## 2 Energy-Based Wear Model and Apparent Friction Energy Density  
Friction energy causes wear effects on a tribocontact (on the contacting surfaces). In the energy-based wear model, [PERSON] [22] defined the term \"apparent friction energy density, \(e_{R}^{*}\),\" in order to describe the relationship between friction and wear. The apparent friction energy density (\(e_{R}^{*}\)) represents the amount of friction work (\(W_{R}\)) in relation to the wear volume (\(V_{V}\)):  
\[e_{R}^{*}=\frac{W_{R}}{V_{V}}. \tag{1}\]  
Assuming that the friction shear stress (\(\tau_{R}\)) can be determined from the friction coefficient (\(f\)) and the nominal contact pressure (\(P_{a}\)):  
\[\tau_{R}=f\cdot P_{a}, \tag{2}\]  
the linear wear intensity (\(I_{h}\)) [33] according to [PERSON]'s fundamental equation of wear can be obtained using:  
\[I_{h}=\frac{\tau_{R}}{e_{R}^{*}}. \tag{3}\]In general, the expended energy consists of several components; therefore, there are also several components of wear [22]. Therefore, the apparent energy density can be extended to:  
\[\frac{1}{e_{R}^{*}}=\frac{\alpha_{1}}{e_{R1}^{*}}+\frac{\alpha_{2}}{e_{R2}^{*}}+ \frac{\alpha_{3}}{e_{R3}^{*}} \tag{4}\]  
by considering the applied lubricant as a friction body. The parameters \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) represent the energy proportion factors, which represent the distribution of the friction energy to the respective friction bodies in a tribological system [34]. If the apparent friction energy density is considered in the steady-state phase of solid-state wear, Equation (1) can be rewritten as follows:  
\[e_{R}^{*}=\frac{W_{R}}{V_{V}}=\frac{F_{R}\cdot s_{R}}{A_{R}\cdot h_{\
u}}= \tau_{R}\cdot\frac{1}{I_{h}}=\frac{\tau_{R}}{I_{h}}, \tag{5}\]  
where \(F_{R}\) is the friction force, \(s_{R}\) is the friction distance, \(A_{R}\) is the frictional contact area (wear area), and \(h_{\
u}\) is the wear depth. [PERSON]'s energetic wear model was based on the so-called hypothesis of energy storage [22; 23]. This hypothesis considered that the formation of a wear particle would occur only after repeated contacting since the critical energy level is usually not exceeded by the energy impulse of contact [22]. According to the energy accumulation hypothesis, a part of the energy of each impulse would be stored in a tribologically stressed material [22]. When the accumulated energy reached the critical energy value, a wear particle would form, i.e., particle separation from the contacting surface would occur and the accumulated energy would dissipate, indicating that the energy of a single contact (\(W_{R_{\
u}}\)) in the friction process partly accumulates and partly dissipates [22]:  
\[W_{R_{e}}=W_{store_{e}}+W_{Diss_{e}}. \tag{6}\]  
[PERSON] et al. [22] described the critical energy level required for the formation of the wear particle by identifying the average fracture energy density with the application of a critical number of contacts (\(n_{k}\)):  
\[\overline{e_{B}}=\zeta_{R}\cdot e_{R_{e}}\cdot(n_{k}-1)+e_{R_{e}}, \tag{7}\]  
where \(\zeta_{R}\) is the energy accumulation number and \(e_{R_{e}}\) is the energy density of a single contact. The previously given equation of the apparent frictional energy density (Equation (1)) can be rewritten [22] as:  
\[e_{R}^{*}=\frac{n_{k}}{\upsilon_{V}}\cdot\frac{\overline{e_{B}}}{1+\zeta_{R} \cdot(n_{k}-1)} \tag{8}\]  
by introducing the wear number (\(\upsilon_{V}\)):  
\[\upsilon_{V}=\frac{V_{V}}{V_{R}}, \tag{9}\]  
where \(V_{R}\) is the frictional volume. Figure 1 shows the graph of the basic energy equation of wear and allows for visualization of the relationship between the apparent energy density (\(e_{R}^{*}\)), the intensity of wear (\(I_{h}\)), and the friction shear stress (\(\tau_{R}\)). [PERSON] [23] divided this plot into five different friction and wear states (areas 0-4) according to the occurring mechanisms, as shown in Figure 1 and Table 1. Range 1 is hydrodynamic lubrication, which leads to negligible wear intensities and zero wear in fluid friction.