# An Interface Pressure Detection Method of Cable Silicone Rubber-XLPE Based on Nonlinear Ultrasound
## 3 Materials and Methods
### Experimental Results and Analysis  
Nonlinear ultrasonic testing of the pressure at the silicone rubber-XLEE interface was carried out using the test setup and the test method described above. The waveform obtained from the test when no external pressure was applied is shown in Figure 12.  
Figure 11: Relationship between the second harmonic amplitude A\({}_{2}\) and A\({}_{1}\)\({}^{2}\).  
Based on the relationship between time, sound velocity, and propagation distance for the interface-reflected waveform verification in Figure 10, according to Equation (10) [25]:  
\[2h=v(t_{2}-t_{1}) \tag{10}\]  
where:  
\(h\) is the thickness of the silicone rubber,  
\(v\) is the speed of sound propagated in the silicone rubber,  
\(t_{1}\) is the front signal time, 0.732 \(\upmu\)s,  
\(t_{2}\) is the reflected signal reception time, 31.448 \(\upmu\)s.  
Through calculation, it can be concluded that the time domain waveform of 30-40 \(\upmu\)s in Figure 12 is the primary echo of the silicone rubber-XLPE interface. The spectrum was obtained by Fourier transforming the silicone rubber-XLPE interface echoes in Figure 12, as shown in Figure 13. The second harmonic signal appeared in the spectrogram after the ultrasonic waves were reflected by the silicone rubber-XLPE interface. This indicates that the interaction of ultrasound with the nonlinear features of the interface produced a nonlinear effect.  
Figure 14a shows the variation curve of the fundamental wave amplitude of the interfacial-reflected waves at different pressures. Figure 14b shows the variation of the second harmonic amplitude of the interface-reflected wave at different pressures. According to Equation (8), the ultrasonic nonlinear coefficient of the interface-reflected wave at different pressures was calculated, and its variation law with pressure is shown in Figure 15.  
Figure 12: Reflected waveform at the silicone rubber\(-\)XLPE interface.  
Figure 13: Reflected wave spectrum at the silicone rubber-XLPE interface.  
As shown in Figure 14a, the fundamental wave amplitude decreased as the contact pressure at the interface increased. This is because the pressure increased the actual contact area of the interface, and the enhancement of the transmitted wave caused the reflected wave to weaken. As shown in Figures 14b and 15, the second harmonic amplitude increased, and the ultrasonic nonlinear coefficient also increased as the interface pressure increased. The reason for this is that the compressive stress reduced the gap between the interfaces, enhancing the generation of second harmonics and the contact nonlinearity at the silicone rubber-XIPE interface.  
As the contact interface studied in this paper consisted of two different materials, the ultrasonic transducer was placed on the XLPE surface and tested again in order to investigate whether the materials had an effect on the test results. The relationship between the ultrasonic nonlinear coefficient and the pressure at the interface is shown in Figure 16. As the interface pressure increased, the relative nonlinearity coefficient tended to increase overall, and the relative nonlinearity coefficient increased more and more rapidly at equal pressure changes, due to the interface microstructure changing its contact state more under pressure.  
Figure 14: Curves of fundamental and second harmonic amplitudes of reflected waves at the silicone rubber-XLPE interface as a function of pressure. (**a**) Fundamental wave amplitude variation curve. (**b**) Second harmonic amplitude variation curve.  
Figure 15: Silicone rubber-XLPE interface ultrasonic nonlinearity coefficient as a function of pressure.