Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

Abstract

SUMMARYA general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three-dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed-form real-variable expressions of the fundamental solution in displacements U ik and in tractions T ik , originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for U ik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50:407-426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U ik , and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U ik, j and the corresponding stress kernel i jk and traction kernel T ik has been developed in the present work. These expressions of U ik , U ik, j , i jk and T ik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex-valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational-symmetry axis. The expressions of U ik , U ik, j , i jk and T ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three-dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials.

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

“…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. Távara et al [2008] have recently derived completely general and unique expressions valid for all possible configurations given in terms of real functions only. These fundamental solutions are presented in what follows.…”

“…The closed-form expressions ofˆ i jk (x) can be found in [Távara et al 2008]. The traction fundamental solution associated with the normal vector n j (x) can be obtained directly using…”

“…The BEM implementation uses the fundamental solutions recently introduced in [Távara et al 2008]. The BEM solution strategy for the fracture problem and the EDI implementation is an extension of that proposed in [Ortiz and Cisilino 2005] for interface cracks in dissimilar isotropic bimaterials.…”

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

“…A comprehensive review of the history of the development of fundamental solutions and their derivatives for general anisotropic materials, and in particular for the transversely isotropic materials, was recently presented in [1,2]. Thus, only the most relevant contributions closely related to the present work will be briefly mentioned.…”

“…The use of U ik (x) expression introduced by Ting and Lee [9] and the transformation rule introduced therein, that was corrected and extended in [1], are crucial for the achievement of this aim.…”