Stress State During High-Speed Cutting of a Ductile Metal

V. A. Govorukhin, G. L. Kufarev

(Presented by the scientific seminar of the Department of “Machine Tools and Metal Cutting”)

G. D. Delem and M. G. Goldschmidt [1, 2, 3] proposed a method for determining all components of the stress tensor from the distortion of a rectangular grid and the distribution of hardness in the chip-formation zone. They used this method in deciphering the stress state during cutting at microspeed.

The stress state at a given point of a continuous medium is characterized by the stress tensor, consisting, in the plane stress state, of the components:

\[ \sigma_x=\sigma-K\cdot\sin 2\Theta,\qquad \sigma_y=\sigma+K\cdot\sin 2\Theta, \]

\[ \tau_{xy}=K\cdot\cos 2\Theta \tag{1} \]

\[ K=\frac{\sigma_i}{\sqrt{3}} \quad \text{— intensity of shear stresses,} \]

\[ \sigma_i \quad \text{— stress intensity,} \]

\[ \Theta \quad \text{— angle between the tangent to the slip line} \]
\[ \text{and the chosen direction, the } x \text{-axis,} \]

\[ \sigma=-K(A_\alpha+A_\beta) \quad \text{— hydrostatic pressure (2),} \]

\[ A_\alpha,\ A_\beta \quad \text{— generalized parameters.} \]

Under complex loading, the best confirmation has been obtained by the relations of the flow theory [4], from which it follows that

\[ \tan 2\Theta=\frac{\dot e_x-\dot e_y}{\dot\gamma_{xy}}, \tag{3} \]

where \(\dot e_x,\ \dot e_y,\ \dot\gamma_{xy}\) are the components of the strain-rate tensor.

The stress state was determined near the conventional shear plane during free cutting at a speed \(V=98.0\) m/min of steel ShKh15. Cutting of a composite specimen, consisting of two polished disks, each 3 mm wide, was carried out on a lathe with transverse feed \(S=0.31\) mm/rev, using a cutter with a T5K10 hard-alloy insert having a rake angle \(\gamma=17^\circ\). On the side surface of the specimen, a coordinate grid was applied in a polar coordinate system with an angular base of 5 minutes and, along the radius, 0.05 mm. The grid was applied with the aid of a simple diamond device on a lathe (Archimedean spiral) and a universal microscope UIM-21 (radial rays). The cutting process was fixed with the aid of a specially made explosive-action device with electro-

by a contact device that made it possible to stop the cutting process at any previously specified point on the circumference of the disk, which made it possible to apply a dividing grid over a small section of the disk surface.

The fixed chip roots were photographed on the PMT-3 instrument; then photographs of the roots with the distorted dividing grid were made at a total magnification of 250–270 times. The dimensions of the grid were measured on a picture of the grid distortion drawn on the reverse side of the photograph. On the PMT-3 instrument, after appropriate surface preparation, the distribution of microhardness (Fig. 1a) in the chip root was measured. From the microhardness distribution, using the calibration graph “stress intensity—microhardness \(H_{200}\),” the distribution of the intensity of tangential stresses \(K\) was determined. The calibration graph was constructed from the results of axial compression tests with lubrication and subsequent measurement of microhardness in the meridional section of cylindrical specimens upset to different degrees of deformation. In this case the stress intensity was calculated from

\[ \sigma_i=\frac{4P\cdot h}{\pi\cdot d_0^2\cdot h_0}, \]

where \(P\) is the compressive force in kg;
\(h_0, h\) are the height of the specimen before and after deformation;
\(d_0\) is the initial diameter of the specimen, and the strain intensity is

\[ e_i=\ln\frac{h_0}{h}. \]

Figures 1b and 1c present the experimental data for calculating the components of the strain-rate tensor, which are determined from the relations

\[ \dot e_x=\frac{V_0}{a_0}\left(\frac{\partial a}{\partial x}\cdot\cos\alpha-a\cdot\sin\alpha\cdot\frac{\partial\alpha}{\partial x}\right), \]

\[ \dot e_y=\frac{V_0}{a_0}\left(\frac{\partial a}{\partial y}\cdot\sin\alpha+a\cdot\cos\alpha\cdot\frac{\partial\alpha}{\partial y}\right), \tag{4} \]

\[ \dot\gamma_{xy}=\frac{V_0}{a_0}\left[\sin\alpha\left(\frac{\partial a}{\partial x}-a\cdot\frac{\partial\alpha}{\partial y}\right)+\cos\alpha\left(\frac{\partial a}{\partial y}+a\cdot\frac{\partial\alpha}{\partial x}\right)\right], \]

where \(V_0\) is the velocity of displacement of the cutter;
\(a_0\) is the grid base;
\(a\) is the distance between neighboring nodes of the dividing grid along the trajectory of particles during deformation;
\(\alpha\) is the angle between the tangent to the trajectory of the particles and the \(x\)-axis, measured in the positive direction.

The field lines \(a\) (Fig. 1c) are drawn through cells of the dividing grid that have an equal distance between the middle grid nodes along the trajectory of particles during deformation. The lines \(\alpha=\mathrm{const}\) (Fig. 1b) were constructed graphically with the aid of tangents drawn to the trajectories of particles during deformation.

To determine the stress state in the vicinity of the conditional shear plane (Fig. 2), the latter is divided into a number of separate points \(M, 1, 2, 3\), etc., and at these points the values of the partial derivatives \(\partial a/\partial x\), \(\partial\alpha/\partial x\), \(\partial a/\partial y\), \(\partial\alpha/\partial y\) are calculated. For this it is necessary to construct graphs of the variation of the values \(a\) and \(\alpha\) along the axes \(x\) and \(y\). The tangent of the angle of inclination of the tangen-

Fig. 1.

a) Scale: 0.1 mm.

b) Axes: \(x\), \(y\); angle \(\alpha^\circ\).

c) Axes: \(x\), \(y\).

tial at the point under consideration, to the corresponding curve, determines the value of the partial derivative.

The generalized parameters \(A_\alpha\) and \(A_\beta\) are determined as follows. Parallel to the line \(OM\) (Fig. 2), at a sufficiently small distance from it, the line \(O'M'\) is drawn. Along it, as well as along the line \(OM\), the values of the angle \(\Theta\) are determined. Then parallel straight lines are drawn between these lines, on which the values of \(\Theta\) are determined by linear averaging.

Fig. 2

At the point \(M'\) the quantities \(A_\alpha\) and \(A_\beta\) are found as on a free boundary, from the equilibrium conditions

\[ \sigma=-K,\qquad A_\alpha=0.5+\Theta,\qquad A_\beta=0.5-\Theta . \]

From this point, at an angle \(\dfrac{\pi}{4}\) to the boundary, a segment of a slip line of the family \(\alpha\) is drawn. If the angle \(\Theta_1\) differs substantially from the angle \(\Theta_{M'}\), the position of point 1 is refined by drawing from point \(M'\) a ray at an angle \(\dfrac{\Theta' + \Theta_{M'}}{2}\) to the \(x\)-axis. At point 1

\[ (A_\alpha)_1=(A_\alpha)_M-\frac{K_b-K_a}{2K_1\,l_{a-b}}\,l_{M'-1} -\frac{(\Theta_{M'}+\Theta_1)(K_M-K_1)}{2K_1}, \]

where \(l_{M'-1}\), \(l_{a-b}\) are the lengths of the segments \(M'-1\) and \(a-b\). The segment \(ab\) is drawn through the midpoint of the segment \(M'-1\).

\(K_b\), \(K_a\), \(K_1\) are the intensities of the shear stresses, respectively, at the points \(b\), \(a\), 1.

The quantity \((A_\beta)_1\) is calculated on the basis of the known \((A_\alpha)_1\) and \(\Theta_1\), from the relation \((A_\beta)_1=(A_\alpha)_1-2\Theta_1\).

Next, from point 1, at an angle \(\dfrac{\Theta_1+\Theta_2}{2}\) to the \(y\)-axis, a segment \(1-2\) of a slip line \(\beta\) is drawn. At point 2

\[ (A_\beta)_2=(A_\beta)_1-\frac{K_c-K_d}{2K_2\,l_{c-d}}\,l_{1-2} +\frac{(\Theta_1+\Theta_2)(K_1-K_2)}{2K_2}, \]

\[ (A_\alpha)_2=2\Theta_2-(A_\beta)_2 . \]

From point 2 a segment of the slip line \(\alpha\), etc., is drawn. Knowing the values \(A_x\) and \(A_z\) at various points of the shear line, one can calculate the value of the hydrostatic pressure at these points from equation (2), and from relations (1) determine the values of the components of the stress tensor in the \(xy\) system. To determine the stresses in a system of axes, one of which is oriented along the conditional shear line, it is sufficient in relations (1) to replace \(\Theta\) by the angle \(\Theta'=\Theta-\varnothing\).

Fig. 3

Figure 3 shows the stress diagram, obtained by the method described, along the conditional shear plane \(OM\). Owing to the use of a dividing grid with a small base, it was possible to obtain the stress diagram near the tool tip. It follows from Fig. 3 that the shear stresses along the conditional shear plane are distributed practically uniformly. The hydrostatic pressure, and consequently also the stresses \(\sigma_{y'}\) normal to the conditional shear plane, remain compressive and increase in absolute value when moving from the free boundary toward the tool tip.

The stress diagrams obtained were checked by comparing the mean values of \(\tau_{x'y'}\) and \(\sigma_{y'}\) with the mean values of the tangential and normal stresses calculated from the cutting forces. The calculated value of the shear stress proved to be 15.8% lower than the actual value, while the normal stress was 1.3% higher.

The underestimated value of the calculated shear stress may be explained by the fact that deformation ends beyond the conditional shear plane. This leads to the stress in the chip exceeding the stress in the conditional shear plane.

As follows from Fig. 2, the conditional shear plane \(OM\) is not the plane of maximum shear stresses, since the slip lines of the \(\alpha\) family (\(M\)—1, 2—3, 4—5, etc.) do not coincide with the plane \(OM\). At the same time it should be noted that the conclusion obtained must be approached with caution, since it was obtained on the basis of a calculation

of a single root. To resolve definitively the question of whether the conditional shear plane is the plane of maximum shear stresses, additional experiments must be carried out.

REFERENCES

1. G. D. Del’. Investigation of plastic deformation by hardness measurement. Izvestiya TPI, vol. 138, 1965.

2. M. G. Goldschmidt. Investigation of the stress-strain state in the cutting zone. Dissertation, Tomsk, 1966.

3. G. D. Del’, M. G. Goldschmidt. Determination of the stress state in the cutting zone. Izvestiya TPI, vol. 162, 1967.

4. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, 1961.