Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30



          



1         j y 

0 0

1 1

0 0

0

0

   

   

   

   

(   



(   



(   







 

 

)

)

)











   



dx

dx

dx

ij

ij

ij







x

x

x

y

x







j

i

j

i

i





   

   

0 k

0 k

1

1 1

0 0

1 1

  

  

 

 

 

(   

(   

(   









u x

u

)

)

)











dx e 

ij 

dy

e

dx

ikl

ikl





i    y

y

y

x

xl







(59)

j

i

l

i





1 k u        l   y 

1 k

0 0

1 1

   

   



(   



(   



u

)

)

0 1       0; , ( ) H

1 0







dx e 

dy

e

i

kl

ikl







x

y

y







i

l

i

with  represents the operator average defined on any Y-periodic function f(y) by:

1

f 





( ) f y dy YY

(60)

In the particular case where 1 1

1 1 , v u and    

we get:



   



0 0

(  

0 k 0 1       1 , v v ( ( )) H ijkl l l a        0 1         , ( ) ij j j x y    H                   3 0 u u x y        0 1 1 k

0 0

v u

)

0

1

   

   

(  

v u

x y      



)

j

j



i

i

dx e 



dx

0;

ijk

(61)





x

x



i

k

k

j





0 k u u x y         1 k

0 0

0 0

   

(   



(   



)

)



dx e 



dx

0;

ikl

(62)





x

x





i

l

l

i

1 0

Consequently, the corresponding homogenized equations in which there are no more fissures then follow:

0        ij j x 

0

(63)





0 k u u x y          1 k l l

0

1

   

   

   

   

     



0

 





a

e

ij

ijkl

ijk

x y

k

k

and

i i D x  

0

 

0  







(64)

   

  

0 k u u x y       1 k l     l   

0

1

  



 

D 

0







e

 

i

ij

ikl

xj    y

j

287