PSI - Issue 5

B. Boukert et al. / Procedia Structural Integrity 5 (2017) 115–122 B.Boukert / Structural Integrity Procedia 00 (2017) 000 – 000

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117

2. Model description

2.1. Displacement Field A third order plate theory developed by Reddy [1][7] is used in this paper through a cubic function of the thickness coordinate, where ( u o , v o , w o ) are the displacement components along the ( x, y, z ) directions, respectively, of a point on the midplane (i.e., z=0 ). ø x and ø y denote rotations about the y and x axes [1-5][11].

w



4

3

0

( , , ) u x y z u z x y    ( , )

z

(

)





x 

x

0

2

x



4 h

3

w



3

0

( , , ) v x y z v z x y    ( , )

z

(

)







(1)

y

y

0

2

y



h

3

( , , ) w x y z w x y  ( , )

0

The strain is given by equation (2)

0                                                          1 3 0 1 3 3 0 1 3 xx xx xx xx yy yy yy yy xy z z            

0                              2 2 0 2 yz yz yz xz xz xz z     

(2)

,

xy

xy

xy

Where ε 0 ,ε 1 ,ε 3 ,γ 0 ,γ 2 are given by

          

          

2

2



      

 

      

w

x           y  y 

u

0 w          x w  



1 . 2 1 .

x 



0

0  

2

x x

x



x





1                        1 1 xx yy xy   

3               3 3 xx yy xy   

0                  0 0 xx yy v x 

  

y 



2

2

w

4 h

0

,

,

 

0

,





y

2

2



y

3

0 0 y u v w w y x x y                          0 0 0 2 . xy

y   

x 

2

y 

w

x     



x y        

0

2.

 

y x

x y

 

0       y  w

0       y  w

0               0 yz xz  

2         yz

 

4

y

y





  

  

  

(3)

,

2

w

w

h





2

xz    

0

0

x   

x   

x  

x  

2.2. Equation of motion

The equation of motion of the third order theory are derived using the principle virtual displacement, the obtained set of equations are presented below.