Boundary integral equation methods for the solution of crack problems

Smith, R.N.L.; Aliabadi, M.H.

doi:10.1016/0895-7177(91)90073-g

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…However, the subtraction of singularity approach (or equivalently TSS) converges more rapidly and requires fewer elements for very accurate stress intensity factors. 1 …”

Section: Singular Fields and The Bemmentioning

confidence: 97%

“…A new numerical scheme based on varying the boundary conditions arising from the known form of singularity is used with the two-step superposition method to predict the dominant eigenvalue 1 for a variety of notch sizes. The numerical values of 1 obtained are in excellent agreement with Williams' analytical results as reported by Smith and Della-Ventura.…”

Section: V-notch Singularitiesmentioning

confidence: 99%

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Some applications of singular fields in the solution of crack problems

Della-Ventura

Smith

1998

Int. J. Numer. Meth. Engng.

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This paper reviews some recent developments in superposition methods for calculating linear elastic stress intensity factors and eigenvalues for cracks and notches, presents some new results for pairs of edge cracks and provides new insights into the nature of the errors in these processes. The procedure requires a numerical solution to the full cracked problem and a second solution on the same mesh using the known form of the singularity in an inÿnite region. This is equivalent to the well-known Subtraction of Singularity (SST) method. The advantages of this procedure over conventional SST are: (1) no modiÿcations need to be made to a standard computer program; (2) multiple crack tips may be analysed without the di culty of unknown rigid body displacements at the crack tips; (3) solutions with di erent boundary conditions on the same mesh may be obtained simply in one step by re-using one singular ÿeld solution; The singular crack tip ÿeld may also be studied independently leading to estimates of the eigenvalues and some insight into mesh-induced errors. The additional computational cost of a two-step procedure is minimal since the solution matrix from step one may be re-used with a new right-hand side. Numerical experiments using the boundary element method demonstrate the accuracy and simplicity of the superposition approach for notches, simple cracks, mixed-mode cracks, two edge cracks of di erent lengths and eigenvalues under various boundary conditions. ? 1998 John Wiley & Sons, Ltd.

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“…However, the subtraction of singularity approach (or equivalently TSS) converges more rapidly and requires fewer elements for very accurate stress intensity factors. 1 …”

Section: Singular Fields and The Bemmentioning

confidence: 97%

Section: V-notch Singularitiesmentioning

confidence: 99%

Some applications of singular fields in the solution of crack problems

Della-Ventura

Smith

1998

Int. J. Numer. Meth. Engng.

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“…In a recent paper [58], comparisons were made between the traction singular quarter point element (TSQP) and the subtraction of singularity technique (SST). The configuration studied was that of a central slant crack in a large sheet shown in figure 5.9.…”

Section: Subtraction Of Singularitymentioning

confidence: 99%

Application of Boundary Element Methods to Fracture Mechanics

Aliabadi¹,

Rooke²

1991

Solid Mechanics and Its Applications

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the point P'(x,-y) has field parameters ~j' £~j' uj and ~, ~ they [,,11,,11,,11] [ ~' ; l{ 2 } + + 1 [,,11,,11,,11 [,,11,,11,,11] Now, since the integration path is taken to be symmetrical with respect to the crack along the x-axis (y=O), the outward normal components (nx,n» at points p(x,y) and P'(x,-y) have the following relationships (5.86) By using this relationship, the traction fields in (5.84) at point P(x,y) can be related to those at P'(x,-y), that is (5.87)Other field parameters are also related as follows:r;,} = { ~'} {:~}=r~J} Therefore using the above relationships the J-integral can be decoupled into mode I and mode II components, hence ~ and Kn can be evaluated separately from (5.81).The J-integral being a energy approach has the advantage that elaborate representation of the crack tip singular fields is not necessary. This is due to the relatively small contribution that the crack-tip fields make to the total J (Le. strain energy) of the body. Further advantage of the J-integral approach is that it can be used as a post-processing procedure for the evaluation of stress intensity factors with the internal stress components oj' evaluated from (4.9) using the boundary values of the displacements and tractioJs. The strain components £ij are Numerical Fracture Mechanics 171 subsequently evaluated using strain-stress relationships in (2.10) for plane strain or (2.12) for plane stress. The partial derivatives of displacement 8U/8X may be evaluated accurately using the formula in (4.8) or alternatively a simple approximation formula [51] It is worth noting that the denominator in (5.91) may become zero, depending on the integration path; in such cases it is possible to use the following relationship2 xy 8yIn order to assess the accuracy of the J-integral approach described above, the problem of a central slant crack of length 2a in a rectangular sheet of width W, height 2W and angle of inclination a.=fI1' (see figure 5.9) subjected to tensile uniform stress was studied since alternative accurate results are available [52]. Table 5.3 shows the percentage error compared to [52] in the calculated values of stress intensity factors for a/W=0.1 and 0.2 using different size circular integration paths of radius r centered at the crack tip. The results were obtained using 58 quadratic elements to model the boundary and 18 internal points. The partial derivatives of displacements were calculated from equation (4.8) So far all the techniques discussed in this chapter, have been based on attempts to model the singular stress behaviour at the crack tip. In contrast, the subtraction of the singularity technique avoids the need for this difficult task; it removes the singular fields completely. This leaves a non-singular field to be modelled numerically-a much simpler task. This approach was first introduced into the two-dimensional boundary element formulation of potential problems by Symm[53] and Papamichel and Symm[54] who used constant elements to model a symmetrical slit. Later Xanthis et al [ll] used th...

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“…It has been pointed out that stress singularity behaviors in a problem must be properly considered to obtain a convergent and accurate numerical solution in different numerical approaches [12]. Therefore, the knowledge of the nature of the stress singularities is found to be very important to achieve greater efficiency and higher accuracy in the numerical computation of various quantities of engineering interest including stresses and stress intensity factors by taking the stress singularity function into consideration in an interpolation function (shape function) of finite element [1,2,6,20,26] and boundary element methods [5,18,19,21,23,30]. Notably, these afore-cited studies reveal that singular elements can be used successfully to investigate the problems with stress singularities by describing the exact order of stress singularities in the vicinity of singular points.…”

Section: Introductionmentioning

confidence: 99%

A revisit to the Williams's solution in bending problem of plates with stress singularities

Boonchareon¹,

Sompornjaroensuk²,

Kongtong³

2013

ijma

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This paper aims to present the details of mathematical derivation for static bending problem of isotropic sectorial plates that involve stress singularities at the vertex of the plates due to geometry and boundary conditions. Following the classical Kirchhoff's plate theory, the general complete solution of 4 th -order partial differential equation governing the plate-bending behaviors can be determined mathematically and expressed explicitly in the form of infinite series through the separation of variables method in terms of polar coordinates. Based on the principle of superposition the solution can be separated into two parts; namely, the particular solution and complementary solution in which the latter describes the local singular behaviors at its sharp vertex exactly. Some cases of circular, annular and sectorial plates with mixed circumferential edge conditions that remain undetermined analytically up to the present time are also suggested.Mathematics Subject Classification: 31A25, 74K20, 35Q74

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Boundary integral equation methods for the solution of crack problems

Cited by 10 publications

References 15 publications

Some applications of singular fields in the solution of crack problems

Some applications of singular fields in the solution of crack problems

Application of Boundary Element Methods to Fracture Mechanics

A revisit to the Williams's solution in bending problem of plates with stress singularities

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