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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8

faces of the crack can be calculated by using results by Sih et al. (1965) as, ∆ u ( x ) = 2 c 11 ℑ  µ 1 µ 2 p 0 + ( µ 1 + µ 2 ) q 0  √ b 2 − x 2 √ b 2 + p 0 +

(58)

∆ v ( x ) = − 2 c 22 ℑ 

1 µ 2 

q 0 µ 1 µ 2 

√ b 2

1 µ 1

11 ℑ  µ 1 µ 2 ( µ 1 + µ 2 ) p 0 + µ 1 µ 2 q 0 

− x 2 = 2 c

− x 2

(59)

where the operator ℑ stands for an imaginary part of the argument. Eq.(59) can be easily concluded using a following relation. µ 1 µ 2 µ 1 µ 2 = (60) which comes from a relationship between solutions and coe ffi cients of the characteristic equation (Lekhnitskii (1968)). c 11 µ 4 − 2 c 16 µ 3 + (2 c 12 + c 66 ) µ 2 − 2 c 26 µ + c 22 = c 11 ( µ − µ 1 )( µ − µ 2 )( µ − µ 1 )( µ − µ 2 ) = 0 (61) By multiplying the magnitude of point force doublets f ( ξ ), g ( ξ ) and h ( ξ ) with the compliance matrix C inEq.(6), the relative displacement between the upper and lower crack faces is obtained. Therefore, the density function of point force doublets f ( ξ ), g ( ξ ) and h ( ξ ) that produce the identical relative displacements in Eqs.(58) and (59) are obtained in the following manner as discussed by Aono and Noguchi (2000).    0 ∆ v ( ξ ) ∆ u ( ξ )    = 2 c 11    0 0 ℑ [ µ 1 µ 2 ( µ 1 + µ 2 )] ℑ [ µ 1 µ 2 ] ℑ [ µ 1 µ 2 ] ℑ [ µ 1 + µ 2 ]     p 0 q 0   b 2 − ξ 2 =    c 11 c 12 c 16 c 12 c 22 c 26 c 16 c 26 c 66       f ( ξ ) g ( ξ ) h ( ξ )    (62) By solving this relation with respect to the density of distributed point force doublets f ( ξ ), g ( ξ ) and h ( ξ ), those of corresponding to the relative displacement between the upper and lower faces of a line crack subjected to uniform internal pressure p 0 or shear stress q 0 in its plane is obtained by combining Eq.(62) with Eqs.(58) and (59) as follows.    f I ( ξ ) g I ( ξ ) h I ( ξ )    = 2 c 11    c 11 c 12 c 16 c 12 c 22 c 26 c 16 c 26 c 66    − 1    0 ℑ [ µ 1 µ 2 ( µ 1 + µ 2 )] ℑ [ µ 1 µ 2 ]    p 0  b 2 − ξ 2 (63) c 22 c 11

for mode I problem and    f II ( ξ ) g II ( ξ ) h II ( ξ )    = 2 c 11

  

  

− 1   

   q 0  b

c 11 c 12 c 16 c 12 c 22 c 26 c 16 c 26 c 66

0 ℑ [ µ 1 µ 2 ] ℑ [ µ 1 + µ 2 ]

2 − ξ 2

(64)

for mode II problem. It should be noted that components of compliance matrix c i j and material parameter µ j in Eqs.(63) and (64) have to be transformed according with the crack inclination. That is, c i j and µ j have to be replaced with those transformed by accounting the k -th crack inclination β k , which will be denoted by adding the superscript k . At the same time, the internal pressure and the shear stresses acting on the crack face are replaced by not a constant but a weighting function W I ( s k ) and W II ( s k ) which may vary with a position on the crack to treat a non-uniform loading along the crack face. Therefore, they are replaced by a function of s k where s k is a local coordinate along the k -th crack line. Accordingly, the complex potential for a point force doublet considering a fundamental property of the relative displacements on the crack line becomes,

11 π i 

I k ( s k ) + B 0 k − s k (cos β k + µ j sin β k )  b 2 II j ( β k ) W II k ( s k )

B I

j ( β k ) W

c k

b k

2 k ds k , ( j = 1 , 2; k = 1 , 2 , · · · , N )

Φ j ( z j ) = −

k − s

(65)

z j − ζ

− b k

ζ 0 k is a complex number expressing a central point of the k -th crack, b k is a half length of the k -th crack and N is a total number of cracks. Constants B I j ( β k ) and B II j ( β k ) are defined as follows.

( 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 ( µ k 1 + µ k 2 )] ℑ [ µ k 1 µ k 2 ] 

j ( β k ) =   

B I

(66)