SUMMARY 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 e⁄ect 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 di⁄erent for the displacement and stress calculated from isotropic solutions. ( 1998 John Wiley & Sons, Ltd.

https://ir.nctu.edu.tw:443/bitstream/11536/32583/1/000074569100002.pdf

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

A strain-gradients theory of elastic dielectrics with spatial dispersion

Elastic closed-form solutions for the displacements and stresses in a transversely isotropic half-space subjected to various buried loading types are presented. The loading types include finite line loads and asymmetric loads (such as uniform and linearly varying rectangular loads, or trapezoidal loads). The planes of transverse isotropy are assumed to be parallel to its horizontal surface. These solutions are directly obtained from integrating the point load solutions in a transversely isotropic half-space, which were derived using the principle of superposition, Fourier and Hankel transformation techniques. The solutions for the displacements and stresses in transversely isotropic half-spaces subjected to linearly variable loads on a rectangular region are never mentioned in literature. These exact solutions indicate that the displacements and stresses are influenced by several factors, such as the buried depth, the loading types, and the degree and type of rock anisotropy. Two illustrative examples, a vertical uniform and a vertical linearly varying rectangular load acting on the surface of transversely isotropic rock masses, are presented to show the effect of various parameters on the vertical surface displacement and vertical stress. The results indicate that the displacement and stress distributions accounted for rock anisotropy are quite different for those calculated from isotropic solutions. Copyright © 1999 John Wiley & Sons, Ltd.

https://ir.nctu.edu.tw:443/bitstream/11536/31546/1/000078652200002.pdf

Elastic solutions for a transversely isotropic half-space subjected to buried asymmetric-loads

In this work, we present the solutions for displacements and stresses along the centerline of a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space with its Young's and shear moduli varying exponentially with depth. The planes of cross anisotropy are assumed to be parallel to the horizontal surface. The presented solutions can be directly integrated from the point load solution in a cylindrical coordinate system, which were derived by the writers. However, the resulting integrals of the circular solution for displacements and stresses cannot be given in closed form; hence, numerical integrations are required. For a homogeneous cross- anisotropic half-space, the numerical results agree very well with the exact solutions of Hanson and Puja, published in 1996. Two examples are given to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement and vertical normal stress in the inhomogeneous isotropic/cross-anisotropic soils subjected to a uniform vertical circular load acting on the surface. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice.

https://blogs.uakron.edu/ernianpan/files/2014/09/092_2006IJGeomechWangPanTzengHanLiao.pdf

Displacements and Stresses Due to a Uniform Vertical Circular Load in an Inhomogeneous Cross-Anisotropic Half-Space

Elastic and piezoelectric fields due to polyhedral inclusions

The methods of images and Hankel transforms are used to construct solution to an axisymmetric boundary value problem of a semi-space of transversely isotropic (granular) material due to a point force applied at a distanceh beneath its stress free plane boundaryz=0. Exact closed form expressions are determined for the components of displacements and stresses throughout the interior of the granular semi-space. The solution is then used to derive the surface displacements due to a uniformly distributed force over a circle of radius ‘a’ with centre at (0, 0, −h) in the planez=−h of the semi-space. By a suitable choice of material constants and through a limit process as α1, α2 approach 1, the granular semi-space becomes isotropic and the corresponding results derived in this particular case agree with those presented in [14].

On the axisymmetric Mindlin's problem for a semi-space of granular material

Variational principles and Mutter's form of the entropy inequality are used to establish the balance laws and constitutive equations for thermorigid dielectrics. An isotropy integrity basis that depends on the polarization-gradient tensor, polarization vector, and temperature-gradient vector is constructed for the free-energy function, and its minimality is verified. Representations for the entropy-flux vector and the heat-flux vector, when substituted into the restrictions derived from the entropy inequality, lead to the conclusions that the coldness function A depends on polarization and that the commonly known relation for isotropic materials, according to which the entropy-flux vector is equal to the ratio of heat flux to temperature, is not valid for polarized materials.

On thermorigid dielectrics

The matrix of fundamental solutions is constructed and used to obtain an integral representation for the displacement vector of the field equations of linear couple stress theory. The dynamic volume potential and single and double layer surface potentials are defined and the analogue of Poisson formula is obtained. The formal solutions to two fundamental boundary value problems are expressed by integral representations involving Green's matrices.

Potential methods in the linear couple-stress theory of elasticity

Constitutive equations for KDP by group theoretic methods

Similarity transformations are constructed and used to obtain an exact static solution for the axisymmetric boundary value problem of a homogeneous isotropic, thermoelastic half-space subjected to a hot pin normal to its surface. The closed form expressions derived for the displacement and the stresses, in the absence of thermal effects and for the case of a concentrated normal force applied to the surface of the semi-space, are found to agree with known results.

K. L. Chowdhury

Papers

On the axisymmetric Mindlin's problem for a semi-space of granular material

On thermorigid dielectrics

Potential methods in the linear couple-stress theory of elasticity

Constitutive equations for KDP by group theoretic methods

A hot pin on a semi infinite solid