A homogenized analysis à la Hashin for cracked laminates under equi-biaxial stress. Applications to laminated glass

We extend, to the case of equibiaxial states of stress, the variational approach originally proposed by Hashin for the analysis of cracked three-layer laminates under uniaxial tensile loading. The type of damage considered here is complementary to that analyzed by Hashin, in the fact that the external plies of the laminate, rather than the inner layer, are those that are supposed to break, now according to a regular 2D crack pattern. This renders the model particularly suitable for the post-breakage response of laminated glass, composed by a polymeric interlayer sandwiched by two brittle glass plies: the resulting shards partially delaminate but remain attached to the polymer, so to impart to the system a residual load bearing capacity. Within a class of admissible stress systems that verify the equilibrium equations, variational minimization of the complementary energy functional is used to find the optimal approximation, which represents a lower bound for the stiffness. The major complication here consists in the definition of such class, which must take into account the biaxial nature of the problem. Comparisons with numerical experiments confirm the very good accuracy of the proposed approach, at least when applied to laminated glass.