# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 3 Results and Discussion on Minimum Undeformed Chip Thickness
Assuming linear cutting force curves, the slope coefficients \(A_{c}\) and \(A_{f}\) were calculated, which allow for determining the critical angle \(\psi_{crit}\) according to Equation (1) [24]. \[\psi_{crit}=acfg\frac{A_{f}}{A_{c}} \tag{1}\]  
Figure 2: Visualization of the force direction in the cross-section of the cut layer. The angular coordinate of the point on the cutting edge rounding, where the uncut chip thickness equals the nominal minimum thickness \(h_{min}\), defines the conditions favorable for chip formation. The ratio in Equation (1) ranges between 0 and 1. The lower and upper curves define the boundary region of angle \(\psi_{crit}\) variability for this ratio range. The upper curve, at a ratio of \(\mu=0\), assumes only normal force is present; the lower curve corresponds to equal tangential and normal forces. The highlighted area in Figure 4 indicates the range of variability in the experimental data. The most probable values of the minimum undeformed chip thickness, corresponding to specific values of the cutting edge radius, are derived from the variability range of the critical angle. A probability distribution analysis was performed for the previously calculated minimum undeformed chip thickness as a function of the cutting edge radius. This analysis takes into account the actual standard deviation values, reflecting the real variability observed in the measurements and enhancing the reliability of the estimated \(h_{min}\). Figure 4: Influence of the \(A_{f}/A_{c}\) ratio on variations in the angle value \(\psi_{crit}\). Figure 3: Algorithm for determining the minimum undeformed chip thickness based on the ratio of directional coefficients of the cutting force components. By transforming the equation for the minimum thickness of the cutting layer presented in [22] and substituting \(h_{min0}=1-\frac{A_{f}/A_{c}}{\sqrt{1+\left(A_{f}/A_{c}\right)^{2}}}\), Formula (2) for determining the minimum thickness of the undeformed chip takes the form of a function depending on the radius of the cutting edge rounding \(r_{n}\). \[h_{min}=\left(1-\frac{A_{f}/A_{c}}{\sqrt{1+\left(A_{f}/A_{c}\right)^{2}}} \right)r_{n}=h_{min0}r_{n} \tag{2}\]  
The value of the coefficient \(h_{min0}\) was determined based on the conducted experiment, and the obtained repetitions allowed for the evaluation of its variability. The variability of the coefficient \(h_{min0}\) in relation to \(h_{min}\) reflects the sensitivity of the minimum undeformed chip thickness to changes in cutting edge geometry and cutting conditions. This coefficient represents a dimensionless ratio between the minimum undeformed chip thickness and the cutting edge radius, and is often used to generalize \(h_{min}\) across different tool geometries or materials. Experimental results indicate that \(h_{min0}\) varies within a certain range depending on material properties. Figure 5 shows that although the mean values of the coefficient \(h_{min0}\) are very similar--approximately 0.313--the scatter is significantly smaller in the case of machining the Ti Grade 2 titanium alloy.