PSI - Issue 13

Yinghao Dong et al. / Procedia Structural Integrity 13 (2018) 1714–1719 Yinghao Dong, Xiaofan He, Yuhai Li / Structural Integrity Procedia 00 (2018) 000–000

1717

4

where σ is the remote tensile stress, a the crack depth, c half the crack length and θ the parametric angle, as shown in Fig. 1. Q is the square of the complete elliptical integral of the second kind, i.e. π / 2 0 a 2 c 2 cos 2 θ + sin 2 θ 1 / 2 d θ , and can be estimated by 1 + 1 . 464( a / c ) 1 . 65 when a / c ≤ 1 Newman and Raju (1983). The expression of f 1 ( θ ) is 1 + (sin θ ) 3 2 . Then, a directly fit of the numerical K n results was performed with assumed functions. The K n expression obtained is as follows:

2 n = 0

A n cos n ( θ − π / 2)

(2)

K n =

where A n = B n , 1 + B n , 2 sin π l a calculated as l =  1

t + B n , 3 cos

a t + B n , 4 sin

a t , n = 0 , 1 , 2; l is the maximum value of a / t and can be

2 π l

π l

c / a ≤ W / (2 t ) W / (2 t · c / a ) c / a > W / (2 t ) ;

   2 j = 0 2 j = 0

W

c a

I n , i , j , 1 + I n , i , j , 2 exp I n , i , j , 3 t J n , i , j , 1 + J n , i , j , 2 exp J n , i , j , 3 t

W 2 t

c a ≤

j

1 ≤

B n , i =

5 , i = 1 , 2 , 3 , 4 where the values of I n , i , j , k and

W

log

c a

c a ≤

W 2 t

j

<

J n , i , j , k ( n = 0 , 1 , 2 , i = 1 , 2 , 3 , 4 , j = 0 , 1 , 2 , k = 1 , 2 , 3) are listed in Table 1.

4. Verification of the K n expression

In this section, the K n expression, Eq. (2), is verified by comparing the K n estimated by Eq. (2) with the numerical results. Eq. (2) enables K n to be determined for an arbitrary point on the crack front. Considering the large amount of the numerical data, we only examine the maximum error in Eq. (2). To spare space, we present a part of comparison results in Fig. 3. The comparison involves only the cracks with a / ( lt ) ≤ 0 . 8 ( l is the maximum a / t value). As shown, for most surface cracks examined, the maximum error in Eq. (2) is less than ± 3%. However, large error occurs when Eq. (2) is used to estimate K n for surface cracks with c / a > 4 . 0 in plates with t / W = 0 . 125. Given the varying accuracy of Eq. (2), we need to determine the application range of Eq. (2). For a given combination of t / W and c / a , we determined the maximum a / ( lt ) value allowing for the accuracy of Eq. (2) better than 3%. It is found that when t / W and c / a of a surface crack with a / ( lt ) ≤ 0 . 8 fall within the following range, the accuracy of Eq. (2) is better than 3%: • 0 . 05 ≤ t / W ≤ 0 . 175 , 1 . 0 ≤ c / a ≤ 2 . 4; • 0 . 175 < t / W ≤ 0 . 5 , 1 . 0 ≤ c / a ≤ 5 . 0. Implementing the same procedure, we determined the range of t / W and c / a allowing for the accuracy of Eq. (2) better than 5% on condition that a / ( lt ) ≤ 0 . 8:

• 0 . 05 ≤ t / W ≤ 0 . 175 , 1 . 0 ≤ c / a ≤ 3 . 2; • 0 . 175 < t / W ≤ 0 . 5 , 1 . 0 ≤ c / a ≤ 5 . 0.

The above ranges cover most surface crack configurations occurring in practice. For cracks falling out of the ranges, the error in Eq. (2) can vary from 1% to 30%, depending on the combination of t / W , c / a , and a / ( lt ). Therefore, attention should be paid when using Eq. (2) to evaluate K I for surface cracks falling out of the ranges.