Stress-strain state of an elastic soft functionally-graded coating subjected to indentation by a spherical punch

The article is devoted to the construction of fields of displacements, stresses and strainsarising in a linearly elastic half-space with a functionally-graded coating subjected to indentation by a punch with a spherical tip. Calculations of displacements, stresses and strains at the inner point of the coating andthe substrate is reduced to the integration on an infinite interval. The integrand is dependent on an unknown function of stresses distribution in the contact region. Contact stresses arising due to the indentation of a rigid spherical punch into an elastic half-space with a functionally-graded coating have earlier been constructed by the authors by solving the problem with mixed boundary conditions. For this purpose, the problem was reduced to the solution of a dual integral equation using the integral transformation technique. For a general case of independent arbitrary variation of Young’s modulus and Poison’sratioin the depth of the coating, the kernel transform of the integral equation can be calculated only numerically from the solution of a Cauchy problem for a system of ordinary differential equations with variable coefficients. Using approximations for the kernel transform of the integral equation by a product of fractional quadratic functions, approximated analytical expressions for the contact stresses and unknown radius of the contact area were constructed. The expressions obtained are asymptotically exact for both small and big values of relative coating thickness and high accuracy of intermediate values can be reached. The method is effective for an arbitrary variation of elastic properties and makes it possible to consider values of Young’s modulus of the substrate with more than two orders of magnitude higher than that in the coating. Series of numerical calculations of elastic displacements and stresses inside the coating and the substrate are provided for a case of soft homogeneous or functionally-graded layer lying on an elastic half-space (foundation). Young’s modulus of the layer is assumed to be constant or linearly varying (increasing or decreasing) in depth. At the layer-foundation interface Young’s modulus of the layer is 100 times as much as that of the foundation. This approach makes it possible to avoid the use of assumption about the non-deformability of the foundation for modeling soft homogenous or functionally-graded layers.