PSI - Issue 2_A

2430 Alireza Hassani et al. / Procedia Structural Integrity 2 (2016) 2424–2431 Author name / Structural Integrity Procedia 00 (2016) 000–000 In the proceeding examples, uniform traction � �� � � � is applied on the layer boundaries. 3.2. Example (2): A crack parallel to the boundaries of a layer The dimensionless plastic region lengths at a crack tip of a horizontal central crack are also shown in table 2. Table 2. The dimensionless plastic zone size of a horizontal crack for different values of � � � � ⁄ � � � � � �.� � � � � � �.� � � � � � �.� � � � � � �.� � � � � � �.� � � � � � �.� �� � ��� � � � � � �.���� � � � � � �.���� � � � �.���� � � � � � �.���� � � � � � �.���� � � � � � �.���� �� � ��� � � � � � �.���� � � � � � �.���� � � � �.���� � � � � � �.���� � � � � � �.���� � � � � � �.���� The common feature of the above two examples is that the plastic region sizes are symmetrical and identical for both crack tips in the sense that the cracks are identical. Also for bigger crack lengths, the plastic region sizes are generally bigger than those of smaller crack lengths. 3.3. Example (2): Two interacting cracks in a layer The last example deals with the interaction between two horizontal cracks. The centers of both identical cracks are located at the points with coordinates ��, ���� and ��� � ����, ����, respectively, in which ��� � �⁄�. The values of the dimensionless plastic region lengths for both cracks are given in table 3. Table 3. The dimensionless plastic zone size for two interacting cracks � � � � � �.� � � � �� � �.��� � � � �� � �.��� � � � �� � �.��� � � � �� � �.��� � � � � � �.�� � � � �� � �.��� � � � �� � �.��� � � � �� � �.��� � � � �� � �.��� As one expects the interaction between the cracks widens the plastic region size. Also because of the symmetry of the problem, we have � �� � � �� and � �� � � �� . We assumed that the plastic region size at each of the adjacent tips of the cracks to be �. In this case, we have ���� � �� � � � � �� � � � � �� . For nonintersecting plastic regions, the condition �� � ���� should be satisfied which according to the table 3 it is fulfilled. 4. Conclusions Based on the solution of screw dislocation integral equations are derived in a layer containing multiple cracks. These equations are solved numerically to determine dislocation density on the surfaces of the cracks. The solution is utilized to determine plastic zone sizes in front of each crack tip under anti-plane deformation. The plastic zone length ahead of the tips of a crack is specified using Dugdale's model. The analysis allows consideration of multiple arbitrary oriented cracks. We observed that interactions between cracks have crucial effect on the size of the plastic region. References Bhargava, R., Hasan, S., 2011. Crack opening displacement for two unequal straight cracks with coalesced plastic zones–A modified Dugdale model. Applied Mathematical Modelling 35, 3788-3796. Dugdale, D., 1960. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8, 100-104. Faal, R.T., Fariborz, S.J., Daghyani, H.R., 2006. Antiplane deformation of orthotropic strips with multiple defects. Journal of Mechanics of Materials and Structures 1, 1097-1114. 7