We investigate unilateral contact problems with cohesive forces, leading to
the constrained minimization of a possibly nonconvex functional. We
analyze the mathematical structure of the minimization problem.
The problem is reformulated in terms of a three-field augmented
Lagrangian, and sufficient conditions for the existence of a local
saddle-point are derived. Then, we derive and analyze mixed finite
element approximations to the stationarity conditions of the three-field
augmented Lagrangian. The finite element spaces for the bulk displacement and
the Lagrange multiplier must satisfy a discrete inf-sup condition, while
discontinuous finite element spaces spanned by nodal basis functions are
considered for the unilateral contact variable so as to use collocation
methods. Two iterative algorithms are presented and analyzed, namely an
Uzawa-type method within a decomposition-coordination approach and a
nonsmooth Newton's method. Finally, numerical results illustrating the
theoretical analysis are presented.