MATHEMATICAL MODELING OF PHYSICAL PROPERTIES OF ANISOTROPIC MATERIALS

Abstract

The problem of selecting a material with an extreme value of its performance using its anisotropy is considered. It is important for specialists of metallurgical profile to be able not only to select the material for realization of the set engineering task, but also to use its anisotropy, and to be able to determine the orientation of the material with the extreme value of its performance. Mathematical modeling and computer analysis of anisotropy of tensor coefficients using the example of thermal expansion coefficient have been performed. Since thermal expansion, like any tensor physical property of crystals, is a continuous function of direction, then in order to determine the directions with a zero value of thermal expansion, the following ratio must be satisfied: αn = 0. This can only happen if the main components of the thermal expansion tensor have different symbols. The Mathcad Prime 6 software complex has defined a function that performs the calculation of the value of thermal expansion coefficients in crystals in any direction, calculated the value and position of extremums of thermal expansion coefficients, and constructed an index surface, a stereographic projection of the index surface and the cross section of the index surface of thermal expansion coefficients X1X3. The lowest and highest values of the thermal expansion coefficient of the crystal have been found.