A class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.2000 Mathematics subject classification: primary 74M10; secondary 74M15, 35J50, 35J66.

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We consider an antiplane model which describes the contact between a deformable cylinder and a rigid foundation, under the small deformation hypothesis, for quasistatic processes. The behaviour of the material is modelled using a viscoelastic constitutive law with long memory and the frictional contact is modelled using Tresca’s law. We focus on the weak solvability of the model, based on a weak formulation with dual Lagrange multipliers.