PSI - Issue 2_A

Alireza Hassani et al. / Procedia Structural Integrity 2 (2016) 2424–2431 Alireza Hassani , Reza Teymoori Faal / Structural Integrity Procedia 00 (2016) 000–000 The stress components (3) behave as � �� ��, � � �� ��⁄��� , �� � � � and � �� ��, ��� � �� ���� � � � � ⁄ , as � � � � . Therefore, the Cauchy-type singularity of stresses at the placement location of the dislocation can be proved. We consider a strip containing � interacting cracks. The crack configurations may be described in parametric form as � � � � � ���, � � � � � ��� in which �� � � � �, � ∈ ��,�, � , ��. The related crack problem is solved by means of the distributed dislocation technique (Hills et al., 2013). Therefore, the solution of screw dislocation is employed and the following Cauchy singular integral equations are constructed � ��� �� � ���, � � ���� � � � � �� ���� �� ��, ���� � �� � ��� (4) where � ��� is the traction on the i‐th crack and � �� ��� is the dislocation density function on the normalized length, �� � � � � of the j‐th crack. From Eqs. (3), kernel of integral equation (4) becomes � �� ��, �� � � � � � � � � � ��� � � � � � ��� � � � � ��� � � � � � ��� � � � � ����������� � ��� � � � ����� � � � � ���������� � ��� � � � ����� �������� � ��� � � � ����� � ������� � ��� � � � ����� � � � � ����������� � ��� � � � ����� � � � � ���������� � ��� � � � ��� � ���� �������� � ��� � � � ����� � ������� � ��� � � � ��� � ���� � (5) For embedded cracks Eq. (4) should be complimented by the following closure condition � ��� � � � ��� � � � � � ����� �� � � � ����� � �, � ∈ ��, ��� (6) The left-hand side of Eq. (4) is � ��� � � �� � ���� � � � �� � ���� � wherein � �� � and � �� � are the stress components caused by the external traction in the intact strip i.e., strip without cracks, and � � is the angle between y‐ axis and normal to the presumed surface of the i‐th crack. The solution to Eqs. (4) and (6) yields the density of dislocation on a crack surface. 2.1. Dugdale's model A simplified model for evaluating the plastic zone size around a crack tip is the Dugdale's model which avoids the complexities of a true elastic-plastic solution. This model is applied to materials with elastic-perfectly plastic behavior which obeys the Teresca's yield criterion. We make the hypotheses that the plastic deformation concentrates in a line in front of the crack which its effective length equals to that of the physical crack plus the length of the plastic zone. Based on the Dugdale's model the length of the plastic zone � is determined from the condition that the shear stress � �� at the tip of the effective crack to be bounded and equal to the yield stress � � . Therefore, the lengths of the plastic zone at two tips of the i-th crack are obtained by imposing the left-hand side of Eq. (4) to the following conditions � ��� �� � ���, � � ���� � � � �� � ���� � � � �� � ���� � � � � ��� � � � � � �� � �� � � �� � ���� � � � �� � ���� � ��� � � � �� � � � � � � �� � � �� � ���� � � � �� � ���� � � � � ��� � � �� � � � � � (7) where � � � � �� � and � � � � �� � are the dimensionless lengths of the plastic zone at the tips corresponding to � � �� and � � �, respectively and � � is the half effective crack length. 2427 4