**1. Introduction**

Accurate contact interface modelling requires a knowledge of interfacial parameters, including interface contact stiffness. For many applications, characterizing and understanding the contribution of the interface to the dynamic response of the system is critical. These include robotic applications [1], grippers [2], micro-bearings [3], adhesive surfaces [4] and, wherever dry contact occurs between solids [5], with specific attention to lightly loaded joints. In the case of structural diagnostic, health monitoring and quality control of components and joints, these are based on the measurement and interpretation of wave interaction with joint interfaces or component defects [6].

Although contacts are common in practical engineering applications, there are certain aspects, such as sensitivity to interfacial parameters, which are not fully understood and modelled. Such sensitivity causes uncertainty in system performances and reliability predictions.

*Lubricants* **2020**, *8*, 73

One of the main methods used to model a contact interface is to use a spring and a viscous damper in parallel. Contact stiffness can be obtained from analytical contact models, for instance, the Herzian contact model for spherical contacts [7]. In the case of rough surfaces in contact, the Greenwood and Williamson [8] statistical model and its successive reformulations [6,9–11] have been used to obtain overall mean stiffness. Experimental values have been extracted using indirect methods [12] or system identification methods [13].

Recently, Jin et al. [14] used a quasi-static model developed within the GW framework, in which all the microscopic geometric features of contact interfaces are extracted directly from high-resolution scanning electron microscopy (SEM) images of real fatigue cracks.

However, the development of increasingly sophisticated numerical models with contact interfaces means that more reliable and fine contact parameters need to be defined. Contact stiffness has been proved to be sensitive to contact conditions such as contact pressure [15–17], third body rheology [18] and the true area of contact [19].

In more detail, the force is supported by surface asperities. As the force increases, more asperities come into contact, while each asperity undergoes flattening deformation. In [20], three contact states can be identified: total sliding, partial slip and contact loss. In the case of partial slip, roughness has been described by Aleshin [20] using the Method of Memory Diagrams (MMD), a model developed to describe partial slip for rough surfaces in contact. The MMD model was then extended to take into account the other two regimes of total sliding and contact loss [21,22]. The contact interface has a further nonlinear behaviour due to asymmetry between traction [23,24] and compression configurations. During compression, the change in the contact interface configuration, as a function of contact pressure, also results in nonlinearity in the interface response.

When these nonlinearities are activated by the interaction between propagating waves and the contact interface, higher-order harmonics are then generated [25]. While these effects have been well studied in the ultrasonic field [26], they also represent a new area of investigation from a vibrational point of view [9]. In particular, the generation and features of second harmonics [27] deserve to be further analysed and exploited.

In this context, the aim of this study is to present a numerical and experimental analysis to provide a basic insight into the nonlinear vibrational response of a contact interface, as a basis for evaluating nonlinear contact through stress-dependent stiffness in compression.

To this end, a numerical model with contact interfaces was developed, considering different nonlinear contact models, with different stress-dependent stiffnesses. A specific contact law is proposed, including a specific evolution of the stiffness for low pressures, and compared to a classical power law, fitting experimental values.

An experimental campaign was then conducted on a specific test bench in order to investigate the nonlinear response of the system, tested under a contact pressure of up to 1 MPa. By comparing experimental and numerical nonlinear responses, the sensitivity of the system response to the contact interface stiffness trend at low pressures was highlighted, where experimental data are missing in the literature.
