PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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9

Fig. 4. Di ff erence of the distribution of weight function for point collocation and resultant force method.

and

( a j 1 cos β k + a j 2 sin β k )(cos β k + µ j sin β k ) ( − a j 1 sin β k + a j 2 cos β k )( − sin β k + µ j cos β k ) a j 1 ( µ j cos2 β k − sin2 β k ) + a j 2 (cos2 β k + µ j sin2 β k )    T    d k 12 d k 16 d k 22 d k 26 d k 26 d k 66     ℑ [ µ k 1 µ k 2 ] ℑ [ µ k 1 + µ k 2 ] 

j ( β k ) =   

B II

(67)

In those expression, superscript T denotes a matrix transpose and d k

i j is a component of transformed sti ff ness matrix

considering the crack inclination β k , which is an inverse of the transformed compliance matrix as,    d k 11 d k 12 d k 16 d k 12 d k 22 d k 26 d k 16 d k 26 d k 66    =    c k 11 c k 12 c k 16 c k 12 c k 22 c k 26 c k 16 c k 26 c k 66    − 1 j is a material parameter according to the crack inclination β k defined as, Here again, µ k

(68)

µ mj cos( α − β k ) + sin( α − β k ) cos( α − β k ) − µ mj sin( α − β k )

µ k

(69)

j =

2.4. Determination of distribution of weight function by resultant force method

In numerical analysis, weighting functions W I

II k ( s k ) in Eq.(65) are discretized using boundary elements

k ( s k ) and W

and nodal values are determined based on boundary conditions. The simplest technique for determining the values of weight function is a point collocation method (Fig.4(a)). In this method, the line to be a crack is divided into n equal-length segments and the weight function is assumed to be constant within each segment, and the value is determined so that the stress boundary conditions at the midpoint C ℓ , ( ℓ = 1 , 2 , · · · , n ) of each segment is satisfied at the same time. On the other hand, the most popular technique for determining unknowns in the BFM is a resultant force method proposed by Isida (1978). In this method, the line to be a crack is equally divided into n several segments as in a same way of point collocation technique, but this time the weight function is assumed a linear function within each division as seen in Fig.4(b). The nodal values of the weight function, which are unknown to be determined, are calculated so that the resultant of traction generated along the interval formed by connecting the midpoints of each adjacent segment D ℓ D ℓ + 1 , ( ℓ = 1 , 2 , · · · , n + 1) satisfy the boundary conditions. In the present study, the resultant force method is used to determine the nodal values of distributed weight function. Once the nodal value of weight function is obtained, the crack tip SIF is calculated promptly from the values of weight function at a crack tip s k = ± b k as, K I ( b k ) = W I ( b k )  π b k , K II ( b k ) = W II ( b k )  π b k (70) for crack tip at s k = b k and K I ( − b k ) = W I ( − b k )  π b k , K II ( − b k ) = W II ( − b k )  π b k (71)