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.
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.
Academician Yaroslav Stepanovych Pidstryhach was a noted scholar and specialist in mechanics, the founder of theoretical modeling in the mechanics of deformable media taking account of their structure and the linking of mechanical and nonmechanical processes. He devoted particular attention to the rote of classical and modern branches of mathematics that are of vital importance in constructing mathematical models for computing, incorporating in those models the latest results in the mechanics of solid media, nonequilibrium thermodynamics, condensed-matter physics and chemistry, electrodynamics, and other related areas. To revive the mathematical schools in L'viv and the Zakhid region of the Ukraine, the L'viv branch of mathematical physics of the Institute of Mathematics was founded and became the basis for the Institute of Applied Problems of Mechanics and Mathematics of the Ukrainian National Academy of Sciences. These institutions became the pillars of the Academy of Sciences in mathematics and mechanics in the region. He invited a number of distinguished scholars in the area of mathematics to work at the Institute.Ya. S. Pidstryhach understood that the high demands of rapid scientific and engineering progress could no longer be satisfied by the results of mechanical studies based on the classical Hooke's law, that the physicomechanical state of individual mechanical structures is determined not only by the mechanical load, but also by a wide range of external actions and interconnected internal processes caused by these actions. As a scholar who reached the intellectual summit of his time, he realized that a more profound knowledge of the hidden laws of nature would require that all scientific researchers understand not just the problems of their own narrow specialties, but also those of related areas and at least the "idiom" of very distant branches of knowledge. This does not mean, of course, that a scientist is required to be a professional in each of these areas; but he must be sufficiently erudite
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