Effective approach to the contact problem for a stratum

A novel effective algorithm for the problem of the circular punch in contact with a stratum rested on a rigid base is suggested in this paper. The problem is reduced to the Fredholm integral equations of the second kind. In contrast to the Cooke–Lebedev method and the moments method, which are traditionally employed, the operators of these integral equations are strictly positive definite even in the limiting case of the zero thickness. The latter provides efficient applications of numerical methods. It is also shown that a special approximation enables to obtain an approximate solution via a finite system of linear algebraic equations. As example, the well-known problem for a homogeneous layer is studied. An approximate analytical solution is found with a certain iterative method for a flat punch. This solution is remarkable accurate and possesses the right asymptotic behavior for both a very thin and a very thick layers. Asymptotic formulas for the thin inhomogeneous stratum indented by an indenter of arbitrary profile are pointed out.