13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- experimental investigations on the propagation and coalescence of micro cracks in rock-like specimens under uniaxial and biaxial compression have been published [7-14]. Some useful and modern numerical models such as FROCK code, damage model, Rock Failure Process Analysis (RFPA2D) are used in this field [5, 15-19]. There has never been a random investigation on the effect of fracturing in brittle substances altering positions of random micro cracks which is under tension. In this research, a numerical model is proposed using a higher order displacement discontinuity code for micro crack analysis (HDDMCR2D code). This model is designed on the basis of the linear elastic fracture mechanics (LEFM) principles so that it is able to simulate the micro crack interaction of specimens containing random micro cracks. In this paper, the numerical analysis of the growth of wing cracks from pre-existing micro cracks in rock-like specimens under uniform normal tensile stress is studied. In order to verify the validity of proposed numerical model, the experimental and numerical results of the wing crack initiation directions (given by Guo et.al. [20]) for different micro crack inclination angles of a center slant micro crack under uniform tension have been used. The crack propagation mechanism of two pre-existing micro cracks are investigated (for a specimen under uniform tension) in which the propagation direction of micro cracks is estimated by using the maximum tangential stress fracture criterion ( -criterion) proposed by Erdogan & Sih [21]. 2. The higher order displacement discontinuity method In the higher order displacement discontinuity method, the boundaries are discretized into multiple segment elements. The formulations of three types of displacement discontinuity variations with: a constant variation of the displacement along the elements, a linear variation, and a quadratic variation were previously mentioned and used in the literature [22-24]. The quadratic element displacement discontinuity is fundamentally based on integrating the quadratic element shape functions over collinear, straight-line displacement discontinuity elements. Fig. 1 shows the displacements distribution at quadratic collocation point ‘m’, which can be calculated as: D form to j xy D m j m j , 1, 3, ( ) ( ) (2) where Dj is the fundamental variable. It should be noted that two fundamental variables are calculated in each collocation point. Using 3 2 1c c c , it can be written: 2 1 1 3 2 1 2 1 2 2 2 1 1 1( ) ( 2 )/8 , ( ) ( 4 )/4 , () ( 2)/8 c c c c c c (3) which are the shape functions of the quadratic collocation point ‘m’. In the quadratic collocations, there are three collocation points for each element, for which the displacements are typically calculated. These collocations are located in the center of the elements (Fig. 1).
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