Issue 73

N. Laouche et alii, Fracture and Structural Integrity, 73 (2025) 88-107; DOI: 10.3221/IGF-ESIS.73.07

         

2

2

d w d w

   du du

b

s





z

f

xx

1

2

2

dx dx dx

dx

df dw dz dz 2

 



w

(4)

zz

z

  

  

  

  

 dw dw dx dx s

    du dw 1

1 2

z



f

xz

2

dz dx

2





                         xx xx zz xz xz zz zz xx 2 2 2 

(5)

where

E





(6)

 2 1







   E 1 1 2  





(7)







Hence and are the Lame constants, and is the Poisson's ratio. The substitution of Eqns. (4)-(7) in Eqn. (3) give:  2 2 2 2

2

 e l 0

  

  

b d w d w dx dx 2 2 2

d w 2

d w

d w

       du dx 1  

du dx

du dx

1 2

b

s

s

b

 U I









I

I

I

2

2

2

2

3

4

5

2

2

2

dx

dx

dx

2

  

  

d w 2

d w 2

d w 2

du dx

  2

s

b

s



 w I 2

 w I 2

I

I w I 2

w

(8)

z

z

z

z

6

7

8

9

10

2

2

2

dx

dx

dx

  



  

2

2

     z dw

dw

dw

dw

z

      s dx

s





dx

I

2

11

dx

dx dx

    



with:

    

    

h

   

        df dz 2 2  

 2





   I 1:7





      2



2 2

  2

 

   

b

z f zf z f 1 1

dz

1, , , , ,

,

s

s

c

s

c

s

1

h



2

(9)

    

     

       s h e 2

   

  

2

       df dz 2







2 2

b e 2

z f zf z f 1 1

dz

1, , , , ,

,

 

s

1

        s h e 2

    

     

       s h e 2

    

    

h

 2

df

df

  I 8:10



 













2

2



 

 s

 

b

z f dz

b e 2

z f dz

1, ,

1, ,

(10)

 

c

s

s

1

1

dz

dz

h

h







e

      s 2

2

91