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

Abstract

A fully coupled poroelastic solution for spherical indentation into a half space with an impermeable surface when the indenter is subjected to step displacement loading is presented. The solution is obtained within the framework of Biot’s theory using the McNamee–Gibson displacement function method. Mathematical difficulties associated with solving poroelastic contact problems are overcome by the use of a series of special functions such as the modified Struve and Bessel functions for evaluating integrals with kernels that oscillate rapidly and the method of successive substitution for solving Fredholm integral equation of the second kind. Expressions for the poroelastic fields on the surface and inside the half space are derived. Effect of poroelasticity on incipient failures in form of tensile crack initiation and plastic deformation are discussed. The theoretical analysis shows that the normalized indentation force relaxation has a relatively weak dependence on material properties through a single derived material constant ω only and the asymptotic behaviors at ω=0 at both early and late times can be expressed in closed-form. Master curves of indentation force relaxation can be constructed by fitting the full solution with an elementary function for convenient use of poroelasticity characterization in the laboratory. In addition, excellent agreement is achieved between the theoretical solution and numerical results from FEM simulations using a hydromechanically coupled algorithm that we had previously benchmarked rigorously.