On Poroelastic and Poro-elasto-plastic Hertzian Contact Problems
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A class of contact problems between a rigid sphere and a linear poroelastic or poro-elasto-plastic half-space is examined through an integrated theoretical and numerical analysis to understand the feasibility of spherical indentation as a testing technique for poroelasticity characterization of geomaterials. Fully coupled theoretical solutions are first derived for spherical indentation into a poroelastic half-space with three distinct cases of surface drainage conditions when the indenter is subjected to step displacement loading. The solutions are obtained within the framework of Biot's theory using the McNamee-Gibson displacement function method. Specifically, we overcome the mathematical difficulties associated with evaluating integrals with highly oscillatory kernels by using techniques such as special functions and the method of contour integration. Moreover, the method of successive substitution, instead of the method of quadrature previously used in the literature, is employed to solve the Fredholm integral equation of the second kind to improve solution accuracy. The theoretical analyses show that the normalized transient indentation force response has a relatively weak dependence on material properties through a single derived material constant only. 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. A hydromechanically coupled finite element method (FEM) algorithm following a mixed continuous Galerkin formulation for displacement and pore pressure and incorporating a frictionless contact scheme is then constructed for modeling of spherical indentation in a poro-elasto-plastic medium. Numerical simulations with the step displacement loading condition realized with or without ramping show that the normalized force relaxation responses from the numerical simulations agree very well with the theoretical solutions if the indentation strain and ramping duration are relatively small. For indentation in a poro-elasto-plastic medium, it is shown that even though plasticity could occur immediately at the undrained limit, if the indentation strain and material strength are such that there is no additional plastic strain accumulation during the transient period, the normalized force relaxation behavior could be approximated as poroelastic. Finally, a combined theoretical and numerical analysis is performed to examine the step force loading condition. Results show that the normalized transient displacement response has a relatively stronger dependence on material properties, suggesting that force-controlled tests may be less reliable than displacement-controlled tests for poroelasticity characterization.