Determination of surface deformation of continuously inhomogeneous thermoelastic half-space under local heating

The axially symmetric quasistatic problem on thermoelasticity for the functionally graded half-space, with elasticity modulus, Poisson ratio, heat conduction, and linear expansion coefficients varying continuously in the near-surface layer is considered. It is supposed that the area in the circle with the known radius is being heated by the heat source with time-constant temperature. Outside the circle, the surface is perfectly insulated. Analytic methods, particularly, Hankel integral transform, are used to solve the problem. At first, the solution is reduced to solving a two-point boundary problem for the ODEs system with varying coefficients of the sixth order. The modulating functions method is used to provide a stable numerical solution to the ordinary differential system. As a result, the solution to the mixed boundary problem is reduced to solving the dual integral equation. The kernel transform properties allow applying a well-established evolutionary bilateral asymptotical method. The heat flow approximation and the half-space surface displacement are determined through this method. The numerical results showing the curvature of the inhomogeneous half-space surface under the influence of the uniform temperature within the unit circle are given for various cases of the mechanical and temperature property changes in the near-surface layer. The following cases are considered: when the coating property value does not differ from the corresponding substrate property value, and when the coating value differs two times (to the higher or smaller side) on the surface, and linearly decreases (grows) in depth up to the property value in the substrate. It is shown that the ultimate effect on the maximum surface riser is caused by the oppositely directed changes in the thermal conductivity and linear expansion coefficients in the coating.