## 3 Surface Roughness Modeling Based on Residual Height
### Surface Roughness under Flank Wear Condition
In addition, \(a\), \(b\), \(c\), \(d\), \(g\), \(e\) in the substitution formulas \(V_{1}\) and \(V_{2}\) of (12) are the parameters representing the material properties of the tool and the workpiece, which can be obtained by reference [34; 35], \(n_{0}\) is a categorical variable, and its value depends on the change rule of tool and workpiece relative hardness with temperature. In addition to geometric and kinematic factors, wear \(VB\) and roughness \(R_{a}\) are also affected by material properties and processing test environment. Model \(VB\)-\(t\) (11) mainly relies on the correction and selection of the above two types of parameters to reflect the influence of tool and workpiece material properties on wear, and based on the wear study, \(R_{a}\)-\(t\) has also introduced the effect of material properties. Through the correction of experimental data, the errors of the machining system and the measuring system are eliminated by the error correction parameters \(K_{0}\) and \(C_{0}\). The correction and selection of the above parameters are completed in the previous wear study [11], which experimental conditions are the same as in this stud. So, the specific parameter values are directly given in Section 5.1. The experiment and simulation involved cutting stress \(\overline{\sigma}=2165\) MPa and temperature \(T_{0}=475\)\({}^{\circ}\)C are selected according to the numerical value of the experiment stable cutting. \(V_{c}\) represents the cutting speed that can be determined by calculation and \(\alpha\), \(\beta\), \(\gamma\), and \(R\) in (12) represents the geometric parameters rake angle, helix angle, flank angle and radius of the tool as shown in Figure 6. The surface roughness determined from this model can directly be applied to and describe the service performance of the tool. Additionally, the performance attenuation law of the tool during service can be predicted with this model and contributes to the control of the tool service performance. \[R_{a}=\frac{K}{\omega}\cdot[H(\theta_{1})-H(0)-\theta_{1}\cdot h(\theta_{1}) ]+C \tag{9}\]  
\[\omega=\varphi+\Delta\varphi \tag{10}\]  
\[t=K_{0}\cdot(\frac{2\cdot V_{2}^{2}\cdot\ln(V_{1}\cdot\sqrt{VB}+V_{2})}{P\cdot [R_{M}\cdot V_{1}^{2}-A\cdot V_{2}^{2}]\cdot V_{1}}-\frac{R(x_{2})\cdot V_{1} \cdot\ln[R_{M}-A\cdot VB]}{P\cdot[R_{M}\cdot V_{1}^{2}-A\cdot V_{2}^{2}]\cdot A }-\frac{2\cdot R_{M}\cdot V_{2}\cdot\arctan h(\sqrt{\frac{A\cdot VB}{R_{M}} })}{P\cdot[R_{M}\cdot V_{1}^{2}-A\cdot V_{2}^{2}]\cdot\sqrt{R_{M}\cdot A}})+ C_{0} \tag{11}\]  
In this research, \(VB\) represents the maximum wear width of the flank wear band, which changes as the milling depth and the part shape change. \(\alpha\) is the flank angle, \(\gamma\) is the rake angle, \(a_{p}\) is the milling thickness, \(\beta\) is the helix angle, and \(V_{c}\) is the integral median of the linear velocity on the cutting edge involved in milling. \(V_{c}\) in (11) is only used for the average temperature estimation in the tool-workpiece contact area of the diffusion wear process as the temperature distribution law on the cutting edge is not the focus of this research. Taking the integral median of the velocity distribution to represent \(V_{c}\) based on the integral mean value theorem is a necessary and effective way to reduce the difficulty of subsequent integral calculation within the correctness and precision. and \(V_{1}\), \(V_{2}\), and \(A\) are given by the following equations. \[\left\{