DETERMINATION OF THE STRESS STATE IN THE CUTTING ZONE BY THE HARDNESS MEASUREMENT METHOD

V. I. KARNOV, G. D. DEL

(Presented by the scientific seminar of the Departments of Machine Tools and Metal Cutting and of Mechanical-Engineering Technology)

A number of works have been devoted to the study of the stress state arising in the process of metal cutting [1, 2]. In these works, the stresses during the formation of continuous chips are considered. Below are given the results of a study of the stress state in the cutting zone during the formation of shear chips. The study was carried out by the hardness measurement method [3], which, in the case of plane deformation, makes it possible to determine all components of the stress state [4]. Brass LS-59-1 was subjected to cutting; its mechanical properties allow it to be classified as a low-plasticity metal. The cutting of the brass was performed at microspeeds (tool travel speed \(V = 19\text{—}98\ \mathrm{mm/min}\)). Since the size of the plastic region in the case under consideration is very small (cutting depth \(t = 0.8\text{—}1.2\ \mathrm{mm}\)), and the gradient of the yield limit is very large, microhardness was measured on a PMT-3 instrument with a load on the indenter of 100 g.

The calibration curve relating hardness \(HV\) to the intensity of tangential stresses \(K\) was obtained from the results of testing the material under study in axial compression. Six specimens 12 mm in diameter and 24 mm high were compressed to various degrees of deformation. The compressed specimens were cut along a meridional section; on the resulting surface, after polishing, hardness was measured at five points. The average of these measurements was compared with the intensity of tangential stress.

\[ \kappa = \frac{4P}{\sqrt{3}\,\pi D^2}, \tag{1} \]

where \(P\) is the deforming force.
\(D\) is the diameter of the compressed specimen.

The stress state was determined for three cases: for the final stage of element formation at rake angles of \(30^\circ\) and \(-10^\circ\), and at 50% penetration of the tool with a rake angle of \(-10^\circ\). The interpretation was carried out according to the method set forth in article [4]. The stress components satisfy the equation:

\[ (\sigma_x - \sigma_y)^2 + 4\tau_{xy}^{\,2} = 4K^2(x,y). \tag{2} \]

In the interpretation, equation (2) is considered together with the equilibrium equations

\[ \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} = 0, \qquad \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_x}{\partial y} = 0. \tag{3} \]

The system of equations (2, 3) is solved by the slip-line method. The relations on these lines [4] are:

\[ \frac{\sigma}{2K}-\Theta=-A_{\alpha}, \qquad \frac{\sigma}{2K}+\Theta=-A_{\beta}, \tag{4} \]

where

\[ A_{\alpha}=\frac{1}{2K}\int_{\alpha}\frac{\partial K}{\partial \beta}\cdot d\alpha, \qquad A_{\beta}=\frac{1}{2K}\int_{\beta}\frac{\partial K}{\partial \alpha}\,d\beta, \tag{5} \]

here \(\sigma\) is the hydrostatic pressure,

\(\Theta\) is the angle between the tangent to the slip line \(\alpha\) and the \(x\)-axis,

\(\dfrac{\partial}{\partial \alpha}\) and \(\dfrac{\partial}{\partial \beta}\) are derivatives along the slip lines \(\alpha\) and \(\beta\). The stresses \(\sigma_x\), \(\sigma_y\), \(\tau_{xy}\) are related to \(\sigma\), \(K\), \(\Theta\) by the relations

\[ \sigma_x=\sigma-K\sin 2\Theta;\qquad \sigma_y=\sigma+K\sin 2\Theta;\qquad \tau_{xy}=K\cos 2\Theta. \tag{6} \]

The contour \(AB\) of the chip element (Fig. 1) is free. The distribution of stresses along the rake face of the cutter is unknown. Accordingly, in decoding, a Cauchy problem is solved in the region \(ABC\), and an initial characteristic problem in the region \(BCD\). The tangential stress at point \(B\) of the contour \(OB\) is taken to be zero.

Fig. 1. Scheme for determining stresses in the chip-formation zone.

The stress state in the region \(OBD\), adjacent to the rake face of the cutter, remains unknown. Along the boundary \(AB\) the following relations are valid:

\[ \begin{aligned} \sigma&=-K, & A_{\alpha}&=\Theta+0.5,\\ \Theta&=\varphi+45^\circ, & A_{\beta}&=0.5-\Theta, \end{aligned} \tag{7} \]

where $\vartheta$ is the angle between the tangent to the contour and the horizontal axis $x$.

Knowing $\Theta$, $A_\alpha$, $A_\beta$ along the boundary $AB$, we determine these same quantities in the next layer. Thus, for example, the position of point 3 is determined as follows. From points 1 and 2 rays are drawn at angles

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

and $\Theta_2$ to the $x$ axis. At the intersection of these rays we find point 3, at which

\[ \begin{aligned} (A_\alpha)_3 &= (A_\alpha)_2 - \frac{1}{2K_3}\cdot \frac{K_6 - K_7}{e_{6-7}}\cdot e_{2-3},\\ (A_\beta)_3 &= (A_\beta)_1 - \frac{1}{2K_3}\cdot \frac{K_4 - K_5}{e_{4-5}}\cdot e_{1-3}, \end{aligned} \qquad \Theta_3 = \frac{1}{2}(A_\alpha - A_\beta). \tag{8} \]

The indices attached to $A_\alpha$, $A_\beta$, and $\Theta$ indicate to which points these quantities refer. By $l_{2-3}$, $l_{1-3}\ldots$ are denoted the distances between the corresponding points. Segments 6—7 and 4—5 are perpendicular to segments 1—3, 2—3 and pass through their midpoints. The solution obtained was refined. To this end, from points 1 and 2 rays were drawn at angles $90^\circ + \dfrac{\Theta_1+\Theta_3}{2}$ and $\dfrac{\Theta_2+\Theta_3}{2}$ to the $x$ axis. At the intersection of these rays point $3'$ was found, for which the quantities $A_\alpha$, $A_\beta$, $\Theta$ were again determined. In this way the solution was constructed over the entire region $ABC$.

Point $B$ is a special point, since, when moving along the boundary, $\sigma$, $\Theta$, and $A_\alpha$ change discontinuously. Through this point passes a fan of the family of slip lines $\alpha$. The integral $A_\beta$ for all these lines at point $B$ is the same and is equal to the integral $A_\beta$ at point $B$ of the contour $AB$. In the absence of friction forces at point $B$, the slip line $DB$ is inclined at an angle of $45^\circ$ to the rake face of the cutter.

Knowing $A_\alpha$ and $A_\beta$ at any point of the plastically deformed zone, one can determine the hydrostatic pressure

\[ \sigma = K(-A_\alpha - A_\beta), \tag{9} \]

and consequently all the stresses from formulas (6).

Experiments at micro-speeds were carried out on a universal milling machine of the Tula plant. Previously ground plates 3 mm thick were subjected to cutting by a tool with various rake angles. When the cutting process was stopped instantaneously, chip roots were obtained at different stages of element formation. The hardness of the chip roots in the deformation zone was measured on a coordinate grid with a square side of 0.1—0.2 mm.

From the hardness measured in the cutting zone, using the calibration graph $HV - K$, lines of equal intensity of tangential stress were constructed. Figure 2 shows a grid of slip lines for one of the three cases investigated (for the final stage of element formation in cutting with a tool having rake angle $\gamma = -10^\circ$). As is seen from the figure presented, the extreme lines of the family $\beta$ converge when moving from the boundary to the cutter tip (the $\beta$ lines issuing from points 1.....5). This confirms the geometry of shear lines adopted in the theory of metal cutting (5).

It is of great theoretical and practical interest to determine the stresses in the shear plane.

In Fig. 2 the line $OO'$ is drawn parallel to the shear line. For small penetrations of the cutter, the angle between the line $OO'$ and the slip lines is close to $45^\circ$. As penetration of the cutting tool increases, these lines increasingly, in their direction, approach the lines

Fig. 2. Slip-line mesh in the zone of formation of an elemental chip.

In-figure labels: \(O_1\), \(O\), \(\beta_1 = 39^\circ\), \(\gamma = -10^\circ\).

sliding \(\beta\). However, even at the moment preceding fracture, their coincidence is far from complete.

Figure 3 presents stress diagrams along the line \(OO'\). These stresses are determined from formulas (6), in which, instead of the angle \(\Theta\), we substitute the angle \(\Theta-\alpha\) (Fig. 1).

Fig. 3. Diagrams of tangential and normal stresses along the shear line.

Analysis of the diagrams shows that, as the penetration of the cutter increases, the distribution of tangential stress becomes increasingly uniform; however, even at the moment preceding spalling of the element, the nonuniformity of these diagrams remains significant. The stresses \(\sigma_x\) are compressive. The normal stresses \(\sigma_y\) in all the cases studied have an identical character: at first, when moving from the boundary toward the cutter tip, these stresses are compressive; then they change sign. Subsequently, however, they again become compressive and increase sharply in the direction toward the cutter tip.

The calculation performed was checked by comparing the tangential stresses found from the dynamometry results during cutting with the calculated ones. According to the dynamometry results, the maximum value of the tangential stress in the shear plane when cutting brass is equal to \(26—27\ \mathrm{kg/mm^2}\) [6]. These stresses are nothing other than the average stress \(\tau_{xy}\) along the line \(OO'\) at the moment preceding the shear of the element.

Since the stresses were not determined over the entire region \(OO'\), it was necessary to confine ourselves to determining the average tangential stresses \(\tau_{xy}\) along the calculated segments of these lines. Comparison of these stresses with the maximum tangential stress during cutting seems possible, since one cannot expect an abrupt change in \(\tau_{xy}\) at the tool tip (owing to the inequality \(\tau_{xy} \leq K\)).

REFERENCES

1. R. Hill. Mathematical Theory of Plasticity. GITTL, 1956.

2. V. I. Sadchikov. The stressed and deformed state in cutting metals. Izv. vuzov. Fizika, No. 6, 1960.

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

4. G. D. Del. Investigation of plane deformation by measuring hardness. Izv. TPI, vol. 138, 1965.

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

6. Metal Cutting and Tools, edited by Doctor of Technical Sciences, Professor A. M. Rozenberg. Mashinostroenie Publishing House, Moscow, 1964.