Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29 (2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 231 interfaces. The theories, through the imposition of interfacial continuity conditions, define a homogenized displacement field which depends on a limited number of unknowns and is able to reproduce through-thickness zigzag patterns due to the material inhomogeneities. Later, the zigzag theories were extended to describe plates and shells with imperfectly bonded, purely elastic interfaces in [11-14]. The extended theories however manifest a number of inconsistencies: (i) they are unable to reproduce the expected transitions in the internal gross stress resultants on varying the stiffness of the imperfect interfaces (first noted in [11,15,16]); (ii) they give rise to unrealistic effects in the transverse displacements, which are larger than those of fully debonded plates in partially bonded plates (this effect was defined shear-locking in [15]); (iii) they show some inconsistencies in plates with clamped edges (noted in [17]). Recently the author formulated in [18,19] a theory which, starting from those in [11-14], extends the formulation to plates and beams with interfaces characterized by affine interfacial traction laws (to describe piecewise linear cohesive functions) and accounts for the energy contribution of the imperfect interfaces in the derivation of the equilibrium equations. This contribution was erroneously omitted in the original theories and in all theories derived later from the original models (see [18] for a list). Corrected formulations of the models [11-14] are presented in the Appendices in [18]. The accuracy of the theory proposed in [18,19] has been verified against exact 2D elasticity solutions in highly anisotropic, simply supported, multilayered plates, with continuous sliding interfaces and deforming in cylindrical bending. The model accurately describes stress and displacement fields in plates with different numbers of interfaces, equally and unequally spaced in the thickness, over the whole range of interfacial stiffnesses, from fully bonded to fully debonded. A limitation of the theory has been observed when dealing with very thick, highly anisotropic plates with compliant interfaces, where the shear deformations are underestimated in the derivation of the transverse displacements as a consequence of the assumed continuity between interfacial tractions and shear stresses. It is expected that this problem could be solved using a shear correction factor which depends on the interfacial properties (work in progress). In this paper, the issues noted in [17] in beams with clamped ends, where the zigzag theory proposed in [9,10] for fully bonded systems shows some apparent inconsistencies in the transverse shear force, are discussed; and the equilibrium equations derived in [19] for plates in cylindrical bending are restated to clarify the problem. The new formulation introduces a generalized transverse shear force, which is the gross stress resultant directly related to the bending moment in the equilibrium equations of multilayered plates with imperfect interfaces and substitutes for the shear force of single- layer theory. Finally, a delaminated cantilevered wide plate with a clamped edge is studied as a preliminary investigation of the applicability of the model to fracture mechanics problems. M ODEL onsider a rectangular multilayered plate of thickness h and in-plane dimensions 1 L and 2  L L , with 1 2  L L . A system of Cartesian coordinates, 1 2 3   x x x , is introduced with the axis 3 x normal to the reference surface of the plate, which is arbitrarily chosen, and measured from it (Fig. 1). The plate consists of n layers exhibiting different mechanical properties and joined by 1  n interfaces, which are described as mathematical surfaces where the material properties and the displacements may change discontinuously while the interfacial tractions are continuous. The layer k, where the index 1  ,.., k n is numbered from bottom to top, is defined by the coordinates 1 3  k x and 3 k x of its lower and upper interfaces,  S ( ) k and  S ( ) k , and has thickness ( ) k h , Fig. 1 (the k superscript in brackets identifies affiliation with layer k ). Each layer is linearly elastic, homogeneous and orthotropic with material axes parallel to the geometrical axes. The displacement vector of an arbitrary point of the plate at the coordinate   1 2 3  x , , T x x x is   1 2 3   v , , T v v v w . The plate is subjected to distributed loads acting on the upper and lower surfaces,  S and  S , and on the lateral bounding surface, B , is in plane strain conditions parallel to the plane 2 3  x x and deforms in cylindrical bending. In addition, the plate is assumed to be incompressible in the thickness direction and the interfaces to be rigid against mode I (opening) relative displacements. This latter assumption, which is often used in the literature, is rigorously correct only in problems where the conditions along the interfaces are purely mode II. The assumption, however, is acceptable in the presence of continuous interfaces, when the interfacial normal tractions are small compared to the tangential tractions and interfacial opening is prevented, e.g. by a through-thickness reinforcement or other means. To describe cohesive delaminations under general mixed mode conditions, the general treatment, which has been proposed in [18], for plates, and in [19], for wide plates in cylindrical bending, and accounts for interfacial opening, must be applied. Based on the C