Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09 117 The deformations related to the displacement-field in Eq. (1) contains only three unknowns   0 0 , , u w  . The linear strains corresponding with the displacement field in Eq. (1) are: 0 1 2 0 ( ) , ( ) x x x x xz xz z f z g z           where         0 0 0 1 2 0 1 1 , , ² , , , , ' , , , , , ² x x x xz u x y t w x y t k A x y t k x y t dx x x                 (2) The integral appearing in the above expressions shall be resolved by a Navier type solution and can be represented as: ' dx A x       (3) where the coefficient ' " " A is depending on the type of solution chosen, in this case via Navier. Therefore, ' " " A and 1 k is expressed as follows: ' 2 1 2 1 , A k      (4) According to the polynomial material law, the effective Young’s modulus E(z)     ( ) 0.5 p m c m E z E E E z h     (5) The constitutive relations of an FG plate can be written as: 11 55 0 0 x x xz xz C C                        (6) where ij C are, the three-dimensional elastic constants given by:     11 55 2 ( ) ( ) , 2 1 1 E z E z C C       (7) The equilibrium equations can be obtained using the Hamilton principle, in the present case yields:   2 1 0 t t U V K dt        (8)   /2 /2 /2 0 /2 , ( ) , ( ) h x x xz xz h e L h h U d d z V q f w d K z u u w w dzdydx                                     (9)