Friction in Metal Forming and Cutting

L. M. Sedokov

(Presented by the scientific seminar of the Department of Strength of Materials)

A characteristic feature of most studies on metal forming is the absence of an organic study of the interrelation between deformations and forces—the very interrelation that takes place in real processes of plastic deformation. Noting this, I. Ya. Tarnovskii, A. A. Pozdeev, and O. A. Ganago [1] point out that in many books and articles on metal forming the effect is considered in isolation from the causes that produced it.

A similar situation may also be noted in many studies of plastic deformation during metal cutting. V. D. Kuznetsov [2] wrote: “In the field of the dependence of cutting characteristics on the physical properties of metals, we have almost a blank spot, and research in this field must be carried out from the very beginning.”

The main reason for the situation that has arisen is that the magnitude of the forces producing plastic deformation is very substantially affected by friction between the tool and the metal being worked. Therefore, consideration of the work of friction and of the ratio of the component parts of all the work expended in carrying out plastic deformation is of interest from both the theoretical and the practical points of view.

Work of Friction in Upsetting a Cylinder

When a cylinder is upset, friction arises on its end faces, the external manifestation of which is distortion of the shape. However, the absence of noticeable barreling of the deformed cylinder does not yet mean the absence of significant work of friction.

Let us consider two variants of deriving a formula for calculating the work of friction. Figure 1 shows a diagram of the process of upsetting a cylinder. An elementary area \(2\pi r\,dr\), as a result of deformation, will increase in thickness by the amount \(e_r\,dr\) (Fig. 1, b), where \(e_r\) is the relative radial strain. Thus it will push apart the outer shaded part of the cylinder, on whose end face there acts the force

\[ \tau \pi (R^2 - r^2), \]

where \(\tau\) is the shear stress on the end face.

Then the elementary work of friction is

\[ dA_{\mathrm{fr}}=\tau\pi (R^2-r^2)e_r\,dr. \]

The total work on both end faces is

\[ A_{\mathrm{fr}}=2\pi\int_0^R e_r\tau (R^2-r^2)\,dr. \]

Assuming uniformity of deformation along the radius of the end face, \(|e_r|=0.5e_z\), and constancy of the shear stresses, we obtain

\[ A_{\mathrm{fr}}=\frac{2}{3}\pi e_z\tau R^3. \tag{1} \]

Formula (1) can be derived in another way. On an elementary area \(2\pi r\,dr\) there act shear stresses \(\tau\). Over a certain period of deformation this area will move through a distance \(re_r\), and the friction force acting on the elementary area will perform the work

\[ dA_{\mathrm{fr}}=2\pi r\,dr\,\tau\,re_r. \]

The total work of friction on both end faces, under the same assumptions, is

\[ A_{\mathrm{fr}}=4\pi\int_0^R \tau e_r r^2\,dr= \]

\[ =\frac{2}{3}\pi e_z\tau R^3. \]

Fig. 1. For the calculation of the work of friction during upsetting of a cylinder

The total specific work of deformation in upsetting of a cylinder \(a\) will be equal to the product of the mean pressure on the end face \(q\) by the magnitude of deformation. The specific work of deformation without friction \(a_{\mathrm{def}}\) will be equal to the product of the mean normal stress acting in the transverse section of the cylinder by the magnitude of deformation \(e_z\),

\[ a=qe_z=\sigma e_z+a_{\mathrm{fr}}. \]

To obtain the specific work of friction, it is necessary to divide the total work of friction by the deformed volume. Then we obtain the widely known formula [3]

\[ q=\sigma\left(1+\frac{\mu}{3}\frac{2R}{h}\right), \tag{2} \]

where \(\mu\) is the mean coefficient of friction on the end faces.

Figure 2 gives the dependence of the ratio of the work of deformation with friction to the work of deformation without friction on the current ratio of the diameter of the specimen to its height according to formula (2), for \(\mu=0.1\) and \(\mu=0.25\). The experimental data, in the form of individual points, are given from the experiments of M. A. Bolshanina and Mendeleev [4] in compression of copper specimens. The stresses were taken from the effective compression curves.

Although the scatter of the data is very considerable, the coefficient of friction falls in all cases within the range from 0.1 to 0.25. In calcu-

it is assumed that at \(h_0 : d_0 = 2.0\) deformation takes place without any substantial influence of friction on the deformation force, which, evidently, has led to a certain underestimation of the work of friction.

Using the method set forth, one can obtain a formula for calculating the deforming stress in upsetting annular blanks

\[ q=\sigma\left(1+\frac{\mu}{2}\frac{R-R_0}{h}\right). \tag{3} \]

Here \(R\) and \(R_0\) are, respectively, the outer and inner radius of the annular blank.

It follows from formula (3) that when the thickness of the blank \((R-R_0)\) is reduced, the deforming stress decreases. Figure 3 gives the results of experiments by A. S. Zhibinov, E. S. Gitman, and A. A. Chapkovich, carried out under the author’s supervision, on the compression of

Fig. 2. Relationship between the ratio of the total work to the work of deformation without friction and the current geometric dimensions of the deforming cylinder

Fig. 3. Compression curves for annular specimens made of steel 3

tubular specimens of equal height and equal cross-sectional area made of steel 3, which confirm the dependence [3]. Formula (3) is considerably simpler than that proposed by L. A. Shofman [6].

Work of friction in upsetting rectangular prisms

In deriving formulas for calculating deforming stresses in the case of upsetting rectangular prisms, the so-called normal kinematic scheme of metal flow is always adopted, according to which the horizontal projection of the displacement trajectory of any point of the deformed body is a normal to the lateral surface of the body [3, 5, 6]. I. Ya. Tarnovskii proved theoretically and experimentally that in real processes the actual displacements do not coincide with the law of normal flow, just as they do not coincide with the law of radial flow [7].

Using the principle set forth above for determining the work of friction and the scheme shown in Fig. 4, we obtain

\[ dA_{\mathrm{fr}}=e_x x \tau b\,dx, \]

Then the total work on both end faces, under the same assumptions, is

\[ A_{\mathrm{fr}}=4\int_0^{l/2}\tau b e_x x\,dx+4\int_0^{b/2}\tau l e_y y\,dy, \]

\[ A_{\mathrm{fr}}=\frac{1}{2}\tau b^2 l\left(e_x\frac{l}{b}+e_y\right). \]

The specific work of friction is

\[ a_{\mathrm{fr}}=\frac{A_{\mathrm{fr}}}{bhl} =\frac{1}{2}\tau\frac{b}{h}\left(e_x\frac{l}{b}+e_y\right). \tag{4} \]

Taking the empirical relation of I. Ya. Tarnovsky [7],

\[ \frac{e_y}{e_z}=\frac{1}{1+\left(\dfrac{b}{l}\right)^{4\mu}}, \tag{5} \]

we obtain

\[ a_{\mathrm{fr}}=\frac{1}{2}\tau\frac{b}{h}\, \frac{1+\left(\dfrac{b}{l}\right)^{4\mu-1}} {1+\left(\dfrac{b}{l}\right)^{4\mu}}\,e_z. \tag{6} \]

Fig. 4. For calculating the work of friction in upsetting a rectangular prism

If \(b:l\to 0\) or \(b:l\to 1\), then the specific work of friction will be the same,

\[ a_{\mathrm{fr}}=\frac{1}{2}\tau\frac{b}{h}e_z. \tag{7} \]

The conclusion that the specific work of friction does not depend on the length of the prism being upset can be proved in a somewhat different way. It is known that the middle part of the prism will be under conditions of plane strain. Therefore the work of friction on this part of the prism can be determined as follows:

\[ A_{\mathrm{fr}}=4\int_0^{b/2}\tau(l-b)y e_y\,dy =\frac{1}{2}(l-b)e_z\tau b^2. \]

The specific work of friction on the middle part of the prism is

\[ a_{\mathrm{fr}}=\frac{A_{\mathrm{fr}}}{(l-b)bh} =\frac{1}{2}e_z\tau\frac{b}{h}. \]

The remaining part of the prism will deform as a square prism. In this case

\[ A_{\mathrm{fr}}=8\int_0^{b/2}\tau b y e_y\,dy =e_y\tau b^3, \]

\[ a_{\mathrm{fr}}=\frac{1}{2}\tau e_z\frac{b}{h}. \]

Consequently, the specific work of friction on both parts of the prism will be equal.

Then the ratio of the work of deformation with friction to the work of deformation without friction is

\[ \frac{a}{a_{\mathrm{def}}}=1+\frac{\mu}{2}\frac{b}{h} \tag{8} \]

and the working deformation stress for any prisms is

\[ q=\beta\sigma\left(1+\frac{\mu}{2}\frac{b}{h}\right). \tag{9} \]

Here \(\beta\) is a coefficient that takes into account the change in the averaged deformed state. According to the Mises–Hencky plasticity condition it varies from 1.0 to 1.15. According to the condition of P. O. Pashkov [8], it varies from 1.0 to 1.33.

Adopting the usual assumption that the middle part of the upset prism \((l-b)\) is under conditions of plane deformation, while the edge parts deform axisymmetrically [3, 5, 6], on the basis of the condition of P. O. Pashkov [8] we obtain

\[ \beta=\frac{2}{3}\frac{2l+b}{e+b}. \tag{10} \]

Formula (9) differs from the generally known formulas for calculating the deforming stress in upsetting prisms [1, 3, 5, 6] by the numerical coefficient in the second term. For a square it is usually [5] taken to be the same as for a cylinder. From a simple comparison it is clear that the share of the work of friction in the overall balance of the work of deformation in upsetting a square prism will be greater than in upsetting the inscribed cylinder \((2R=b)\), which is also taken into account by formula (2).

Ratio of the Component Parts of the Work of Cutting

As investigations show [9], the coefficient of friction is a kinematic parameter of chip formation, uniquely related to the degree of plastic deformation of the layer of metal being removed,

\[ \mu=1-\operatorname{tg}(\beta_1-\gamma). \tag{11} \]

Here \(\beta_1\) is the shear angle and \(\gamma\) is the rake angle of the cutting tool.

Dependence (11) makes it possible to obtain an analytical expression for the total specific work of chip formation,

\[ a=\tau\left[\operatorname{ctg}\beta_1+ \frac{1}{1-\operatorname{tg}(\beta_1-\gamma)+\operatorname{tg}^2(\beta_1-\gamma)} \right]. \]

Taking into account that, in the cutting process, the work of plastic deformation without friction is equal to [10]

\[ a_{\mathrm{def}}=\tau\varepsilon, \]

where \(\tau\) is the shear stress in the shear plane;

\(\varepsilon\) is the relative shear,

\[ \varepsilon=\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma), \]

we obtain

\[ a_{\mathrm{fr}}=\tau\left[ 1-\frac{\operatorname{tg}^3(\beta_1-\gamma)} {1-\operatorname{tg}(\beta_1-\gamma)+\operatorname{tg}^2(\beta_1-\gamma)} \right]. \tag{12} \]

In many cases the second term in equation (12) is close to zero. Then one can obtain a very simple expression for the ratio of the component parts of the cutting work:

\[ \frac{a_{\mathrm{fr}}}{a_{\mathrm{def}}}=\frac{1}{\varepsilon}, \tag{13} \]

\[ \frac{a_{\mathrm{def}}}{a}=\frac{\varepsilon}{\varepsilon+1}. \tag{14} \]

In Fig. 5 the solid line shows dependence (14), and the individual points show the results of experiments by A. M. Rosenberg [11].

As verification against experimental data shows, dependence (14) is general for all materials that, under the given cutting conditions, produce a continuous chip, and for any ordinary geometry of the cutting tool.

The ratio of friction work to the total work expended in carrying out plastic deformation in metal forming depends on the geometric dimensions of the body being deformed [8]. Friction work is performed on the end faces of the specimen and is a function only of the transverse dimensions of the body. The work expended only on deformation of the body is a function of all three of its dimensions.

Fig. 5. Dependence of the ratio of deformation work (without friction) to the total cutting work on relative shear

The ratio of friction work to the total work of plastic deformation in this case does not depend on the degree of deformation (Fig. 2). This is because the transverse deformations are rigidly connected with the longitudinal deformation, while the component parts of the work are proportional to the corresponding deformations.

In the cutting process, friction changes the degree of plastic deformation [11]. An increase in chip compression reduces the relative displacement of the metal along the rake face and decreases the share of friction work in the overall balance of cutting work.

Knowing the ratio of the component parts of the work in the processes of plastic deformation in metal forming and cutting, and the nature of the change in this ratio, one can study the laws of plastic deformation in full interrelation with the stresses and forces that carry out this deformation.

LITERATURE

1. I. Ya. Tarnovskii, A. A. Pozdeev, O. A. Ganago. Deformations and Forces in Metal Forming. Mashgiz, 1959.
2. V. D. Kuznetsov. Physics of the Solid Body. Vol. III, Materials on the Physics of Metal Cutting, Tomsk, “Krasnoe znamya,” 1944.
3. S. I. Gubkin. Plastic Deformation of Metals. Vol. III, Theory of Plastic Working of Metals, Metallurgizdat, 1961.

1. V. D. Kuznetsov, M. A. Bolshanina. Physics of the Solid State. Vol. II, Tomsk, “Krasnoe Znamya,” 1941.

2. M. V. Storozhev, E. A. Popov. Theory of Metal Forming. Mashgiz, 1957.

3. L. A. Shofman. Fundamentals of Calculating Stamping and Pressing Processes. Mashgiz, 1961.

4. I. Ya. Tarnovskii. Shape Change in the Plastic Working of Metals. Metallurgizdat, 1954.

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

6. L. M. Sedokov. Stresses and deformation in the process of metal cutting. Problems of Metal Cutting, Conference Proceedings, Moscow, 1963.

7. A. M. Rozenberg, A. M. Eremin. Elements of the Theory of the Metal-Cutting Process. Mashgiz, 1956.

8. A. M. Rozenberg. Components of the cutting work in high-speed machining of steels. Proceedings of the TPI, vol. 85, 1957.