## 4 Failure assessment of laminates exposed to impacts  
To assess the capability of the proposed FE-SPKF approach to provide accurate failure forecasts for layered composites subject to low velocity impacts and to simultaneously calibrate the inter-laminar constitutive laws, two experiments reported in the literature have been considered: an impact test on a 7-layer composite plate (experiment FY06001 in [13]), and an impact test on a 11-layer composite plate backed by another 11-layer composite plate (experiment 1 in [14]). In both cases, plane impacts caused the propagation of compressive waves behind the impact location; upon reflection at the rear surface of the plates, a spall-like failure governed by tensile tractions perpendicular to the inter-laminar surfaces took place. Because of the experimental set-up, shear tractions do not play any role in the failure events; in what follows, for ease of notation we therefore avoid using subscript \(n\) to denote (mode I) opening/closing displacement discontinuities along interfaces. Figure 3: Impact on a 7-layer composite [13]: space-time diagram (the vertical dashed line here represents the possibly debonding surface of a brittle, homogeneous material). Figure 4: Impact on a 7-layer composite [13]: (a) comparison between experimental and tracked free surface velocity records; (b-c) estimated values of inter-laminar openings \([u]_{j}\), \(j=1,...,6\) (b: PWL law; c: L-E law). As far as space discretization is concerned, owing to the major role played by dilatational waves the full three-dimensional problem can be approached as a one-dimensional one featuring constrained deformation in the plane perpendicular to the wave propagation direction. Accordingly, one-dimensional meshes of constant strain elements have been adopted for the bulk; the characteristic size (length) of FEs in the through-the-thickness direction was determined by a trade-off between accuracy requirements and computational economy (leading to around 10 elements to discretize each lamina). In the first experiment, each lamina was 1.37 mm in thickness, and was made of a balanced 5-harness satin weave E-glass and LY564 epoxy. The overall wave speed in the through-the-thickness direction was measured as 3.34 km/s, while the mass density was \(\varrho=1885\) kg/m\({}^{3}\); additional details and a thorough discussion of the results from an experimental perspective can be found in [13]. The specimen was stricken by a 12.5 mm thick aluminum impactor, flying at velocity \(v=71\) m/s, which led to dynamic delamination. The free surface velocity profile \(\dot{u}_{r}\) was measured during the test with a velocity interferometer for any reflector (VISAR). Figure 5: Impact on a 7-layer composite [13]: evolution in time of estimated inter-laminar (a) strength \(\tau_{M}\) and (b) toughness \(G\). The space-time diagram of this test is shown in Figure 3: in this diagram, wave reflection and dispersion due to the inter-laminar phases [45] have been neglected. Moreover, since failure is actually caused by delamination, the possibly debonding surface highlighted in the diagram is here to be considered as a reference one: even in the presence of wave dispersion, the actual failure of the laminate is expected to occur close to this reference surface. Figure 4 shows a comparison between the estimated and the experimentally observed free surface velocity \(\dot{u}_{r}\), and the time evolution of the tracked inter-laminar openings \([u]_{j}\), \(j=1,...,6\); filter results are shown for both linear and exponential softening models. Before \(t\approx 7\)\(\mu\)s information on the mechanical properties of the failing inter-laminar surface are not filtered, since they are not yet available through measurables in \(\mathbf{y}\) (see Figure 3), and some discrepancies between experimental and numerical graphs of \(\dot{u}_{r}\) are shown (see Figure 4(a)).