# An Interface Pressure Detection Method of Cable Silicone Rubber-XLPE Based on Nonlinear Ultrasound
## 2 Finite Element Simulation  
### Theory  
As the surface of objects have a certain degree of roughness, the contact interface is actually the contact of the rough peaks of the object surface. The uncontacted part of the interface is the air gap. This is shown in Figure 1. When pressure is applied, the contact interface undergoes intense collision and deformation, resulting in elastic and plastic deformations [17]. When the ultrasonic longitudinal wave propagates in one dimension at the contact interface, the reflected wave expression is as outlined in the equations below [23].  
\[U_{R}=-\frac{K_{2}A_{0}^{2}}{K_{1}(1+4K_{1}^{2}/(\rho c\omega)^{2})}-\frac{A_{0} \sin(\omega t+\delta_{1})}{\sqrt{1+4K_{1}^{2}/(\rho c\omega)^{2}}}\]  
\[+\frac{K_{2}A_{0}^{2}\sin(2\omega t-2\delta_{1}+\delta_{2})}{\rho c\omega(1+4K _{1}^{2}/(\rho c\omega)^{2}))\sqrt{1+K_{1}^{2}/(\rho c\omega)^{2}}}\]  
\[\delta_{1}=\tan^{-1}\frac{\rho c\omega}{2K_{1}}\]  
\[\delta_{2}=\tan^{-1}\frac{K_{1}}{\rho c\omega}\]  
\[K_{1}=CP^{m}\]  
\[K_{2}=\frac{1}{2}mC^{2}P^{2m-1}\]  
where:  
\(A_{0}\) is the incident wave displacement amplitude,  
\(K_{1}\) is the interfacial linear stiffness,  
\(K_{2}\) is the interfacial nonlinear stiffness,  
\(c\) is the longitudinal wave speed,  
\(\rho\) is the material density,  
\(\omega\) is the incident angular frequency,  
\(P\) is the pressure,  
\(C\) and \(m\) are the constants associated with the surface roughness of the interface.  
Figure 1: Ultrasonic wave propagation at the pressure-bearing interface.  
The expressions for the fundamental wave amplitude and the second harmonic amplitude in the reflected wave are:  
\[A_{1}=\frac{A_{0}}{\sqrt{1+4K_{1}^{2}/\left(\rho c\omega\right)^{2}}}=\frac{A_{0} }{\sqrt{1+4C^{2}P^{2m}/\left(\rho c\omega\right)^{2}}} \tag{6}\]  
\[\begin{array}{ll}A_{2}&=\frac{K_{2}A_{0}A_{1}}{\rho c\omega\sqrt{1+4K_{1}^{2}/ \left(\rho c\omega\right)^{2}}\sqrt{1+K_{1}^{2}/\left(\rho c\omega\right)^{2} }}\\ &=\frac{\frac{1}{2}mc^{2}P^{2m-1}A_{0}A_{1}}{\rho c\omega\sqrt{1+4C^{2}P^{2m}/ \left(\rho c\omega\right)^{2}}\sqrt{1+C^{2}P^{2m}/\left(\rho c\omega\right)^{2 }}}\end{array} \tag{7}\]  
where:  
\(A_{1}\) is the fundamental amplitude in the frequency domain of the reflected wave,  
\(A_{2}\) is the second harmonic amplitude in the frequency domain of the reflected wave.  
The relative nonlinear coefficient of ultrasonic-reflected waves is defined as the ratio of the second harmonic amplitude to the square of the fundamental wave amplitude, as follows:  
\[\beta=\frac{A_{2}}{A_{1}^{2}}=\frac{K_{2}}{\rho c\omega\sqrt{1+K_{1}^{2}/\left( \rho c\omega\right)^{2}}}=\frac{\frac{1}{2}mc^{2}P^{2m-1}}{\rho c\omega\sqrt {1+C^{2}P^{2m}/\left(\rho c\omega\right)^{2}}} \tag{8}\]  
where:  
\(\beta\) is the ultrasonic relative nonlinear coefficient.  
From Equation (6) to (8), when the ultrasonic frequency, material density, and ultrasonic propagation velocity are constant, the change of pressure will cause the change of the ultrasonic fundamental wave amplitude, second harmonic amplitude, and the ultrasonic relative nonlinear coefficient. Therefore, the nonlinear coefficient can be used to reflect the magnitude of the interface pressure.