Regularized boundary integral representations for dislocations and cracks in smart media

Abstract

This paper presents a complete set of singularity-reduced boundary integral relations for isolated discontinuities embedded in three-dimensional infinite media. The development is carried out within a broad context that allows the treatment of a well-known class of smart media such as linear piezoelectric, linear piezomagnetic and linear piezoelectromagnetic materials. In addition, resulting boundary integral representations are applicable to general discontinuities of arbitrary geometry and possessing a general jump distribution. The latter aspect allows the treatment of two special kinds of discontinuities: dislocations and cracks. The most attractive feature of the current development is that all integral relations for field quantities such as state variables and their gradients, the body flux, and the generalized interaction energy produced by dislocations are expressed only in terms of line integrals over the dislocation loops and, for cracks, the key governing boundary integral equation is established in a symmetric weak form and contains only weakly singular kernels of O(1/r ). Results for the former case are fundamental and useful in the context of dislocation mechanics and modeling while the resulting weakly singular, weak form integral equation constitutes a basis for the development of a well-known numerical technique, called a symmetric Galerkin boundary element method (SGBEM), for analysis of cracked bodies. The weakly singular nature of such an integral equation allows low order interpolations to be used in the numerical approximation.The key ingredient for achieving such development of integral representations is the use of certain special decompositions in the derivative-transferring process via Stokes's theorem. Existence of such decompositions is ensured by a careful consideration of the singularity nature of the kernels, and a particular solution of the weakly singular functions involved is obtained by solving a system of partial differential equations via a method of Radon transforms. The final results, for general anisotropy, are given in a concise form in terms of an equatorial line integral that is suitable for numerical evaluation. As part of the verification, a numerical experiment is carried out for isolated crack problems via use of a weakly singular SGBEM and results exhibit only mild dependence on the mesh refinement and excellent agreement with existing analytical solutions.

SGBEM–FEM coupling for analysis of cracks in 3D anisotropic media

“…This problem is used to validate the formulation and implementation within the context of conforming discretizations, and then (for more complicated boundary value problems that do not admit an analytical solution) the results obtained using conforming discretizations are used to validate the procedure in the context of nonconforming discretizations. We remark that the results for nonconforming discretizations are reported only for the case in which the constraints given by (40) are utilized (i.e. for the case in which the reference interface coincides with S IF ); we have carried out extensive numerical experiments and have found that, at least for all cases considered, the choice of reference interface has little influence upon the numerical results (see [33]).…”

BIEs for modelling of discontinuities in linear multi-field half-space

“…[15]) along with the symmetrical and antisymmetrical conditions. It should be emphasized that results presented here are limited only to the case that a material constituting the half-space is homogeneous and possesses the plane x 3 = 0 as a plane of material symmetry.…”

“…where 1 S  ξ x for certain classes of multi-field materials can be found in the work of [15]. Due to the singularity nature of ( ) …”

“…Other essential data such as the intensity factors contained in the singular term and the first non-singular term in the expansion of the body flux in the neighborhood of the boundary of discontinuity surface can also be extracted. In particular, techniques analogous to those proposed by [15] and [16] are employed along with the solved data ˆJ u  and ˆP u   to compute the intensity factors and the first non-singular term, respectively.…”

“…The regularization procedure analogous to that used by Li [6] was extended to model cracks in a whole space and finite bodies [7,8] and to take into account the material anisotropy [13,14], but the extension for the case of a half-space has not been found. In the present study, the procedure used by [6,7,[13][14][15] is further generalized to derive a set of regularized integral equations for a half-space containing discontinuities and made of multi-field materials. In addition, a solution scheme based on a well known, weakly singular boundary integral equation method is also established.…”