Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation

A stress formulation for frictionless contact problems between deformable bodies is proposed. Linear compatibility equations are assumed, while the constitutive relations are supposed nonlinear, yet Reversible, i.e., ruled by a convex strain potential. The relevant contact rules are formulated in terms of concave conjugated potentials, whose superdifferentials yield the constitutive laws for the unilateral contact interface. Generalization of the mixed Hellinger-Reissner functional, and of the functionals of total potential energy and of complementary energy are formulated. The last one is used for numerical developments. The functional is regularized by means of an augmented Lagrangian function. Solution to the saddle point problem arising from the regularization is obtained in the subspace of self-equilibrated stresses only, using equilibrium equations for condensing out the complementary stresses. In the paper, some examples of more complex unilateral contact relations are also presented.