The Quadratic Curve and the Trajectory in the Shear Zone in Metal Cutting

The quadratic curve was investigated as an approximation of the streamline in the steady-state shear zone during metal cutting. Experimentally, a very fine grid of parallel lines (spacing 25 μ, depth 6 μ) was inscribed on the work face through a simple method which eliminates the problems associated with the photographic and etching techniques. Quick stops were taken using an explosive device for cutting speeds of 50, 150, and 250 m/min. Next, several quadratic curves were fitted to the experimental streamline, which was measured at six points along its path. The two tangents at the start and end of the shear zone were also measured. Of the three types (ellipse, hyperbola, and parabola) studied, the hyperbola and, specifically, the rectangular hyperbola (x² − y² = c²) gave the best fit with experiment. Trajectory deviation in these cases is <1 percent and the strain-rate deviation is <6 percent. These are in the order of magnitude of the experimental errors.