Issue 59

E.S.M.M. Soliman, Frattura ed Integrità Strutturale, 59 (2022) 471-485; DOI: 10.3221/IGF-ESIS.59.31

G OVERNING EQUATIONS

I

n this study, the displacement and stresses near the crack apex are described. Consequently, the co-ordinate systems and contour around the crack apex are adopting as depicted in Fig. 2.

Figure 2: Crack apex co-ordinate systems

The displacement and stress fields near the tip of the crack with traction-free faces known as Williams expansion, originally introduced by [19-21] and can be written as [22]:

n

      2 r

2 n

n n

n

  

 

  

  

  

n

 

    0 0

  









u u y

a n k

1 cos

cos

2

 

2 2 2





n

1 2



(1)



2 n

 2 2 2 n n n    

  

 

n

  

  

  

 

    b n k

  

1 sin

sin 2



n

      2 r

2 n

 2 2 2 n n n    

  

 

  

  

n

 

  

  

 sin 2

   0 0 x



a n k

1 sin

 





n

1 2



(2)



2 n

n n

n

  

 

n

  

  

  

  

 

    b n k



   2



1 cos

cos

2 2 2

 

n

1

       2 n r a n

2 n

2 n

        1 2   n

2 n

  

 

     

     

  

n

 







  



2

1 cos

1 cos

3

 

x n



1 2



(3)



2 n

2 n

  

 

2 n

        2   n

n

  

  

  

  

  

  

 

  

    2 b n

1 sin 1

1 sin 3



n

1

       2 n r a n

2 n

2 n

        1 2   n

2 n

  

 

     

     

  

n

 







  



2

1 cos

1 cos

3

 

y n



1 2



(4)



2 n

2 n

  

 

2 n

        2   n

n

  

  

  

  

  

  

 

  

    2 b n

1 sin 1

1 sin 3



473