Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body

We consider the axisymmetric contact problem of a multi-elastic layer with various elastic constants bonded to an elastic semi-infinite substrate indented by rigid flat-ended cylindrical and spherical indenters. The transfer matrix method is applied to each elastic layer, and dual integral equations are reduced to an infinite system of simultaneous equations by expressing the normal contact stress at the surface elastic layer as an appropriate series with Chebyshev orthogonal polynomials. Numerical results demonstrate the effects of the elastic constant of each elastic layer and the semi-infinite elastic substrate on the radial distribution of the normal contact stress and normal displacement of the free surface of the elastic layer, stress singularity factor at the edge of the cylindrical indenter, and axial load of a rigid indenter which penetrates the multi-layer material to a constant depth. The results of axial load are in good agreement with previously reported results. The numerical results are given for several combinations of the shear modulus of each elastic layer and the substrate. These results will contribute to the establishment of indentation tests for composite materials and serve as guidelines for the design of appropriate mechanical properties of layered materials.