On the Contact Pressure Near the Cutting Edge of a Tool
Abstract Generated abstract
This paper examines the difficulty of determining contact pressure on a cutting tool rake face in the small region adjacent to the cutting edge, where existing photoelastic and split-tool methods lack sufficient resolution. It reviews a prior slip-line based estimate that assumes through slip lines in the cutting zone, then argues from earlier photoelastic evidence that such lines are generally absent and that the resulting pressures should be treated as lower bounds. A contrasting slip-line model is developed for a localized plastic region near the cutting edge, especially relevant to cutting low-ductility metals with brittle chip separation. The analysis indicates that contact pressure near the cutting edge may be 3 to 4 times greater in the localized plastic case than under the through-slip-line assumption, defining broad theoretical limits for this pressure.
Full Text
ON THE ISSUE OF CONTACT PRESSURE NEAR THE CUTTING EDGE OF A TOOL
M. F. POLETIKA
The known experimental methods for studying contact pressure on the rake face of a tool (the photoelastic method and the split-tool method) make it possible, with greater or lesser reliability, to record the regularity of the distribution of contact pressures over almost the entire length of contact between the chip and the tool, with the exception of a small region adjacent to the cutting edge. For determining the contact pressure within this region, the resolving power of the named methods proves insufficient. This compels one to seek ways of theoretically determining the contact pressure near the cutting edge.
One such method was proposed by T. N. Loladze [1]. Using slip-line theory and assuming the existence of “through” slip lines beginning at the tool tip, crossing the cutting zone, and emerging at its free surface, T. N. Loladze obtained the following relation for the contact pressure \(p_0\) in the vicinity of the cutting edge:
\[ 2k(1.3-\gamma)>p_0>2k(1-\gamma),\ldots \tag{1} \]
where
\[ \begin{aligned} k&\text{ — plasticity constant;}\\ \gamma&\text{ — rake angle of the tool.} \end{aligned} \]
The upper limit in formula (1) corresponds to the case when there is no sliding on the rake face; the lower limit corresponds to the case when the coefficient of friction on the rake face is zero.
Earlier photoelastic investigations of the stressed state in the workpiece [2, 3] showed that “through” slip lines are absent in the cutting zone. This means that expression (1) gives underestimated values of the contact pressure. Assuming that in particular cases “through” slip lines may nevertheless exist, we shall regard the values of \(p_0\) found from formula (1) as the minimum possible ones.
We shall obtain the maximum possible values of the contact pressure in the case where the plastic region is localized in the vicinity of the cutting edge and is separated from the free surface of the cutting zone by an elastically stressed region. The appearance of such a localized plastic region is possible when cutting low-ductility metals, when plastic deformation in the primarily deformable
in the volume is absent, and chip separation occurs by brittle fracture.
Figure 1 shows a scheme of slip lines in the localized plastic region for the case when the coefficient of friction on the rake face is equal to zero. For any of the slip lines shown in the scheme, the relation holds:
Fig. 1.
\[ \sigma_A=\sigma_B+2k\alpha_{AB},\ldots \tag{2} \]
where
\(\sigma_A;\ \sigma_B\) are the mean normal stresses at points A and B;
\(\alpha_{AB}\) is the angle of rotation of the slip line.
Since point B is located on a free surface, at it \(\sigma_B=k\). It is significant that, in contrast to the analogous case of initial plastic flow considered in the specialized literature [4], the stress \(\sigma_B\), as shown by experimental investigation [2], will be positive. According to the scheme, \(\alpha_{AB}=\pi+\gamma\). Consequently, from equation (2) we obtain:
\[ p_0=2k(\pi+\gamma)=2k(3.14+\gamma). \tag{3} \]
In the absence of slip on the rake face, we obtain
\[ \alpha_{AB}=\frac{3\pi}{4}+\gamma, \]
and after substitution into (2):
\[ p_0=2k\left(-\frac{1}{2}+\frac{3}{4}\pi+\gamma\right)=2k(1.85+\gamma). \tag{4} \]
For a numerical comparison of the contact pressures calculated by formula (1) and by formula (2), it is convenient to transform the latter to the form:
\[ (1.3-\gamma)>\frac{p_0}{2k}>(1-\gamma), \tag{1'} \]
\[ (3.14+\gamma)>\frac{p_0}{2k}>(1.85+\gamma). \tag{5} \]
Comparison of expressions (1) and (2) shows that the contact pressure in the vicinity of the cutting edge, in the case of the existence of a localized plastic region, will be 3–4 times greater than in the presence of through slip lines. In the general case, the magnitude of this pressure will lie within rather wide limits established by expressions (1) and (5).
REFERENCES
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T. N. Loladze, Wear of Cutting Tools. Moscow, Mashgiz, 1958.
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M. F. Poletika, M. Kh. Uteshev. Investigation of the cutting process by the polarization-optical method. Izv. TPI, vol. 114, 1964.
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M. F. Poletika. On the problem of the stressed state in the shear zone during cutting. Proceedings of the Irkutsk Polytechnic Institute, issue 36, 1967.
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A. D. Tomlenov. Mechanics of Metal-Forming Processes. Moscow, Mashgiz, 1963.