RESISTANCE TO DEFORMATION IN METAL CUTTING

G. L. Kufarev, V. A. Govorukhin

(Presented by the scientific seminar of the departments of machine tools and metal cutting and of mechanical-engineering technology)

In [1] it is shown that, in metal cutting, each particle of the material being machined, upon entering the chip-formation zone, is deformed at every point of this zone under a different temperature–velocity regime. This circumstance must leave an imprint on the flow curve $\sigma_i - e_i$ during cutting.

In order to calculate the cutting forces, it is necessary to know, first of all, the maximum tangential stress acting on the upper boundary of the chip-formation zone. This is possible only if the actual flow curve during cutting and the degree of deformation of the material in the chip are known. Determination of the flow curve in the cutting process is possible and is shown in [2], but it is laborious and painstaking work. A second possible way of determining the maximum tangential stress in cutting is to carry out mechanical tests of the material being machined under specified temperature–velocity deformation regimes. However, this way is also laborious because of the countless number of experiments required in mechanical testing, and because at present it is impossible to carry out mechanical tests at such deformation rates as accompany the cutting process. In principle, this way is promising in the future and, evidently, will find wide application with certain assumptions. One of the assumptions of this solution path is the transfer of relationships obtained under other deformation schemes to the cutting process.

In the theory of metal forming under pressure, at different temperatures and deformation rates, the following relationships have found wide application [3, 4]:

\[ \sigma_V = \sigma_0 + c \cdot \ln \frac{\dot e_i}{\dot e_{i_0}}, \tag{1} \]

\[ \sigma_t = \sigma'_0 \cdot e^{-a \cdot T}. \tag{2} \]

Relationship (1) is at present considered the most acceptable for incomplete cold deformation.

$\sigma_V$ is the resistance to deformation at the given deformation rate $\dot e_i$.

\(\sigma_0\) — resistance to deformation at the strain rate \(\dot e_{i0}\),

\(\sigma_t\) — resistance to deformation at the deformation temperature \(T^\circ\mathrm{K}\).

\(a\) — temperature coefficient, depending on the type of material and the strain rate.

\(c\) — rate coefficient.

\(\sigma'_0\) — resistance to deformation extrapolated to \(0^\circ\mathrm{K}\).

\(e\) — base of natural logarithms.

If it is further assumed that the plasticity modulus, when the strain rate changes within the range of deformation temperatures occurring in cutting, remains constant, then the stresses in the cutting process can be determined by the formula

\[ \sigma_i=\sigma_0 \cdot e^{-aT}+c\cdot \ln \frac{\dot e_i}{\dot e_{i0}}, \tag{3} \]

where \(\sigma_0\) is the resistance to deformation at the given degree of deformation under conditions of static compression at room temperature, determined from the flow curve, which is well approximated by the equation

\[ \sigma_0=A\cdot e_i^k, \]

where \(e_i\) is the intensity of deformation,

\(A, k\) are constant coefficients depending on the material.

One of the possible ways of solving equation (3) is the variant described below.

Let us assume that the influence of strain rate on the stress during cutting can be assessed by tensile or compression tests at the same level of loading rate. By loading rate is meant the change in the applied stress per unit time.

We shall further assume that the influence of rate in the cutting process is determined by its maximum value in the chip-formation zone, since the degree of deformation also increases most rapidly at the moment when the strain rate is maximal.

Yu. Ya. Voloshenko-Klimovitskii [5], in impact tension of Armco iron and steels St. 45, St. 30KhGSA, St. 40Kh, St. ShKh15 at an impact velocity of \(8.7\ \mathrm{m/sec}\) (the loading rate was constant in the tests and amounted to \(\sigma_i=3\cdot 10^6\ \mathrm{kg}/\mathrm{mm}^2\mathrm{sec}\)), obtained a decrease in the dynamic coefficient \(\eta\) with an increase in the static yield limit. By the dynamic coefficient is meant the ratio of the dynamic yield limit \(\sigma_d\) to the static \(\sigma_0\). It may be assumed that the metals used in the present study satisfy the dependence \(\eta=f(\sigma_0)\) obtained by Yu. Ya. Voloshenko-Klimovitskii. Mathematical processing of this dependence showed that it is very well described by the equation

\[ \sigma_d=27\cdot \sigma_0^{0.35}. \]

The loading rate in the cutting process was estimated from the increase in stress intensity per unit time. The stress intensity at each point was determined from the known distribution of microhardness in the plastic region and from a calibration graph relating stresses to hardness and obtained in mechanical compression tests of the material. The results of calculating the rate

the loading rates for steels St. 40, St. 2Kh13, St. ShKh15, and brass L62, machined under different cutting conditions, are presented in Fig. 1.

It turned out that the loading rate is almost independent of the physicomechanical properties of the material being machined, the feed, and the tool geometry, and is determined mainly by the cutting speed. It follows from Fig. 1 that when the cutting speed changes from 40 m/min to 250 m/min, the maximum loading rate in the chip-formation zone changes within the range from \(1 \cdot 10^5\ \mathrm{kg/mm^2\,sec}\) to \(7 \cdot 10^5\ \mathrm{kg/mm^2\,sec}\). Therefore it may be considered that, at cutting speeds used in practice, the loading rate on average has a value of \(4 \cdot 10^5\ \mathrm{kg/mm^2\,sec}\).

Fig. 1

The process of cutting metals is a process of plastic deformation, and it must obey the laws of plastic deformation established in other deformation processes, although the deformation scheme may introduce certain peculiarities into this dependence. If one disregards the influence of the temperature factor in the cutting process, then the relation \(\sigma_d = 27 \cdot \sigma_0^{0.35}\), obtained in dynamic tension, can be transferred to the cutting process at equal loading rates. It may then be assumed that in the metal-cutting process the dynamic yield strength must have the same value. The transfer of the dependence \(\sigma_d = f(\sigma_0)\), obtained for dynamic tension with a loading rate of \(3 \cdot 10^6\ \mathrm{kg/mm^2\,sec}\), to the cutting process with an average loading rate of \(4 \cdot 10^5\ \mathrm{kg/mm^2\,sec}\), is entirely justified, since works [6, 7] showed that a change in deformation rate within one order of magnitude has no substantial effect on the change in stresses. Consequently, no large error is introduced when this dependence is transferred to the cutting process.

Then the stresses in the cutting process, without taking account of the temperature correction, are determined from the expression:

\[ \sigma_p = \sigma_0 + \left(27 \cdot \sigma_{0.2}^{0.35} - \sigma_{0.2}\right)\ \mathrm{kg/mm^2}, \tag{4} \]

where

\(\sigma_0\) is the stress intensity in the process of static compression at the given degree of deformation.

\(\sigma_{0.2}\) is the conventional yield strength in static compression. It is determined from the flow curve in compression at \(e_i = 0.2\).

The expression \(\left(27 \cdot \sigma_{0.2}^{0.35} - \sigma_{0.2}\right)\) takes into account the increase in stresses at a loading rate of \(3 \cdot 10^6\ \mathrm{kg/mm^2\,sec}\).

To determine the correction to the stress caused by rest, a series of experiments was carried out on cutting materials differing in chemical composition—

by purpose and service properties—carbon steel (St. 10, St. 20, St. 60, St. U10A), structural alloy steel (St. 30KhGSA, St. 45Kh), corrosion-resistant heat-resistant steel (St. 2Kh13, St. 4Kh13, St. Kh12), and die chromium-vanadium steel (St. 3Kh2V8).

All the materials being machined were annealed to relieve internal stresses and were mechanically tested in static compression at room temperature to obtain the flow curve.

The stresses in the conditional shear plane were determined from the forces measured during cutting, and the degree of deformation from the chip shrinkage.

\[ \sigma_i=\sqrt{3}\,\frac{P_\tau\cdot\sin\Phi}{a\cdot b}, \tag{5} \]

\[ e_i=\frac{1}{\sqrt{3}}\,\frac{\xi^2-2\xi\cdot\sin\gamma+1}{\xi\cdot\cos\gamma}, \tag{6} \]

where

\(\Phi\)—the angle determining the position of the conditional shear plane, in degrees.

\(\gamma\)—the rake angle of the tool, in degrees.

\(P_\tau\)—the force acting along the conditional shear plane, in kg.

\(a, b\)—the thickness and width of the layer being cut, in mm.

\(\xi\)—chip shrinkage.

The calculation of stresses was carried out only for continuous chips in the range of cutting conditions where there was no built-up edge on the cutter.

The results of cutting and compression experiments, presented in Fig. 2 as an example for 5 grades of steels, provide ample opportunity for determining weakening in the cutting process. As follows from the graphs presented, the stresses in the cutting process \(\sigma_i\) are less than \(\sigma_p\) by the amount \(\Delta\sigma_t\), caused by recovery during deformation; consequently, the difference between \(\sigma_p\) and the actual stresses \(\sigma_i\) in the cutting process determines the value \(\Delta\sigma_t\)

\[ \Delta\sigma_t=f(T,e_i)=\sigma_p-\sigma_i. \tag{7} \]

Calculations carried out by formula (7) at the values of \(e_i\) occurring in the cutting process of each of the materials showed that the magnitude of the stresses \(\Delta\sigma_t\) is determined primarily by the degree of deformation and is well described by an equation of the form:

\[ \Delta\sigma_t=B\cdot e_i^{\,b}, \tag{8} \]

where

\(B, b\)—coefficients constant for each material.

Analysis of the obtained coefficients \(B\) and \(b\) showed that there is no linear relationship between them and the thermophysical characteristics. It turned out that such a relationship exists with the conditional yield strength \(\sigma_{0.2}\). Linear approximation of the values \(B\) and \(b\) by the least-squares method made it possible to obtain equations for determining their magnitude as a function of the conditional yield strength:

\[ B=80.4-0.44\,\sigma_{0.2}; \tag{9} \]

\[ b=0.0035\cdot\sigma_{0.2}. \tag{10} \]

Fig. 2

Then the equation for calculating the final stresses in the cutting process in the conditional shear plane, taking into account the temperature–speed regime, has the form

\[ \sigma_i=\sigma_0+\Delta\sigma_v-\Delta\sigma_t. \tag{11} \]

In order to calculate the stresses by formula (11), it is necessary to know:

1. The flow curve \(\sigma_0=A\cdot e_i^k\), obtained under static compression of specimens of the given material at room temperature.

2. The final strain intensity of the material in the chip.

Equation (11) is suitable only for materials that, during cutting, produce a continuous chip, and for a cutting process close to the plane-strain scheme.

Equation (11) was checked against the results of experiments by N. N. Zorev [8]. A comparison of the calculated and experimental values of the stress intensity in the conditional shear plane is presented in Fig. 3. In these graphs, through the experimental points, the flow curve under compression or tension has been drawn as a solid line. In ob-

Fig. 3

…region of high values of deformation \(e_i\); it is drawn with a dashed line according to the approximation of N. N. Zorev. The crosses on the graph denote the experimental values of \(\sigma_i\) and \(e_i\), determined in cutting experiments. The solid line here shows the results of calculating \(\sigma_i\) from equation (11).

Thus, the obtained equation (11) agrees satisfactorily with the experimental data for the cutting of 18 materials of widely differing characteristics and may be recommended for practical use.

REFERENCES

1. V. A. Govorukhin, G. L. Kufarev. Temperature–velocity regime of deformation in metal cutting. Izv. TPI, vol. 224.

2. V. A. Govorukhin, G. L. Kufarev. On the coincidence of the flow curve in cutting and compression processes. Izv. TPI, vol. 209, in press.

3. N. P. Ageev, S. I. Karatushin. Mechanical Testing of Metals at High Temperatures and Short-Time Loading. Metallurgiya, Moscow, 1968.

4. I. E. Tarnovskii, A. A. Pozdeev et al. Mechanical Properties of Steel during Hot Pressure Working. Metallurgizdat, Sverdlovsk, 1960.

5. Yu. Ya. Voloshenko-Klimovitskii. Dynamic Yield Limit. Nauka, Moscow, 1965.

6. L. M. Shestopalov. Deformation of Metals and Plasticity Waves in Them. Academy of Sciences of the USSR, Moscow–Leningrad, 1958.

7. G. I. Pogodin-Alekseev. Dynamic Strength and Brittleness of Metals. Moscow, Mashinostroenie, 1966.

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