The paper examines the axisymmetric problem of the indentation of a poroelastic halfspace that is reinforced with an inextensible permeable/impermeable membrane located at a finite depth by a rigid indenter. The constitutive behavior of the poroelastic halfspace is described by the three‐dimensional theory of poroelasticity proposed by M.A. Biot. The contact conditions between the indenter and the poroelastic halfspace are varied to accommodate both adhesive/frictionless contact and impermeable/permeable conditions. The formulation of the mixed boundary value problems uses the stress function approaches applicable to semi‐infinite domains. Successive applications of Laplace and Hankel integral transforms are used to reduce the mixed boundary value problems to sets of coupled Fredholm integral equations of the second kind. These integral equations are solved using numerical approaches, applicable both for the solution of the systems of coupled equations and for Laplace transform inversion, to examine the time‐dependent displacement of the rigid indenter. The analytical‐numerical estimates for the time‐dependent displacements of the rigid indenter are compared with results obtained using a finite element approach.

中文翻译：

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包含嵌入式不可拉伸膜的多孔弹性半空间的接触问题

本文研究了多孔弹性半空间的压痕的轴对称问题，该半空间由刚性压头固定在有限深度的不可扩展的渗透/不渗透膜增强。MA Biot提出的多孔弹性三维理论描述了多孔弹性半空间的本构行为。压头和多孔弹性半空间之间的接触条件是变化的，以适应粘合剂/无摩擦接触和不渗透/可渗透条件。混合边值问题的制定使用了适用于半无限域的应力函数方法。拉普拉斯和汉克尔积分变换的连续应用被用于将混合边界值问题简化为第二类耦合的Fredholm积分方程组。这些积分方程是使用数值方法求解的，该方法既适用于耦合方程组的求解，也适用于拉普拉斯变换反演，以检查刚性压头的时变位移。将刚性压头随时间变化的解析数值估计与使用有限元方法获得的结果进行比较。