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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3. Some examples In this chapter, some typical two-dimensional crack problems are solved using the abovementioned method. 3.1. Crack close to bimaterial interface Consider a mode I crack the tip of which is close to the bimaterial interface as shown in Fig. 2. The shear moduli and the Poisson’s ratios are denoted by and , respectively, where =1 and =2 correspond to the cracked material side and non-cracked one, respectively. The applied stresses 1 and 2 are assumed to satisfy the condition that the average strain in the -direction of each material is the same, i.e., 1 +1 1 1 = 2 +1 2 2 ∙∙∙∙∙ ( 22 ) where =3−4 ( =1, 2) for the plane strain condition. The traction free condition on the crack surface can be described as the following singular integral equation using the exact solution of an edge dislocation close to a bimaterial interface obtained by Dundurs (1969). 2 1 ( 1 + 1)(1 − 2 ) ∫ { 1− 2 − + + 2 + + 2( − )(1 − ) ( − ) ( + ) 3 } ( ) + − + 1 =0 ( − < < + ) ∙∙∙∙∙(23) where and are so-called Dundurs ’s parameters, i.e., = ( 1 +1)−( 2 +1) ( 1 +1)+( 2 +1) , = ( 1 −1)−( 2 −1) ( 1 +1)+( 2 +1) , = 2 1 . ∙∙∙∙∙(24) The stress component on the crack surface is given by the following equation. = 2 1 ( 1 + 1)(1 − 2 ) ∫ { 1− 2 − + + 2 + − 2(1 + ) + − 2( − )(1 − ) (3 + ) ( + ) 3 } ( ) + − ( − < < + ) ∙∙∙∙∙(25)

Fig. 2. Crack close to a bimaterial interface.