PSI - Issue 26

Merdaci Slimane et al. / Procedia Structural Integrity 26 (2020) 35–45 Slimane et al. / Structural Integrity Procedia 00 (2020) 000 – 000

41

7

Where λ= mπ/a and μ= nπ/b, « m » and « n »are mode numbers and U mn , V mn , W bmn , and W smn are arbitrary parameters. Can be combined into a system of equations as:     ( )     2 K ω M Δ 0 − = (19)

Where [K] and [M], stiffness and mass matrices, respectively, and represented as:

λI μI

λI μI

-I

0

      

      

      

     

11 12 13 14 a a a a a a a a a a a a a a a a 12 22 23 24 13 23 33 34 14 24 34 44      

U V

0 0 0 0       =          

    

mn  

1

2

4



0 -I

(20)

 

1

2

4

 −

mn

2

2 2 ω λI μI - I + I (λ +μ ) -I (λ + μ ) 2 2

 

 

 

W W

  

2

2

1 3

5

bmn

2 2 -I (λ + μ )

2 2 -I (λ + μ )

λI μI

smn

4

4

5

6

(

)

(

)

2

2

2 2 ) ]; λ μ A A a λ B λ B 2B μ a λ B λ B 2B μ ; [ ( [ ( = − + = + + = + + s 2 s s 2 ) ]

A λ A μ a ; +

a

= −

11

11

66

12

12

66 13

11

12

66

14

11

12

66

(

)

(21)

2 2 s 2 A λ A μ a μ B 2B λ B μ a μ B 2B λ B μ ; [( ) ]; [( ) + = + + = + + 2 2 2 s s

a a a ; =

]

= −

21 12 22

66

22

23

12

66

22

24

12

66

22

= − (

)

(

)

4

2 2 λ μ D μ a H λ 2 H 2H λ μ H μ A λ A μ ( ) ; ( ( ) + + = − + + + + + + 4 s 12 66 22 34 11 12 s 4 s s 2 2 s 4 s 2 s 2 2 D 2D

s 4 D λ 2 D 2D λ μ D μ ) + + s 2 2 s

4

11 D λ

31 13 32 23 33 a a a a a ; ; = =

+

66

22

(

)

41 14 42 24 43 34 44 a a a a a a a ; ; ; = = =

= −

11

12

66

22

55

44

7. Numerical results and discussion

In this section, some numerical illustrations are carried out and discussed to prove the efficiency and accuracy of the proposed theory in the free vibration responses of simply supported isotropic homogeneous FG plates. For numerical results, an Al/Al 2 O 3 functionally graded plate FGP which composed of Aluminum (as metal) and Alumina (as cerami c) is considered. The Young’s modulus, density and Poisson ratio of Aluminum are E c =70GPa, ρ c =2702kg/m 3 and υ=0.3, respectively, and that of Alumina E m =380GPa, ρ m =3800kg/m 3 and υ=0.3, respectively. The non-dimensional entities were used as the following formulas:

m m ω=ωh ρ E / ,

c c ω=ωa h ρ E ² / /

(22)

In table 1 and table 2, another comparison is made of the fundamental frequencies of a square plate FG (b=a=1) and rectangular plate Al/Al 2 O 3 (b=2a). The same observation is observed. It can be seen that the results are in excellent agreement with those of sinusoidal plate theory given by Zenkour (2005) and saidi et al. (2019) for a square plate FG and another comparison is exposed that of the fundamental frequencies of a rectangular plate with those of Mouaici et al. (2016). The same observation is observed. There is an excellent agreement between the results given by this model and those of the literature. In the above, and after the different comparisons of the results, we can say that the present method is reliable for the study of the plates presenting manufacturing defects to know the porosity. Table 1. Comparison of fundamental frequency ( ) ω for square (b=a=1) FG plate Al/Al 2 O 3 (P=1). Model ξ=0 ξ=0.1 ξ=0.2 a/h=5 a/h=10 a/h=20 a/h=5 a/h=10 a/h=20 a/h=5 a/h=10 a/h=20 Zenkour (2005) 0,3204 0,0868 0,02222 0,3144 0,08498 0,02174 0,3062 0,08256 0,0211 Saidi et al. (2019) 0,3206 0,0868 0,02222 0,3146 0,08504 0,02174 0,3062 0,08256 0,0211 Present 0,3205 0,08682 0,02221 0,3082 0,08340 0,02133 0,2953 0,07980 0,02040