CONSIDERATION OF GEOMETRIC NONLINEARITY IN MATHEMATICAL MODELING OF PROBLEMS OF THE THEORY OF SPRING

Abstract

Solutions to many problems that are important for practice that arise in modern technology cannot always be obtained by traditional methods of the theory of analytical functions or by means of integral transformations. This applies, for example, to contact problems in which the finite dimensions of the region are taken into account in at least one direction, or media with curvilinear anisotropy are studied, etc. The means of the mathematical theory of elasticity are not very effective for the study of such problems. In this case, it is advisable to use the achievements of the potential theory. The use of asymptotic methods at the same time, even in complex cases, makes it possible to obtain well-founded approximate equations, clarify qualitative regularities, and obtain analytical solutions to problems. This paper presents a generalization of the perturbation method, which makes it possible to reduce the study of complex problems of geometrically nonlinear elasticity theory (in the plane and spatial formulation) to the consistent solution of simpler boundary value problems of the potential theory. The geometrically nonlinear theory of elasticity contains some features that make it different from the classical (linear) theory. The main difference is to take into account the difference between the geometry of the undeformed and deformed states of the body under study, when there are movements that cause significant changes in the geometry of the body. At the same time, the equilibrium equation must be drawn up taking into account changes in the shape and size of structures. Taking into account finite deformations, which when creating mathematical models leads to significant difficulties in solving problems, but at the same time brings the model closer to the real problem.