PSI - Issue 17

B. Boukert et al. / Procedia Structural Integrity 17 (2019) 37–43 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

39

3

where:

N z

k

1 Q z z z z z +

(3)

2 3 4 6 (1, , , , , )

1 k z = =  

( , , A B D E F H , , ,

)

ij

ij

ij

ij

ij

ij

ij

k

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

xx N N x y Nxy N x y Q Q        y



xy

0

+ =



yy

0

+ =

(4)

2

2

xy P P 

2





P



4 (

yy

x

xx

q + =

2

)

0

+ +

+

+

2

2

2

x  

y

x y

 

h x 

y

3



2 M M Q x y M M Q x y Where M M P Q Q R h h = =   + − =     + − =   − − 2 0 0 4 4 : , 3 xx xy x xy yy y ij ij ij ij ij

ij

The development of Navier solutions of simply supported antisymmetric cross-ply laminates using the third order theory are presented below, where the boundary conditions are satisfied by the following expansions.         0 1 1 0 1 1 ( , ), ( , ) , .cos( ).sin( ) ( , ), ( , ) , .sin( ). s( ) x mn mn n m y mn mn n m u x y x y U X x y v x y x y V Y x co y         = =   = = = =    

n m     = =

(5)

w x y

mn W x

.sin( ).sin( ) y  

( , )

=

0

1 1

where

/ ,     = = m a

/ m b

:

The mechanical transverse load q(x,y) is also expanded in double Fourier series

0 0 4 ( , ) sin( ) sin( ) a b q x y x ab    

mn Q

y dxdy

(6)

=

The coefficients ( U mn ,V mn , W mn , X mn , Y mn ) of the Navier solution are obtained through the resolution of the system given by (7) . 1 1 11 12 13 14 15 2 2 T C mn mn mn T C N N U S S S S S     − −            −

 

12 22 23 24 25 13 23 33 34 35 14 24 34 44 45 15 25 35 45 55 S S S S S S S S S S W Q S S S S S X S S S S S Y       =        −          mn mn mn mn V

N

N





−

      

mn

mn

(7)

    

mn

T

C

1

1

2 mn M M M M   − − T mn



mn

C

2

−





mn