Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24 289 For elastic and isotropic FGMs, the constitutive relations can be expressed as: 11 12 12 22 66 44 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x y y xy xy yz yz xz xz C C C C C C C                                                                (6) where ( x  , y  , xy  , yz  , xz  ) and ( x  , y  , xy  , yz  , xz  ) are the stress and strain components, respectively. Using the material properties defined in Eq. (1), stiffness coefficients, ij C , can be written as 11 22 2 ( ) , 1 E z C C     12 2 ( ) , 1 E z C       44 55 66 ( ) 2 1 E z C C C      (7) E QUATION OF MOTION amilton’s principle is herein employed to determine the equations of motion: (8) where U  is the variation of strain energy; V  is the variation of work done; and K  is the variation of kinetic energy. The variation of strain energy of the plate is computed by 0 0 0 0 0 0 x x y y xy xy yz yz xz xz V b b b b b b x x y y xy xy x x y y xy xy A s s s s s s s s x x y y xy xy yz yz xz xz U dV N N N M k M k M k M k M k M k S S dA                                                           (9) where A is the top surface and the stress resultants N , M , and S are defined by     /2 /2 , , 1, , h b s i i i i h N M M z f dz     ,   , , i x y xy  and     /2 /2 , , h s s xz yz xz yz h S S g dz      (10) The variation of the potential energy of elastic foundation can be calculated by 0 e A V f w dA     (11) where e f is the density of reaction force of foundation. For the Pasternak foundation model [43-53]. (12) H 2 2 2 2 2 1 y w K x w KwK f S S W e        0 0 ( ) t U V K dt       