Solutions to the problem of a point load acting in the interior of a full-space are called the fundamental solutions or the elastic Green’s function solutions (Pan et al., 1976 and Tarn et al., 1987). In the problems of infinite media, Willis (1965) estimated the elastic interaction energy of two infinitesimal dislocation loops in transversely isotropic magnesium and zinc media. There were two reasons to choice this medium, the first being the case of presentation of the results afforded by the axial symmetry, facilitating a ready comparison with the isotropic results. The second one was that to find the closed-form expressions for fundamental elasticity tensor for such a medium were possible.
Ting and Lee (1996) obtained the solution of Green’s function for three-dimensional space of general anisotropic inclined medium subjected to a unit point force. It was expressed in terms of the Stroh eigenvalues pv (v=1, 2, 3) on the inclined plane, and it remained valid for the degenerate cases when p1=p2, and p1=p2=p3. The Stroh eigenvalues pv were the roots with positive imaginary part of a sextic algebraic equation.
The Green’s function was simple when the sextic equation was a cubic equation in p2. This was the case for any point in a transversely isotropic material and for points on a symmetric plane of cubic, and monoclinic materials.
These solutions in exact closed-form have always played an important role in applied
mechanics and in particular numerical formulations of boundary element methods.
Many investigators have presented analytical solutions for displacement under a point load in a transversely isotropic full-space, whose the transversely isotropic planes are parallel to the horizontal loading surface. A summary of the existing solutions is given in Table 2.1.
Table 2.1 Existing solutions for a transversely isotropic full-space subjected to a point load
Author Analytical methods Type of loading Presented solutions Chowdhury (1987) methods of images and
Hankel transforms
vertical all displacements
Pan (1989) vector functions 3D all displacements Willis (1965) Fourier transforms vertical all displacements Elliott (1948) potential functions vertical all displacements Chen (1966) potential functions vertical
horizontal
all displacements all displacements Pan and Chou (1976) potential functions 3D all displacements Fabrikant (1989) potential functions 3D all displacements Sveklo (1969) complex variables vertical all displacements Tarn and Wang (1987) Fourier and Hankel
transforms
3D all displacements
Lu (1991) Fourier and Hankel transforms
3D all displacements
Liao and Wang (1998) Fourier and Hankel transforms
3D all displacements
Sheu (2004) Fourier and Hankel transforms
3D all displacements
Table I indicates the analytical methods, the type of loading and the presented results. To the best of the authors’ knowledge, no closed-form solutions for the displacement have been obtained in cases in which the planes of transverse isotropy inclines to the full-space subjected to 3D point loads (Px, Py, Pz), as displayed in Figure 3.1. In this thesis, the methods proposed by Willis (1965) for a transversely isotropic medium are followed. That is, the triple Fourier transforms are adopted to obtain the integral expressions of Green’s displacement;
then, the triple inverse Fourier transforms and residue calculus are performed to integrate the contours. However, Willis’s expressions for Green’s function are only valid when the elastic constants fulfill conditions that enable inverse transforms to be carried out (Tarn et al., 1987). Notably, the main difference between Willis’s approach (1965) and that proposed herein was the use of orthogonal vectors. In the former, two axes were on the transversely isotropic plane, and the third was parallel to the axis of rotation associated with elastic symmetry. Accordingly, a state of plane strain was assumed in that procedure.
2.2 Three-Dimensional Elastic Solution for Displacements and Stresses in a Transversely Isotropic Half Space
A point load solution is the basis of complex loading problems. For an isotropic solid, it has been studied by Kelvin (Thompson, 1848) for a full-space, Boussinesq
(1885) and Cerruti (1888) for a half-space with a vertical and horizontal point load, respectively. In the case of a single concentrated force acting in the interior of a half-space, Mindlin (1936) proposed closed-form solutions for an isotropic medium using the principle of superposition of eighteen nuclei. Mindlin derived analytical solutions following the Kelvin's (1848) approach and satisfying the condition of vanishing traction on a plane boundary. However, the calculation of nuclei for a half-space is very difficult (Mindlin and Cheng, 1950). Dean et al. (1944) recommended another approach for the same problem by the method of images. Some of their solutions can be extended to anisotropic media, whereas others are difficult.
For the displacements and stresses in transversely isotropic media subjected to a point load, analytical solutions have been presented by several investigators. Some of the solutions were directly derived using the approaches for isotropic solutions (Michell, 1900; Wolf, 1935; Koning, 1957; Barden, 1963; de Urena et al., 1966; Misra and Sen, 1975; Chowdhury, 1987; Pan, 1989). Nevertheless, others employed complex mathematics techniques, such as Fourier transformations (Kröner, 1953; Willis, 1965;
Lee, 1979), potential functions (Lekhnitskii, 1940; Elliott, 1948; Shield, 1951; Eubanks and Sternberg, 1954; Lodge, 1955; Hata, 1956; Chen, 1966; Pan and Chou, 1976; Pan and Chou, 1979; Okumura and Dohba, 1989; Fabrikant, 1989; Lin et al., 1991; Hanson and Wang, 1997) and complex variables (Sveklo, 1964, 1969), etc. The summary of the existing solutions is given in Table 2.1. Table 2.1 indicates the type of analytical space, the load, and the results presented in their solutions. Because of mathematical difficulty or oversimplification for solving the problems, these solutions were limited to three-dimensional problems with partial results of displacements (Michell, 1900; Shield,
1951; Barden, 1963) and stresses (Michell, 1900; Shield, 1951; Barden, 1963; Misra and Sen, 1975; Pan and Chou, 1979; Chowdhury, 1987; Fabrikant, 1989), or axially symmetric problems invariant with the tangential co-ordinate, θ (Lekhnitskii, 1940;
Elliott, 1948; Koning, 1957; Sveklo, 1964; de Urena et al., 1966; Misra and Sen, 1975).
Neglecting the θ, the solutions cannot be extended to solve a half-space problem subjected to asymmetric loads. Pan and Chou (1979) proposed a more general solution using potential functions. In their solution, the buried loads can be vertical or horizontal with respect to the boundary plane. However, only the stress components related to the z-direction were given (i.e., σzz, σzx, σzy), and the expressions for the solution are quite lengthy.
Table 2.2 Existing solutions for transversely isotropic media subjected to a point load Author Space Type of loading Solutions
Michell (1900) half vertical vertical surface displacement, and partial stresses
(inapplicable to boundary value problems) Wolf (1935) half vertical all displacements and stresses
(oversimplified the elastic constants) Koning (1957) half vertical all displacements and stresses Barden (1963) half vertical vertical surface displacement,
and stresses on load axis de Urena et al.
(1966)
half vertical all displacements and stresses
Misra and Sen (1975)
half vertical all displacements, and stresses on load axis (oversimplified the elastic constants) Chowdhury (1987) full
half
vertical buried, vertical
all displacements, and stresses on load axis all displacements, and stresses on load axis Pan (1989) full 3-D all displacements and stresses Kröner (1953) half vertical all displacements (dimensionally incorrect)
Willis (1965) full vertical all displacements (cumbersome and inaccurate) Lee (1979) half buried, vertical all stresses (complicated)
Lekhnitskii (1940) half vertical all stresses (incomplete)
Elliott (1948) full vertical all displacements and stresses (incomplete) Shield (1951) half buried, vertical all displacements and stresses at the surface
Eubanks and Sternberg (1954)
half vertical (completeness of Lekhnitskii’s method)
Lodge (1955) (transformed anisotropic problem
into isotropic one, inapplicable to general boundary value problems) Hata (1956) half vertical (rederived the Elliott’s and Lodge’s solution) Chen (1966) full vertical
horizontal
all displacements and stresses all displacements Pan and Chou
(1976)
full 3-D all displacements and stresses
Table 2.2. Existing solutions for transversely isotropic media subjected to a point load (continued)
Author Space Type of loading Solutions
Pan and Chou (1979)
half buried, vertical buried, horizontal
all displacements, and stresses on load axis all displacements, and partial stresses (potential functions assumed are lengthy) Okumura and
Dohba (1989)
half vertical all displacements
Fabrikant (1989) full half
3-D 3-D
all displacements, and partial stresses all displacements, and partial stresses (solution of the shear stress is wrong) Lin et al. (1991) half vertical, horizontal all displacements and stresses Hanson and Wang
(1997)
half buried, 3-D (only the potential functions listed)
Sveklo (1964) half vertical all displacements
Sveklo (1969) full half
vertical buried, vertical
all displacements all displacements
Following the method proposed by Tarn and Wang (1987), Lu (1991) presented analytical solutions for the displacements in a full or half soil space (transverse isotropy) under a long-term consolidation. However, a part of the solutions might be error in handwriting. Utilizing the approaches proposed by Lu (1991), closed-form solutions for displacements and stresses in a transversely isotropic half-space subjected to a point load are rederived and parts of the results are published (Liao and Wang, 1998).
However, the solutions are limited to the cases of planes of transverse isotropy parallel to the horizontal loading surface.
The solution of the stresses and displscements in a half-space or a layered solid with transverse isotropy is fundamental to the development of the theory of elasticity and is of importance to many engineering applications. Ding (1987) presented a
unified solution for a point force applied on the surface/in the interior of a half-space.
The solution could be extended to the problem of layered media using the state-space and Fourier transform methods. Hence, to solve the problems of semi-infinite media, Ding considered a transversely isotropic medium in a Cartesian co-ordinate system (x, y, z), whose z-axis is perpendicular to the isotropic plane of material. Any point force
(or concentrated force) applied in the body can be resolved into three components T, Q and P in x-, y-, and z-direction, respectively. Ding assumed that an arbitrary point force was applied at the origin. It can be decomposed the problem into three sub-problems by using the principle of superposition; namely, the solution corresponding to a vertical force, P, in the positive z-direction, the solution to a tangential force, Tx=T, in the x-direction, as well as the solution to a tangential force, Ty=Q, in the y-direction. The last solution can be obtained from the second solution by a co-ordinate transform with x replaced by y, and y by –x, respectively. However, it is clear that Ding’s solutin has not yielded the analytical solutions of displacements and stresses for an inclined transversely isotropic material owing to three–dimensional point loads. All the fundamental solutions of literature for transversely isotropic materials being the case of axisymmetric problem.
CHAPTER Ⅲ BASIC THEORY
To derive the solutions for stresses, strains, and displacements in finite domains with simple geometry, the method of separation of variables is usually applicable for the partial differential equations to reduce to the ordinary differential equations (Chiang, 1997). Then the solutions can be constructed by superposition of eigenfunctions.
However, considering the domain is infinite or semi-infinite, it is hard to achieve a similar reduction. Thus, the integral techniques include the Fourier, Laplace, Hankel, and Mellin transforms are often utilized to attain the goal. Among them, the Fourier and Laplace transforms are basic and most useful. Generally, the governing equations for infinite or semi-infinite solids are partial differential equations. The Fourier and Laplace integral transforms are efficient methods for solving the partial differential equations and corresponding boundary value problems. Employing the two methods can reduce the problem from solving partial differential equations to ordinary differential equations or algebraic equations.
3.1 Basic Equations for Elastic Boundary Value Problems