Issue 42

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

and









     









H h 

 

( ) ( e H h u h w dy     ) kh kl i ikl

ij 

( ) ( 

) h w dy 

0

j

j

i

Y

Y

(77)







*

w V

YC

0

0

i i D D  can be written as follows:







Therefore,

and

ij

ij









  





( ) a H h u e H h    ijk k k ijkl kl kl

( ) 

ij

(78)









D H h   

( ) 



( ) H h u 

e

i

ij

j

j

ikl

kl

kl

So, the homogenized (strain, electric potential)-(stress, electric displacement) law is characterized by the function defined by:

D    

, H h kl

(

) ( ,

)

(79)

kl

ij

i

6 3  R R towards R , by:

Nevertheless, for the study of (79), let us introduce the following functions, defined from

1



 







( ) a H h u H h u dy   ( ) lm lm ijkl ij ij

( W H H , sm s

)

Y

2

Y

(80)

1

 









ijk i e H h 

( ) H h u dy  jk jk

( ) 

i

Y

2

Y

1

 





*





 Y e H h u H h dy Y            ( ) ( ) 2 1 ( ) ( ) ij i i j j Y ikl ik ik l l Y H h H h

( W H H , sm s

dy

)

(81)



2

Moreover, the proposition that follows presents the main result of this analysis:

and D    satisfying the

Proposition 4. The functions defined above, through (80) and (81), are of class C 1 , positive;

following relations:

1 2

*       

ij    W H 

ij

(82)

1 2



W D H  

i

i

 

  i i

6 3 ( . ) resp R R

Proof. As u and  are continuous functions of

, defined from

H H 

and H H 

, 1,2,3 ij i j 



1,2,3

* W and W are then of class 0 . C Let us now introduce:

( . YC V resp V  u YC

),

towards

* h h u H and h h u H          * ( ) ( ) ij ij i i

ij

(83)

i

290