## 4 Results and Discussion  
Figure 8a shows the reflection coefficient \(R\) for incident waves of excitations 90 V, 140 V and 210 V as a function of the nominal contact pressure, determined from the amplitude of the first order reflection from the solid interface divided by that from the solid-air interface according to Equation (36). They were measured after 10 loading/unloading cycles to remove any asperity plasticity effects. It can be seen that the reflection coefficient decreases with increasing the nominal contact pressure due to increase in the real contact area. The data of Figure 8a was used to determine the first order linear interfacial stiffness \(K_{1}\) using Equation (37) (Figure 8b). Higher pressures cause a stiffer interface. As expected, the reflection coefficient and linear stiffness determined at any voltage is identical. The amplitude of the ultrasonic wave does not change the linear stiffness component. Figure 6: Schematic diagram of voltage to meter conversion apparatus. Figure 7: Voltage to meter conversion at ultrasonic incident amplitudes 70 V, 90 V, 140 V, 210 V, 280 V and 350 V.  
A best fit curve to the linear interfacial stiffness shown in Figure 8b is a fourth-order polynomial expression [22]:  
\[K_{1}(p_{0})=\sum_{n=0}^{4}a_{n}{p_{0}}^{n} \tag{42}\]  
where \(a_{n}\) is a constant coefficient. This coefficient will depend on factors such as surface topography and elastoplastic deformation of the asperities in contact. The nominal contact pressure \(p_{0}\) can be expressed in terms of the relative surface approach \(Y\) of the contacting surfaces. To do this, the linear interfacial stiffness \(K_{1}\) (Equation (42)) was substituted into Equation (27) creating a first order nonlinear homogeneous differential equation:  
\[\frac{dp}{dY}+K_{1}(p_{0})=0\,\ p(0)=0 \tag{43}\]  
The initial condition indicates that in the absence of elastic wave (ultrasound), there is zero relative surface approach. The homotopy perturbation method (HPM) was employed to solve Equation (43) with respect to relative surface approach:  
\[p_{0}(Y)=\sum_{n=0}^{2}b_{n}Y^{n} \tag{44}\]  
where \(b_{n}\) is a constant coefficient. This coefficient, as with \(a_{n}\), will also depend on the surface asperity and material properties. A numerical differential equation (ode45 function in MATLAB [48]) of the data in Figure 8b according to Equation (43) was used to determine the nominal contact pressure-relative surface approach relationship, as shown in Figure 8c. The shape of the nonlinear nominal contact pressure-relative surface approach curve demonstrates the stiffening behaviour of the interfacial spring under increasing contact pressure. In other words, increasing the nominal contact pressure leads to a reduction in the gap, more solid contact, and hence a stiffer interface. In the absence of an elastic wave, the relative surface approach was zero (\(Y=0\)) under nominal contact pressure \(p_{0}=0\) MPa (nominal contact pressure generated by the weight of upper body is negligible). The negative sign of the relative surface approach \(Y\) indicates the reduction of the interface separation (greater deflection in the asperities). The gradient of the curve increases with increase in nominal contact pressure demonstrating stiffer interfacial contact. Figure 9a shows the second order nonlinear parameter \(\gamma\) at excitations 90 V, 140 V, and 210 V as a function of the nominal contact pressure. This nonlinear parameter is  
Figure 8: The influence of nominal contact pressure \(p_{0}\): (**a**) reflection coefficient; (**b**) linear interfacial stiffness; (**c**) pressure-relative surface approach derived from the linear interfacial stiffness at incidents amplitude excitations of 90 V, 140 V and 210 V.  
determined from the amplitude of the second harmonic divided by the square of the first harmonic (Equation (40)). To evaluate the second order nonlinear stiffness \(K_{2}\), the experimental results of the second order nonlinear parameter \(\gamma\) were substituted into Equation (39) (Figure 9b).