treatment and thus opens an increasing number of regions to practical interest and application, which were formerly reserved to theoretical analysis. One might expect that this would diminish the interest in theoretical analysis. Fortunately, this is not the case. On the contrary, numerical studies call for a deeper analysis and thus open new fields in continuum mechanics to mathematical analysis. In a sense, by generating this renewed interest in continuum mechanics, the computer is leading mathematical analysis back to its origins in the studies of Newton, Euler and the Bernoullis.This book also has its origins in these developments. To utilize the capacity of fast computers and of the finite element analysis in computing the deformation and stress distribution in metallic structures, constitutive equations are needed which model the properties of metals with sufficient accuracy. In mechanics and the engineering sciences, therefore, the often rather complex constitutive equations, which have been developed in the last twenty years, are of great interest. The numerical solution of the initialboundary value problems containing these constitutive equations is now an important and central part in many engineering problems. My work in the investigation of these initialboundary value problems with the means of mathematical analysis started, when colleagues from the engineering sciences brought these problems to my attention, and the results contained in this book have been obtained in close cooperation with them.Of course, investigations of problems from plasticity and viscoelasticity have a long tradition in mathematical analysis, and a small number of investigations of initialboundary value problems to these newly developed constitutive equations already exist. Generally however, in these studies the methods of the investigations and of the proofs strongly depend on the constitutive equations under consideration, and the results are often far from the results, which one might optimistically hope to obtain. The aim of this book is to improve this situation and to contribute to the development of a general mathematical theory of initialboundary value problems to these constitutive equations.Accordingly, the book is mainly directed to mathematicians interested in studying constitutive equat...