CONSIDERATION OF NONLINEAR PROPERTIES OF MATERIALS IN MATHEMATICAL MODELING

Abstract

The relevance and demand of analytical and numerical-analytical approaches in the problems of the modern theory of elasticity, related to taking into account the nonlinearity of materials, does not cause any doubts. This behavior is characteristic of composite materials of various types (reinforced concrete, fiberglass). Non-linearity must be taken into account, for example, with complex loading, the influence of the external environment, extreme bending of the structure, etc. From this point of view, the development of analytical methods is very important for the development of modern design and construction technologies, as they immediately allow you to see reasonable approximate results, and can also serve to check numerical calculations. The authors in their previous works [1] already considered planar and spatial problems taking into account geometric nonlinearity. It should be noted that the proposed approach can be applied to solving problems in which residual deformations play a significant role (bending of thin plates and shells). A very important point is the verification of the obtained results, assessment of the adequacy and accuracy of the developed methods. The authors of the work always paid special attention to these issues. Proposed by A.V. Pavlenko and the approach developed by his students [2-4] has been repeatedly tested on problems of different levels of complexity. Based on the ideas of asymptotic integration for a small parameter related to the physical characteristics of the material, the method made it possible to reduce the problems of the linear theory of elasticity to the sequential solution of the problems of the potential theory, which is the most developed branch of mathematical physics. The application of this approach makes it possible to solve a number of new complex problems, among the advantages it is possible to specify the possibility of analyzing the stress-strain state of multilayer bodies with reinforcing elements. In the proposed article, using the approach developed by the authors, problems are solved for physically nonlinear materials in which the laws of deformation do not correspond to Hooke's law, that is, the dependence between stresses and deformations is nonlinear. In addition, the cylindrical anisotropy of the material of the interaction bodies is taken into account.