3D fracture analysis by the symmetric Galerkin BEM

Abstract

The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method.Keywords Fracture, Boundary element method, Galerkin approach IntroductionIn the numerical modelling of linear elastic fracture mechanics problems, boundary element methods (BEM) have distinct advantages over domain approaches, especially when cracks are directly represented as displacement discontinuity loci and the traction integral equation is employed to enforce static conditions on the crack itself. The displacement discontinuity method [6], the dual BEM [11,13,21] and the symmetric Galerkin BEM [4] share the above features and permit single domain formulations for problems with single or multiple cracks embedded in finite bodies or in the infinite medium. However the SGBEM differs from the others in that it is based on a variational (weak) version of the integral equations, thus entailing double integrations, and leads (through an appropriate discretization scheme) to matrix operators which exhibit symmetry and sign-definiteness. A 1998 review paper on the SGBEM [4] shows that the method has been the subject of extensive investigations since it was first proposed in 1979 [24] and the most recent literature gives evidence of a lively interest towards this methodology (see e.g. [10,17,19]).While for 2D linear elasticity a number of implementations of the SGBEM into computer codes have been presented (see, e.g. [7,25]), some of which also addressing crack problems, only few 3D implementations for elastic problems, with or without cracks, have been described in the literature [15,27]. The evaluation of the double surface integrals in the singular cases, not an easy task especially if general isoparametric elements are employed, represents probably the main obstacle which has hampered the application of the method in the 3D context.The present work addresses the application of the SGBEM in the context of 3D linear elastic fracture mechanics using a ''regularized'' symmetric formulation which is essentially the same as that expounded in [5,8,18]; a first fairly general implementation of this approach is documented in [15] where, however, no details are given on the adopted integration techniques. On the contrary here focus is set on the development of efficient algorithms for the crucial singular double surface integrals, according to the new schemes introduced in [1, 9, 23] and further expanded so as to permit the combined use of quadrilateral and triangular quadratic isoparametric elements. With reference to both cracks in the unbounded space and in finite bo...

High-resolution brittle fracture simulation with boundary elements

“…There is a wide range of methods available for the deformation step, such as mass-spring networks [Norton et al 1991;Hirota et al 1998;Hirota et al 2000;Smith et al 2001;Levine et al 2014], finite element methods (FEM) [O'Brien and Hodgins 1999;Müller et al 2001;O'Brien et al 2002;Bao et al 2007;Wicke et al 2010;Zheng and James 2010;Koschier et al 2014] and extended FEM (X-FEM) Gravouil et al 2002;Abdelaziz and Hamouine 2008;Kaufmann et al 2009;Mousavi et al 2011], mesh-less or particle-based methods [Pauly et al 2005] such as the material point method (MPM) [Stomakhin et al 2013;Stomakhin et al 2014], and boundary element methods (BEM) [Aliabadi 1997;James and Pai 1999;Wilde and Aliabadi 1999;Frangi et al 2002;Sutradhar et al 2008;Kielhorn 2009;Messner and Schanz 2010;Sauter and Schwab 2011;Keeler and Bridson 2014].…”

“…This means that fractures need to be connected to existing surfaces where they meet. However, when reducing the problem to computing only crack-opening displacements (instead of absolute displacements of both sides of the crack), LEFM problems can be solved using a single trianglesheet per crack [Frangi et al 2002].…”

“…Our method builds on the BEM formulation presented by Frangi et al [2002], who have shown that it accurately computes stress intensity factors (SIF) in fractured objects. While it is known that SIF are underestimated near boundaries (which would not happen in a DualBEM [Wilde and Aliabadi 1999]), it has the advantage that adding new fracture surfaces can be done simply by inserting triangles into the mesh; no changes need to be made to previously existing elements.…”

Sliding frame-solid interaction using BEM/FEM coupling

“…The main strength of the BEM for boundary-value problems governed by linear, homogenous, and elliptic differential equations with constant coefficients is the reduction of the dimensionality of the problem by one unit for linear constitutive relations Brebbia, 1978;Brebbia, 1980 . Particularly, BEM is useful to model special situations such as very large or unbounded domains, geometrical singularities e. g. cracks or to obtain very accurate results in regions of complicated shape Aliabadi, 1997;Bonnet, 1999;Frangi et al, 2002 . Thus, coupling the BEM and the FEM allows exploiting their complementary advantages.…”

Boundary integral equations for general laminated plates with coupled stretching–bending deformation

“…Among these boundary elements, the development of plate bending problems (Stern 1979;Hartmann & Zotemantel 1986;Shi & Bezine 1988) started relatively later than that of twodimensional or three-dimensional elastostatics (Rizzo 1967;Brebbia et al 1984). *chwu@mail.ncku.edu.tw For example, even though vast advances have been made in boundary element methods for two-and three-dimensional fracture problems (Li et al 1998;Frangi et al 2002;Rungamornrat & Mear 2008a,b), relatively few studies have been done on boundary element formulation for cracks in plates subjected to transverse loading or bending moments. As to the stretching-bending coupling analysis of composite laminates, as far as the author is aware, no associated boundary integral equation has ever been derived in the literature.…”