Issue 55

S. Merdaci et alii, Frattura ed Integrità Strutturale, 55 (2021) 65-75; DOI: 10.3221/IGF-ESIS.55.05

h

/2















2





, , ij A B D ij

z z Q dz

, i j

1, ,

1, 2, 6

ij

ij



h

/2

h

/2















s

s

s

2





, B D H ,

( ), ( ), ( ) f z z f z f z Q dz

, i j

(12a)

1, 2, 6

ij

ij

ij

ij



h

/2

h

/2





  s ij A









2







( ) g z Q dz

, i j

,

4, 5

ij



h

/2

h

3

E z

E z

( )

( )



n

1

 n h   1

   ( )

2

( ) 11 n

s

s



 

Q

44 A A

g z dz

(12b)

,

 2 1

55

2









1

n

Thus the balance equations associated with the newest shear deformation theory can be obtained,

 

x N N

xy



 

u

:

0

(13a)





x

y



   y

xy N N



v

:

0

(13b)





x

y

b

b

2

2



 M M

b x

2



M

xy

y





  q



w

(13c)

:

2

0

b

2

2

 

x y





x

y

s

s

s

s

2

2



 M M S S  

s

2



M

xy

y

xz

yz

x





   



w

q

(13d)

:

2

0

s

2

2

 





x y

x

y





x

y

FG PLATES ANALYTICAL SOLUTIONS

O

n the side edges of the FG plate are placed the following easily endorsed boundary conditions:

          b s b s w w v w w

 N M M and x a 0 0,

(14a)

b

s

x

x

x

0

  y

y

          b s b s w w u w w



(14b)

0 N M M and y

b

0,

b

s

y

y

y

0

 

x x

The external force can be expressed as per the Navier solution:

      1 1 m n

 sin( )sin( ) mn x y 

q x y

q

( , )

(15)

where λ = m π /a and μ = n π /b , « m » and « n » are mode numbers. Where q 0 represents the intensity of the load at the plate center. For the case of a sinusoidal distributed load, we have

 Q q m n 0 , 11

 

1

(16)

69