## 2 Modeling the Frictional Energy of the Microstructure-Dominated Chip-Contact Interface
### Theoretical Modeling of the Weaving Structure of the Rear Blade Surface  
During milling, plastic deformation occurs on the frictional surfaces of the tool, storing strain energy in deformed regions. As the original surface wears away, new surfaces formed during subsequent machining generate additional deformation. These accumulated strain energies are predominantly converted into heat during the milling process [12]. During milling, the front and back cutting faces exhibit similar frictional interactions with the workpiece, but the back cutting face demonstrates two distinct friction regimes:one characterized by tight-contact dominance, as shown in the AB segment deviating from Coulomb friction theory, and another partially obeying classical sliding friction laws where frictional force correlates linearly with normal pressure as shown in the BD segment presented in Figure 5 illustrating back cutting face microstructures [13]. Assuming the relative sliding velocity between the back cutting face and the machined surface over the entire contact zone equals the shear velocity \(v_{s}\) of the workpiece, the friction coefficient \(\mu_{(x)}\) on the back cutting face can be expressed as Equation (16), where \(a\), \(c\), and \(k\) are undetermined parameters satisfying \(0<k<1\). As shown in Figure 5, Point A represents the intersection of the back cutting face and the cutting edge arc, where the tangential contact stress is zero. The total length of the back cutting face friction zone \(AD\) is denoted as \(l_{AD}\). \[\mu(x)=\left\{ \begin{array}{ll}\frac{\alpha e^{-CV_{s}}}{\sigma_{(x)}^{2}} &0\leq x\leq B\\ \mu_{0}&B\leq x\leq D\end{array} \right. \tag{16}\]  
In the contact zone \(AB\) between the back cutting face and the machined workpiece, normal stress induces plastic strain during tool-workpiece contact, achieving tight contact. Embedding microstructures in zone \(AB\) effectively reduces frictional interaction. The normal contact stress \(\sigma_{(x)}\) in the friction region of the back surface of the tool is distributed in an exponential form as shown in Figure 5; in the contact region near the cutting edge of the tool, the material will flow plastically and the normal contact stress of the tool will change drastically, which can be expressed as Equation (17). \[\sigma_{(x)}=\sigma_{max}\Big{(}1-\frac{x}{l}\Big{)}^{\xi} \tag{17}\]  
where \(\xi\) is the stress distribution coefficient of the tool surface, \(\sigma_{max}\) is the maximum positive stress at the cutting edge of the tool, and \(x\) is the distance from any point in the friction region to point A.  
\[\tau_{(x)}=\mu_{(x)}.\sigma_{(x)} \tag{18}\]  
The sectional distance of the friction area \(AB\) is \(l_{AB}\), the sectional distance of the friction area \(BD\) is \(l_{BD}\), the total friction length is \(l_{AD}\), the friction force in the close contact area between the rear blade surface of the micro-weaving tool and the workpiece is \(F_{fa}\), and the normal force in the total friction area of the rear blade surface of the tool is \(F_{fa}\). The  
Figure 5: Back cutting face microstructures during milling operations (A,B,D are milling positions). relationship between the two and the tangential contact stress and the normal contact stress is represented by Equations (19) and (20). \[F_{fa}=\int_{A}^{l_{AB}}\sigma_{(x)}\cdot\mu_{(x)}=a\cdot e^{-CVs}\cdot\sigma_{ max}^{1-k}\cdot\frac{l_{AB}}{-\xi k+\xi+1} \tag{19}\]  
\[F_{f\sigma}=\int_{O}^{l_{AD}}\sigma_{max}\cdot\left(1-\frac{x}{l_{AD}}\right)^ {\xi}\,dx=\sigma_{max}\cdot\left(\frac{l_{AD}}{\xi+1}\right) \tag{20}\]  
The expression for the friction energy \(U_{C}\) on the back face was calculated as (21), employing the identical energy model for the chip-contact interface on the front face. \[U_{c}=\left(F_{fa}+F_{f\sigma}\times\mu_{0}\right)\cdot V_{s} \tag{21}\]  
Analysis based on Equation (19) indicates that the frictional force on the back cutting face is proportional to the contact length \(l_{AB}\) in the tight-contact zone.