Issue 51

S. Merdaci et alii, Frattura ed Integrità Strutturale, 51 (2020) 199-214; DOI: 10.3221/IGF-ESIS.51.16 202 0 0 ( , , ) ( , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( , ) ( , ) b s b s b s w w u x y z u x y z f z x x w w v x y z v x y z f z y y w x y z w x y w x y                 (4) The number of unknown functions is only four, while five or more in the case of other shear deformation theories (Tab. 1). The strains associated with the displacements in Eqn. (5) are 2 2 2 2 ( ) 1 3 '( ) ( ) 1 '( ) 4 4 5 5 df z f z and g z f z dz z z h h        (5) 2 2 0 2 2 2 2 0 2 2 2 2 0 0 ( ) ( ) 2 ( ) 2 ( ) ; ( ) 0 b s x b s y b s xy s s yz xz z u w w z f z x x x v w w z f z y y y u v w w z f z y x x y x y w w g z g z and y x                                                                (6) For elastic and isotropic FGMs, the constitutive relations can be written as 11 12 44 12 22 55 66 0 0 0 , 0 0 0 x x yz yz y y zx zx xy xy Q Q Q Q Q Q Q                                                             (7) where ( x  , y  , xy  , yz  , yx  ) and ( x  , y  , xy  , yz  , yx  ) are the stress and strain components, respectively. Using the material properties defined in Eqn.(1), the stiffness coefficients, ij Q , can be expressed as   11 22 11 22 12 44 55 66 2 2 2 ( ) ( ) ( ) ( ) , , , 2 1 1 1 1 E z E z E z E z Q Q Q Q Q Q Q Q                  (8) E QUILIBRIUM EQUATIONS he static equations can be obtained by using the principle of virtual displacements. It can be stated in its analytical form /2 /2 0 h x x y y xy xy yz yz xz xz h d dz q W d                                   (9) where  is the top surface . By substituting Eqns. (6) and (7) into Eqn. (9) and integrating through the thickness of the plate, Eqn. (14) can be rewritten T