Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29(2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 236 02  : v 22 2 2   B N n N or 02 02   v v (19a) 2  : 22 2 2   b bB M n M or 2 2     (19b) 0  : w 2 2 3   B g Q n N or 0 0   w w (19c) 0 2  , : w   22 2 22 2 2 2      z S zB SB M n M n M M or 0 2 0 2   , , w w (19d) where   3 1 3 3 1 2 3 , for , k k n x k B B i i x k N F dx i       (20a)   3 1 3 2 2 3 3 1       , k k n x k bB B x k M F x dx (20b)    3 1 3 1 2 2 22 3 1 1           , k k n k x k SB B i x k i M F dx (20c)       3 1 3 1 1 2 2 22 3 3 3 1 1 ; , k k n k x k i zB B i x k i M F x x dx            (20d) Terms with the tilde define prescribed values of generalized displacements and gross forces and couples applied to B . Equilibrium and boundary conditions can be expressed in terms of the homogenized displacement variables using the constitutive and compatibility Eq. (1), (12), (14) and (15). The equations are presented in [19]. Eq. (14a) and (9) show that the transverse shear stress, 23  , obtained from the shear strains, 23  , through the constitutive Eq. (1), is constant in the thickness and related to the transverse shear force, Eq. (18d), through 23 2   / b Q h . This stress does not describe the effective status of the material, but for the limit case of a system with perfectly bonded interfaces and layers with the same elastic constants, where 1 2 22 2 22 2 2 0 , , ˆ z z S Q M M      and 2 2  b g Q Q . In the presence of imperfect interfaces, 23  follows the dependence of the interfacial tractions on the stiffness of the interfaces, due to the imposed continuity, Eq. (7)-(10), and progressively goes to zero when the stiffness of the interfaces decreases; in fully bonded systems with a multilayered structure, where 2 2 2 22 2    , b z z g Q Q Q M , 23 2   / b Q h again does not describe the actual stress distribution but for the special case of layers with the same elastic constants where 2 22 2 0   , z z Q M . Based on the observations above, a generalized transverse shear stress can be introduced, which is the relevant internal stress for strength predictions, 23 2 g g Q h   . The generalized transverse shear stress, 23 g  , averages the actual shear stress distribution which can be obtained a posteriori from the bending stresses by satisfying local equilibrium, 22 2 23 3 0     ( ) ( ) , , k k post , so that 3 1 3 23 23 3 1 1        ( ) / k k n x k post g x k h dx . Similarly, the transverse shear strain, which is related to the transverse shear stress through the constitutive Eq. (1), 23 23 55 2 / C    , only partly describes the shear deformations of the plate whose correct measure within this model is given by a generalized shear strain 23 23 55 2 / g g C    . In order to account for the correct shear deformations in the solution of the differential equations, a shear correction factor, 2 K , can be introduced such that 2 23 23 55 2 /( ) K C    . 2 K is equal to 5/6 in fully bonded unidirectionally reinforced plates, to account for the approximated constant distribution of 23  in the thickness, and it becomes a problem dependent parameter in multilayered plates, e.g. [22]; in [9] it was shown that, for simply supported plates with common layups and geometrical/loading conditions, the homogenized zigzag theory with 2 K = 1 leads to accurate predictions of the displacement field. In plates with imperfect interfaces, 2 K must depend on the stiffness of the interfaces. Work in in progress on the derivation of 2 K and results are presented here for 2 K =5/6 (see also [18-19]).