PSI - Issue 13

Yaroslav Dubyk et al. / Procedia Structural Integrity 13 (2018) 1502–1507 Dubyk at al./ Structural Integrity Procedia 00 (2018) 000–000

1505

4

   sin 2 1 2 k s L   where L is the total length of the strip contour L AB BC CO    and s characterises the position of any point along the boundary of the body counted from A to O. The realization of the boundary condition by ortogonalizations of them over these functions leads to the least accurate and often diverging results. They are not presented here. The second approach is based on the least squares method (LSM). We write the functional of the stress I on the boundary as: 

N 

N 

   

2         

   

s 



b

b

2

S 

s W s 

(2)

( ) W s   n n 

n n  

ds I 

( )





n

n

1

1

 is normal Williams stress and

n s W  is tangential Williams stress on the boundary while b  and b 

where n s W

are stresses defined on the boundary and S is the boundary. The numerical integration is performed by employing the rule of parabola. We obtain the values of I K for N =40, 50, 60 dividing each side of the strip by m points of numerical integration, where m is taken to be between 1000 and 2000. Calculation results for the relative crack depth 0.4 a t  and 0.6 a t  differ from the ones given in literature (Fett, 2008) by 0.4% and, for particular values of m up to 0.8%. Worst results were obtained for 0.2 a t  . In this case, I K values turned out to be smaller than known results from the literature and the average error reached up to 4.5%-5.5% while in particular cases they go up to 10%. So we observe that for 0.2 a t  the error becomes one order of magnitude larger that for 0.4 a t  . The reason is obvious and has already been discussed above. Williams functions are defined in polar coordinates which makes the method less employable as the ratio of distances between the nearest to the farthest points on the contour decreases. The next approach employed is associated with normalisation of the stress expressions with respect to the Williams functions themselves. Here and below, we present results only for 0.2 a t  . Now we write the boundary conditions in the integrally averaged form over the Williams functions themselves This approach is generally and technically similar to the previous one and leads to similar results. Better results are obtained if the normalization functions are taken in form   1 / 2 k k W W   . The values of I K differ from those given in Fett (2008) by 3%-4% in case of N =40, 50, 60. In particular cases of slightly different intervals of integration, for example, when N=60, for 2500, 2501 and 2502 points of integration the deviation is equal to 4.6%, 0.05% and 4.1%, respectively. Such a fluctuation of results with small changes in numerical procedure indicates the instability of the method itself. Our additional idea for improving the results is associated with the introduction of the requirement of equality of calculated and applied main vector of forces on OY axis and the main moment with respect to point A . Note that calculated integral force on OX axis is equal to zero because of symmetry of Williams functions. The system (2) is thus supplemented by two additional equations: 1 xy y xy W ds W ds W ds ds           2 1 k W s W ds        ( ) b s s n n 1 S n     0 N        2 1 k W s W ds        ( ) b s s n n 1 S n     0 N        (3)









n n

n n

n n

AB

BC

CO

BC

(4)









y

xy

xy W ds















n n W sds b  

n n W ds t  



1  

W sds

sds

n n x

n n

AB

BC

BC

CO

BC

The deviation of the obtained results from the ones given in Fett (2008) depends on N and varies between 3% and 5% for 20 30 N   and between 0.2% to 1% for 35 46 N   . For 50 N  the results diverge when the number of integration points change, that indicates instability of the method. Nevertheless, the error is quite moderate compared to the previous approaches and supports the need and effectiveness of integral equilibrium conditions on the boundary. It is apparent that equilibrium conditions stabilise solutions for SIF. Although similarly to the previous approaches calculated stress on the boundary still demonstrates major fluctuation from the predefined one, integrally they are in equilibrium.