CALCULATION OF STRESSES IN THE CUTTING PROCESS

V. A. Govorukhin, G. L. Kufarev

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

It is generally known that shear stresses during cutting at high speeds, as deformation increases, may increase, decrease, or remain unchanged. The decrease in stresses cannot be explained by the influence of anisotropy, since in cutting at microspeeds we obtain a continuous increase in stresses as the degree of deformation increases to large values (see Fig. 1 in the article “On the coincidence of the flow curve in cutting and compression processes” in this collection). The explanation for this should be sought in the change of the temperature–speed deformation regime in the cutting process.

To clarify the question of the influence of the temperature–speed regime on stresses in the cutting process, special experiments were carried out on free cutting of steels ShKh15 and 2Kh13 in the as-delivered and annealed states, steel 40, and brass L62. The cutting speed was varied from 5 to 250 m/min, the feed from 0.1 to 0.39 mm/rev, and the rake angles within the range \(\gamma = 0 \div 22^\circ\). The workpieces consisted of 2 paired disks, each of thickness \(b = 3\) mm; on the inner surface of one of the disks a coordinate measuring grid was applied in a polar coordinate system with a base of 0.05 mm along the radius and 5 minutes along the polar angle. The study of the cutting process was carried out on chip roots obtained with the aid of a special device. One of the chip roots was used to measure distortion of the coordinate measuring grid, the second for preparing a microsection and measuring microhardness over the chip-formation zone. Calculation of deformations was performed on the “Minsk-1” electronic digital computer according to a method developed for 19 chip roots obtained under various cutting conditions and on various materials. From the distribution of microhardness values in the chip-formation zone, lines of equal hardness values—hardness isoclers—were drawn.

Assuming that the cutting process is a steady-state process (continuous chip), and that the continuity of the material is not disrupted in the chip-formation zone, one can calculate the time during which each material point travels from one node of the grid to another along the flow line, according to the relation

\[ \Delta t = \frac{a_0}{V_0}, \]

where \(a_0\) is the base of the measuring grid, or the distance between two adjacent rays of the measuring grid at the given radius.

$v_0$ is the cutting speed.

Having the hardness field, a dividing grid, and the time $\Delta t$, one can calculate the loading rate $\dot{\sigma}$ in the given process over the chip-formation zone:

\[ \dot{\sigma}=\frac{\Delta\sigma_i}{\Delta t}\ \text{kg}/\text{mm}^3\cdot\text{s}, \]

where $\Delta\sigma_i$ is the increment in the stress intensity in a portion of the chip-formation zone.

The transition from microhardness to stress intensity was carried out on the basis of the independence of the relation between hardness and $\sigma_i$ from the type of stress state, and of the calibration curve $\sigma_i=f(H)$ obtained under static compression of each material.

As a result of the investigation it was established that the loading rate in the plastic zone is approximately constant. When the cutting speed is increased from the microspeed level to $10\ \text{m}/\text{min}$, the loading rate increases sharply to a value of the order of $1\cdot10^4\ \text{kg}/\text{mm}^2\,\text{s}$. With a further increase in cutting speed within the range $50\div250\ \text{m}/\text{min}$, the loading rate increases monotonically from $1\cdot10^5$ to $7\cdot10^5\ \text{kg}/\text{mm}^2\,\text{s}$. Yu. Ya. Voloshenko-Klimovitskii [1] established that, under impact tension, the ratio of the dynamic yield strength $\sigma_g$ to the static value $\sigma_0$ decreases as the strength of the material increases. Processing the experimental data of Yu. Ya. Voloshenko-Klimovitskii at a loading rate $\dot{\sigma}=3\cdot10^6\ \text{kg}/\text{mm}^2\,\text{s}$, which almost coincides with the loading rates in the cutting process, showed that they are well approximated by the equation $\sigma_g=27\cdot\sigma_0^{0.35}$.

Consequently, in the cutting process the dynamic yield strength should obey this dependence, since the cutting process is one type of plastic deformation. However, the experiments carried out on cutting the metals indicated above showed that the final stresses calculated from the measured forces in the cutting process and the known deformation, in most cases, decrease as the degree of deformation increases (Fig. 1). Obviously, this can be caused only by the influence of the temperature factor, which leads to a decrease in stresses during deformation. This is confirmed by the results of experiments by K. A. Nassonov [2] on cutting preheated EI1T steel at various temperatures. He found that, at the same preheating temperature, the tangential stresses in the cutting process decrease the more strongly, the higher the degree of deformation and the longer the time of action of this heat in the cutting zone on the material.

On this basis, taking into account the rate and temperature factors, the dependence for calculating stresses in the cutting process should have the form

\[ \sigma_p=B\cdot e_i^c+\left(27\cdot\sigma_{0.2}^{0.35}-\sigma_{0.2}\right)-\sigma_t, \]

where $B\cdot e_i^c=\sigma_i$ is the stress obtained in static tests at room temperature; $\left(27\cdot\sigma_{0.2}^{0.35}-\sigma_{0.2}\right)$ is the correction for the increase in stresses due to the rate factor; $\sigma_{0.2}$ is the conventional yield strength in compression; $\sigma_t$ is the amount of stress reduction due to the influence of temperature.

If this is so, then $\sigma_t$ can be calculated from the results of cutting experiments if the compression flow curve under room-temperature conditions is known.

\[ \sigma_t=\sigma_p-B\cdot e_i^c-\left(27\cdot\sigma_{0.2}^{0.35}-\sigma_{0.2}\right). \]

In a similar manner, all the results of our tests and the experiments of [3] were processed. It turned out that, for one and the same material, the temperature correction is the larger, the greater the degree of deformation. For each material the temperature correction was approximated by the equation \(\sigma_t=A\cdot e_i^b\). It turned out that the most reliable correlation of the coefficients

Fig. 1

\(A\) and \(b\) exists only with the conventional yield strength (Fig. 2). On the basis of Fig. 2 it was found that the coefficients \(A\) and \(b\) obey the following dependences:

\[ A=80.4-0.44\cdot\sigma_{0.2}, \]

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

As a result, the final form of the equation for calculating stresses in the cutting process was established:

\[ \sigma_p=B\cdot e_i^c+\left(27\cdot\sigma_{0.2}^{0.35}-\sigma_{0.2}\right)-\left(80.4-0.44\cdot\sigma_{0.2}\right)\cdot e_i^{0.0035\cdot\sigma_{0.2}}. \]

To verify the equation obtained, tests were used

Dependence of the coefficients \(A\) and \(b\) on the conditional yield strength

Fig. 2

○ — L62, ● — st. 002, + — st. 10, × — st. 40, □ — annealed steel 2×13,
■ — st. 2×13, △ — st. 30Kh, ▲ — st. 9Kh, ▽ — st. 4KhVS, ▼ — st. 50,
○ — st. ShKh15, ○ — st. 35Kh3MN, ○ — st. 18KhNV.

Fig. 3

G. L. Kufareva’s experiments on cutting steel 40 over a wide range of cutting conditions and N. I. Khovakh’s experiments [4] on cutting steel ShKh15 in the annealed and hardened-to-hardness \(HB = 500\) states. In Fig. 3, the flow curve under static compression is drawn with a solid line, and the stress values in the cutting process according to the equation are drawn with a dashed line. The good agreement of the calculated values with the experimental data confirms the validity of the equation obtained.

REFERENCES

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

2. K. A. Nassonov. High-Speed Milling of Steel EYa1T in the Heated State. Dissertation, Tomsk, 1954.

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

4. N. I. Khovakh. Calculation of Cutting Forces in Turning Hardened and Strengthened Steel and the Influence of Hardness on the Regularities of This Process. Dissertation, Tomsk, 1967.