PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 584–591 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

586

0 ( , ) + ( , ) 0 ( , ) + ( , ) 0 ( , )

3

( , , ) = ( , , ) = ( , , ) =

(1) (2)

(3) where ( 0 , 0 , 0 , , ) are unknown functions to be determined (called the generalized displacements). As before, ( 0 , 0 , 0 ) denote the displacements of a point on the plane = 0 . Moreover, and indicate the rotations of a transverse normal about the y - and x -axes, respectively. Notice that , denotes the rotation of a transverse normal about the y -axis and denotes the rotation about the x -axis and they do not follow the right-hand rule. For thin plates, i.e., when the plate in-plane characteristic dimension to thickness ratio is on the order 50 or greater, the rotation functions and should approach the respective slopes of the transverse deflection as = − 0 and = − 0 . The standard system of motion equations for homogenous laminates is extensively described by Reddy (Reddy 1997). As a special case of isotropic single-layered configurations and neglecting the thermal and piezoelectric effects, the simplified motion equations in terms of displacement ( 0 , 0 , 0 , , ) used in this study take the form: 11 ( ∂ 2 0 2 + ∂ 0 ∂ 2 0 2 ) + 12 ( ∂ 2 0 + ∂ 0 ∂ 2 0 ) + 66 ( ∂ 2 0 2 + ∂ 2 0 + ∂ 2 0 ∂ 0 + ∂ 0 ∂ 2 0 2 ) = 0 (4) 66 ( ∂ 2 0 + ∂ 2 0 2 + ∂ 0 ∂ 2 0 2 + ∂ 0 ∂ 2 0 ) + 12 ( ∂ 2 0 + ∂ 0 ∂ 2 0 ) + 22 ( ∂ 2 0 2 + ∂ 0 ∂ 2 0 2 ) = 0 (5) 55 ( ∂ 2 0 2 + ) + 44 ( ∂ 2 0 2 + ) + = 0 (6) 11 ( ∂ 2 2 ) + 12 ( ∂ 2 ) + 66 ( ∂ 2 2 + ∂ 2 ) − 55 ( 0 + ) = 0 (7) 11 ( ∂ 2 2 ) + 12 ( ∂ 2 ) + 66 ( ∂ 2 + ∂ 2 2 ) − 44 ( 0 + ) = 0 (8) For a single isotropic layer with material constants , and , shear correction coefficient, Young’s modules and Poisson ratio, receptively and considering the shear modulus as = 2(1 + ) ⁄ and the plate thickness h , the nonzero laminate stiffnesses are: 11 = 1 − ℎ 2 12 = 11 22 = 11 44 = 55 = 66 = 1 − 2 11 11 = ℎ 3 12(1 − 2 ) 12 = 11 22 = 11 66 = 1 − 2 11 The shear correction factor, , for a general laminate depends on lamina properties and lamination scheme. This factor is the ratio of strain energies due to transverse shear stresses which gives = 5/6 (Reddy 1997) . In this study, since thermal and piezoelectric effects are not present, the stress resultants ( ′ and ′ ) are related to the generalized displacements (Reddy 1997); = 11 + 12 (9) = 11 0 + 12 0 (10) = 12 + 22 (11) = 12 0 + 22 0 (12)