Mathematical simulation of the stress-strain state structures from 3d-periodic MATERIALS UNDER THERMAL loads

Abstract

Nowadays 3D-periodic composite materials are widely used in the design of the bearing and enclosing structures. Such materials are multifunctional, one of the important functions of their application is to protect the bearing structures and vital components of machines and mechanisms from strong thermal action. To assess the strength of the structure under thermal action, it is necessary to be able to calculate the thermal stresses. To find the stresses occuring in the constructions made of 3D-periodic materials, the method of asymptotic splitting (also called the method of cellular functions), developed by the authors together with prof. Yu. V. Nemirovsky. Two scales of description are introduced: the scale of the periodic cell and the scale of the entire construction, and three levels of description: the real level, the macro level and the level of the periodic cell. The problem of minimization of the conditional functional for the macro environment, the corresponding boundary value problems for the macro continuum and the cell boundary value problems are obtained, the solution of which makes it possible to calculate deformations and stresses for the real level of the description from the effect of thermal loads. It is established that the components of the thermal expansion tensor for the macro environment depend both on the values of the thermal expansion coefficients for the matrix and the inclusions, and on the elastic properties of the matrix and inclusions. In addition, they depend on the shape of the inclusions, their location within the periodic cell, the volume content of inclusions and the temperature distribution inside the cell. An example of heating a multilayer periodic wall is considered.