Poroelastic response of spherical indentation with step displacement loading is investigated numerically using a hydromechanically coupled finite element method (FEM) algorithm following a mixed continuous Galerkin formulation for displacement and pore pressure. A stabilization scheme, which permits equal-order interpolation for the displacement and pore pressure fields and suppresses pore pressure oscillation in the incompressible or nearly incompressible limit, is incorporated in this FEM algorithm. A frictionless contact scheme based on the penalty method that establishes the contact condition on the Gaussian points is implemented to account for the interaction. Numerical simulations with and without the contact scheme are compared with our theoretical solutions which are derived following the Hertzian assumption with only the normal displacement over a fixed contact area being prescribed. We show that after frictionless contact is incorporated, the normalized force relaxation response from the numerical simulations can still be very well captured by the theoretical solutions when the indentation strain is relatively small. Insights gained from this numerical analysis are valuable in supporting the use of the theoretical master curves for poroelasticity characterization for geomaterials.
Material characterization via contact with a rigid tool has its root in the study of Hertz on frictionless contact between two bodies (Hertz, 1881). Since then normal indentation with tools of various shapes has evolved to become standard testing techniques for measuring material properties such as hardness, strength and toughness (Johnson, 1987; Lawn, 1993). Recently, experiments on fully saturated soft and biological materials such as polymeric gels and hydrated bones show that spherical indentation could also be an effective technique for poroelasticity characterization (Hu et al., 2010; Kalcioglu et al., 2012; Oyen, 2008; Galli and Oyen, 2008; Lai and Hu, 2017; Lai and Hu, 2018; Islam and Oyen, 2021). In theory, for a fully saturated porous medium consisting of incompressible constituents, if the indenter is subjected to either step displacement or step force loading, elastic constants can be determined from the undrained and drained limits according to the Hertzian contact solution, while hydraulic diffusivity or the coefficient of consolidation can be obtained from the transient response by matching the measured indentation force or displacement as a function of time against a master curve. Attempts have been made to establish this type of master curves numerically for step displacement loading (Hu et al., 2010; Lai and Hu, 2017) and semianalytically for step force loading (Agebuge and Deresiewicz,1974; Oyen, 2008).