Indentation of an elastic layer by a rigid cylinder

Abstract

The Green’s functions for the indentation of an elastic layer resting on or bonded to a rigid base by a line load are found efficiently and accurately by a combination of contour integration with a series expansion for small arguments. From the form of the equations it is clear that the function is oscillatory when the layer is free to slip over the base, but for the bonded layer, the function simply decays to zero after a single overshoot.The deformation due to pressure distributions of the form of the product of a polynomial with an elliptical (“Hertzian”) term is calculated and the coefficients chosen to match the indentation shape to that of a cylindrical indenter. The resulting pressure distributions behave much as in Johnson’s approximate theory, becoming parabolic instead of elliptical as the ratio b/d of contact width to layer thickness increases, or, for the bonded incompressible (ν=1/2) layer, becoming bell-shaped for very large b/d.The relation between the approach δ and the contact width b curves has been investigated, and some anomalies in published asymptotic equations noted and, perhaps, resolved.A noticeable feature of our method is that, unlike previous solutions in which the full mixed boundary value problem (given indenter shape / stress-free boundary) has been solved, the bonded incompressible solid causes no problems and is handled just as for lower values of Poisson’s ratio.