Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29 (2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 237 A PPLICATIONS Highly anisotropic, simply supported, multilayered wide plates with imperfect interfaces he homogenized model for multilayered plates with imperfect interfaces has been verified against exact 2D elasticity solutions in [18,19]. Simply supported, highly anisotropic multilayered plates, loaded quasi-statically, with one or more weak layers have been examined on varying the properties of both layers and interfaces. The whole transition between fully bonded and fully debonded plates has been considered. The diagrams in Fig. 2 show exemplary results taken from [19] and refer to a highly anisotropic, thick plate, with two imperfect interfaces having different stiffnesses. (a) (b) (d) (e) Figure 2. Simply supported wide plate with L/h = 4 subjected to a sinusoidal transverse load,   0 2   sin q q x L . Stacking sequence: three layers, (0/90/0), elastic constants: 25 / T L E E  , 50 / LT L G E  , 125 / TT L G E  and 0 25 . LT TT     [1]. (a) Longitudinal displacements and (b) transverse shear stresses at the end support, (c) bending stresses and (d) transverse normal stresses at mid-span. Stresses are shown through the thickness (transverse shear/normal stresses derived a posteriori from equilibrium). Lower interface, 1 4 S T B E h  (very compliant), with 2 1 / ( ) T T TT E E    , upper interface, 2 4 1 1 10 S S B B    (almost fully bonded). (modified after [19]). Wide plates with clamped edges The zigzag theory for fully bonded plates [9,10] was applied in [17] to a multilayered cantilever beam subjected to a concentrated force F at the free end. The authors noted that the shear force was not constant along the beam length, as they would have expected given the linear distribution of the bending moment. In addition, the shear force increased from zero (at the clamped edge) to an asymptotic value higher than F. These apparent inconsistencies are explained by Eq. (17b), (18c) and (19c), which show that the internal gross stress resultant related to the bending moment, through Eq. (17b), is the generalized transverse shear force, Eq. (18c), which is statically equivalent to the vertical equilibrant of the external forces acting on the portion of the plate to the right of each arbitrary cross section, 2 g Q F  . The diagrams in Fig. 3 refer to a cantilevered wide plate of length L/h = 10 made with two unidirectionally reinforced layers with elastic constants, 25 / T L E E  , 50 / LT L G E  , and 0 25 . LT TT     , connected by a linearly elastic weak layer with 1 1 1 23 2 ˆ ˆ ˆ S S K v     . Diagram (a) depicts the transverse shear force, 2 b Q Eq. (18d), along the plate length for different values of the elastic interfacial stiffness and highlights the apparent inconsistency noted in [17] for a fully bonded multilayered plate. Note that 2 b Q is forced to be zero at the boundary by the geometric boundary conditions at 2 0 x  , 0 2 0 2 0     , w w Eq. (19c,d), which yield 23 23 2 0 b Q      . Diagram (b) highlights that the generalized transverse shear force in Eq. (18c), 2 g Q , correctly coincides with the external applied force F at all coordinates and 1 2 22 2 2 , ˆ b S Q F M     (in the example 2 22 2 0 , z z Q M   since the layers are equals; in [17] 2 2 2 22 2 , b z z g Q Q Q M    and 1 22 2 2 0 , ˆ S M    since the plate is multilayered and fully bonded). Diagram (c) compares the values of the interfacial tractions along the plate length, obtained a posteriori from local equilibrium, with those obtained with a classical discrete layer approach [6-8] and shows the existence of a boundary region near the clamped edge where the interfacial tractions predicted through the homogenized approach are not correctly described (bending stresses, not shown, are accurately T