Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

Abstract

Unique, explicit, and exact expressions for the first-and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997, "The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids," Q. J. Mech. Appl. Math. 50, pp. 407^26) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues Pa fa = 1,2,3j on the oblique plane with normal coincident with the position vector; second, it remains welt-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

“…Like the stress fundamental solutions or the first derivatives of the displacement fundamental solutions presented in the last section, the present unified explicit expressions for the derivatives of the stress fundamental solutions or the second derivatives of the displacement fundamental solutions have the following distinct features compared with some previous works (e.g. [20,26,28]):…”

“…Compared with the expressions derived by Lee [20,26], the new expressions may be more suitable for the applications since the expressions of Lee [20] are not valid in degenerate and nearly degenerate cases while the expressions of Lee [26] may cause numerical errors in the nearly degenerate cases. Buroni & Sáez [28] also presented unified explicit expressions by taking the derivatives of the expressions given by Ting & Lee [21] in the spherical coordinate system. However, it seems that our expressions are more explicit and clear than those presented by Buroni & Sáez [28] in the sense that our expressions without terms like p i − p j in the denominators can be more evidently identified.…”

“…Based on the explicit displacement fundamental solutions of Ting & Lee [21] and using the spherical coordinate system, Lee [26] and Shiah et al [27] derived explicit expressions for the first and second derivatives of the displacement fundamental solutions by performing the spatial differentiations with respect to the spherical coordinates. Unified and explicit expressions for the displacement fundamental solutions and their derivatives based on the displacement fundamental solutions of Ting & Lee [21] have been recently presented by Buroni & Sáez [28], who also performed the spatial derivatives in the spherical coordinate system and argued that their expressions are valid for both non-degenerate and degenerate cases. In any case, care should be taken by using fundamental solutions containing terms like p i − p j in the denominators, which may either destroy the validity of the fundamental solutions or cause considerable numerical errors and instability in nearly degenerate cases, where p i is very close to p j [24,25,29,30].…”

“…The resulting explicit fundamental solutions are then recast into their equivalent but more convenient forms, which are unified, in closed-form and valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. In contrast to some previous works [26][27][28], in this analysis the partial derivatives of the displacement fundamental solutions are performed with respect to the Cartesian coordinates instead of the spherical coordinates, which is easier, especially for the second derivatives of the displacement fundamental solutions. To the best of the authors' knowledge, the present unified and closed-form analytical fundamental solutions for three-dimensional anisotropic elasticity are novel and have not yet been reported elsewhere.…”

On two accurate methods for computing 3D Green׳s function and its first and second derivatives in piezoelectricity

“…Both the Stroh formalism and the residue calculus methods can lead to explicit expressions of the 3D anisotropic elastic Green's function and its derivatives. The residue calculus method has been further applied by Ting and Lee [5], Phan et al [6,7], Lee [8,9], Buroni and Sáez [10] and Xie et al [11], among others.…”