MATHEMATICAL MODELING IN GEOMETRICALLY NONLINEAR ELASTICITY THEORY PROBLEMS

Abstract

The resolve of many important problems in practice that arise in modern technology cannot always be obtained by traditional methods of analytic function theory or by means of integral transformations. This applies, for example, to contact problems, which take into account the finite size of the region in at least one direction, or investigate environments with curvilinear anisotropy and the like. The means of 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 theory of potential. The use of asymptotic methods, even in complex cases, allows to obtain reasonable approximate equations, to clarify the qualitative patterns and to obtain analytical solutions. This paper presents a generalization of the perturbation method, which allows us to reduce the study of complex problems of geometrically nonlinear elasticity theory (in plane and spatial formulation) to the sequential solution of simpler boundary value problems of potential theory. Geometrically nonlinear theory of elasticity contains some features that make it different from classical (linear) theory. The main difference is to take into account the difference between the geometry of the undeformed and deformed states of the studied body, when there are displacements that cause significant changes in the geometry of the body. The equilibrium equation must be made taking into account changes in the shape and size of structures. Taking into account the final deformations, which in the creation of mathematical models leads to significant difficulties in solving problems, but at the same time brings the model closer to the real problem. The perturbation method, which is used to solve nonlinear equations in partial derivatives, has theoretical and practical significance. It is universal and can be used to analyze various problems of mathematical physics. The developed approach can be applied to solve problems in which residual deformations play a significant role. For example, the bending of thin plates and shells. In the considered model problem it was possible to allocate influence of geometrical nonlinearity on a stress-strain state of the investigated body. That is why the results of the presented work have both theoretical and applied significance, and the study is relevant.