Existence and uniqueness of solutions of the equilibrium equations for a class of nonlinearly elastic materials

This paper is motivated by an industrial project concerning structural optimization problems for nonlinearly elastic materials arising from car industry. The investigated class of material laws covers a type of nonlinear materials, especially gray cast iron that is used in car engines. For such materials we prove the existence and uniqueness of solutions of the equilibrium equations. Based on this work, algorithms for solving sizing optimization problems for nonlinearly elastic materials will be presented in a subsequent paper. Although the results presented here are connected to structural optimization problems, they are also of interest on their own in the existence theory of nonlinear continuum mechanics.