## 4 Results and Discussion
It is seen from Figure 9b that the second order nonlinear stiffness increases as the nominal contact pressure increases up to 2.5 MPa. However, beyond this value the nonlinearity decreases, so the nonlinear stiffness decreases. The comparison shows that \(K_{2}\) is independent of the amplitude of the incident wave. A best fit curve to the second order nonlinear interfacial stiffness \(K_{2}\) shown in Figure 9b is a fourth-order polynomial expression [22]:  
\[K_{2}(p_{0})=\sum_{n=0}^{4}c_{n}p_{0}{}^{n} \tag{45}\]  
where \(c_{n}\) is a constant coefficient. This coefficient depends on some factors such as surface topography and elastoplastic deformation of the asperities in contact. However, further studies are required. As before, the nominal contact pressure \(p_{0}\) is expressed in terms of the relative surface approach \(Y\) of the contacting surfaces. To do this, the nonlinear interfacial stiffness \(K_{2}\) (Equation (45)) was substituted into Equation (28) creating a second order nonlinear homogeneous differential equation:  
\[\frac{1}{2}\frac{d^{2}p}{dY^{2}}-K_{2}(p_{0})=0\,\ \frac{dp}{dY}=0\,\ p(0)=0 \tag{46}\]  
The initial conditions indicate that in the absence of an elastic wave (ultrasound), there is zero relative surface approach and linear interfacial stiffness. The homotopy perturbation method (HPM) was again employed to solve Equation (46) with respect to relative surface approach. A numerical differential equation (ode45 function in MATLAB [48]) of the data in Figure 9b according to Equation (46) was used to determine the nominal contact pressure-relative surface approach relationship, as shown in Figure 9c. The result was the same as Equation (44). The comparison between the pressure-relative surface approach derived from linear and nonlinear interfacial stiffness is presented in Figure 10. Figure 10a shows the pressure-relative surface approach derived from the present experimental data. Figure 10b is included for comparison, and shows the pressure-relative surface derived from the  
Figure 9: The influence of nominal contact pressure \(p_{0}\): (**a**) second order nonlinear parameter for the reflected ultrasound from the interface \(\gamma\); (**b**) nonlinear interfacial stiffness; (**c**) pressure-relative surface approach derived from the nonlinear interfacial stiffness at incidents amplitude excitations of 90 V, 140 V and 210 V.  
experimental data from [22]. It should be noted that the experiment presented in [22] was through transmission while the current paper considered reflected signal. Both approaches yield a similar relationship. It is seen that the pressure-approach relationship derived from the linear stiffness \(K_{1}\) and second order nonlinear stiffness \(K_{2}\) are very similar for all excitation voltage cases. As expected, the pressure-approach response should not depend on whether it was determined from either linear or nonlinear stiffness measured data. This gives confidence in the robustness of the measurement method and the analysis approach. The interfacial spring can be modelled with a stiffness described by a Taylor expansion with first and second order components (Equation (26)). Further, an experimental measurement of either one of these components, using ultrasonic reflection, is sufficient to deduce the other and hence characterise the spring stiffness.