# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 4 Results and Discussion on Forces and \(h_{min}\) in Oblique Cutting  
In oblique cutting, the chip formation mechanism is inherently more complex due to the three-dimensional nature of material flow and the inclination angle between the cutting edge and the direction of motion. This configuration introduces additional components of force and deformation, which significantly affect both the value and variability of the minimum undeformed chip thickness. To evaluate this variability, a detailed statistical analysis was conducted based on experimental data obtained under controlled cutting conditions. During constrained oblique cutting, cutting force measurements were performed at a constant cutting temperature, focusing on turning operations with a small uncut chip thickness. Example measurement data for cutting force components for a feed rate of \(f=~{}0.018\) mm/rev and a depth of cut \(a_{p}=0.3\) mm is presented in Figure 6.  
Figure 7 shows the cutting force \(F\) over time. After an initial rise during tool engagement, the force stabilizes at about 43 N, indicating steady-state cutting. This stable behavior confirms proper chip formation without clogging or tool issues, ensuring consistent material removal under the given cutting conditions. The average amplitude of the cutting force signal DC was determined after isolating the conditions corresponding to steady-state chip formation, when the cutting process had stabilized. The recording of force components  
Figure 5: The variability of the coefficient \(h_{min0}\) in relation to \(h_{min}\): (**a**) Ti Grade 5, (**b**) Ti Grade 2.  
began prior to cutting, capturing the tool's entry into the material and the onset of material removal, followed by continuous monitoring throughout the cutting process. Figure 8 shows the cutting force components as a function of the feed rate. The cutting force components depend linearly on the feed rate for higher values. As the feed rate decreases below a dozen or so micrometers, this relationship ceases to be linear. This is particularly evident in the case of Ti Grade 2. The nonlinear dependence of the cutting force components on the cross-section of the cutting layer at a small cross-section of the cutting layer influences the dynamics of the cutting process and thus the surface created. Figure 8: Cutting force values as a function of uncut chip cross-section (feed): (**a**) Ti Grade 5, (**b**) Ti Grade 2. Figure 6: Cutting force components during machining of Ti Grade 5 at a feed rate of 18 μm/rev. : (**a**) main cutting force \(Fc\), (**b**) back cutting force \(Fp\), (**c**) feed cutting force \(Ff\). Figure 7: Cutting force during machining of Ti Grade 5 at a feed rate of 18 μm/rev. : (**a**) initiation of the process (A—no contact, B—contact between the tool and the workpiece), (**b**) stable process. The basic model for surface roughness in turning is the kinematic-geometric model (\(f^{2}/(8r_{\epsilon})\)) [67], which serves as a theoretical foundation for more advanced models [68; 69; 70; 71] that include scale effects and additional factors to better reflect real machining conditions. These more advanced models rely on the concept of minimum undeformed chip thickness \(h_{min}\). Estimating \(h_{min}\) from simplified material and geometric assumptions, supported by experiments, helps capture the stochastic nature of chip formation. The value of \(h_{min}\) depends strongly on machining conditions, tool geometry, and material properties. Figure 9 shows the \(h_{min}\) values determined, taking into account \(h_{min}\) for Equation (2) and the experimentally determined \(r_{n}\). The \(h_{min}\) values at individual points of the experiment were not constant, but covered a certain range of values due to the variability of the \(h_{min0}\) coefficient and the uncertainty of the measurement of the cutting edge rounding radius \(r_{n}\). In the analysis of the cross-section of the machined layer, the decohesion phenomenon occurs within a region representing only a fraction of the cutting edge radius \(r_{n}\).