CALCULATION OF DEFORMATIONS IN THE CUTTING PROCESS

V. A. GOVORUKHIN, G. L. KUFAREV

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

The study of the deformed state of the chip-formation zone has until now been carried out either at very low cutting speeds, or at cutting speeds not exceeding \(25\ \text{m/min}\), with the investigations being performed on machines with rectilinear motion of the tool or of the workpiece [1, 2, 3].

As the dividing grid, systems of circles or straight lines were used with a base (the distance between two parallel lines or between the centers of two adjacent circles) of not less than \(0.1\ \text{mm}\). In this case, for a sufficiently reliable description of the deformed state it was necessary to work with a cut thickness exceeding \(0.5\text{--}8\ \text{mm}\), which in the practice of medium machine building is encountered rarely. Therefore it is necessary to investigate deformations at cutting speeds and feeds most often used in production. Such cutting speeds can be obtained on machines with rotational motion of the tool or workpiece, but in this case a dividing grid in a polar coordinate system must be applied to the part under investigation. At present there are no calculation formulas for determining deformations from such a grid.

Figure 1 shows an element of such a grid during deformation. Suppose that the infinitely small element \(abcd\) undergoes shear deformation in the direction \(of\). Let the position of any fiber that has received the greatest elongation after deformation be denoted by \(ak_1\). Let the position of this same fiber before deformation be denoted by \(ak\). Then, according to [4, 5], the true deformation of segment \(ak\) is equal to

\[ l=\ln \frac{ak_1}{ak}. \]

From Fig. 1,

\[ ak=\sqrt{(ab)^2+(bk)^2} \]

\[ ak_1=\sqrt{(b_1k_1)^2+(ab_1)^2-2(b_1k_1)\cdot(ab_1)\cdot\cos[180^\circ-(\beta+\alpha)]}. \]

Let us denote

\[ ab=d\rho,\quad ad=a_0,\quad b_1c_1=a_x, \]

\[ ab_1=l_y,\quad b_1k_1=n\cdot a_x, \]

Fig. 1

Here \(d\rho, a_0\) are the dimensions of the element before deformation;
\(a_x, b_y\) are the dimensions of the element after deformation;
\(n\) is a parameter characterizing the direction of the greatest deformation.

\[ bk=n\cdot(\rho+d\rho)\cdot d\theta . \]

Whence

\[ l=\frac{1}{2}\ln \frac{ \left(\frac{a_x}{d\rho}\right)^2\cdot n^2+ \left(\frac{b_y}{d\rho}\right)^2+ 2n\cdot\cos(\beta+\alpha)\cdot\frac{a_x\cdot b_y}{d\rho^2} }{ 1+n^2\cdot\left(\frac{\rho+d\rho}{d\rho}\cdot d\theta\right)^2 }. \tag{1} \]

Let us denote

\[ A=\left(\frac{a_x}{d\rho}\right)^2,\qquad B=\left(\frac{b_y}{d\rho}\right)^2,\qquad C=2\cdot\cos(\alpha+\beta)\cdot\frac{a_x\cdot b_y}{d\rho^2}, \]

\[ D=\left(\frac{\rho+d\rho}{d\rho}\cdot d\theta\right)^2 . \]

Then equation (1) takes the form

\[ l=\frac{1}{2}\ln\frac{A\cdot n^2+C\cdot n+B}{1+D\cdot n^2}. \tag{2} \]

To determine the values of \(n\) corresponding to the greatest deformations, it is necessary to examine equation (2) for a maximum.

\[ n_{1,2}=-\frac{B\cdot D-A}{D\cdot C}\pm \sqrt{\left(\frac{B\cdot D-A}{D\cdot C}\right)^2+\frac{1}{D}} . \tag{3} \]

The direction of the greatest deformations, characterized by the angle \(\varphi\), is determined by the sine theorem from triangle \(ab_1k_1\).

\[ \frac{n\cdot a_x}{\sin(\alpha-\varphi)} = \frac{b_y}{\sin(\varphi+\beta)} . \]

The solution of this equation gives

\[ \tan\varphi= \frac{b_y\cdot\sin\alpha-n\cdot a_x\cdot\sin\beta} {b_y\cdot\cos\alpha+n\cdot a_x\cdot\cos\beta}. \tag{4} \]

Fig. 2

The position of the fiber before deformation, determined by the angle \(\mu\), is found from triangle \(abk\), neglecting the curvature of side \(bk\).

\[ \mu=\varphi_1-d\theta . \tag{5} \]

Here the angle \(\varphi_1\) is determined from

\[ \tan\varphi_1=\frac{(\rho+d\rho)\cdot d\theta\cdot n}{d\rho}. \]

The magnitude of the greatest deformation \(e_3\) is determined from the condition of incompressibility.

\[ e_1+e_2+e_3=0. \tag{6} \]

The intensity of deformations is calculated by

\[ e_i=\frac{2}{\sqrt{3}}\sqrt{e_3^2-e_1\cdot e_2}. \tag{7} \]

The pseudomaximum shear \(q'\) for the case of plane deformation [6]

\[ q' = e_1 - e_2 . \tag{8} \]

The maximum shear of an infinitely small element is

\[ \varepsilon = 2 \cdot \operatorname{sh}\frac{q'}{2}. \tag{9} \]

The direction of shear, characterized by the angle \(\beta_1\), can be calculated for each element from Fig. 2.

From triangle \(ac_1f\), by the sine theorem we find

\[ \frac{c_1 f}{\sin[90^\circ-(\alpha+\beta)]} = \frac{ac_1}{\sin[90^\circ-(\beta-\beta_1)]}; \]

\[ \frac{c_1 f}{\cos(\alpha+\beta)} = \frac{ac_1}{\cos(\beta-\beta_1)}, \]

but

\[ c_1 f = \frac{ac \cdot \cos \beta_1}{\cos(\beta_1-\beta)}, \]

then we obtain

\[ \frac{ac}{ac_1} = \cos\alpha - \sin\alpha \cdot \operatorname{tg}\beta \quad \text{and} \quad \operatorname{tg}\beta_1 = \operatorname{ctg}\alpha - \frac{ac}{ac_1 \cdot \sin\alpha}. \tag{10} \]

The magnitude \(ac_1\) and the angle \(\alpha\) are measured experimentally. Thus, for each element it is possible to determine all the deformation characteristics.

As deformation characteristics one may use both the true deformations discussed above and the principal deformations, between which there exists a known functional relationship.

REFERENCES

1. V. A. Krivoukhov et al. Machinability by Cutting of Heat-Resistant and Titanium Alloys. Mashgiz, 1961.

2. M. G. Goldshmidt. Investigation of the Stress-Strain State in the Cutting Process. Dissertation. Tomsk, 1966.

3. G. L. Kufarev. Deformation of Metal in the Cutting Zone during the Formation of Continuous Chips. Dissertation, Tomsk, 1958.

4. P. O. Pashkov. Plasticity and Fracture of Metals. Sudpromgiz, 1950.

5. Ya. B. Fridman et al. Study of Plastic Deformation and Fracture by the Method of Scribed Grids. Oborongiz, 1962.

6. V. G. Osipov. On the characteristics of finite deformations. In: Problems of Metallurgy, Moscow, 1953.