Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25

1       



0

    

  

1

  C



1 0 0 0 2  

.

E

 

Based on the constructed relations for stresses (5) and strains (6), the left part of Eqn. (1) determining the virtual work of strains can be represented as follows:

T

        1 0 T c k k p R   

1 1 k m     

       1 0 c m m t x C t x r y dV R p     ( ) ( ) ( ) T c c k m m 



T



  T

 x y dV



 









( , ) x y

k r y

( ,

( )







V

V

       1 0 T T c k k p R   

       1 0 a m m t x C t x r y dV R p     ( ) ( ) ( ) T c a k m m 



T

 





k r y

( )







V

       1 0 T T a k k p R   

       1 0 a m m t x C t x r y dV R p     ( ) ( ) ( ) T a k a m m 



T

 





k r y

( )







V

   

       1 0 T T a k k p R   

       1 0 c m m t x C t x r y dV R p     ( ) ( ) ( ) T a k c m m 



T

 



k r y

(7)

( )







V

If in Eqn. (1):

 

 

  c k 

  a k 

  T 

 

 

 

 

* k t x c

* k t x a





( )

( )





k

k

1

1

where

k 

k 

x

x

cos

0

0 0

0 0 0

sin

0

0 0

0 0 0

    

     

    

     

k 

k 

x

x

0 sin

0 cos

,

c      

a      

* k

* k

t

t

k 

k 

x

x

0 0

0 cos

0 0

0 sin

k 

k 

x

x

0

0 sin

0

0 cos

then its right-hand part that determines the external virtual work is given as:

   ( ) t x U x dS   * k ( ) T a

   

 

    T 

    ( ) a k t x U x dS p    * k ( ) T c

  c k p 

T

T

S 

S 

S 

 

 



T U dS

(8)





k

1

Substituting of (7), (8) into variational Eqn. (1), makes it possible to use the standard procedures of the finite element method: integrating for the relevant areas and boundaries of a layer; constructing the local compliance matrices; proceeding to global coordinates; combining the generated matrices and, finally, forming the system of independent algebraic equations for the coefficients of the force vector decomposition at the boundary of each layer:       k k k S p F  (9) where   k S is the global compliance matrix, which has a band structure,   k F is the k -th harmonic of a given displacement vector along the boundaries of the layers.

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