PSI - Issue 45

Koji Fujimoto et al. / Procedia Structural Integrity 45 (2023) 74–81 Koji Fujimoto / Structural Integrity Procedia 00 (2019) 000 – 000

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Next example is an infinite row of equally spaced parallel cracks as shown in Fig. 3. The traction free condition is described as the following equation. 2 ( + 1) ∫ {2 coth ( − ) − ( − ) cosech 2 ( − ) } ( ) − + =0 (− < < ) ∙∙∙∙∙(26) The stress component on the crack surface is given by = 2 ( + 1) ∫ ( − ) cosech 2 ( − ) ( ) − (− < < ) ∙∙∙∙∙(27) and T -stress can be calculated. Even in this problem, the convergence is very excellent though the numerical details like Table 1 is omitted here. Converged values of the T -stresses are shown in Fig. 4 together with the stress intensity factors. It should be noted that / → −1 when / → ∞ and that / → −0.5925408 when / → 0 . 3.3. Two parallel cracks Figure 5 shows a model of two parallel cracks subjected to uniform tension at infinity. This example is a mixed mode problem of I and II. Denoting the dislocation densities along the crack surface AB by 1 ( ) and 2 ( ) , the Burgers vectors of which are in the - and the -directions, respectively, the traction free conditions =0 and = 0 on the crack surface AB are expressed by the following equations. 2 ( + 1) ∫ [ 1 − − ( − + ){( − + ) 2 +3 2 } {( − + ) 2 + 2 } 2 ] 1 ( ) − + 2 ( + 1) ∫ {( − + ) 2 − 2 } {( − + ) 2 + 2 } 2 2 ( ) − + =0 (− < < ) ∙∙∙∙∙(28)

2 ( + 1) ∫ {( − + ) 2 − 2 } {( − + ) 2 + 2 } 2 1 ( ) − + 2

( + 1) ∫ [ 1 − − ( − + ){( − + ) 2 − 2 } {( − + ) 2 + 2 } 2 ] 2 ( ) −

=0 (− < < ) ∙∙∙∙∙(29)

IB /( √ )

/

IIB /( √ )

B /

Fig. 6. Relation of T -stress, stress intensity factors at the crack tip B and the eccentricity of cracks in the -direction ( d/a =0.1 ).

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Fig. 5. Two parallel cracks.