Issue 64

M. A. Kenanda et alii, Frattura ed Integrità Strutturale, 64 (2023) 266-282; DOI: 10.3221/IGF-ESIS.64.18

2

3

N





u

w

2

  N



θ

 0 + =D -D +D x y t t x w + =D -D +D x y t t y         xy N N     1 2 2 2 2 3 0 0 1 2 2 2 xy x y 0 v

x

δ u :

4



0

2

t

2





θ

y

δ v :

4



0

2

t

2

2





  

  

2

2

3

3

xy M M M









w

u

v

y

x

0 D D +

0

0

+2

+ =

+

δ w :

1

2

 









 t x t y   2 2

x y

0

x

y

t

2

2

2

   

   

3



θ

  

  

4

4

3

2









w

w

θ

θ

y

x

z

D

D

D

0

0

-

+

+

+

+

(11)

3

5

7



 t x t y   2 2





 t x t y   2 2 2 2

2

t

2

3

2

        x xy S S x y S S x y

u

w







θ

xy

x

0 D D -

D

0

+ -Q =

+

x δ : θ

xz

4

5

6







t

 t x 2

t

2

2

2



θ

2

3

w





v

y

y

0 D D -

D

0

+ -Q =

+

y δ : θ

yz

4

6

5

 





t

 t y 2

t

2

2



2

2





xz Q Q

w

θ

yz

z

0 D D +

+ -N =

z δ : θ

z

7

8









x

y

2

2

t

t

ij ij ij N ,M ,S andQ ), and the moments of inertia

ij (D ) are defined in Appendix

where the moment, the stress resultants ( ij

A.

A NALYTICAL SOLUTION

A

ccording to Navier’s solution approach for simply supported FG plate, the analytical solutions of motion’s Eqns. (11) are expanded relying on double Fourier series:

                        cos λ x sin μ y e sin λ x cos μ y e sin λ x sin μ y e U cos λ x sin μ y e V sin λ x cos μ y e W sin λ x sin μ y e               mn mn mn x mn y mn z mn

i ω t

          

0 u (x,y) v (x,y) w (x,y) θ (x,y) θ (x,y) θ (x,y) 0 0 x y

         

         

i ω t

i ω t

 

 

=

(12a)

i ω t

m n =1 =1

i ω t

i ω t

z

in which

λ =m π a ;

μ =n π b

(12b)

where ω is the natural frequency, (m, n) are the frequency mode numbers, mn U , mn V , mn W , x  mn ,  y mn and  z mn are coefficients must be determined. According to the Eqns. (11 and 12), the eigenvalue problem can be expressed in algebraic form as:           2 ij ij j j S - ω m Δ = 0 (i,j=1,6) (13a)

in which

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