Mathematical and Numerical Analysis of Elastoplastic Material with Multi-Surface Stress-Strain Relation

Acknowledgments It is my pleasure to express my thanks to my supervisors Professor Dr. C. Carstensen for suggesting this topic to me, for many valuable discussions, and for their supervision during my stays in Kiel and Munich. I wish to thank all of my colleagues at the Lehrstuhl für Wissenschaftliches Rechnen and Dr. G. Grammel for scientific advice. I am grateful to Dr. J. Alberty and R. Klose for constructive discussions on plasticity and PD Dr. A. Prohl and S. Bartels for their useful advice concerning numerical analysis. Summary The aim of this thesis is the mathematical and numerical analysis of a multi-yield (surface) model in elastoplasticity. The presented Prandtl-Ishlinskii model of play type generalizes the linear kinematic hardening model and leads to a more realistic description of the elastoplastic transition of a material during a deformation process. The unknowns in the quasi-static formulation are displacement and (several) plastic strains which satisfy a time-dependent variational inequality. As for the linear kinematic hardening model, the variational inequality consists of a bounded and elliptic bilinear form, a linear functional, and a positive homogeneous, Lipschitz continuous functional; hence existence and uniqueness of a weak solution is then concluded from a general theory. Our time and space discretization consists of the implicit Euler method and the lowest order finite element method. For any one-time step discrete problem, the vector of plastic strains (considered on one element) depends on the (unknown) displacement only. In contrast to the linear kinematic hardening model, the dependence can not be stated explicitly, but has to be calculated by an iterative algorithm. An a priori error estimate is established and shows linear convergence with respect to time and space under the assumption of sufficient regularity of the solution. A MATLAB solver, which includes the nested iteration technique combined with an (ZZ-) adaptive mesh-refinement strategy and the Newton-Raphson method, is employed for solving the two-yield material model. Various numerical experiments support our theoretical results and give more insight to complex dynamics in elastoplasticity problems. Lösung des " two-yield " Problems herangezogen. Zahlreiche numerische Experimente bele-gen die theoretischen Resultate dieser Arbeit und führen zu einem besseren Verständnis des Materialmodells.