Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 231 the thickness direction; the half of the fissure was discretized using 6 cubic modules. The material properties and the main model characteristics are presented in Fig. 2b. Fig. 3 shows the constitutive law adopted for the material, considered homogeneous in the vertical direction of the plate. The critical step time is related to the time that an elastic wave takes to pass through the normal elemental bar. In the present case, for homogeneous material, the step critical time was Δtcrit =1.82 E-8 s. The main difference between Paulino and Zhang’s [2] analysis and the present study is the Poisson coefficient value, in DEM model when one works with the cubic arrangement (see Fig.1) one is limited to work with Poisson’s coefficient of 0.25, if the goal is to model an elastic isotropic and homogeneous material. If one wishes to model elastic and isotropic materials with other Poisson’s coefficient, it will be necessary to employ another kind of geometric arrangement. In Fig.2d the case characteristics studied by Paulino and Zhang [2] are presented. The uniaxial constitutive law used in this work in the case 1 is also presented in Fig 2.c. Obtained Results Case 1, Homogeneous case: The homogeneous material is considered in this first case. Thus, in this case, there isn’t any spatial variation in the elements constitutive law in DEM model. The Fig. 4(i) shows a fracture configuration obtained by Paulino and Zhang [2] (a), the main crack begins to branch when it reaches a length of 1.05mm, with an approximate angle of 29° in relation to the horizontal direction. In the case of the LDEM simulation (b) the branching begins when the main crack reaches a length of 1.0mm with an angle of 32° in relation to the horizontal direction. In both simulations it is possible to verify another characteristic that is seen in Fig.4(i) (a) and (b) which is the similarity between the two final configurations, for example, it is possible to observe a secondary branching in both figures. The lack of symmetry in the final configuration of LDEM simulation characterizes an unstable propagation process. This method has been verified sensitive to disturbances that occur during the process. These disturbances could be produced by rounding errors in the calculus and due to little changes in the step integration time. Case 2, Graded G c : In the present case only the toughness parameter c G is graded in vertical direction. The cohesive interface of the Finite Element Model used by Paulino and Zhang [2] is characterized by three parameters that are: the normal maximum tensile in the interface max n T , the critical opening displacement  n and the area closed by the curve that is proportional to the toughness c G . The grade in the interface properties that was implemented to modify the curve is illustrated in Fig. 3(a). The graded material proposed in case 2, for LDEM simulation, was carried out modifying the uniaxial curve as indicated in Fig. 3b. The Elastic Modulus that is directly linked with the initial slope is maintained constant. The c G graduation is obtained modifying the critical strain  p , maintaining  r constant to facilitate the comparison with the cohesive model. The material properties in the bottom boundary are: G c = 176.1 N/m and  p = 0.0125,  r = 0.06, these values grade linearly up to G c = 528.4 N/m ,  p = 0.0375 and  r = 0.06 in the top boundary. As the minimum value of toughness occurs in the bottom boundary and the maximum one occurs in the top boundary, it is expected that crack propagation take place in the inferior region of the plate, as it is possible to observe in Fig. 3(ii) (a) and (b). The comparison between Paulino and Zhang’s [2] results and LDEM, in terms of final configuration, show a significant similarity. -0.15 0.00 0.15 0.30 0.45 0.60 0.75 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 CER for -1/2W CERmidle CER for 1/2W F (N)  (%) (a) (b) (c) Figure 3 : a) The cohesive interface laws used by Paulino and Zhang [2]; The elemental constitutive law used in LDEM implementation: (a) for the graded material in case 2; (b) for the graded material in case 3. -0.15 0.00 0.15 0.30 0.45 0.60 0.75 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 CER for -1/2W CER midle CER for 1/2W F (N)  (%)