Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29 (2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 235 plates with arbitrary stacking sequences, mixed mode interfaces and under arbitrary loadings have been derived in [18]; a particularization of the equations to plates deforming in cylindrical bending have been presented in [19]. The equilibrium equations for wide plates with sliding only interfaces and quasi-static loading with 2 0    , S S F are presented here in a form that highlights similarities and differences with respect to single-layer theory: 02  : v 22 2 0  , N (17a) 2  : 22 2 2 0   , b g M Q (17b) 0  : w 2 2 3 3 0      , S S g Q F F (17c) where 22 N and 22 b M are normal force and bending moment in the 2 x direction and 2 g Q is a generalized transverse shear force which is statically equivalent, at any arbitrary sections of the plate with outward normal n =   2 0 1 0   , , T n , to the vertical equilibrant of the external forces acting on the portion of the plate to the right of the sections (Fig. 1a):  normal force:   3 1 3 22 22 3 1      k k n x k x k N dx (18a)  bending moment:   3 1 3 22 22 3 3 1      k k n x k b x k M x dx (18b)  generalized shear force:   1 2 2 2 22 2 22 2 2       , , ˆ b z z S g Q Q Q M M (18c) where  transverse shear force:   3 1 3 2 23 3 1 k k n x k b x k Q dx      (18d)  gross resultants and couples associated to the multilayered structure:     3 1 3 1 1 2 23 22 3 1 1           ; , k k n k x k i z x k i Q dx       3 1 3 1 1 22 22 22 3 3 3 1 1            ; k k n k x k i z i x k i M x x dx (18e)  gross resultants and couples associated to the cohesive interfaces:     3 1 3 1 22 22 22 3 1 1 k k n k x k S i x k i M dx           ,     1 1 2 22 1          ˆ ˆ n l l l S S l t (18f) The terms in the Eq. (18e) vanish in unidirectionally reinforced systems and those in Eq. (18f) vanish in fully bonded systems. The generalized shear force then coincides with the resultant of the transverse shear stresses, 2 2 b g Q Q  , when the layers have the same elastic constants, namely when   1 22 0 ; j   so that 22 0 z M  and 2 0 z Q  , and the interfaces are perfect, namely when 0 k S t  , k S B  0 and 2 ˆ k v  0 so that 22 0 S M  and 1 2 0 ˆ   ; in this case the equilibrium equations coincide with those of single layer theory. In all other cases, the generalized shear force depends on the multilayered structure, through 2 z Q and 22 z M , and on the status of the cohesive interfaces, through 22 S M and 1 2  ˆ , and the classical relation between bending moment and shear force of single-layer theory is modified as in Eq. (17b). The mechanical/geometrical boundary conditions on C , at 2 0  , x L , with n =   2 0 1 0   , , T n the outward normal, are: