Poroelastic response of spherical indentation into a half space with a drained surface via step displacement

Abstract

Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’ s theory. We seek the response of the indentation force and the field quantities as functions of time when a rigid pervious indenter is loaded instantaneously to a fixed depth. The particular case where the surface of the semi-infinite domain is permeable and under a drained condition is considered. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee–Gibson displacement function method is adopted in this work. One of the difficulties in solving poroelastic contact problems theoretically is in evaluating integrals with kernels that oscillate rapidly. We show that such issues can be overcome by using alternative integral representations with exponentially decaying functions in the kernels. Special functions, such as the modified Struve functions and the modified Bessel functions, and the method of contour integration can be utilized to aid the removal of the oscillation. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and onset of plastic deformation are then discussed. An interesting outcome from this analysis is that the transient response of the dimensionless indentation force shows only weak dependence on one derived material constant and can in fact be fitted by a simple elementary function, which can then be conveniently used for material characterization in the laboratory.