## 2 Materials and Methods  
### Virtual Material-Based Acoustic Model  
The thin layer of the rough surface is assumed as a virtual material with thickness \(h\), elastic modulus \(E_{0}\), shear modulus \(G_{0}\), Poisson's ratio \(v_{0}\), and density \(\rho_{0}\), as shown in Figure 1. The thin layer is rigidly connected with the contacting body to ensure the continuous propagation of elastic waves. Therefore, the contacting bodies and the equivalent thin layer can be assumed as a multilayer system (see Figure 1). When the longitudinal wave incident is normal compared to the equivalent thin layer, the reflection coefficient at the contact interface can be expressed as [18]  
\[R=\frac{\rho_{0}c_{0}\sin(\omega h/c_{0})-\left(Z^{2}/\rho_{0}c_{0}\right)\sin( \omega h/c_{0})}{\rho_{0}c_{0}\sin(\omega h/c_{0})+2 iZ\cos(\omega h/c_{0})+ \left(Z^{2}/\rho_{0}c_{0}\right)\sin(\omega h/c_{0})} \tag{1}\]  
where \(\rho_{0}\) is the density of the virtual material, \(c_{0}\) is the longitudinal wave velocity of the virtual material, \(Z\) is the acoustic impedance of the contacting solid, \(\omega\) is the angular frequency of ultrasound, \(h\) is the layer thickness, i.e., surface separation, and \(i\) is the imaginary unit. Since the thickness of the virtual thin layer is much smaller than the incident wavelength, Equation (1) can be simplified by asymptotic expansion and truncated to the first order [19]. Therefore, the reflection coefficient can be expressed as  
\[R=\frac{\rho_{0}h\omega-\left(Z^{2}\omega h/\rho_{0}c_{0}^{2}\right)}{\rho_{0 }h\omega+\left(Z^{2}\omega h/\rho_{0}c_{0}^{2}\right)+2 iZ} \tag{2}\]  
where \(\rho_{0}h\) represents the influence of the mass on the virtual thin layer. However, the mass term is effective only when the thickness of the thin layer is large [20]. For general rough contacts, the interface contains low-density air gaps, and the virtual interface layer is very thin, which means that the interface mass term has almost no effect on the reflection coefficient. Therefore, the mass term, \(\rho_{0}h\), can be ignored, and Equation (2) can be written as  
\[R=\frac{-Z\omega}{Z\omega+2i\rho_{0}c_{0}^{2}/h} \tag{3}\]  
where \(\rho_{0}c_{0}^{2}/h\) represents the normal contact stiffness \(K_{c}\) per unit of nominal contact area. In this study, the virtual thin layer is assumed to be elastically deformed so that the contact stiffness \(K_{c}\) is real and can be written as Equation (4), which is the stiffness expression form of the quasi-static spring model [10]. \[K_{c}=\frac{Z\omega}{2}\sqrt{\frac{1}{|R|^{2}}-1} \tag{4}\]  
According to Equation (4), the thickness of the thin layer \(h\) can be deduced as  
\[\frac{1}{h}=\frac{Z\omega}{2\rho_{0}c_{0}^{2}}\sqrt{\frac{1}{|R|^{2}}-1} \tag{5}\]  
Figure 1: Rough interface represented by a thin layer of virtual material. For the equivalent virtual thin layer, the normal contact stiffness \(K_{n}\) equals the ratio of elastic modulus \(E_{0}\) to thickness \(h\). Therefore, combining Equations (4) and (5), the interfacial normal contact stiffness can be expressed as [15]  
\[K_{n}=\frac{E_{0}(3G_{0}-E_{0})}{G_{0}(4G_{0}-E_{0})}\frac{Z\omega}{2}\sqrt{ \frac{1}{\left|R\right|^{2}}-1}=\frac{E_{0}(3G_{0}-E_{0})}{G_{0}(4G_{0}-E_{0}) }K_{c} \tag{6}\]  
The rough contact interface is a mixed distribution of asperities and air gaps. Therefore, the equivalent material properties of the virtual thin layer need to be determined by integrating the mechanical properties of single asperities based on the statistical microcontact theory. As the asperity interaction will affect the asperity's deformation, the average interface separation and the interface stiffness, the solution of the mechanical properties of a single asperity needs to consider the effect of the asperity interaction. As shown in Figure 2, the total interference of a given asperity is \(\delta\), which includes two parts: deformation due to the contact load \(w\) and deformation due to asperity interactions \(u_{g}\).