In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of “approximate-homogeneous” state in the functionally graded half-space are developed.

Multilayered and functionally graded coatings are extensively used for protection against wear of the working surfaces of mechanisms and machines subjected to sliding contact. The paper considers the problem of wear of a strip made of a functionally graded material, taking into account the heating of the sliding contact from friction. Wear is modeled by a moving strip along the surface of a hard abrasive in the form of a half-plane. With the help of the integral Laplace transform with respect to time, the solutions are constructed as convolutions from the law of the introduction of an abrasive into the strip and the original in the form of a contour integral of the inverse Laplace transform. The study of the integrands of contour quadratures in the complex plane allowed determination of the regions of stable solutions to the problem. Unstable solutions of the problem lead to the concept of thermoelastic instability of the contact with friction and formed regions of unstable solutions. The solutions obtained made it possible to determine a formula for the coefficient of functionally graded inhomogeneity of the coating material and to study its effect on the occurrence of thermoelastic instability of the contact taking friction into account, as well as on its main characteristics: temperature, displacement, stress and wear of the functionally graded material of the coating. The effects of the abrasive speed, contact stresses and temperature on wear of the coating with the functionally graded inhomogeneity of the material by the depth were investigated.