Mykhalchuk H. Contact problems of motion of deformable bodies along hard surfaces

The object of research is the quasistatic process of elastic bodies motion along the hard surfaces based on coupling concept. The purpose of the study is to formulate the new contact problems of deformable bodies motion along the hard surfaces, to create an effective numerical method, and to investigate the features of the motion. The research methods are the variational inequality approach combining with finite element method and the mathematical programming algorithms. The formulations of the problem under the form of a variational inequality and under the form of the extremal variational problem were obtained. An equivalence of these formulations was proved. Also, it is proved that the solution of variational problem is equivalent to the solution of the problem in the initial (differential) statement. The uniqueness of the variational problem solution was proved. The numerical algorithm was developed with using the finite element method and method of successive over-relaxation. The numerical investigation of bodies motion along horizontal and vertical surfaces and along surfaces with roughnesses was carried out. Scientific novelty. The new class of contact problems of elastic bodies motion along hard surfaces was considered. For the first time, the conditions of interaction between the body and the motion surface in the form of the inequalities that consider uncertainty of contact and coupling areas and limitation of the contact stress were formulated. For the first time, the variational formulations of the problem of the elastic bodies motion along the surface were received and proved. The numerical method was developed and new scientific data on modeling of the quasistatic motion of elastic bodies along different surfaces were obtained after computing experiment. The areas of application are mechanical engineering, biomechanics, and the learning process.