Weak solutions via bipotentials in mechanics of deformable solids

We consider a displacement-traction boundary values problem for elastic materials, under the small deformations hypothesis, for static processes. The behavior of the material is modeled by a constitutive law involving the subdifferential of a proper, convex, and lower semicontinuous map. The constitutive map and its Fenchel conjugate allow us to construct a bipotential function. Based on this construction, we propose a weak formulation of our mechanical problem. Furthermore, we prove the existence of at least one weak solution and we investigate the uniqueness of the weak solution. We also comment on the relevance of our variational approach, by considering three significant examples.