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 Making use of the relations ��� � � � ��� � � � ����� � ���������� � ����� and � �� ���� � � ���� ⁄ � � �1⁄����� ��|������⁄2�� � �����⁄2� ������⁄2� � �����⁄2�� ⁄ | the above integrals may be calculated analytically. Therefore for the condition � � � � � � �⁄2, the Eq. (12) is rewritten as below � � ��� � � � 2 � ��� � � � � ���������| ��� � 2 � ��� � 2 ��� � 2 � ��� � 2 |� � � �� � � ����� � ������| � � �� � � �� � ��������| ��� � 2 � ��� � 2 ��� � 2 � ��� � 2 |� �� � � ����� � ������| �� � � ��� � � � � ������ ��� � ����| ��� � 2 � ��� � 2 ��� � 2 � ��� � 2 |� ���� � ���| ��� � 2 � ��� � 2 ��� � 2 � ��� � 2 |� � � � �� � ������ � ������| � � � �� (13) Finally we have � � ��� � �� � �� �� � 2 � � � �� � 2� � ����� � 1 2 ������| ����� � �� ����� � �� |� (14) By satisfying the condition � � �� � �1� � � or � � �� � � � � � � � we arrive at � 2 � � � ��� �� � � � ��� �� � � � �� � 2� � (15) wherein the effective half crack length equals to the actual half crack length plus � . The above relation is exactly that of obtained from reference (Shi, 2015), which analytically validates the employed method. 3. Numerical Results We now furnish the paper with three examples to demonstrate the suitability of the developed solution for evaluation of plastic zone size. In all of the numerical examples, crack length is 2� � ��� , unless we introduce a new value. 3.1. Example (1): A crack perpendicular to the boundaries of a layer The dimensionless plastic zone sizes for a central crack �� � � �, � � � ��2� which is perpendicular to the boundaries of a layer and subjected to uniform anti-plane traction � �� � � � are tabulate in table 1. Table 1. The dimensionless plastic zone size of a vertical crack for different values of � � � � ⁄ � � � � � ��1 � � � � � ��2 � � � � � ��� � � � � � ��� � � � � � ��� � � � � � ��� 2� � ��� 1 � � � � ���12� 1 � � � � ������ � � � ��1��� 1 � � � � ��1�1� 1 � � � � ��2�2� 1 � � � � ���122 As it can be seen from table 1, by increasing the applied traction, that is, the ratio of � � � � ⁄ , the plastic zone size is extended. 2429 6