DETERMINATION OF THE STRESS STATE IN THE CUTTING ZONE

G. D. Del, M. G. Goldshmidt

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

The method of investigating the stress state in the plastic region by measuring hardness [1] made it possible to determine stresses in a considerable part of the cutting zone during the formation of a shear chip [2]. The application of this method to the analysis of continuous-chip formation does not make it possible to determine the stresses in the main deformation zone even in the case where the boundary conditions are known.

In connection with this, a new procedure has been developed that makes it possible to obtain the stress state in the cutting zone during the formation of a continuous chip. This procedure can also be used in the study of other technological problems in the theory of plasticity.

The process of formation of a continuous chip is steady. As a result, lines that before deformation were parallel to the cutting plane coincide with the trajectories of displacement of the particles located on them relative to the cutter. The velocity of displacement of the particles is directed along the tangent to the trajectory. If marks 1, 2, 3, … are placed at equal distances on these lines, then in their motion relative to the cutter point 1 occupies the position of point 2, point 2 the position of point 3, etc., over one and the same time interval

\[ \Delta t=\frac{a_0}{v_0}, \tag{1} \]

where \(v_0\) is the velocity of displacement of the cutter, and \(a_0\) is the distance between the indicated points before deformation. The relative velocity of displacement of the particles is

\[ v=\frac{a}{\Delta t}=v_0\frac{a}{a_0}, \tag{2} \]

where \(a\) is the distance between the same points during deformation. The components of the velocity of displacement of the particles relative to the specimen are

\[ v_x=v\cos\alpha-v_0,\qquad v_y=v\sin\alpha . \tag{3} \]

Here \(\alpha\) is the angle between the tangent to the particle trajectory and the \(x\)-axis, measured in the positive direction.

It is best to determine the velocity field from the distortion of a square dividing grid, placed so that one family of lines is oriented parallel to the cutting plane. In this case \(a_0\) is the grid base, and \(a\) is the distance between neighboring nodes along the particle trajectory during deformation.

Components of the strain-rate tensor [3]:

\[ \begin{aligned} \dot{\varepsilon}_x &=\frac{\partial v_x}{\partial x} =\frac{v_0}{a_0}\left(\frac{\partial a}{\partial x}\cos\alpha -a\sin\alpha\,\frac{\partial \alpha}{\partial x}\right),\\ \dot{\varepsilon}_y &=\frac{\partial v_y}{\partial y} =\frac{v_0}{a_0}\left(\frac{\partial a}{\partial y}\sin\alpha +a\cos\alpha\,\frac{\partial \alpha}{\partial y}\right),\\ \dot{\gamma}_{xy} &=\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x} =\frac{v_0}{a_0}\left[ \sin\alpha\left(\frac{\partial a}{\partial x}-a\,\frac{\partial \alpha}{\partial y}\right) +\cos\alpha\left(\frac{\partial a}{\partial y}+a\,\frac{\partial \alpha}{\partial x}\right) \right]. \end{aligned} \tag{4} \]

As is known, under complex loading the flow-theory relations have received the best experimental confirmation [3]. According to these relations, the components of the stress deviator are related to the components of the strain rate:

\[ s_x=\frac{2}{3}\frac{\sigma_i}{\dot{\varepsilon}_i}\dot{\varepsilon}_x,\qquad s_y=\frac{2}{3}\frac{\sigma_i}{\dot{\varepsilon}_i}\dot{\varepsilon}_y,\qquad \tau_{xy}=\frac{1}{3}\frac{\sigma_i}{\dot{\varepsilon}_i}\dot{\gamma}_{xy}, \tag{5} \]

where \(\sigma_i\) is the stress intensity, which can be determined experimentally by hardness measurement. The strain-rate intensity is

\[ \dot{\varepsilon}_i=\frac{\sqrt{2}}{3} \sqrt{(\dot{\varepsilon}_x-\dot{\varepsilon}_y)^2+\dot{\varepsilon}_x^2+\dot{\varepsilon}_y^2+\frac{3}{2}\dot{\gamma}_{xy}^{\,2}}. \tag{6} \]

As is evident from relations (5), the constant multiplier \(\dfrac{v_0}{a_0}\) in relations (4) cancels; therefore, in what follows it is expedient to pass to simplified strain-rate characteristics:

\[ \begin{aligned} \bar{\varepsilon}_x &=\frac{\partial a}{\partial x}\cos\alpha -a\sin\alpha\,\frac{\partial \alpha}{\partial x},\\ \bar{\varepsilon}_y &=\frac{\partial a}{\partial y}\sin\alpha +a\cos\alpha\,\frac{\partial \alpha}{\partial y},\\ \bar{\gamma}_{xy} &=\sin\alpha\left(\frac{\partial a}{\partial x}-a\,\frac{\partial \alpha}{\partial y}\right) +\cos\alpha\left(\frac{\partial a}{\partial y}+a\,\frac{\partial \alpha}{\partial x}\right). \end{aligned} \tag{7} \]

In this case expressions (5) and (6) retain their previous form \((\dot{\varepsilon}_x,\ \dot{\varepsilon}_y\) and \(\dot{\gamma}_{xy}\) are replaced respectively by \(\bar{\varepsilon}_x,\ \bar{\varepsilon}_y,\ \bar{\gamma}_{xy})\).

The hydrostatic pressure \(\sigma\) can be determined by integrating the differential equilibrium equations

\[ \frac{\partial \sigma_x}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}=0,\qquad \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \sigma_y}{\partial y}=0. \tag{8} \]

The stress state was determined in the chip-formation zone during free cutting at a speed \(v_0=23.5\ \text{m/min}\) of L-62 brass (the proposed method may also prove acceptable in studying the stress state under conditions of higher cutting speeds). The cut thickness was \(0.97\ \text{mm}\), and the rake angle of the tool was \(+30^\circ\). Carbon tetrachloride was used as the lubricant. The specimen was assembled from two plates; on the side surface of one of them, after copper plating, a square grid with a base of \(0.1\ \text{mm}\) was scratched. The grid was applied by means of a simple

the fixture on a toolmaker’s microscope and was placed in the midsection of the specimen, whose total width was 3.5 mm. The fixed chip root was photographed; then the grid was drawn on a projector at a total magnification of 70×. On a PMT-3 instrument, after appropriate preparation of the surface, the distribution of microhardness in the chip root was measured. From the distribution of microhardness, using the calibration graph “intensity of shear stresses \(\kappa\)—microhardness \(H_{200}\),” the distribution of \(\kappa\) was determined. The calibration graph was constructed from the results of tests in axial compression with lubrication and subsequent measurement of microhardness in the meridional section of cylindrical specimens upset to various degrees of deformation.

Figure 1 presents the experimental data for the calculation: the distribution of the intensity of shear stresses (Fig. 3, a) and the fields \(a\) (Fig. 1, b) and \(\alpha\) (Fig. 1, c) obtained by measuring the elements of the deformed grid. The lines \(\alpha=\mathrm{const}\) (Fig. 1, c) were constructed using a fan of straight lines drawn at various angles to the \(x\)-axis.

According to the method described above, the epure of shear stresses along the line \(CM\), hereinafter conventionally called the shear line, was determined. For this purpose the line \(CM\) was divided into a number of segments by the points \(M, 1, 2, 3\), etc. (Fig. 1, c). Through these points straight lines parallel to the coordinate axes were drawn, along which graphs of the variation of \(a\) and \(\alpha\) were constructed. The values of the partial derivatives \(\dfrac{\partial a}{\partial x}\), \(\dfrac{\partial a}{\partial y}\),

\[ \frac{\partial a}{\partial x},\quad \frac{\partial a}{\partial y}, \]

entering expression (7), were determined as the tangent of the angle of inclination of the tangent at the point under consideration to the corresponding curve. From relations (7) the components of the strain-rate tensor were determined, and the angle \(\Theta\) was calculated from the formula

\[ \tg 2\Theta=\frac{\bar{\varepsilon}_{y}-\bar{\varepsilon}_{x}}{\dot{\gamma}_{xy}}, \tag{9} \]

obtained under the assumption, in flow theory, of coaxiality of the stress tensors and strain-rate tensors. The shear stresses may be calculated from the relation

\[ \tau=\kappa\cos 2\Theta. \tag{10} \]

The shear stresses acting along the line \(OM\) are obtained with allowance for the inclination of this line at the angle \(\beta_1\) to the \(x\)-axis:

\[ \tau=\kappa\cos 2(\Theta+\beta_1-180^\circ). \tag{11} \]

It was not possible to determine the normal stresses by the method described above: for approximate integration of the differential equilibrium equations it is necessary to have a grid of higher accuracy. In studying the cutting process, satisfying this requirement is difficult because of the comparatively small dimensions of the base of the dividing grid. The normal stresses were determined in the following way. The epure of shear stresses along the shear line, together with the free boundary, was used as a boundary condition. By the procedures described in [1], the Cauchy problem was solved in the region \(AMB\) (Fig. 2); \(MBDC\) was a mixed problem, and \(CDFE\) an initial problem. In the course of the solution, a grid of lines was constructed—

Fig. 1. Experimental data for calculating stresses in the cutting zone:
a — lines \(\kappa = \mathrm{const}\), b — lines \(a = \mathrm{const}\), c — lines \(\alpha = \mathrm{const}\).

slip. Beginning at point \(E\), the angle between the slip lines of family \(\alpha\) and the line \(OM\) exceeds \(90^\circ\); therefore the solution of the mixed problem in the zone \(EOF\) is impossible. The dimensions of such a zone, determined in the present example by the position of point \(E\), depend on the conditions of the cutting process and on the orientation of the line along which the stresses are determined. In those cases where the undetermined—

Fig. 2. Stress state in the cutting zone

Fig. 3. Determination of stresses along the shear line

—zone is small, it is expedient to carry out the calculation by the method described, since as a result of the decoding the stresses are determined throughout the entire zone of plastic deformation below the line \(OM\).

For the case when the segment \(OE\) proves to be significant, the following procedure is proposed. Parallel to the line \(OM\), at a sufficiently short distance from it, the line \(O'M'\) is drawn (Fig. 3). 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, and the values of \(\Theta\) on them are determined by linear averaging. At point \(M\), the quantities \(A_\alpha\) and \(A_\beta\) [1] are found as on a free boundary:

\[ A_\alpha = 0.5+\Theta,\qquad A_\beta = 0.5-\Theta . \tag{12} \]

From this point, at an angle of \(45^\circ\) 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 \(\Theta_M\), then the position of point 1 is refined by drawing from point \(M\) a ray at an angle

\[ \frac{\Theta_1+\Theta_M}{2} \]

to the \(x\)-axis. At point 1:

\[ (A_\alpha)_1 = (A_\alpha)_M -\frac{1}{2\kappa_1}\frac{\kappa_b-\kappa_a}{l_{a-b}}\,l_{M-1} -\frac{(\Theta_M+\Theta_1)(\kappa_M-\kappa_1)}{2\kappa_1}, \tag{13} \]

where \(l_{M-1}\) and \(l_{a-b}\) are the lengths of the segments \(M-1\) and \(a-b\) (the latter is drawn through the midpoint of the segment \(M-1\)). The quantity \((A_\beta)_1\) is calculated by the formula

\[ (A_\beta)_1 = (A_\alpha)_1 - 2\Theta_1 . \tag{14} \]

Next, from point 1, at an angle

\[ \frac{\Theta_1+\Theta_2}{2} \]

to the \(y\)-axis, the segment \(1-2\) of the slip line \(\beta\) is drawn. At point 2:

\[ (A_\beta)_2 = (A_\beta)_1 -\frac{1}{2\kappa_2}\frac{\kappa_c-\kappa_d}{l_{c-d}}\,l_{1-2} +\frac{(\Theta_1+\Theta_2)(\kappa_1-\kappa_2)}{2\kappa_2}, \tag{15} \]

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

From point 2, a segment of the slip line \(\alpha\) is drawn, and so on. Knowing the quantities \(A_\alpha\) and \(A_\beta\) at various points of the shear line, one can calculate the magnitude of the hydrostatic pressure at these points:

\[ \sigma = -\kappa(A_\alpha + A_\beta) \tag{16} \]

and, from the relations

\[ \sigma_x=\sigma-\kappa\sin 2\Theta,\qquad \sigma_y=\sigma+\kappa\sin 2\Theta,\qquad \tau=\kappa\cos 2\Theta \tag{17} \]

determine the stress values in the \(xy\) system. To determine the stresses in a system of axes one of which is oriented along the shear line, it is sufficient in expressions (17) to replace \(\Theta\) by the angle \(\Theta+\beta-180^\circ\). By the described procedure, in the region near the tool tip, segments of the slip lines \(\alpha\) and \(\beta\) adjoining the shear line were constructed; by solving the initial characteristic problem, a network of slip lines was constructed in the region \(ELNF\) (Fig. 2). The resulting network of slip lines in the cutting zone confirms the scheme of plane chip formation of N. N. Zorev [4].

Figure 2 gives the stress diagrams along the shear plane. The tangential stresses are distributed practically uniformly. The hydrostatic pressure, and consequently the stress \(\sigma_y'\) normal to the shear line, changes sign when moving from the free boundary to the tool tip, becoming positive (tensile). This corresponds to the results of analytical studies.

Oxley [5] and a number of other authors. It proved impossible to determine the stresses near the tool tip because the pattern of the deformed grid was not sufficiently clear. Determination of the stresses in this region is also made difficult by a whole series of other reasons (a considerably higher strain gradient, the great sensitivity of the stress state to the geometry of the tip, etc.). Point O is singular; the hydrostatic pressure in it changes discontinuously. The study of all these questions should apparently be a task for further investigations.

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. In doing this, the stress diagrams were extrapolated to the tool tip. The calculated values of the quantities proved to be lower than the actual ones by 6 and 15%, respectively.

The discrepancy in the mean stresses is partly explained by the fact that, when recalculating the dynamometric results, the forces on the rear face of the tool were not separated out.

Having the diagram of $\sigma_{y'}$, it is possible, by the method described in article [6], to construct the diagram of normal stresses on the front face of the tool.

REFERENCES

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

2. M. G. Goldshmidt, G. D. Del, G. L. Kufarev. The stress state during chip formation by cutting. Izvestiya TPI, vol. 139, 1966.

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

4. N. N. Zorev. Problems in the Mechanics of the Metal-Cutting Process. Mashgiz, 1956.

5. Oxley. Mechanisms of metal cutting with a variable yield stress. Design and Technology of Machine Building, vol. 85, series B, No. 4, IL, 1963.

6. N. N. Zorev. On the interdependence of processes in the chip-formation zone and in the contact zone of the front surface of the tool. Vestnik Mashinostroeniya, No. 12, 1963.