Application of Boundary Element Methods to Fracture Mechanics

Abstract

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...

Residual opening of hydraulic fractures filled with compressible proppant

“…There are a number of mathematical approaches that can be employed to solve the problem presented above, such as the Distributed Dislocation Technique (DDT) [21,22], formulations based on integral equations with hypersingular kernels [34][35][36][37] and numerical techniques based on the Boundary Integral Equation Method [38,39]. In this work we applied the DDT as it has proved to be a very efficient method for solving Fracture Mechanics problems.…”

Analytical integrations and SIFs computation in 2D fracture mechanics

“…Further to the numerical approximation of problem (10), several methods [49] allow for the evaluation of SIFs, deducing them from related quantities (J or M-integrals [25,50]) or from asymptotic relationships that hold for the exact solution (one or two point displacement correlation techniques [51], traction correlation technique [52]). By exploiting analytical integrations of Section 4, a direct and coherent approximation of SIFs can be obtained by the asymptotic analysis of representation formula (6) by means of the evaluated numerical approximation y h .…”

A coupled BEM/scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics

“…Methods existing to calculate analytically the SIFs are limited to simple geometries. As a result, numerous numerical methods have been developed, each with distinct and overlapping advantages and disadvantages, including the finite element method (FEM), the boundary element method (BEM), the scaled boundary finite element method (SBFEM) [1], the dual boundary element method (DBEM) [2] and the eXtended finite element method (XFEM) [3].…”