Generalized solution to the anisotropic Mandel's problem

• Published in: International journal for numerical and analytical methods in geomechanics 2020-12, Vol.44 (17), p.2283-2303
• Main Authors: Liu, Chao, Abousleiman, Younane N.
• Format: Article
• Language: English
• Subjects: analytical solutions; Anisotropy; Axes (reference lines); Deviation; Direction; Elasticity; Exact solutions; Mandel‐Cryer effect; Mathematical analysis; Parameter estimation; Plane strain; Pore pressure; poroelasticity; Sedimentary rocks; Shale; Strain; Symmetry; Vertical loads
• ISSN: 0363-9061; 1096-9853
• DOI: 10.1002/nag.3128

Summary Existing solutions to Mandel's problem focus on isotropic, transversely isotropic, and orthotropic materials, the last two of which have one of the material symmetry axes coincide with the vertical loading direction. The classical plane strain condition holds for all these cases. In this work, analytical solution to Mandel's problem with the most general matrix anisotropy is presented. This newly derived analytical solution for fully anisotropic materials has all the three nonzero shear strains. Warping occurs in the cross sections, and a generalized plane strain condition is fulfilled. This solution can be applied to transversely isotropic and orthotropic materials whose material symmetry axes are not aligned with the vertical loading direction. It is the first analytical poroelastic solution considering mechanical general anisotropy of elasticity. The solution captures the effects of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction on the responses of pore pressure, stress, strain, and displacement. It can be used to match, calibrate, and simulate experimental results to estimate anisotropic poromechanical parameters. This generalized solution is capable of reproducing the existing solutions as special cases. As an application, the solution is used to study the responses of transversely isotropic and orthotropic materials whose symmetry axes are not aligned with the vertical loading direction. Examples on anisotropic shale rocks show that the effects of material anisotropy are significant. Mandel‐Cryer's effects are highly impacted by the degree of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction.