The goal of this paper is to study a mathematical model of a nonlinear static frictional contact problem in elasticity with the mixed boundary conditions described by a combination of the Signorini unilateral frictionless contact condition, and nonmonotone multivalued contact, and friction laws of subdifferential form. First, under suitable assumptions, we deliver the weak formulation of the contact model, which is an elliptic system with Lagrange multipliers, and which consists of a hemivariational inequality and a variational inequality. Then, we prove the solvability of the contact problem. Finally, employing the notion of H-convergence of nonlinear elasticity tensors, we provide a result on the convergence of solutions under perturbations which appear in the elasticity operator, body forces, and surface tractions.