Solution of the Quasi‐Static Contact Problem of Upsetting an Opaque Indenter in a Liquid‐Saturated Poroelastic Biot Medium

ABSTRACT We consider the contact problem of the settling of a rigid strip indenter, opaque for liquid, with a flat base shape into a liquid‐saturated poroelastic Biot medium in the form of a half‐space. Mixed boundary conditions of the problem are formulated separately for the elastic and liquid phases. The solution of the contact problem is reduced to a system of two two‐dimensional integral equations of the first kind regarding the contact stresses and the contact pressure of pore liquid. A system of two‐dimensional integral equations is reduced to an equivalent system of two one‐dimensional integral equations of the first kind with respect to the Laplace transforms of unknown functions. After selecting special parts of the integral equation kernels (including singular ones), the system of integral equations is reduced to a triangular form by the elimination method. Sequential inversion of integral operators on the left side leads the system of integral equations of the first kind to a system of integral equations of the second kind. To solve the latter one, a scheme of the method of successive approximations is organized, through which the Laplace transforms of its zero approximation are determined. After inverting the found Laplace transforms, a zero approximation of the solution of the original two‐dimensional system of integral equations (contact stresses and contact pressure of pore liquid) is determined. A numerical analysis of the solutions obtained is presented.