Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24

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

x                 y xy yz xz     

x                   y xy yz xz     

11 C C C C 12

0 0 0 0 0 0

       

       

12

22

C

  

0 0

0 0

(6)

66

C

0 0 0

0

44

C

0 0 0 0

55

 ,

 , y

y  , xy

 , yz

 , xz

 ) and ( x

 , xy

 , yz 

, xz  ) are the stress and strain components, respectively. Using the

where ( x

C , can be written as

material properties defined in Eq. (1), stiffness coefficients, ij

E z

( )



E z

2 ( ) , E z

2 ( ) ,

  

C C C

 



(7)

11 C C

C

 2 1



44

55

66

22

12













1

1

E QUATION OF MOTION

H

amilton’s principle is herein employed to determine the equations of motion:

t

0 0 (  

) U V K dt     

(8)

U 

V 

K 

is the variation of strain energy;

is the variation of work done; and

is the variation of kinetic

where

energy. The variation of strain energy of the plate is computed by



 

 



x x   

  

xy   

yz   

xz   











U

dV

y

y

xy

yz

xz

V

A 

0   x x N N N M k M k M k            0 y   0 b b b b b y xy xy x x y y xy

b

(9)



xy



s

s

s

s

s

s

s

s  

0

0

x x M k M k M k S       y y xy xy

 





0  

S

dA

yz

yz

xz

xz

where A is the top surface and the stress resultants N , M , and S are defined by

h

h

/2

/2





 , , i x y xy  and 











, 

h   

h   

b

s

s S S

, s xz yz

dz 





, N M M ,

z f

g

dz

1, ,

,

(10)

i

i

i

i

xz yz

/2

/2

The variation of the potential energy of elastic foundation can be calculated by

0 A V f w dA     e

(11)

f is the density of reaction force of foundation.

where e

For the Pasternak foundation model [43-53].

2

2

2 x w KwK f S    2 2 y w K

  

(12)

1

S W e



289