PSI - Issue 13

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

1507

6

2



0

N

N

N

           

  

u     u I  II r r n n W n n W   

u

II

r n W x  

n 

d   



cos

0







n

n

n

1

1

1

0

2



0

N

N

N

  

u     u I  II n n W n n W     

u

II

n W x   

d I 

n 

  

cos

d

0







n

n

n

1

1

1

0

2

2





0

0

N

N

N

N 

N

N

  

   

  

I     II r  r  n n W n n W   

II   r  n n W   

r W d   n 

r  



II

I 

II

0 

0 

r n W d   



n 



n 







n n W













n

n

n

n

n

n

1

1

1

1

1

1

(6)

2

2

N

N

N

N

   

  

   

  





II   x n n W   

II   xy n n W  

n 

II

II

xy





x W ds  

n 

n 







n W ds  









n

n

n

n

1

1

1

1

AB

AB

2

2

2

N 

N 

N

N

N 

   

  

   

  

   

  









II   y n n W  



I

xy

II

II

y

II n n W

xy

I







x W ds 

n 



n 





 





1 W ds 

n W ds  



n n

n











n

n

n

n

n

1

1

1

1

1

BC

PO

BC

2

2

2

N

N

N

N

N 

   

  

   

  

   

  







II   x n n W   

II   xy n n W  

n 

II

II

xy

II

xy







x W ds  







W ds I  

n 

n 



n W ds  

n n

s











n

n

n

n

n

1

1

1

1

1

CP

CP

PO

Here 0  is an angle between x axis and intersection between the circle and boundary CO labelled with P . See details in Fig. 1. From (6) we obtain a system of linear equations with respect to three sets of unknown coefficients I n  , II n  , II n  ( n=1…N ) and displacement 0 x . Therefore, the total number of unknowns is 3N+1 . The smallest error is reached for N=26-28 and is equal to 0.2%-0.5%. Increasing the number of unknowns leads to dependence of the results on number of integration points, indicating divergence of the solution. 4. Conclusions This paper discusses the possibility of application of internal and external Williams functions to calculate SIF in plane bodies and have more of educative rather than practical meaning, and it summarises all the discussions about the Williams functions method as a tool to address boundary value problems. 1. The application of the Williams functions becomes problematic for non-circular geometries. The bigger is the ratio between the distances from the farthest point in the region to the nearest point in the region the higher is the error of the method. The best results are obtained in this case when the boundary conditions are fulfilled by the least squares method or by minimization by Williams functions themselves. However, even employing these methods the error in SIF values for the above ratio greater than 7 reaches 5%. 2. The additional application of the global equilibrium conditions leads to the better convergence of the results which yields better accuracy up to 0.2-0.5% for the geometry discussed in section 3. 3. The artificial separation of the noncircular body into sub regions and introduction of both the internal and external Williams functions in them does not lead to much improvement in accuracy. References Bouledroua, O., Meliani, M.H., Pluvinage, G. 2016. A review of T-stress calculation methods in fracture mechanics. Nature & Technologie. A Sciences fondamentales et Engineering N1, V5. Dolgov, N.A., Soroka, E.B., 2004. Stress singularity in a substrate-coating system. Strength of Materials 6, 636–642. Fett, T. 1997. A semi-analytical study of the edge-cracked circular disc by use of the boundary collocation method. Engineering Fracture Mechanics 56(3), 331-346. Fett, T. 2008. Stress Intensity Factors, T-Stresses, Weight Functions. University of Karlsruhe; Karlsrue Malíková, L., Seitl, S., 2017. Application of the Williams Expansion near a Bi-Material Interface. Key Engineering Materials 754, 206-209. Pei-Qing, G. 1985. Stress intensity factors for a rectangular plate with a point-loaded edge crack by a boundary collocation procedure, and an investigation into the convergence of the solutions. Engineering Fracture Mechanics 22(2), 295-305. Savruk, M.P. 1988. Fracture mechanics and strength of materials, Vol. 2: Stress intensity factor in cracked bodies. Naukova Dumka, Kiev. Williams, M.L., Pasadena, C., 1957. On the Stress Distribution at the Base of a Stationary Crack. J. Appl. Mech. 24(1), 109–114.