Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09

The deformations related to the displacement-field in Eq. (1) contains only three unknowns 



0 0 , , u w  . The linear strains

corresponding with the displacement field in Eq. (1) are:

0

1 x

2 ( ) , 

0



z     







f z

g z

( )

x

x

x

xz

xz

where













, , u x y t

² w x y t , ,











0

0

0

1 x

2

0



 











' k A x y t 



, , k x y t dx 

,

,

, , ,

(2)

x

x

xz

1

1





x

x

²

The integral appearing in the above expressions shall be resolved by a Navier type solution and can be represented as:

 





' dx A  

(3)

x

' " " A is depending on the type of solution chosen, in this case via Navier. Therefore,

' " " A and 1 k

where the coefficient is expressed as follows:

2 1 ,

'

2

 





A

k

(4)

1



According to the polynomial material law, the effective Young’s modulus E(z)

 





p

( ) m E z E E E z h     0.5 m c

(5)

The constitutive relations of an FG plate can be written as:

x     xz  

x       xz  

C

0

  

  

11

  

(6)

C

0

55

ij C are, the three-dimensional elastic constants given by:

where

E z

E z

2 ( ) 

( )





C

C

(7)

,





 2 1



11

55







1

The equilibrium equations can be obtained using the Hamilton principle, in the present case yields:

t

2







U V K dt     



(8)

0

t

1

h

/2

/2     h

 

 



x x xz    

d d z 



  

U

,

xz

L h    

( V q f   

) , w d  

(9)

e

/2







( ) z u u w w dzdydx      



K

0 /2 h 

117