## 2 Fundamentals of the APofRCI Method, Taking into Account Torus Milling Cutter Cutting Blade Wear
### The Torus Milling Cutter Wear and Life Model  
The wear rate of a torus milling cutter reflects the wear rate at any point on the active cutting edge, which is a part in common with the torus action surface and depends on the inclination angle. Importantly, it changes during machining. Figure 6 shows a typical tool wear curve as a function of the cutting time at a certain constant cutting speed. It presents a classic three-phase course. During phase I--the initial phase--tool wear increases dramatically and reaches its baseline value in a short period of time. In phase II, wear and tear of a quasi-linear and difficult-to-predict nature can be observed. Phase III is characterized by accelerated wear until a critical value associated with blade failure is reached. Furthermore, for wear modeling (dotted green line), i.e., using regression analysis [38; 39; 40], phase I can be neglected, and the tool wear at any cutting point on the torus action surface can be considered as a variable over time (such as in accelerated motion) and described by the model of an exponential function, which is expressed by Equation (10). \[VB_{Bmax}=b_{0}\cdot b_{1}^{t_{c}}, \tag{10}\]  
where \(VB_{Bmax}\)--maximum abrasion width, an important indicator in multi-axis finish milling; \(b_{0}\) and \(b_{1}\)--coefficients determined for a given material pair, i.e., round cutting insert--workpiece, and machining conditions; \(t_{c}\)--cutting time. The \(b_{0}\) and \(b_{1}\) coefficients are constituted on the basis of machinability tests and non-linear estimation analysis. The tool is used for machining with the declared fixed machining parameters until the tool wear reaches the adopted critical value \(VB_{Blim}\) at time \(T_{t}\). The value of time \(T_{t}\), i.e., the tool life, is expressed by Equation (11):  
\[T_{t}=C_{t}\cdot v_{c}^{p}\cdot f^{q}, \tag{11}\]  
where \(T_{t}\)--the tool life; \(v_{c}\)--the cutting speed at the selecting cutting point; and \(f\) is the feed rate. \(C_{t}\), \(p\), and \(q\) are certain constants for a given material pair, i.e., the material of the round cutting insert material of the workpiece. These constants are constituted on the basis of calibrated machinability tests. On this basis, the tool life is determined from which the \(C_{t}\), \(p\), and \(q\) factors are calculated. When milling with a torus milling cutter, with the spindle speed set to an arbitrary and constant value, the cutting speed varies along the cutting edge from the cutter tip point (TCP) towards the tool attachment point in the spindle. The cutting speed \(v_{c,g}\) at a point with height \(g\) from the apex of the TCP torus milling cutter is expressed by Equation (12):  
\[v_{c,g}=\frac{2\pi n\Big{(}R_{T}+\sqrt{2r_{p}g-g^{2}}\Big{)}}{1000}, \tag{12}\]  
where \(n\) is the spindle speed; \(R_{T}\) is the radius from the rotational axis of the tool to the center point of the round cutting insert; and \(r_{p}\) is the radius of the round cutting insert. As shown in Figure 7, the cutting speed varies significantly given the parameter \(g\) along the torus milling cutter axis. The cutting speed values \(v_{c,g}\) in this case were simulated for a cutting angle in the range \(0^{\circ}\leq\psi_{cri}\leq 23.4411^{\circ}\) in the tool base plane coinciding with the tool's feed speed plane \(v_{f}\). Figure 6: Typical tool wear rate at a constant cutting speed. Note that, assuming that both feed rate and the spindle speed are constant, the cutting speed \(v_{c,g}\) depends only on the latitude \(g\) of the adopted cutting point, e.g., the contact point \(CP_{i}\) that falls within the length of the active cutting edge segment \(b\). The latitude \(g\) is described in Equation (13). \[g=r_{p}\bigg{[}1-\cos\bigg{(}\frac{\pi}{2}-\sin^{-1}\bigg{(}1-\frac{a_{p}}{r_{p }}\bigg{)}\bigg{)}+\delta\bigg{]}, \tag{13}\]  
where \(r_{p}\) is the radius of the round cutting insert; \(a_{p}\) is the depth of cut; and \(\delta\) is the inclination angle.