Poroelastic spherical indentation has been developed as an experimental technique to characterize poroelasticity for fully saturated soft and biological materials such as polymeric gels and hydrated bones at both microand macro-scale. Theoretical basis for generalizing the testing methodology to include geomaterials, where compressibility of the constituents is no longer negligible, has been established in our recent works. However, to successfully apply such a concept in the laboratory for geomaterials, we need an experimental parameter space where the material response can indeed be described by the theoretical solutions derived within the framework of Biot's theory and are not strongly affected by factors such as plastic deformation, finite sample size, indenter size, depth of penetration, and loading rate. To this end, we develop a fully coupled finite element algorithm for poro-elasto-plasticity. The porous medium is assumed to be isotropic and elasto-perfectly plastic, following a Drucker-Prager yield criterion with an associative or non-associative flow rule. Galerkin's method is used to convert the governing equations into weak form and the Newton-Raphson method with the tangent stiffness scheme is adopted to deal with plasticity in the solid skeleton. 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. This finite element algorithm is implemented using MATLAB and benchmarked with analytical solutions to the problems of Terzaghi, Mandel, Cryer and De Leeuw. The numerical solution for poroelastic spherical indentation is then compared with our theoretical solution. Effect of plastic deformation is also examined.

1. INTRODUCTION

Poroelastic spherical indentation via step-displacement or step-force loading has been developed as an experimental technique to characterize poroelasticity for fully saturated soft and biological materials such as polymeric gels and hydrated bones at both micro- and macro-scale (Hu et al., 2010; Kalcioglu et al., 2012; Oyen, 2008; Galli and Oyen, 2008). 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 early and late time response 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. Such master curves for various indenter shapes under step displacement loading have been previously constructed through finite element simulations (Hu et al., 2010) and also semi-analytically for spherical indentation with step force loading (Agbezuge and Deresiewicz, 1974; Oyen, 2008). A unique feature of such a testing method is that the testing duration scales with the contact area. As such, significant time saving can be achieved if the size of the spherical indenter and the depth of penetration are chosen appropriately.

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