Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media

Rungamornrat, Jaroon; Mear, Mark E.

doi:10.1016/j.ijsolstr.2007.09.030

Cited by 30 publications

(27 citation statements)

References 26 publications

(41 reference statements)

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“…To obtain a set of governing integral equations for B , we employ the pair of weakly singular, weak-form displacement and traction integral equations developed by Rungamornrat and Mear [16]. When specialized to the BEM region B , these integral equations take the form…”

Section: Governing Equations For Bmentioning

confidence: 99%

“…For anisotropic media, Xu [15] generalized the work of Xu and Ortiz [11] to allow treatment of isolated cracks, and Rungamornrat and Mear [16,17] generalized the work of Li et al [12] to obtain a SGBEM applicable to cracks in a finite domain. We remark that it has been demonstrated that the weakly singular SGBEM is capable of yielding highly accurate stress intensity factors with relatively coarse meshes.…”

Section: Introductionmentioning

confidence: 99%

“…The key features of the technique are the use of a pair of weakly singular, weak-form displacement and traction integral equations developed by Rungamornrat and Mear [16,17], and a weak-enforcement of continuity of displacement and traction across the interface separating the region modeled by the SGBEM and that modeled with the FEM. We remark that the weak-enforcement of the continuity conditions is done in a fashion which differs from that discussed by Frangi and Novati [30], and that the formulation and implementation apply to very general boundary value problems for fractures in anisotropic media.…”

Section: Introductionmentioning

confidence: 99%

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

Rungamornrat¹,

Mear²

2010

Numerical Meth Engineering

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SUMMARYA weakly singular symmetric Galerkin boundary element method (SGBEM) is coupled with the standard finite element method (FEM) in order to establish an accurate and efficient numerical technique for analysis of fractures in three-dimensional, anisotropic, linearly elastic media. In the strategy, the weakly singular SGBEM developed by Rungamornrat and Mear (Int. J. Solids Struct. 2008; 45:1283-1301 Comput. Methods Appl. Mech. Engrg 2008; 197:4319-4332) is utilized to model a small-scale region containing the crack while the (possibly large and complex) compliment region is treated by the FEM. The coupled technique exploits the positive features of both methods; the SGBEM proves to be a convenient and highly accurate method for obtaining mixed-mode stress intensity factors along the crack front, whereas the FEM is very efficient for modeling large-scale problems in the absence of cracks. An important aspect of the formulation and implementation of the technique is that continuity of displacement and traction across the interface between the SGBEM and FEM regions is enforced in a weak sense. This allows the two regions (one modeled by the SGBEM and the other by the FEM) to be discretized independently without the need for the resulting meshes to conform on the interface separating the regions, and this flexibility in the discretization process leads to a significant reduction in the modeling effort. To demonstrate the utility and accuracy of the technique, several boundary value problems involving both embedded and surface breaking cracks are treated, and it is shown that the coupled technique yields highly accurate stress intensity factors that exhibit only a slight dependence upon mesh refinement.

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Section: Governing Equations For Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Rungamornrat¹,

Mear²

2010

Numerical Meth Engineering

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“…As mentioned previously, for crack problems it is typical that t J = 0, and for this reason the term in (4) involving the stress fundamental solution could be dropped without substantive loss. Nonetheless, we choose to retain this term and demonstrate a process for its regularization; the purpose in doing so is to provide a basis to treat the displacement integral relation (3) which, while not of particular relevance to the current investigation for isolated cracks, is important to the development of a SGBEM for treatment of cracks in a finite domain (see Rungamornrat and Mear 2008a).…”

Section: Standard Integral Relations For Isolated Crackmentioning

confidence: 99%

Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

Rungamornrat

Mear

2008

Int J Fract

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A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity factors for isolated cracks in threedimensional, generally anisotropic, piezoelectric media. The method is based upon a weakform integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its integrand contains only weakly singular kernels of O(1/r ) (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high accuracy (even with relatively large elements), and it has the important feature that the mixedmode intensity factors can be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method, special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement.

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“…However, his formulation was limited to an isotropic case and the implementation of those equations to solve half-space problems has not been recognized. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Tan¹,

Rungamornrat²,

Pansuk³

et al. 2015

WIT Transactions on Modelling and Simulation

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This paper presents an efficient and accurate boundary integral equation method for solving a linear, multi-field, half-space containing a surface of discontinuity and subjected to symmetrical and anti-symmetrical conditions on the boundary. Responses of the half-space are governed by a set of linear partial differential equations which are formulated in a general framework allowing the treatment of Laplace equation, linear elasticity problems, and problems involving multi-field materials such as piezoelectric, piezomagnetic, and piezoelectromagnetic solids. In the formulation, a systematic regularization procedure via the integration by parts, symmetrical and anti-symmetrical properties, and special representations of strongly singular and hyper singular kernels is employed to derive a set of singularity-reduced boundary integral relations. A pair of weak-form boundary integral equations involving both the sum and relative crack-face state variable and surface flux across the discontinuity surface is finally established and they contain only weakly singular kernels. A standard symmetric Galerkin boundary element method (SGBEM) is then implemented to solve those weakly singular integral equations for unknown data on the discontinuity surface. In numerical implementations, continuous local interpolation functions are employed in the approximation of solutions and an efficient means for both the kernel evaluation and the numerical integration is adopted to enhance the accuracy and computational efficiency of the developed scheme. The proposed numerical technique is then verified with various, reliable benchmark cases and a selected set of results is presented to demonstrate its capability and robustness.

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Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media

Cited by 30 publications

References 26 publications

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

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

Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

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

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