Solution of the gradient thermoelasticity problem for a cylinder with a heat-protected coating

The investigation of the stress-strain state of an infinitely long thermoelastic cylinder is carried out taking into account the scale effects. A thermal protective coating is applied to the outer side surface of the cylinder, the thermomechanical characteristics of which are functions of the radial coordinate. Thermal boundary conditions of the first kind are set on the stress-free side surfaces of the cylinder. To take into account the scale effects, the one-parameter gradient theory of thermoelasticity proposed by Aifantis is used. Additional boundary conditions and conjugation conditions for couple stresses are specified. The displacements and stresses are represented as the sum of the solutions of the thermoelasticity problem in the classical formulation and the gradient parts. After finding the radial temperature distribution, the thermoelasticity problem in the classical formulation with respect to radial displacements and stresses is solved numerically by the shooting method. Additional boundary layer terms for radial displacements at small values of the gradient parameter are found using the asymptotic method for solving linear differential equations with spatially varying coefficients (the Wentzel-Kramers-Brillouin method - WKB method). Calculations of radial displacements, Cauchy stresses, couple stresses and total stresses in the case of both homogeneous and inhomogeneous coatings are carried out using specific examples. It has been found that the Cauchy stresses and total stresses experience a jump at the boundary between the cylinder and the coating. The couple stresses at small gradient parameters are much less than the total stresses. An increase of the scale parameter reduces the values of radial displacements and total stresses. Deformations are continuous in both cases of coating (homogeneous and inhomogeneous). A comparative study of the influence of the value of the inhomogeneity parameter on the distribution of displacements and total stresses is carried out.