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 $\omega$ only and the asymptotic behaviors at $\omega =0$ 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.

提出了一种完全耦合的多孔弹性解决方案，用于当压头受到阶梯位移载荷时，将球形压入具有不渗透表面的半空间中。该解是在 Biot 理论的框架内使用 McNamee-Gibson 位移函数方法获得的。与求解多孔弹性接触问题相关的数学难题通过使用一系列特殊函数来克服，例如用于评估具有快速振荡内核的积分的修正 Struve 和 Bessel 函数以及求解第二类 Fredholm 积分方程的连续代入方法. 导出了表面和半空间内部的多孔弹性场的表达式。讨论了多孔弹性对拉伸裂纹萌生和塑性变形形式的初始破坏的影响。理论分析表明，归一化压痕力松弛通过单一的导出材料常数对材料特性的依赖性相对较弱$\omega$ 只有和渐近行为 $\omega =0$在早期和晚期都可以用封闭形式表达。压痕力松弛的主曲线可以通过用基本函数拟合完整解来构建，以便在实验室中方便地使用多孔弹性表征。此外，理论解与 FEM 模拟的数值结果之间实现了极好的一致性，使用我们之前严格进行了基准测试的流体力学耦合算法。