Various numerical techniques such as finite element and boundary element methods are commonly used to analysis engineering problems. These methods encounter mesh-related difficulties in dealing with fracture mechanics problems. To overcome these difficulties, a number of meshless methods have been developed in recent years. In this paper the Element Free Galerkin method based on the linear elastic fracture mechanics is used to model the jointed rock medium under axial loads. The stress intensity factors are calculated on the tip of the joints by using J-integrals. The visibility criterion and a cubic spline weight function are applied to model rock fractures. In addition, the Lagrange multipliers method is employed to enforce the boundary conditions. To verify the computational capability and accuracy of the method, a couple of examples of jointed samples in mode I as well as mixed mode are considered and the stress intensity factors are determined. The obtained results by this technique, in comparison with analytical methods, show a good accuracy and denote that the Element Free Galerkin method can be used as a proper tool in rock fracture mechanics.
Rock mass commonly contains fractures in the forms of joints and microcracks, and their failure strongly depends on the propagation of these pre-existing flaws. Propagation of rock mass discontinuities is studied in rock fracture mechanics. Stress intensity factors (SIF) in linear elastic fracture mechanics are the main parameters capable to characterize the stress field in the vicinity of the crack tip. These factors depend on the geometry of the fracture, applied stresses and the initial fracture length. Based on the loading type that a material is subjected to, there are three basic crack propagation modes in a fracture process (Fig. 1), namely: Mode I (extension, opening), Mode II (in-plane shear), and Mode III (out-of-plane shear). Any combination of these modes may occur as a mixed mode. When the stress intensity factors reaches a critical values at some point in a structure, a fracture will initiate and propagate. Therefore determination of stress intensity factors is an essential task: it can be obtained from the stress field, the displacement field or from energy quantities . In practice, because of the mechanical and geometrical complexity of most of the problems, commonly a numerical method such as the finite element or boundary element methods is employed to calculate stress intensity factors [2, 3]. Finite element and boundary element methods encounter mesh-related difficulties in dealing with fracture mechanics problems. To alleviate these difficulties, various mesh free methods such as element-free Galerkin method (EFGM) was developed . In numerical studies, the stress intensity factors is calculated by methods such as displacement extrapolation method , stress extrapolation method , J-integral , Griffith's energy calculations , and the stiffness derivative technique .