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We formulate the basic relations of a mathematical model of mechanics of elastic deformable systems that describes the formation of a near-surface inhomogeneity caused by both the process of local mass displacement and dissipative processes. On this basis, we solve the problem of the stationary stressstrain state of an infinite hollow cylinder. It is shown that the near-surface inhomogeneity of the distribution of stresses and chemical potential is characterized by two parameters. One of them is related to the local mass displacement, and the other is a consequence of dissipative processes in the body.

We propose to apply energy and thermodynamic approaches to the construction of mathematical models describing thermomechanical processes in elastic deformable dissipative systems. Basic relations for the description of both a local thermodynamic state and dissipative processes are obtained. On this basis, we propose a version of the statement of boundary-value problems of optimization of the stress state under given integral restrictions imposed on the functions of external force loading and a scheme of their solution.

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