On the uses of special crack tip elements in numerical rock fracture mechanics

Abstract

Numerical methods such as boundary element methods are widely used for the stress analysis in solid mechanics. These methods are also used for crack analysis in rock fracture mechanics. There are singularities for the stresses and displacements at the crack tips in fracture mechanics problem, which decrease the accuracy of the numerical results in areas very close to the crack ends. To overcome this, higher order elements and isoperimetric higher order elements have been used. Recently, special crack tip elements have been proposed and used in most of the numerical fracture mechanics models. These elements can drastically increase the accuracy of the results near the crack tips, but in most of the models only one special crack tip element has been used for each crack end. In this study the uses of higher order crack tip elements are discussed and a higher order displacement discontinuity method is used to investigate the effect of these elements on the accuracy of the results in some crack problems. The useful shape functions for two special crack tip elements, are derived and given in the text and appendix for both infinite and semi-infinite plane problems. In this analysis both Mode I and Mode II stress intensity factors are computed . Some example problems are solved and the computed results are compared with the results given in the literature. The numerical results obtained here are in good agreement with those cited in the literature. For the curved crack problem, the strain energy release rate, G can be calculated accurately in the vicinity of the crack tips by using the higher order displacement discontinuity method with a quadratic variation of displacement discontinuity elements and with two special crack tip elements at each crack end.

Kinked crack analysis by a hybridized boundary element/boundary collocation method

“…Following the procedure explained by Marji et al (2006) the complete solution of the integrals given in Eq. (23) are given in Appendix D. It should be noted that for this case also two degrees of freedom are used for each node (boundary collocation points) at the center of each sub-elements.…”

“…Based on the linear elastic fracture mechanics (LEFM) principles, the Mode I and Mode II stress intensity factors K I and K II can be easily deduced (Sanford, 2003;Phan et al, 2003;Marji et al, 2006). A crack tip element of length 2a is considered so that the two points of the crack faces which correspond to the midpoint of the crack tip element on each crack surface for h ¼ AE180 degrees, then the stress intensity factors with respect to the normal and shear displacement discontinuities (assuming plane strain condition), can be determined as:…”

“…Kinked and curved cracks may occur in cracked bodies (Shou and Crouch (1995), Marji et al (2006)). The proposed method is applied to the problem of a 45°circular crack under far field biaxial tension shown in Fig.…”

“…Although the constant displacement discontinuity elements are simple and widely used for analyzing engineering problems, they cannot predict the stresses and displacements for field points closer than approximately one element length from the boundary and also for the crack analysis a higher accuracy of the results near the crack tips is needed. Therefore, the higher order variation of the displacement discontinuities together with special crack tip elements are usually used for the treatment of crack problems (Scavia, 1995;Shou and Crouch, 1995;Tan et al, 1996;Marji et al, 2006;Fatehi Marji et al, 2007;Hosseini Nasab and Fatehi Marji, 2007;Fatehi Marji et al, 2009). Kinked cracks in brittle rocks under compressive loading are studied by Bobet (2000), Kishen and Sing (2001); Sagong and Bobet (2002), Wong et al (2001Wong et al ( , 2002 and Tang et al (2005).…”

Numerical Crack Analysis of Blunt Rock Indenters by an Indirect Boundary Element Method

“…Results showed that discontinuum-based DEM has the potential in simulating rock indentation and fragmentation by TBM cutters when rock joints are taken into consideration [14]. The use of boundary element method for analysis and simulation of cracks can be traced in some studies [15][16][17][18][19][20][21]. Displacement discontinuity method is an indirect boundary element method which has been used for the analysis of crack problems.…”