Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29(2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 234   2 0 2 2 22      , ˆ k k k k S S v w B t (10) with     1 1 22 55 22 1 1              ; k j k k k S j C B (11) Eq. (8) and (10) can then be inserted into Eq. (5) to obtain the homogenized displacement field. The displacement components within layer k are:      1 2 02 2 3 0 2 2 22 1           , k k k i i S S S i v v x w R B t (12a)   0  k w w (12b) where:      1 1 22 22 3 22 3 3 22 1                ; ( ) k i k k i i S S i R R x x x (13) Eq. (12) highlight that the displacement field is fully defined by the 3 displacement variables, 02 0 2  , , v w , which describe the global part of the field, and are underlined in the equations, and by parameters which depend on the elastic constants of the material, the layup and the geometry (no line) and parameters depending on the properties of the interfaces through the assumed interfacial traction laws (curved line on top). For perfectly bonded layers, when  k S B 0 for k = 1..n-1 , all terms with the curved line on top vanish and the equations are those of first order zig-zag theory [9,10]. The strain components in the layer k in terms of the homogenized displacement variables are:           1 1 23 0 2 2 22 3 0 2 2 22 1 2 1 1 ; , , , k k i k S i w R w                    (14a)   22 02 2 2 2 3 2 2 0 22 22 , , , , ( ) k k S v x w R        (14b) The interfacial tractions at the ( ) k  S interface at the coordinate 3 k x , in terms of the homogenized displacements are:    23 0 2 2 22 , ˆ ˆ k k k k S S K w        (15) Equilibrium equations Equilibrium equations and boundary conditions are derived in weak form through the Principle of Virtual Works:     1 22 22 23 23 2 3 3 3 3 1 2 0                               V S S S B ( ) ˆ ˆ k n k k k S S B S S i i k dV t v dS F v dS F v dS F v dB (16) where V is the volume of the plate and i = 2,3; the virtual displacements are assumed to be independent and arbitrary and to satisfy compatibility conditions. The first term on the left hand side defines the strain energy in the volume of the body; the second term, with the flat line on top, the energy contributions due to the interfacial tractions on the n -1 interfaces. The last terms define the work done by the external forces, with 3 S F  (top), 3 S F  (bottom) and B i F (lateral) the components of the forces acting along the bounding surfaces of the plate,  S ,  S and B . Tangential forces acting on  S and  S have been assumed to be zero (refer to [18,19] for more general loading conditions). The term related to the interfacial tractions was not present in the models where this approach was first proposed for plates with linear-elastic interfaces [11-14]. The terms were also missing in all subsequent models which extended the theories to different problems (see list in [18]). It has been recently proved in [18] that, in the absence of these terms, the solutions are accurate only in the limiting cases of fully bonded and fully debonded interfaces. Virtual strains and displacements in Eq. (16) are defined as functions of the global displacement variables using Eq. (10), (12) and (14). Then, by applying Green’s theorem wherever possible and after lengthy calculations, Eq. (16) yields the equilibrium equations and boundary conditions. Dynamic equilibrium equations and boundary conditions for multilayered