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This paper presents the weak formulation of a quasi-static evolution model for two deformable bodies with uni-directional adhesive unilateral contact on which external loads act. Small deformations and linearized elastoplasticity with hardening are assumed. The adhesion component is rate-dependent or rate-independent according to the choice of the viscosity coefficient of the glue; elastoplasticity is considered rate-independent. The weak formulation is expressed as a doubly non-linear problem with unbounded multivalued operators, as a function of internal and boundary displacements, plastic and symmetric strain tensors, and the bonding field and its gradient. This paper differs from other formulations by coupling the equations defined inside and on the boundary of the solids in functional form. In addition to this novelty, we verify the existence of solutions by a path other than that displayed in similar articles. The existence of solutions is demonstrated after considering a succession of doubly non-linear problems with an unbounded operator, and verifying that the solution of one of the problems is also a solution to the objective problem. The proof is supported by previous results from non-linear Partial differential equations theory with monotone operators.

Formulation and existence of weak solutions for a problem of adhesive contact with elastoplasticity and hardening