PSI - Issue 63

Kamila Kotrasova et al. / Procedia Structural Integrity 63 (2024) 27–34

29



   ,  

  x y z x y ,  

(1)

x y z , ,



with

.

  

  

x u

y v

y u

x v



 



 

,

,











and

  

  

  

  

x w

y w

 x y ψ , 

 y x ψ  













(2)

ψ ,

γ

κ

,



















again vary linearly through the thickness h and are given by

x u

x u



 





y v

y v



 

 



z









z









x

y

x





y



  

  

y u

x v

y u

x v

 

 



 



 y x    

z













xy





.

x w

z u

x w







y w

z v

y w



 



(3)

























xz

yz











The stresses in the k th layer in the local coordinate system (1, 2, 3) can by expressed by             . 0 . 0 0 0 0 0 0 0 0 23 13 12 22 11 66 55 44 22 12 11 23 13 12 22 11                                                                L L L L L L L L L E E sym E E E E    E The elastic matrix in the global coordinate system ( x , y , z ) can by expressed by               T E T E T E T        L T T L     1 1

(4)

(5)

and analogue

    

   E T E T  T

   E T E T  tT t

  

(6)

t L

t

L

with transformation matrix

       

       

 sc sc c s sc s c sc c s 0 0 0 0 0 0 2 2 2 2 2 2 2 

s c c s 0 0 0 0 0 0

(7)

  

T

0

   

  

T 

  



2



  

T 0 T

t



where

      1     T T T  

(8) The stresses  11 ,  22 and  12 vary linearly through a layer thickness, the stresses  13 ,  23 are constant through the thickness. There is no stress continuity through the laminate thickness but stress jumps from ply to ply at their interfaces depending on the reduced stiffness. Internal forces can by written