# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 3 Results and Discussion on Minimum Undeformed Chip Thickness  
Cutting material thicker than the minimum undeformed chip thickness and any width greater than 2 \(h_{min}\) results in chip formation. Depending on the deformation rate and material properties, the material is either carried forward of the cutting edge as a chip or displaced laterally. The minimum undeformed chip thickness is a critical parameter affecting both chip formation and surface generation. Using the methodology in [31], a finite element plate model was developed and defined for the nominal cross-section of the cutting layer, then subjected to a compressive load perpendicular to its plane. The propagation of the pressure in individual segments of the cross-section allows the determination of the chip flow directions. As the uncut chip thickness decreases and approaches \(h_{min}\), a noticeable stagnation zone of material begins to form in front of the cutting edge. This zone increases in width with a decreasing chip thickness, indicating that a larger portion of the material undergoes elastic or plastic deformation rather than being sheared off as a chip. This phenomenon significantly affects the surface integrity and efficiency of the cutting process, particularly in precision and micro-machining operations. As shown in Figure 2, for a given width of the cutting layer, a decrease in its thickness leads to reduced material removal efficiency. When the cutting layer thickness is too small for chip formation, the tool merely rubs against the workpiece due to insufficient penetration. Very thin cutting layers also limit the material's ability to flow laterally toward the free surfaces, leading to the formation of a stagnation zone beneath the cutting edge. In this zone, the material undergoes plastic deformation. Depending on the ratio of the cutting layer thickness to the cutting edge radius, the affected (burnished) zone may range from negligible to covering the entire contact length of the cutting edge. Figure 1: Data flow algorithm for the acquisition and analysis of cutting force components. With an increasing cutting layer thickness, plastic deformation gradually transitions from burnishing to ploughing, causing lateral flow of the workpiece material. Only after exceeding the minimum undeformed chip thickness does the machining mechanism shift from ploughing to shearing. At this stage, the effects of friction and ploughing become negligible, and the material is efficiently removed as chips through a shearing action. Under machining conditions with a nose radius of 0.4 mm and \(h_{min}\approx 10\) um, cutting below \(h_{min}\) results in an intermittent, free-form cutting process, where chip formation only occurs once \(h_{min}\) is exceeded. It is also possible to estimate the stagnation zone. The smallest model-based stagnation angle (\(\sim\)33\({}^{\circ}\)) is slightly lower than the theoretically maximum possible value (\(\sim\)36\({}^{\circ}\)), due to a \(\sim\)1.5\({}^{\circ}\) flow reduction on each free side. This 1.5\({}^{\circ}\) difference corresponds to the angular equivalent of the minimum undeformed chip thickness. Based on this observation, the depth of cut in the orthogonal cutting tests was selected to ensure effective chip formation without excessive stagnation. Figure 3 presents the algorithm used to determine the minimum undeformed chip thickness. During the tests, cutting force components were recorded at a constant cutting temperature while varying the cutting layer thickness from 25 to 100 um. This approach enabled the analysis of force variations as a function of chip cross-sectional area and facilitated the determination of linear regression coefficients. These coefficients were then used to estimate the critical point where the machining mechanism shifts from ploughing to shearing, thereby identifying the minimum undeformed chip thickness. A linear relationship was observed between the cutting force \(F_{c}\) and \(F_{f}\) and the cutting layer thickness \(h\) at a constant cutting temperature.