## 1 Introduction  
Composite structures are often exposed to impact loadings during their life cycle [1]; this kind of loading conditions may become a serious issue, since composites can be very sensitive to imperfections beyond a critical threshold load. Depending on the impact energy, i.e. on the momentum of the impact when it strikes the structural component, two different failure scenarios can be expected: (_i_) in case of so-called low velocity (and, therefore, low energy) impacts, stress waves propagate inside the composite leading to diffused damage processes that progressively decrease or even annihilate the residual strength and stiffness of the structure; (_ii_) in case of high velocity (and, therefore, high energy) impacts, local perforation of the laminate occurs, giving rise to very localized failure mechanisms. In the former case the response of a laminated plate to the impact is characterized by forced vibrations of the whole structure; in the latter case, the response of a plate and the damaging processes are very localized around the impact location, and deformation modes of the structure as a whole are only marginally excited. Having as a target the development of a real-time health monitoring (and, hopefully, control) strategy for composite structures, in this work we focus on low velocity impact conditions. In such cases, the aforementioned diffused degradation processes consist in inter-laminar debonding (i.e. delamination) and intra-laminar damage, the latter being related to fibers failure, matrix cracking and fiber/matrix decoherence. Typically, a major role is played by delamination; therefore, throughout the whole paper we shall assume the laminae of the composite to behave elastically, while reduction of the structural load-carrying capacity may occur only because of inter-laminar damage. To simplify modeling, the small ratio between the thickness of each inter-laminar phase and the thickness of the whole laminate is exploited by numerically treating inter-laminar phases as lumped interfaces. The relevant constitutive laws locally link the tractions acting upon the surfaces of the two adjacent layers to the displacement jump occurring across the interface, allowing for both opening and sliding (or shear) deformation modes. This lumped interface formulation has to be considered as a coarse grained description of the actual inter-phase behavior, as obtained via homogenization along the thickness direction (i.e. in the direction perpendicular to the stacking plane); the displacement jump thus accounts for both the elastic response of the inter-phase material and the possible discontinuity in the displacement field due to micro-cracking. To model micro-cracking processes and the relevant reduction of the inter-laminar strength, nonlinear constitutive laws are adopted; beyond the attainment of peak tractions, delamination inception is modeled through a softening regime. Since inter-phase materials are usually quasi-brittle, dissipative phenomena due to yielding can be disregarded. Moreover, viscous and rate-dependent properties of the composite phases are not considered here; therefore, only rate-independent constitutive laws will be adopted. Within this framework, laminated plates are treated as a stacking sequence of laminae and interfaces. This scheme, sometimes referred to as meso-scale modeling (see, e.g. [2]), was proved to furnish accurate outcomes at the length scale of the whole laminate (also in terms of delamination prediction), provided the interface constitutive laws are appropriately calibrated. Since they are typically formulated on a pure phenomenological basis and contain several parameters to be tuned, this calibration task turns out to be difficult, also because of the lower dimensionality of interfaces (two-dimensional loci in a three-dimensional setting) and because of shadowing effects due to the global laminate response, see e.g. [3].