Thermal stresses in multi-component anisotropic materials

Abstract

This paper represents a continuation of the previous author’s work which deals with an analytical model of thermal stresses which originate during a cooling process of an anisotropic solid elastic continuum. This continuum consists of anisotropic spherical particles which are periodically distributed in an anisotropic infinite matrix. The particle is or is not embedded in an anisotropic spherical envelope. The infinite matrix is imaginarily divided into identical cubic cells with central particles. With regard to the analytical modelling, this multi-particle-(envelope)-matrix system represents a model system which is applicable to two- and three-component materials of precipitate-matrix and precipitate-envelope-matrix types, respectively. Particle and envelope radii, particle volume fraction, and cubic cell dimension, which is equal to inter-particle distance, represent parameters of the cubic cell as well as microstructural parameters of the two- and three-component materials. The thermal stresses, originating as a consequence of different thermal expansion coefficients of components of the model system, are determined within the cubic cell, and consequently represent functions of these microstructural parameters. The analytical modelling results from fundamental equations of continuum mechanics for solid elastic continuum (Cauchy’s, compatibility and equilibrium equations, Hooke’s law). This paper presents suitable mathematical procedures which are applied to the fundamental equations. The fundamental equations are derived within spherical coordinates, and these mathematical procedures lead to a final differential equation for a radial displacement. The radial, tangential and shear thermal stresses are thus determined as functions of the radial displacement. The mathematical procedures presented in this paper are not as extensive as those presented in the author’s previous work, Results of this paper are then more suitable for numerical determination of the thermal stresses in real two- and three-component materials with anisotropic components.