Poroelastic Response of Spherical-Tip Indentation

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 as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth. We consider the particular case when the surface of the semi-infinite domain is permeable and under a drained condition. 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. One of the difficulties in solving this class of problems analytically is in evaluating integrals with oscillatory kernels over an unbounded interval. We show that such issues can be overcome by the use of a series of special functions. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and plastic deformation are then discussed. An interesting outcome from this analysis is that if the indentation forces at the undrained and drained limits are known, relaxation of the indentation force with time can be used to determine the diffusion coefficient of a porous medium.