# Delamination Assessment in Composite Laminates through Local Impulse Excitation Technique (IET)
## 2 Materials and Methods
### Finite Element Model of Local-IET  
The local-IET test was replicated in an LS-Dyna environment through a modal analysis. To compute the power amplitude of the first resonance frequency \(A_{mode1}\) and the power amplitude of the second resonance frequency \(A_{mode2}\), the FRF due to nodal excitation was computed by considering a damping ratio coefficient \(\zeta\) equal to 0.002 as a result of the logarithmic decrement determined in the IET tests of the bar specimens.  
Four nodes of Belytschko-Tsai shell elements with one integration point for each layer of laminate were used with a mesh size of 1 mm, according to a convergence analysis [18], to model the region within the internal seal. The *PART_COMPOSITE keyword was exploited to build the material stacking sequences. All of the retained materials were modeled with an elastic orthotropic material law (*MAT_ORTHOTROPIC), whose material properties are reported in Table 1. For the specimen with delamination, two layers of shells, each composed of half of the total laminate plies, were modeled and connected through solid adhesive elements to simulate the matrix phase. In correspondence with the delamination, the elements were removed, thus preventing any transfer of forces or moments between the two layers. The material behavior of the resin was assumed to be a linear isotropic elastic (*MAT_ELASTIC), whose properties were taken from [30].  
Ideally, the extremities of the investigated region are perfectly clamped, which assures the isolation of the mechanical vibration [18]. In order to ensure that the actual boundary conditions can diverge from the ideal clamping, two FE approaches were pursued here, both involving springs at the extremities. In the first approach, the out-of-plane displacement was released and governed by the stiffness of the translational springs, which aim to replicate the noninfinite stiffness of the internal seal. In the second approach, in order to simulate the boundary conditions between the simply supporting and the clamping conditions, the in-plane rotational degrees of freedom were released and governed by the stiffness of the rotational springs.  
Figure 5a shows the FE model for the undamaged plate and with the boundaries simulated with springs at the extremities. Figure 5b shows the FE model developed for the simulation of the delamination.  
LS-Dyna allows for the type of springs to be specified, i.e., translational or rotational, through a dedicated card variable [31; 32]. Furthermore, the degree of freedom affected by the spring can be specified through a vector defined with respect to the global reference system [31]. Therefore, according to Figure 5, in the case of the rotational springs model, the red springs affect the rotation around the \(x\) axis, and the blue springs govern the rotation around the \(y\) axis.  
For both FE approaches, the stiffness of the springs was identified for each configuration of material and device by considering the first resonant frequencies acquired on the  
Figure 5: The FE model of the local-IET test: (**a**) springs at the extremities to replicate the compliant constraint provided by the seal of the local-IET device; (**b**) the FE model for the delaminated plates.  
undamaged plates. The FE model that better replicates the local-IET test was identified by comparison with the acquired modal displacement of the first mode. Finally, a relationship between the spring stiffness and the main parameters affecting the vibrational response was identified with an optimization process, which was used to compute the stiffness of the springs for the fifth material in Table 1 and can be exploited to simulate the responses of other materials.