Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24 292 0 0 0 1 1 cos( )sin( ) sin( )cos( ) sin( )sin( ) sin( )sin( ) i t mn i t mn i t m n mn i t mn U e x y u v V e x y w W e x y X e x y                                                (19) where / m a    and / n b    ,  is the frequency of free vibration of the plate, 1 i   the imaginary unit. Substituting Eqs. (19) into Eq. (15) and collecting the displacements and acceleration for any values of m and n , the following problem is obtained: 11 12 13 14 11 13 14 12 22 23 24 22 23 24 2 13 23 33 34 13 23 33 34 14 24 34 44 14 24 34 44 0 0 0 0 0 0 mn mn mn mn S S S S m m m U S S S S m m m V S S S S m m m m W S S S S m m m m X                                                                      (20) where 2 2 11 11 66 S A A     ,   12 12 66 S A A    ,   2 2 2 13 11 12 66 2 S B B B         ,   2 2 2 14 11 12 66 2 s s s S B B B         2 2 22 66 22 S A A     ,   2 2 2 23 11 12 66 2 S B B B         ,   2 2 2 24 11 12 66 2 s s s S B B B         , (21) 4 2 2 4 2 2 33 11 12 66 22 2( 2 ) ( ) w s S D D D D K K              , 4 2 2 4 34 11 12 66 22 2( 2 ) s s s s S D D D D         , 4 2 2 4 2 2 44 11 12 66 22 55 44 2( 2 ) s s s s s s S H H H H A A             11 22 0 m m I   , 13 1 m I    , 14 1 m J    , 23 1 m I    , 24 1 m J    , 2 2 33 0 2 ( ) m I I      , 2 2 34 2 ( ) m J     , 2 2 44 2 ( ) m K     (22) Eq. (20) is a general form for buckling and free vibration analysis of FG plates resting on elastic foundations under in- plane loads. The stability problem can be carried out by neglecting the mass matrix while the free vibration problem is achieved by omitting the in-plane loads. N UMERICAL EXAMPLES AND DISCUSSION n this section various numerical examples are examined to check the accuracy of the present formulation in predicting the free vibration behaviours of simply supported FG plates resting on elastic foundation. Two types of FG plates of Al/Al 2 O 3 are employed in this investigation. The material characteristics of FG plates are presented in Tab. 1. For convenience, the following non-dimensional parameters are employed: ˆ / m m h E     Proprieties Aluminium (Al) Alumina (Al 2 O 3 ) Young’s modulus (GPa) 70 380 Poisson’s ratio 0.3 0.3 Mass density kg/m 3 2702 3800 Table 1 : Material properties employed in the FG plates. In order to validate the present formulations that predict the vibration of FGM plates, various illustrative examples are presented. The present dimensionless frequency of the square FGM plate without a porosity are compared with those of I