Computational mechanics with its various numerical methods is being used to analyze different problems with various kinds of boundary conditions. Basically, the main advantage of meshless methods is to eliminate model meshing stage and to distribute nodal points throughout the model domain instead of meshing. In the Element Free Galerkin Method (EFGM) using moving least squares (MLS) approximation, a linear combination of basic functions is fit to data by a weighted function. In this paper, the kirsch problem is being analyzed to validate the Element Free Galerkin Method accuracy. In order to simulate fracturing process, stress intensity factors and crack initiation angles are computed for some rock mechanical test techniques such as Uniaxial Tension Test and 3-Point Bending Test and the crack trajectory is depicted using maximum circumferential stress method. In continuation, the obtained results of the EFG-code are comprised with other numerical (Boundary Element Method) and experimental researches data.

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