The Generalized Lamé Problem—Part I: Coupled Poromechanical Solutions

Abstract

Cylindrical geometries are known to present special problem simulation capabilities in engineering design. (For example, solid and hollow cylinder tests are routinely studied in soil and rock mechanics to gain insights into the geomechanical properties and to assess the stability of boreholes and cylindrical openings in the project design geomedia.) This paper identifies a unified and universal solution to all the three recognized right-cylindrical problem objects in poromechanics. A closed-form solution to the problem of the finite, homogeneous, isotropic, fully saturated, thick-walled hollow cylinder subjected to various loading modes is readily presented and described. The assumed loading modes encompass arbitrary temporal functions of uniformly distributed inner/outer pore pressure, inner/outer confining pressure, inner/outer deviatoric stress, and end axial compaction or extension. The time-dependent response derivations are outlined within the frameworks of the Biot’s theory of linear poroelasticity and facilitated by the governing generalized plane-strain (GPS) principle. The (as presented) solution is shown to converge asymptotically to those of the two essential problem setups in geomechanics: the finite solid cylinder and the borehole core in an infinite medium. As such, a complete/explicit solution to a generalized statement of the Lamé problem is presented. The solution utilizes a fairly simple loading decomposition scheme which leads to two basic problem forms: a generalized poroelastic axisymmetric problem and a generalized, plane-strain, poroelastic deviatoric problem.