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We consider the problem of contact interaction between two elastic bodies one of which has a coating in the form of a thin Timoshenko-type shell connected with the body through a nonlinear Winkler layer. We present a weak formulation of this problem in the form of a nonlinear variational equation. We propose a class of iterative domain decomposition methods that reduce the solution of this equation to the solution, in each iteration, of independent linear variational equations corresponding to problems of elasticity for massive bodies and a problem of the Timoshenko theory of shells for the coating with Robin boundary conditions imposed on the contact boundaries. The conditions of weak convergence of these methods are established. We analyze the numerical efficiency of the obtained algorithms with the use of finite-element approximations.

On the basis of the penalty method, we propose a number of continuous parallel domain decomposition schemes for the solution of the problems of perfect mechanical contact of elastic bodies. For some of these schemes, we prove the theorems on convergence. We study the problem of optimal choice of the iterative parameters. The relationship between the proposed schemes and the domain decomposition methods without penalty is established.

We study the domain decomposition methods for the numerical solution of the problems of frictionless unilateral contact of many elastic bodies of finite sizes. By using the finite-element approximations, we solve the problems of the unilateral contact of three elastic bodies compressed by rigid plates and the contact of three fixed bodies one of which is subjected to the action of an external load. The distributions of normal contact and equivalent stresses in the bodies are analyzed.