PSI - Issue 13

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

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decreases considerably for the bodies with non-circular geometry. The same is related to Williams, the terms with n r  are neglected when the vicinity of the center of coordinates is considered. However, if 0 r R  , where 0 R is some arbitrary value, the n r  can be taken into consideration. This leads to the first main idea of the study, i.e., we combine internal ( n r , i.e. n W ) and external ( n r  , n W  ) Williams functions. 2. The formulation of the system of equations using non-orthogonal functions leads to nearly absolute linear dependence between equations. Therefore, the solutions demonstrate considerable fluctuations from the boundary values triggered by the fact that terms with large n do not asymptotically tend to zero. In fact, these terms may lead to a violation of the body equilibrium. The use of the orthogonal functions, on the other hand, is very ineffective in Cartesian and polar coordinates for the typical geometries. Only the circular-shape geometries in polar coordinates for r const  yield a system of orthogonal functions cos , sin n n   . In all other cases, non-orthogonal functions are typically used. Therefore, motivated by the need to boost the accuracy of the solutions, we introduce here the conditions for global equilibrium in addition to minimization of errors at the boundary. 2. Application of the internal Williams solutions Consider a symmetric strip with edge crack loaded by a unit loading  on the boundary BC (Fig. 1) while lateral boundaries AB and OC are kept load-free. Here a is the crack length, b is half of the strip height, t is the width of the strip and x and y are Cartesian coordinates counted from the crack tip. In Cartesian coordinates the Williams expressions n x W  , y n W  , xy n W  for stresses are conveniently expressed as:

2 

n

2

2  2 

2  2 

2  2 

2 n 2 n

2 n 2 n

n

n

n

2

2

6

     

     

     

   

n



 





n x W r 

  



2

1 cos 

cos

2 

n

2

n

n

n

2

2

6



n

y

 

(1)





W r 

  



2

1 cos 

cos

n

2 

n

2

2 

2 

2 

2 2 n n      

n

n

n

2

2

6

  

 



n

xy

 





W r 

  



1 sin 

sin

n

y



C

B

II

r

b

II



I



P

I

0 

А

O

x

a

t

Fig. 2. Infinite body with edge crack, loaded with uniform stresses at 1 r 

Fig. 1. The strip loaded by uniform loading

Note, that here and further the integrally averaged boundary conditions are used (Pei-Qing, 1985). Taking into consideration that calculation results very often diverged and were highly sensitive to the number of integration sections and unknowns of series expansion, several types of practical implementation of boundary conditions are considered here. This is done to investigate if either some limitations of the Williams functions method exist or the method has been incorrectly implemented so far. The first approach consists of introduction of orthogonal functions     cos 2 1 2 k s L   and