Traction BIE solutions for flat cracks

Polch, E. Z.; Cruse, T. A.; Huang, Chaofan

doi:10.1007/bf00296420

Cited by 80 publications

(24 citation statements)

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“…For all cases, the crack is traction free for = All the above 3D results are really 'quasi-2D' because the problem being considered is a rectangular crack with uncoupled crack modes. As discussed by Polch et al [19], the in-plane modes II and III are in fact coupled in the general 3D case for planar cracks, while the crack opening mode I is not coupled to the in-plane displacements. A rotation of the co-ordinate system around the axis normal to the crack would give the more general coupled results for the in-plane modes, and the crack surface would still be traction free, as obtained above.…”

Section: D Traction-bie Results For the Buried At Crackmentioning

confidence: 91%

“…Similar asymptotic log-singular behaviour for some integrals in the self-regular traction-BIE is also observed in this work. Polch et al [19] also found similar log terms when integrating contour integrals, approximated as straight line elements, on a patch of elements containing the source point.…”

Section: Introductionmentioning

confidence: 76%

“…For collocation at points near the crack contour, this formulation might encounter numerical di culties. Polch et al [19] observed such numerical di culties when collocating a traction-BIE (obtained using integration by parts) at points close to the crack contour, and careful evaluation of the path integral near the collocation point was required.…”

Section: New Bie Formulation For Near-crack Contour Collocationmentioning

confidence: 99%

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Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

Jorge

Cruse

Fisher

2003

Numerical Meth Engineering

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SUMMARYThis work contains an analytical study of the asymptotic near-crack contour behaviour of stresses obtained from the self-regular traction-boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The at crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self-regular traction-BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near-crack contour self-regular traction-BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log-singular stresses are identiÿed and extracted from the BIE. Log-singular stress terms are revealed for the free integrals written as contour integrals and for the self-regularized integral with the integration region divided into sub-regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the ÿnal BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identiÿed, and the singular information is all contained in the region near the crack contour.

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Section: D Traction-bie Results For the Buried At Crackmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 76%

Section: New Bie Formulation For Near-crack Contour Collocationmentioning

confidence: 99%

See 1 more Smart Citation

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

Jorge

Cruse

Fisher

2003

Numerical Meth Engineering

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“…Hence we can take advantage of this approach only after weakening the hypersingularity of the kernel, or only after "regularizing" it. As a matter of fact, some of such attempts can be found in the articles by Sladek and Sladek [1], Bui [2], Bonnet [3], Polch et al [4], Nishimura and Kobayashi [ 5] who used the collocation method and in Nedelec [6], Bamberger [7] where the variational method has been used.…”

Section: Introductionmentioning

confidence: 99%

“…However, not all of these regularizations are universal because some of them work only with collocation methods, and because others may destroy the causality in the time-domain. For example the authors of [1]- [4] seem to have devised their techniques mainly with collocation methods in mind. Also the formulation in [6] may not be very useful in the time-domain because it will produce kernels which violate the causality.…”

Section: Introductionmentioning

confidence: 99%

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

1993

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Non-Singular Somigliana Stress Identities in Elasticity

Cruse

Richardson

1996

Int. J. Numer. Meth. Engng.

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SUMMARYThe paper presents two M y equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is CIsa continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at comers and edges. These new findings specifically focus on the comer problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity.The resulting regularized and fmite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to C',' continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for comers is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.

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Traction BIE solutions for flat cracks

Cited by 80 publications

References 9 publications

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

Non-Singular Somigliana Stress Identities in Elasticity

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