PSI - Issue 13

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

1506

5

3. Application of the external Williams solutions Let us study an infinite plane with a crack (Fig. 2) loaded at the part of its surfaces by a unit load σ. If we surround the crack with a circle of a unit radius two zones are obtained. The first zone lies within the unit circle (I) and the second zone is the region outside the circle (II). The centre of the polar coordinates system lies in the centre of the circle. We write the integrally averaged boundary conditions on the circumference using the system of normal functions. The boundary conditions imply equality of stress and displacements taking into account rigid body displacement of zone I along x axis.



N

N

               

  





r    n n 

r 





0 

1  

cos W W k    n n 





d  

1/ 2

0





n

n

1

1



N

N

  





r     n n W   

r 





n n W k    

sin 1/ 2 



d  

0





n

n

1

1

0

(5)



N

N

  





u   r n n 

u









0   0

r

1    

n n W W x    





cos cos  

d  

k

1/ 2

0

0





n

n

1

1



N

N

   





u   n n  

u





n n W W x     



sin 1/ 2 k 

d  



cos

0

0





n

n

1

1

Here 0 x is the unknown rigid body displacement along x axis. Expressions (5) represent a system of linear equations with two sets of unknown coefficients I n  and II n  ( 1, ..., n N  ) and 0 x . The values for I K obtained by solving this system are presented in Table 1 along with the error between these values and mathematically accurate value from (Savruk, 1988). From Table 1 we observe that starting only from 8+1 unknown coefficients, the results demonstrate very good accuracy.

Table 1. The value of the SIF for the problem in Fig. 2

%

N

I K (Savruk, 1988)

I K

2 4 6

-2.16 1.5958 1.5958

0.019

1.5955

0.019 Motivated by success of the Williams functions in the latest example we investigate the possibility of using our latest approach to improve the results of the geometry shown in Fig. 1. The main challenge of this geometry is that distance between the crack tip to the farthest point on the contour was approximately 7 times bigger than the distance between crack tip and the nearest one, i.e. / 7 B O    . We have tried to overcome this challenge by introducing the circle of radius 0.5 around the crack tip. Therefore, we have divided the initial area into two regions I and II , shown in Fig. 1. In such a way, we have considerably reduced the variation in distances from any point within region I and II to the centre of coordinates. The variation of the distance in the region I equals 0.5/0.2=2.5 and the variation of the distance in the region II equals ≈1.4/0.5=2.8. Similarly, to the previous problem, we assume that region I is described by internal Williams functions only, while the region II is described by a both internal and external Williams functions. Obviously, we need to impose continuity conditions on the boundary between the regions. Here we employ the least squares method to write the boundary and continuity conditions. This lead to minimisation of displacement functional ( d I ) with respect II n  and 0 x and to minimisation of stress functional ( s I ) with respect to I n  and II n  as shown in (6)