PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

637 13

for the bottom and top edges, where 0

x t and

0 y t are the tractions specified on the boundary, and

( ) ( ( ) ( ( ) ( ( ) ( F E knkn F E knkn F E k n k n F E k n k n F E k n k n F E k n k n FE knkn FE knk                                     = + = + = + = + = + = + = + = + , , , , , , , , ( ) 2 . x n ) ( ) ( ( ) ( ( ) ( ) ) ) ) ( ) 11 12 1 11 2 21 1 12 2 22 21 22 1 21 2 11 1 22 2 12 11 12 1 11 2 21 1 12 2 22 21 22 1 21 2 11 1 22 2 1 , , , , u x y u x y u x y u x y v y x v y x v y x v y

,

)

,

)

,

(53)

4 ( 2) m M N =  + + equations in total along four boundary. However, the number

Therefore, for one block, there are

of linear equations from the domain integral is n M N =  +  + , and the number of unknown coefficients is also equal to n , which means that the number of equations is larger than the number of. The Singular Value Decomposition (SVD) is applied to determine all the unknown coefficients  . 4. Variation technique with independent integral contour method The most effective method to calculate the mixed mode Stress intensity factor is the crack opening displacement method (COD), which is an approximate engineering method in elastic-plastic fracture mechanics that uses the COD as a fracture parameter to judge the start of crack growth. In this manner, in the case of plane strain (Rooke, 1976), the Stress intensity factor can be determined by 2 ( 1) ( 1)

tip 2 ( ) ( ) A r 

tip 1 ( ) ( ) A r 

E z

E z

2

2





,

,

, r a x x a = −  , A A A

K

K

(54)

=

=

I

II

2 8(1 )  −

2 8(1 )  −

r

r

A

A

where tip ( ) E z denotes the Young’s modulus at the crack tip tip z , A r is the radial distance of the evaluation point A to the crack-tip, and ( ) ( ) ( ) A r u z u z     + − = − is displacement discontinuity. Another method to evaluate the stress intensity factor is Singular Stress Method (SSM), by

I   2 K r r K = ( ), A y A I

I   ( ), r r r x a x a = −  , , A xy A A A A

2

(55)

=

II

( ) I y A r  and

( ) I xy A r  denote the normal stress and shear stress, respectively. Although COD and SSM are the

where

most direct and simple, the accuracy and sensitivity of these methods require more attention, such as the distance between the calculation point and the crack tip should be correctly selected. To overcome this difficulty, the independent contour integral method is proposed in this section. Variation technique and static issue We consider a variation a  of crack length at crack tip ( , ) c c c x y = x along the extension of crack surface with slant angle  under a static load in homogenous media (as shown in the Fig. 2). For static problem, taking variation on both sides of the governing equation (40) gives

K

α

α

K

s

s

s

s









1

s −   K

s α K +

α

s

s

(56)

0,

.

=

=−  

a

a

a

a









 K

st



In the above variation equation, the derivative

consists of following derivatives

a