# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 4 Results and Discussion on Forces and \(h_{min}\) in Oblique Cutting
A significant challenge in modeling is that the transition from the elastic stress at the cutting edge interaction to the volume of the removed chip does not occur along a single plane but rather within a defined volume of material. Although orthogonal cutting primarily involves two-dimensional material flow, minor lateral deformation also occurs. In turning, the constrained lateral flow on one side can increase surface roughness due to the asymmetric force distribution around the tool nose radius, which may lead to burr formation. This phenomenon is influenced by the minimum undeformed chip thickness [21; 72], material spring-back effects as observed in [73; 74], and plastic side flow [75; 76]. Additionally, surface integrity is affected by the strain rate; higher deformation speeds tend to reduce plastic flow and alter surface characteristics. As \(h_{min}\) increases, the stagnation point shifts, enlarging the angular portion of material that flows beneath the cutting edge (Figure 10). This simultaneous increase in both \(h_{min}\) and the angular flow fraction amplifies the influence of lateral material flow on surface formation and the development of surface roughness. As \(h\ \rightarrow\ h_{min}\), the material no longer flows freely into the chip; instead, it undergoes significant elastic-plastic deformation under the tool edge, increasing the normal stress on the tool rake and flank faces. Below \(h_{min}\), chip formation ceases, and specific forces rise sharply due to the dominance of ploughing and frictional effects. The cutting force components are then governed not only by the tool geometry and workpiece material properties but also by localized strain hardening and stagnation zones ahead of the tool. Figure 11 presents a visualization of force directions within the cross-section of the cut layer during constrained oblique cutting. The arrows indicate the direction and distribution of the resultant cutting forces. The highlighted region identifies the segment of the uncut  
Figure 9: Comparison of \(h_{min}\) values as a function of feed rate: (**a**) Ti Grade 5, (**b**) Ti Grade 2.  
chip thickness where material flow is restricted, leading to increased compression beneath the tool edge. This zone corresponds to the stagnation area, where the material neither forms a chip nor escapes laterally, thereby playing a critical role in chip formation and surface generation. Figure 11: Visualization of force direction in the cross-section of the cut layer during oblique cutting for \(h_{min}=6\)\(\upmu\)m/rev. The dashed lines represent the boundaries of the stagnation zone. Figure 10: Theoretical distribution of specific forces in the cross-section of the cut layer for a feed rate of 25 \(\upmu\)m and various \(h_{min}\). The dashed line indicates the theoretical separation of the cut layer. Due to the highly dynamic nature of material separation during the transformation of the allowance in the chip, the existence of a limited minimum thickness of the machined layer becomes apparent. As the cutting edge approaches the workpiece, a characteristic noise occurs before visible chips are formed. This noise indicates contact between the blade and the workpiece without actual chip formation--i.e., cutting in the region below the minimum undeformed chip thickness. This ratio \(f/h_{min}\) serves as a key indicator of chip formation efficiency and the prevailing cutting regime. When this ratio falls below approximately 1 for feed rates of 3 \(\upmu\)m and 6 \(\upmu\)m (Figure 12), the feed per tooth is smaller than the minimum thickness required to effectively shear the material. As a result, instead of producing a clean chip, the material tends to deform elastically or plastically beneath the cutting edge. This deformation leads to surface burnishing, where the material is rubbed or pressed rather than cut away, which can negatively impact surface finish quality.