## 2 Methods and Materials
### Characterization of Impact AE Frequency Response  
Theoretically impact AE frequency response signals can be written as follows:  
For convenience, we consider the crushed plastic scraps approximately as circular flakes with different diameters, which are equivalently determined by their particle sizes through fine sieving. According to vibration principles, the equation of motion for flexural waves in a uniform thin plate along the transverse axis \(z\) is:  
\[\frac{\partial^{2}z}{\partial t^{2}}+\frac{Eh^{2}}{12\rho(1-v^{2})}\
abla^{4}z=0 \tag{1}\]  
where \(E\) is the material Young's modulus, \(\rho\) is the material density, \(v\) is the Poisson's ratio of material, and \(h\) is the flake thickness.  
For harmonic solutions, \(z=Z(x,y)e^{i\omega t}\), where \(\omega\) is the angular flexural vibration frequency:  
Figure 1: Stone plate for impact test and impact acoustic emission (AE) acquisition chamber.  
\[\
abla^{4}Z-\frac{12\rho(1-v^{2})\omega^{2}}{Eh^{2}}Z=\
abla^{4}Z-k^{4}Z=0 \tag{2}\]  
where:  
\[k^{2}=\frac{\sqrt{12}\omega}{h}\sqrt{\frac{\rho(1-v^{2})}{E}}=\frac{\sqrt{12} \omega}{c_{\rm p}\,h} \tag{3}\]  
\(c_{\rm p}\) is the velocity of pressure wave in an infinite flake. In the case of a circular flake of radius \(b\) and thickness \(h\), with fixed boundary, the frequency of flexural vibration is determined by equation [26]:  
\[f_{m,n}=\alpha_{m,n}\sqrt{\frac{E}{\rho(1-v^{2})}}\frac{h}{b^{2}} \tag{4}\]  
where \(m\) and \(n\) are integers (beginning with zero), \(\alpha_{m,n}\) is the coefficient associated with the corresponding flexural vibration mode (\(m,n\)), which depends on the impact position and shape of impact bodies. Equation (4) means that multiple-order flexural resonance could be generated by one flake, the frequency and amplitude of which depend on the impact position and the shape of physical void. In this research, all the falling scraps are of a flake structure and the experiment facility was designed to ensure that the impacts always occur on the surface edge of the falling scraps. Additionally, the second passive impact body is also fixed for all scrap impacts. Hence for the scraps of a specific material, the parameter \(\alpha_{m,n}\) is a constant.  
According to Equation (4), the material property parameters of \(\rho\), \(E\), \(b\), \(h\) and are known, \(f\) can be obtained from impact AE signals. The Poisson's ratio \(\
u\) is a dimensionless quantity. It can be deduced that the unit of coefficient \(\alpha_{m,n}\) should be mm/s or m/s, which are the units of velocity.  
From Equation (4), the coefficient \(\alpha_{m,n}\) could be further determined as a dimensionless quantity. Hence in this research, the impact frequency response for specific material depends only on the thickness and radius of single scraps. Combining the operation of all constants as \(k\):  
\[f=k\,\frac{h}{b^{2}}\,,\,\,\,\,\,(k=\alpha_{m,n}\sqrt{\frac{E}{\rho(1-v^{2})}}) \tag{5}\]  
Obviously \(k\) is also a constant for the same material. The frequency response has been simplified to be a function of the particle size and thickness of single scrap particles and Equation (5) could be further normalized by standard thickness \(h_{\rm norm}=1\) mm as: \(\frac{f}{h}=k\,\frac{1}{b^{2}}\,\,\) and further:  
\[f_{\rm norm}=k\,h_{\rm norm}\,\frac{1}{b^{2}}=k_{\rm C}\,\frac{1}{b^{2}}\,, \,\,\,\,\,(k\,h_{\rm norm}=k_{\rm C}\,,\,h_{\rm norm}=1\,\,\mbox{mm}) \tag{6}\]  
The coefficient \(k_{\rm C}\) has its units of mm\({}^{2 Equation (6) also shows that for scraps from a specific kind of plastic, \(k_{\rm C}\) is constant and their impact frequency response depends only on the thickness and particle size of a single scrap. The value of \(k_{\rm C}\) varied according to the different plastic materials. Hence we could avoid direct measurement of the property parameters of specific materials, such as Young's modulus, density, Poisson's ratio _etc._ Different kinds of materials could be distinguished by determining the value of \(k_{\rm C}\) by using fitting analysis of impact frequency responses.