## 2 Modeling the Frictional Energy of the Microstructure-Dominated Chip-Contact Interface  
### Theoretical Modeling of the Weaving Structure of the Front Blade Surface  
A microstructured ball-end milling cutter with planar-edge microtextures was selected for modeling. To simplify the analysis of the cutting process, the energy generated during metal chip formation was categorized into two types, shear energy in the first deformation  
Figure 1: Carp scales.  
Figure 2: Three-dimensional modeling diagram.  
zone and frictional energy in the second and third deformation zones, as illustrated in Figure 3. According to the functional mechanism of microstructures, their friction reduction and wear resistance primarily manifest in the second and third deformation zones.  
The cutting energy \(U\) per unit of time is divided into two main parts, the shear energy per unit of time \(Us\) and the friction energy per unit of time \(U_{f}\)[11], as shown in Equations (1)-(3).  
\[U=U_{s}+U_{f} \tag{1}\]  
\[U_{s}=F_{s}\times V_{s} \tag{2}\]  
\[U_{f}=F_{f}\times V_{c} \tag{3}\]  
where \(Fs\) is the shear force; \(Vs\) is the shear velocity; \(F_{f}\) is the friction force; \(Vc\) is the chip velocity. The microwaving cutting model of the front surface of the blade is shown in Figure 4.  
The shear force \(Fs\) in Figure 4 can be expressed as follows:  
\[F_{s}=\tau_{s}A_{s} \tag{4}\]  
where \(\tau_{S}\) is the shear strength of the workpiece; \(A_{s}\) is the shear area.  
According to the micro-weave cutting model of the front blade face in Figure 2, the relationship between the shear force \(F_{s}\), friction force \(F_{f}\), and the combined force \(F\) is shown in Equations (5)-(7).  
\[F_{s}=F\cos(\varphi+\beta-\gamma_{0}) \tag{5}\]  
\[F_{f}=Fsin\beta \tag{6}\]  
Figure 4: Microtexture cutting model of rake face.  
Figure 3: Cutting deformation zone.  
\[F=\sqrt{F_{s}^{2}+F_{f}^{2}} \tag{7}\]  
where \(\beta\) is the friction angle; \(\gamma_{o}\) is the precession angle Equation; \(\varphi\) is the shear angle.  
The combination of Equations (4)-(6) results in \(F_{f}\):  
\[F_{f}=\tau_{s}A_{s}\bigg{[}\frac{sin\beta}{cos(\varphi+\beta-\gamma_{0})} \bigg{]} \tag{8}\]  
The effective frictional area between the tool and chip is set as \(Q_{p}\), and the relationship between \(Q_{p}\) and \(F_{f}\) is given by Equation (9).  
\[F_{f}=\tau_{s}Q_{p} \tag{9}\]  
From Equations (8) and (9), the expression for \(Q_{p}\) is changed to Equation (10).  
\[Q_{p}=A_{s}\frac{sin\beta}{cos(\varphi+\beta-\gamma_{0})} \tag{10}\]  
The shear velocity \(V_{s}\) and chip velocity \(V_{c}\) are related to the cutting speed \(V\), lead angle, and shear angle, as shown in Equation (11) and Equation (12), respectively.  
\[V_{s}=\frac{Vcos\alpha}{cos(\varphi-\gamma_{0})} \tag{11}\]  
\[V_{c}=\frac{Vsin\varphi}{cos(\varphi-\gamma_{0})} \tag{12}\]  
Based on Equation (3) and Equation (9), the expression for the friction energy at the front face during cutting is as follows:  
\[U_{f}=\tau_{s}Q_{p}V_{c} \tag{13}\]  
The microtexture is placed in the chip contact area of the ball-end mill, and the effective friction area of the actual contact after the weaving is placed is called \(Q_{p}^{1}\).  
\[Q_{p}^{1}=Q_{p}-nS \tag{14}\]  
where \(S\) is the area of a single microtome, \(Q_{p}\) is the effective friction area, and \(n\) is the number of microtomes.  
In summary, the energy of new friction between the front face of the microtexture configuration ball-end mill and the chip is as follows:  
\[U_{f}=\tau_{s}Q_{p}^{1}V_{c} \tag{15}\]  
Equations (14) and (15) show that incorporating the microtexture structure on the face of the ball-end mill reduces the effective frictional surface between the tool and the chip, thus reducing the frictional energy to a certain extent.