Nonlinear Cauchy Problem And Identification In Contact Mechanics: A Solving Method Based On Bregman-Gap

This paper proposes a solution method for identification problems in the context of contact mechanics when overabundant data are available on a part Gamma(m) of the domain boundary while data are missing from another part of this boundary. The first step is then to find a solution to a Cauchy problem. The method used by the authors for solving Cauchy problems consists of expanding the displacement field known on Gamma(m) toward the inside of the solid via the minimization of a function that measures the gap between solutions of two well-posed problems, each one exploiting only one of the superabundant data. The key question is then to build an appropriate gap functional in strongly nonlinear contexts. The proposed approach exploits a generalization of the Bregman divergence, using the thermodynamic potentials as generating functions within the framework of generalized standard materials (GSMs), but also implicit GSMs in order to address Coulomb friction. The robustness and efficiency of the proposed method are demonstrated by a numerical bi-dimensional application dealing with a cracked elastic solid with unilateral contact and friction effects on the crack's lips.