Finding physically justified partial solutions of the equations of the thermoelasticity theory in the cylindrical coordinate system

Abstract

The paper considers the linear model of three-dimensional isotropic body of theories of thermoelasticity in the cylindrical coordinate system. We consider the case when the stationary temperature satisfies the Laplace equation. After substituting thermoelastic stresses into the equilibrium equation, the system of Navier’s differential equations were obtained. Its general solution is presented as the sum of homogeneous and partial solutions. The partial solution of the system of Navier’s equations, which is clearly determined by the stationary temperature and does not contain elastic displacements, is called the temperature solution. The physical and mathematical features of the thermoelastic stress state were taken into account and it was proved that in the temperature solution the sum of normal stresses is zero and the volume deformation is equal to . The found dependencies were used and the new temperature solution of the system of Navier’s equations were constructed in the cylindrical coordinate system, when the temperature does not depend on the axial variable. Simple formulas for expressing temperature stresses are given. The general solution of the equations of the theory of thermoelasticity by four harmonic functions is recorded.