PSI - Issue 2_A

Alireza Hassani et al. / Procedia Structural Integrity 2 (2016) 2424–2431 Author name / Structural Integrity Procedia 00 (2016) 000–000

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Stress fields for embedded cracks in isotropic media are singular at crack tips with square root singularity, thus for the embedded cracks the dislocation density functions are represented by (Faal et al. (2006)) � �� ��� � � �� √1 � �� � � � � �1 � � � 1 (8) The stress intensity factors for the embedded cracks are (Faal et al., 2006) � � ����� � � 2 ��� � � ��1�� � � �� � � ��1�� � � � � � �� ��1� � ����� � � � 2 ��� � � �1�� � � �� � � �1�� � � � � � �� �1� (9) The solution to Eqs. (4) and (6) in conjunction with the conditions � ����� �� � �1� � � ����� �� � 1� � � (for example see (Theocaris, 1983)) leads to the density of dislocation on a crack surface and also � � and � � . Viewing the Eqs. (9), the above-mentioned conditions may be given by the relations � �� �� � �1� � � �� �� � 1� � �. The Eqs. (4) and (6) are solved numerically using the procedure introduced in (Faal et al., 2006). In this procedure, the integral equations are discretized at � � � ����� �2� � 1�⁄2�� � � � 1�2� � � � and � � � ����� �⁄�� � � � 1�2� � � � � 1 in which � is the number of discrete points. Using these discrete points the integral equations (4) and the uniqueness requirement of the displacement field on the cracks borders i.e. (6) are discretized. We assumed that the points � � � � � � � ⁄ and � � � � �� � are identical with one of the discrete points � � � ����� �⁄��. The trial and error solution algorithm is chosen such that two discrete points � � � ����� �⁄�� and � � � ����� �⁄�� � �� � � 1�2� � � � � 1 as the representatives for the points � � � � � � � ⁄ and � � � � �� � are chosen and the discretized forms of the Eqs. (4) and (6) are solved simultaneously and then the conditions � �� �� � �1� � � �� �� � 1� � � are checked. The analytical validation of the work is done for an infinite plane with a single horizontal crack with effective length 2�. The plane is under a remote traction � � which leads to the traction � �� � � � and � �� � � � � on the crack surfaces. To this end, Eqs. (4) and (5), (7) are simplified by setting � � � and changing the location of coordinate system as follows � 2 � � � � � ����� �� � ��√1 � � � � �� � � �� � � � � ��� � 1 � � � ���� �� � ��� � ��� � � � ��� �� � � � � ��� ��� � � � 1 (10) Because of the symmetry of the problem we assume that �⁄� � �⁄� � � � � 1. The analytical solution to the above equation is given as below(Polyanin and Manzhirov, 2008) � � ��� � � � 2 � �� �� � � � � �√1 � � � � � � �� �� � �� � � � � √1 � � � � � � �� � � �� � � � �� � � � � �√1 � � � � � � �� � � � � (11) The above integral can be calculated analytically by use of the change of variables � � ����� � � � ���� and � � ���� . Using these changes of variables we arrive at � � ��� � � � 2 � ��� � � � � � � ��� � ���� � � � ��� �� � � � � � � � � ��� � ���� � � � ��� �� � �� � �� � � � � � � ��� � ���� � � � ��� �� � � � � (12)