Boundary integral equation solution of three-dimensional elastostatic problems in transversely isotropic solids using closed-form displacement fundamental solutions

Abstract

SUMMARYThe boundary integral equation method is used for the solution of three-dimensional elastostatic problems in transversely isotropic solids using closed-form fundamental solutions. The previously published point force solutions for such solids were modiÿed and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples.

Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

“…This fact will guarantee the non-singular character of the modulation function expressions presented. Finally, bringing together Equations (1), (4), (12), (15) and (20)(21)(22)(23), an explicit and completely general expression for the fundamental solution U (x) in a transversely isotropic material is obtained. The form of this expression suitable for a computational implementation obtained by performing explicitly the transforms indicated in (12) is given in [1].…”

“…The completely general and closed-form expressions of U ik;j ( x) can now be easily obtained by substitution of (24) with (27-28) and (19)(20) into (9) considered for a point x (i.e. θ = 0).…”

“…Loloi [20] and Ariza and Domínguez [23] presented the expressions of the strongly singular and hypersingular kernel, respectively, obtained from Pan and Chou's [17] solution. This solution [17] has several features which make somewhat cumbersome its implementation covering all possible cases: i) expressions depending on the values of △ = √ C 11 C 33 − C 13 − 2C 44 (positive, negative or zero) and in particular its complex-variable character for △ < 0; ii) a loss of precision and/or a division by zero for the spherical angle φ = π or φ = 0.…”

“…In the previous expressions it is assumed that the x 3 -axis is the rotational symmetry axis of the transversal isotropic material. Although the difficulty with the degeneracy problem at φ = π has been solved by Loloi [20] by means of an ad hoc approach, the mentioned features may still cause some difficulties in using this expression in further analytic deductions and in BEM development. Other authors as, Ding et.…”

Three-dimensional BEM analysis to assess delamination cracks between two transversely isotropic materials

“…There are several expressions for the fundamental solutions for transversely isotropic materials; see, for example, [Pan and Chou 1976;Loloi 2000]. However, these solutions could be cumbersome to implement in a BEM code because of the multiple cases to consider due to all the possible material orientations and the relative positions of the source and field points.…”