13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- disc cutters using numerical simulations. Their results are in good agreement with those obtained from experimental studies (LCM). Propagation and coalescence mechanism of micro cracks due to indentation of TBM disc cutters into the rock and production of chips in indent process is highly complex and important. The mechanism of micro crack propagation and coalescence has not been widely studied and its real reasons (micro crack propagation and coalescence) have not been fully found. Because of the complexity of the micro crack propagation and coalescence problem under TBM disc cutters, nowadays, numerical or analytical-numerical approaches are mostly used for simulating the rock breakage mechanism [17]. A numerical model, the higher order semi-infinite displacement discontinuity (HOSDD2D) code, a two-dimensional code based on the linear elastic fracture mechanics (LEFM) which uses quadratic displacement discontinuity formulation with three special crack tip elements at each micro crack initial and end is used to simulate the rock breakage mechanism. Based on this numerical method, stresses near the crack tip and distribution of displacements can be clearly defined to determine the accurate strain energy release rate and stress intensity factors. There are three important fracture initiation criteria, which are applicable in action: the maximum tangential stress (σθ-criterion), the minimum energy density criterion (S-criterion) and the maximum energy release rate (G-criterion) or any modified form of those mentioned issues, (e.g. F-criterion which is A modified energy release rate criterion) has mostly been used to study the breakage mechanism of rock [18-24,14]. Although this criteria act prosperously for predicting the crack initiation under TBM disc cutters, the maximum tangential stress criterion is used here to predict the direction of micro crack initiations resulted from artifact cracks of TBM disc cutters. In the present research, the rock breakage mechanism under eroded disc cutters of tunnel boring machines (TBMs) is modeled and studied by the proposed method. A comparison of results between eroded and non-eroded disc models is presented. 2. Higher Order Displacement Discontinuity In a Half-plane The semi-infinite displacement discontinuity method (SDDM) is an indirect boundary element method that is solved problems on the basis of fracture mechanics principles given the boundary conditions and calculates stresses and displacements at discontinuities in all boundary elements. cruch and starfield [25] used the analytical solution of a constant element displacement discontinuity, over the line segment x a, y= 0 in the semi-infinite area y 0 as shown in Fig. 1. For complete computaion of displacements and stresses in semi-infinite body, due to the real displacement discontinuity, its portrait and its resulting from the supplementary solution are denoted by ui R and ij R, ui P and ij P, ui S and ij S, respectively. The complete solution for the semi-infinite plane y 0 can be expressed as: S ij P ij R ij ij S i P i R i iu u u u and (1) x y, and x, y are local coordinates and global coordinates, respectively, that can be transformed by the following two formulas: )cos ( ( )sin ( )sin )cos ( y x y x y c y x c y c x x c (2)
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