# An Interface Pressure Detection Method of Cable Silicone Rubber-XLPE Based on Nonlinear Ultrasound
## 2 Finite Element Simulation
### Silicone Rubber-XLPe Contact Interface Model  
Considering that the roughness of the object surfaces on both sides of the interface affects the actual contact area, the surface roughness of the silicone rubber is measured based on the white-light interference test, and its surface micromorphology is shown in Figure 2. A straight line in the x-direction is selected to measure the surface wave crest of up to 4 \(\upmu\)m, and the distance between the wave crests is 20 \(\upmu\)m. The surface of silicone rubber is represented by a series of semicircles with an interval of 20 \(\upmu\)m and a radius of 4 \(\upmu\)m to characterize its interfacial micromorphology [24]. Compared to silicone rubber, the XLPE surface roughness is much smaller, so it can be regarded as a smooth plane.  
Figure 2: Silicone rubber surface micro-formation.  
Figure 3 is the 2D simulation model of silicone rubber-XLE. The top model is silicone rubber, and the bottom model is XLEP. The uncontracted part of the silicone rubber-XLEP interface is the gap, where the radius of the gap is 4 \(\upmu\)m. In order to examine the ultrasonic nonlinear effects at the contact interface more intuitively and to reduce the computational effort, the size of the silicone rubber is 3 mm \(\times\) 1 mm, and the XLEP is 3 mm \(\times\) 1 mm. The modulus of elasticity of silicone rubber is 4.5 MPa, the density is 1050 kg/m\({}^{3}\), and the Poisson's ratio is 0.488. The modulus of elasticity of XLEP is 130 MPa, the density is 930 kg/m\({}^{3}\), and the Poisson's ratio is 0.32.  
The expression of the excitation source signal is shown in Equation (9). Figure 4a is the time domain waveform of the excitation source, and Figure 4b is the spectrum of the excitation source.  
\[\text{an1}(t)=A_{0}\text{sin}(2\pi ft)\times\text{e}^{(-((t-5T)/(T/0.5))^{2})} \tag{9}\]  
where:  
\(A_{0}\) is the amplitude of the excitation source,  
\(f\) is the center frequency of the excitation source,  
\(t\) is the time  
\(T\) is the signal period of the excitation source.  
Figure 4: Excitation source spectrum. (**a**) Time domain waveform of the excitation source. (**b**) Excitation source spectrum diagram.  
Figure 3: Silicone rubber-XLEP 2D simulation model.  
The finite element analysis of the silicone rubber-XLPE interface was carried out by COMSOL. The coupling of the acoustic and stress fields is carried out in two ways, direct coupling and sequential coupling. Direct coupling refers to the combined application of both pressure and ultrasonic waves into the model, facilitating transient calculations. In contrast, sequential coupling involves an initial static force analysis performed within a steady-state study. A deformation model is derived from this analysis, and it is used as the initial geometric configuration for sound field simulation calculations. The direct coupling approach in the finite element analysis of the pressure at the silicone rubber-XLPE interface was applied in this paper. A displacement load with a central frequency of 1.5 MHz was added to the upper surface of the silicone rubber to simulate the ultrasonic signal, and an upward boundary load was added to the lower surface of the XLPE to generate contact pressure on the silicone rubber-XLPE interface. Contact pairs were added to the silicone rubber-XLPE interface. Low-reflection boundaries were placed on either side of the model to reduce interference with the received signal. The maximum grid size of the simulation model is 0.02 mm, the time step is 0.05 \(\upmu\)s, and the total simulation time is 16 \(\upmu\)s.