Assessment and correction of theories for multilayered plates with imperfect interfaces

Abstract

The response of multilayered plates with imperfectly bonded interfaces and delaminations to quasi static and dynamic loadings is conveniently studied by assuming the displacements as two length-scales fields, given by global displacements, which are continuous in the thickness, and local perturbations or enrichments, which account for the multilayered structure and the jumps at the interfaces. The a priori imposition of interfacial continuity conditions then leads to a homogenized field defined by the global variables only. The number of displacement unknowns is independent of the number of layers and is equal to five or six, for sliding or mixed mode interfaces, respectively. Dynamic equilibrium equations are derived using variational principles. This effective idea, which was originally proposed in the zigzag theories for fully bonded plates, was first applied to plates with imperfect interfaces by Cheng, Jemah and Williams, J. Appl. Mech., 1996, 63, 1019-1026, Schmidt and Librescu, Nova J. Mathematics, Game Theory and Algebra, 1996, 5, 131-147, and Di Sciuva, AIAA Journal, 1997, 35(11), 1753-1759. These pioneering models, as all subsequent models which extend the theories to various problems, suffer from a common omission in the treatment of the interfacial energy in the derivation of the equilibrium equations, which leads to solutions which are accurate only for fully bonded/debonded plates. In this paper a linear, first order model which resolves the inaccuracy and extends the theories to more general interfacial traction laws is presented along with the corrected formulations of the original models. The formulations are validated using exact 2D elasticity solutions for multilayered plates in cylindrical bending

Influence of boundary conditions on the response of multilayered plates with cohesive interfaces and delaminations using a homogenized approach

“…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 assumptions above, the displacement components then simplify in 1 …”

“…Corrected formulations of the models [11][12][13][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.…”

“…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][12][13][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).…”

“…Recently the author formulated in [18,19] a theory which, starting from those in [11][12][13][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][12][13][14] are presented in the Appendices in [18].…”

“…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][12][13][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.…”