Existence, uniqueness and regularity of solutions to elastic-plastic problems with the traction boundary condition

We consider the existence, uniqueness and regularity of the weak solutions to some elastic-plastic problems with the traction boundary condition in Rⁿ. It is shown that all models can be formulated equivalently as variational inequalities or minimization problems. In the first linear elastic-plastic model of Havner-Patel type, the existence of weak solutions is obtained by using an elliptic regularization method. Moreover, the solution is shown to be unique up to an infinitesimal rigid motion. Furthermore, the regularity of the weak solutions is discussed by employing various techniques such as Nirenberg difference-quotient method, reverse Holder inequality, freezing the coefficients, Morrey space and Campanato space methods. We eventually show that the weak solutions lie in (C¹;,α(Ω);) x (C^0,α(Ω))^m;. Then a nonlinear elastic-plastic model is proposed and shown that solutions exist. A partial regularity result of weak solutions is obtained by using a perturbation argument and L^p; estimate. It is shown that the singularity set has measure zero. Next a holonomic elastoplasticity model with linear hardening is considered. A convex minimization method along with a convergence procedure is used to prove the existence of weak solutions. There exists a unique solution up to an infinitesimal rigid motion. We also obtain an interior regularity result for a regularized problem. Finally a similar problem but with nonlinear hardening is shown to possess solutions by pseudomonotone operator theory where an elliptic regularization method is used as well. The structure and regularity of solutions are considered.