b. Contact law

Contact between two interfaces is generally modelled by a relationship between the interfacial gap and the contact pressure. The contact laws can be extracted analytically from various statistical models of rough surfaces. Drinkwater [11] and Baltazar et al. [32] attempted to link the roughness topography to the contact stiffness, transmission/reflexion coefficients, measured ultrasonically, and frequency.

However, it is generally difficult to take into account all the detailed information about third body [33] features, i.e., local deformations and interactions within a real interface and roughness [34,35].

As an alternative approach, the desired function can be modelled as a nonlinear contact stiffness, with respect to the applied contact pressure. When two rough surfaces are pressed together, the stiffness *K<sup>c</sup>* per unit of area of the interface is given by the rate of change in nominal pressure σ with the average interfacial gap. Contact stiffness K<sup>c</sup> can be determined in general from [9]

$$K\_{\mathbb{C}} = -\frac{\partial \sigma}{\partial \mathbf{u}}\tag{6}$$

where σ is the nominal contact pressure and u is the average interfacial gap.

In the literature, for higher contact pressures (greater than 0.14 MPa, which is the lowest value of pressure used for the experimental estimation of contact stiffness, see Table 2), contact stiffness is often assumed to follow a power law [15]. This latter model gives the relationshi between the contact stiffness and the applied pressure as follows

$$K\_{\mathbb{C}} = -\frac{\partial \sigma}{\partial \mathbf{u}} = -\mathsf{C}\sigma^m \tag{7}$$

where C and m are positive constants.

In this study, the contact stiffness of the tested samples was previously determined by experimental measurements, as reported in [18]. From preliminary dynamic tests at different contact pressures, the contact stiffness between 0.14 and 1 MPa could be estimated. Table 3 shows the results for contact stiffness as a function of the average contact pressure for the aluminum–aluminum interface, with a surface roughness of Ra = 1 µm.


**Table 3.** Normal contact stiffness as a function of the average contact pressure in sticking condition [15].

The data highlight how the contact stiffness increases with the rise in the average contact pressure. The contact stiffness values range from 1.15 × 1012 to 2.46 × 1012 Pa/m when the contact pressure increases from 0.14 to 1 MPa.

Even if a contact interface is expected to show hysteresis, with a slightly different stiffness during loading and unloading phases, the authors considered this effect negligible, or at least not measurable. The complexity of an interface, including third body particles, chemical bounds and oxides, leads to the need for an overall approximation, which is here represented by the stiffness parameter. Of course, such approximation can have different implications, as a function of the wished phenomena to be investigated.

These experimental results are useful for defining the numerical contact stiffness within the tested range of contact pressures. Nevertheless, for implementing the contact law in the numerical simulations, it is necessary to define the stiffness for the whole pressure range [0; 1 MPa], in which the contact pressure will vary.

A first approximation within this pressure range was obtained by approximating the experimental measurements by a power law function, as in Equation (3). The least squares method allows a good agreement with the experimental data (Table 3), which are available from pressure 0.14 MPa. This agreement is obtained for C = 1.81 × 10<sup>10</sup> Pa/m and m = 0.35. The power law ("PL" in Figure 3a) is thus defined to approximate the experimental points (Table 3).

**Figure 3.** Normal contact stiffness as a function of contact pressure in compression conditions. (**a**) Experimental results (red cross); Modified PL (blue triangle); PL (black circle). (**b**) Modified PL (blue triangle) and experimental data from the literature [6] (black square)

Table 3 While this power law has been built with consideration to the measurements and the literature dealing with higher pressures, the approximation of the trend at lower pressures is completely arbitrary. In order to model the contact stiffness trend for lower contact pressures, other experimental observations in the literature [6] have been exploited. These experimental results show the existence of an inflexion point for low-contact pressures (Figure 3b). From experimental results (Table 3) and the literature [6]

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

it is then possible to propose a different overall contact stiffness trend, as a function of the contact pressure, including the inflection point between 0.14 and 0 MPa ("Modified PL" in Figure 3.

Finally, the stiffness laws, which will be investigated in the following, are:


The relationship between the contact pressure and the interfacial gap can then be extracted from the implemented nonlinear relation between stiffness and pressure. For example, the results derived from the "Modified PL" are shown in Figure 4, highlighting the nonlinear response of the interface.

**Figure 4.** The corresponding contact pressure-interfacial gap trend for the 'Modified PL'.

In the following, the experimental nonlinear response of the system to an impulsive excitation force will be compared with the nonlinear response obtained by the simulations with the different laws analysed. The comparison will allow for a discussion of the different laws considered to simulate the effective interface stiffness nonlinearity.

## **3. Experimental and Numerical Comparison**
