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The semicircular specimen under three-point bending (SCB) has been widely used to investigate mode I, mode II, and mixed mode I/II fracture behavior in brittle rocks. Compared to the other numerical methods, the extended finite element method (X-FEM) models a crack independently of the finite element mesh without any remeshing step in fracture propagation. In this regard, a numerical code called MEX-FEM, based on X-FEM, has been developed for modeling crack propagation in rock material. Since dimensionless stress intensity factors (i.e., Y I and Y II ) in crack modeling of SCB specimen depend on crack length, crack angle, and span ratio, these factors and crack propagation trajectory are determined for different combinations of crack angle, crack length ratio, and span ratio. A very good agreement exists between the calculated factors Y I and Y II in this research work and those reported in the literature. The results show that the values of Y I and Y II increase by increasing the span ratio. The angle in which the pure mode II occurs decreases with increasing of the crack length ratio and it increases when the span ratio increases. The crack growth is along the initial crack when the specimen is subjected to pure mode I (i.e., the crack angle is 0), whereas in mixed mode, the crack is deviated toward the upper loading point. The results of this study demonstrate the utility and robustness of the X-FEM to simulate the crack growth in rock materials.

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The crack initiation angle and propagation path for two different disc shaped test specimens (i.e., SCB and CBD specimen) are investigated experimentally and theoretically. The Maximum Tangential Stress (MTS) criterion does not calculate the crack initiation angle in SCB and CBD specimens correctly. Moreover, at the angles after occurrence of pure mode II, where the stress intensity factor of mode I becomes negative, this criterion is not applicable. Therefore, in this research work, Improved MTS (IMTS) criterion which has been implemented in the extended finite element method and is applicable under tensile and compressive loading conditions to examine the crack propagation path in the aforementioned disc shaped specimens. Furthermore, an experimental study on a cracked Brazilian disc specimen has been conducted at different angles. Results of IMTS criterion in these specimens show that the crack propagation path and the crack initiation angle can be predicted theoretically by using IMTS criterion.