Algorithm for Solving Discrete Contact Problems for an Elastic Strip

Discrete contact problems between an elastic strip and a rigid punch with previously 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 strip boundary. To approximate this operator, the 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 formulated. As a result of its approximation, a quadratic programming problem with constraints in the form of equalities and inequalities is obtained. To solve the problem numerically, an algorithm based on the conjugate gradient method is used. The algorithm takes into account the specifics of the set of constraints. One-parameter families of punches with a surface relief are constructed the parameter of which is the number of microprotrusions. As a result of computational experiments, the existence of a single envelope of contact pressure, a single envelope of normalized contact tractions, 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 external load parameters and the ratio of the size of the nominal contact area to the strip thickness.