We used computational mechanics consisting of various numerical methods to analyze different problems under various boundary conditions. The main advantage of the element free Galerkin (EFG) method is that the model meshing stage is not required and the nodal points are distributed throughout the model domain instead of being meshed. Nodal points are distributed within the model domain to capture a stress singularity around the crack tip. In this paper, various 2D problems in the field of linear elastic fracture mechanics were analyzed to validate the accuracy of the EFG code that was developed in a Matlab environment. To simulate the fracturing process based on the maximum hoop stress criterion, stress intensity factors and the angle of crack propagation were calculated under different loading conditions and the crack trajectory was determined. The obtained results of the developed EFG-code were compared with available experimental data and other numerical (boundary element method and finite element method) methods.