Algorithm for Solving Discrete Contact Problems for an Elastic Layer

The problems of discrete contact between an elastic layer and a rigid punch with unknown areas of actual contact are considered. A variational formulation of the problems is obtained in the form of a boundary variational inequality using the Poincaré-Steklov operator, which maps normal stresses into normal displacements on a part of the elastic layer boundary. To approximate this operator, a two-dimensional discrete Fourier transform is used, for the numerical implementation of which algorithms of the fast Fourier transform are used. A minimization problem equivalent to the variational inequality is presented. As a result of its approximation, a quadratic programming problem with constraints in the form of equalities and inequalities is obtained. To numerically solve this problem, we used an algorithm based on the conjugate gradient method, which takes into account the specifics of the set of constraints. Two-parameter families of punches rectangular in plan with surface relief are constructed. As a result of computational experiments, the existence of a single envelope of contact pressure, a single envelope of normalized contact forces, and a single envelope of the relative values of the actual contact areas of microprotrusions has been established for each family of punches. The shape and position of these envelopes for a family of punches depend on the parameters of the external load and the ratio of the dimensions of the nominal contact area to the layer thickness.