# Delamination Assessment in Composite Laminates through Local Impulse Excitation Technique (IET)
## 2 Materials and Methods
### Local-IET Setup and Test
According to [PERSON] [28], the first resonant frequency of an orthotropic clamped plate can be written as follows:  
\[f=\frac{\lambda}{2\pi}\sqrt{\frac{1}{\rho\cdot h}}.\sqrt{\frac{D_{11}}{a^{4}}+ \frac{D_{22}}{b^{4}}+\frac{0.605\cdot H}{a^{2}\cdot b^{2}}} \tag{1}\]  
where \(|\rho\) is the material density, \(h\) is the plate thickness, \(a\) and \(b\) are the plate dimensions, and \(D11\), \(D22\) and \(H\) are the flexural stiffness terms, defined as  
\[ \begin{array}{l}D_{11}=\frac{E_{11}\cdot h^{3}}{12(1-v_{12}v_{21})}\\ D_{22}=\frac{E_{22}\cdot h^{3}}{12(1-v_{12}v_{21})}\\ H=v_{12}\cdot D_{22}+\frac{G_{12}\cdot h^{3}}{6}\end{array}  \tag{2}\]  
Considering the square shape of the devices of Figure 2, the formula of Equation (1) was rewritten as  
\[f=\frac{\lambda}{2\pi a^{2}}.\sqrt{\frac{1}{\rho\cdot h}}.\sqrt{TrD} \tag{3}\]  
where the flexural stiffnesses were substituted by the trace of the flexural matrix \(D\), which is proportional to the sum of the flexural stiffnesses \(D_{11}\), \(D_{22}\) and \(\frac{G_{12}}{6\cdot h^{3}}\) and to the Tsai modulus of the material through the plate thickness [27]. The \(\lambda\) parameter of Equation (3) defines the boundary conditions of the rectangular plate. Its value ranges from 1 (for simply supported boundaries) to 22.4 (for fully clamped boundaries). As the device aims to isolate the vibrational response, the desired value of \(\lambda\) for vacuum clamping should be as near as possible to 22.4. Therefore, by investigating the \(\lambda\) parameter, the clamping effectiveness, which is the ability of the device to isolate the vibrational response, can be assessed. The modal displacement of the first mode was acquired with a sampling frequency of 20 kHz. The laser was mounted on a linear micrometric support which allows its position be to precisely adjusted and set. Measurements were acquired along a line with a constant interval between each other. A step size of 10 mm was considered, except for the  
Figure 3: Experimental setup of the local-IET tests for modal displacement investigations. 50 \(\times\) 50 clamping device, whose step size was 5 mm. The vertical displacement was also measured in correspondence with the extremities in order to investigate the behavior of the plate in correspondence with the internal seal. For the twill and plain weave laminates, only one measurement line was acquired, given the symmetry of the system. On the contrary, in the case of the unidirectional plate, two measurement lines were considered in order to determine the modal displacements along the two main material directions. It is worth mentioning that, to provide the impulse impact in the center of the investigated region, the measurements of the modal displacement were acquired along a line located between the center and the boundary of the device. Concerning the fifth material, according to the literature on nonlinear acousto-ultrasonic NDTs [1; 29], a relevant factor for the detection of delamination damage in composites is the ratio between the power amplitude of the second resonance frequency \(A_{mod\ell 2}\) and the squared power amplitude of the first resonance frequency \(A_{mod\ell 1}\), thus defined as  
\[\beta=\frac{A_{mod\ell 2}}{A_{mod\ell 1}^{2}} \tag{4}\]  
The \(\beta\) factor is usually referred to as a second-order nonlinear coefficient and represents a measure of the nonlinear behavior of the material in the presence of damage. In particular, due to the nonlinearities, the participation of the second mode and, accordingly, the \(\beta\) factor increases in the presence of damage. For the experimental investigations, the delamination was positioned in the center of the 100 \(\times\) 100 mm device.