A variational approach via bipotentials for unilateral contact problems

We consider a unilateral contact model for nonlinearly elastic materials, under the small deformation hypothesis, for static processes. The contact is modeled with Signorini’s condition with zero gap and the friction is neglected on the potential contact zone. The behavior of the material is modeled by a subdifferential inclusion, the constitutive map being proper, convex, and lower semicontinuous. After describing the model, we give a weak formulation using a bipotential which depends on the constitutive map and its Fenchel conjugate. We arrive to a system of two variational inequalities whose unknown is the pair consisting of the displacement field and the Cauchy stress field. We look for the unknown into a Cartesian product of two nonempty, convex, closed, unbounded subsets of two Hilbert spaces. We prove the existence and the uniqueness of the weak solution based on minimization arguments for appropriate functionals associated with the variational system. How the proposed variational approach is related to previous variational approaches, is discussed too.