Elastic solutions for a transversely isotropic half-space subjected to a point load

Abstract

We rederive and present the complete closed‐form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half‐space. The half‐space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co‐ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half‐space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half‐space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the effect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite different for the displacement and stress calculated from isotropic solutions. © 1998 John Wiley & Sons, Ltd.

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

“…As opposed to the nonlinear Boussinesq problem, the linear solution to the transversely isotropic, incompressible tensile point load problem is easily discerned from the analogous compressive load linear solution. The displacement and stresses for the latter was determined by De Urena et al [5] and confirmed by Liao and Wang [6] and others. From [5], the deformation in the tensile load case can be expressed as…”

“…This paper will generalize to transverse isotropy, work begun by Simmonds, Polignone Warne, and Warne [1][2][3], using asymptotic methods to study nonlinearly elastic deformations of a general isotropic hyperelastic half-space subjected to a tensile point load acting normal to the undeformed surface of the half-space. Applications of the Boussinesq problem are found in many fields of engineering and mechanics, but the most direct and common applications are found in rock and soil mechanics, see, e.g., [4][5][6][7] and references cited therein. As soil and many rocks demonstrate properties of a transversely isotropic material (see, e.g., [7][8][9]), investigations such as this considering both material and geometric nonlinearity have many important implications in these and other fields.…”

Elastic and piezoelectric fields due to polyhedral inclusions

“…In this formulation the problem is reduced to calculation of the Green tensor. Much work has been done in the elasticity theory to calculate the Green tensor in a transversely isotropic (Chowdhury, 1987;Liao and Wang, 1998) and non-uniform (Wang et al, 2003) media. We restrict ourselves to a uniform isotropic half-space, where the Green tensor is given by Eq.…”

“…The history of the problem and explicit expressions are presented by Pan and Chou (1979a,b);Walker, 1993;Yu et al (1994Yu et al ( , 1995; Yue, 1995;Yu and Rath, 2000. Liao and Wang (1998 review solutions for displacements, strains, and stresses in a half-space subjected to different types of loads.…”

The analysis of a rigid rectangular plate on a transversely isotropic multilayered medium

“…Refs. [1][2][3][4][5][6][7] provided the fundamental solutions of the transversely isotropic half-space and full-space subjected to different loads. In many situations, soils present the stratification property, so it is more realistic to regard soils as transversely isotropic multilayered media.…”