PSI - Issue 26

Merazi Mohamed et al. / Procedia Structural Integrity 26 (2020) 129–138 Merazi et al. / Structural Integrity Procedia 00 (2020) 000 – 000

133

5

For elastic and isotropic FGMs, the constitutive relations can be written as:

 

            

       

          

       

55 0 0 0 0 Q 0 0 0 Q 0 0 0 Q 0 0 0 0 0 Q Q Q Q 0 0 0 44 22 12 12 11

       

       

y x

y x

(10)

=

xy xz yz

xy xz yz

66

) +  2 1 E(z ) ns

( 1 E(z ) −  ) 2 ns

( ) ns 11 ns 12 Q (z ) Q z =  ,

( )

( ) ( =

,

(11)

= Q (z ) Q z Q z ns 55 ns 44 = ns 66

ns Q (z ) 11

=

2.4 Equilibrium equations

The governing equations of equilibrium can be derived by using the principle of virtual displacements. The principle of virtual work in the present case yields:       − − −  +  −  = + + + 0, / 2 / 2 d dz q wd h d h d xz xz yz yz xy xy y y x x            (12)

Where  is the top surface and q is the applied transverse load.

    

    

s y b y M M M M M M N N N , , , , , , s x b x y x

1

    

    

xy

h d − / 2

(

)



(13)

s xy b xy

f z z

dz

x y xy , ,

  

=

ns

( ) ns

h d / 2

− −

h d − / 2

(

)

(

) ( ) ns ns g z dz

(14)



s xz S S ,

s yz

,

=

yz  

xz

h d / 2

− −

The governing equations of equilibrium can be derived from Eq. (12) by integrating the displacement gradients by parts and setting the coefficients zero 0 u  , 0 v  , b w  , and s w  separately. Thus one can obtain the equilibrium equations associated with the present new hyperbolic shear deformation theory:

N





y

: 0 x u N 

0



=

+

x

y



(15)

N

N





xy

y

: 0 v

0



+

=

x

y





b xy

b y

2   

2

x y M

M



b

2



: dx w M b 2

+ = q

2

0



+

+

x

2 2

2

dy

s yz

s xy

s y

2   

2

x y M

M

S





s xz

s

2

S





: dx w M s 2

+ = q

2

0



+

+

+

+

x

2 2

2

x

y

dy



