RESISTANCE TO PLASTIC DEFORMATION IN METAL FORMING AND CUTTING

L. M. SEDOKOV

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

In the processes of metal forming and cutting, substantial degrees of plastic deformation are produced. Therefore, even for metals that harden only weakly, strain hardening cannot be neglected.

The hardening of real materials is determined by the flow curve in tension or compression, which may be written analytically in the general form

\[ \tau_{\max}=F(g), \tag{1} \]

in the form of a power function

\[ \tau_{\max}=Ag^m, \tag{2} \]

or of a linear function

\[ \tau_{\max}=\tau_0(1+2\alpha g). \tag{3} \]

Here \(\tau_{\max}\) is the maximum shear stress, equal to one half of the true principal normal stress;

\(g\) is the current value of the maximum logarithmic shear;

\(A\) is a metal constant, numerically equal to the maximum shear stress at \(g=1\);

\(m\) is the strain-hardening exponent of the metal;

\(\tau_0\) is the conventional yield point of the metal that would actually occur if dependence (3) were valid from the very beginning of plastic deformation;

\(2\alpha\) is the strain-hardening characteristic of the metal, numerically equal to the ratio of the increment in shear stress over the period of plastic deformation from \(g=0\) to \(g=1.0\) to the conventional yield point.

Approximation of the flow curve by the linear function (3) is very convenient, but it is applicable only to a certain range of deformations: the metal characteristics \(\tau_0\) and \(2\alpha\) for very large and very small deformations will differ somewhat. Therefore, the flow curve is sometimes replaced by a broken straight line [1].

Table 1 gives the tensile-resistance characteristics of a number of machine-building metals for calculation by equations (2) and (3). The data of N. N. Zorev [2] were adopted as the basis for calculating the characteristics \(\tau_0\) and \(2\alpha\).

Figure 1 gives a comparison of the actual values of the maximum shear stresses at the moment of neck formation (at the relative uniform reduction \(\psi_p\)) for a number of metals, according to the data of Yu. E. Bondarev [3], with the general linear dependence (3). The corresponding metal characteristics are given in Table 2.

Table 1

Characteristics of the resistance of metals to plastic compression and tension, according to N. N. Zorev

Table 2

Characteristics of the resistance of metals to tension, according to Yu. E. Bondarev

\[ \frac{\tau_{\max}}{\beta \tau_0} \]

\[ \frac{\tau_{\max}}{\beta \tau_0}=\frac{S_B}{2\tau_0} \]

\[ \frac{\tau_{\max}}{\beta \tau_0}=1+2\alpha g_{\max} \]

Figure 1. Comparison of the actual stresses at the moment of neck formation with those calculated by equation (3).

The flow curve and the functions approximating it are applicable only to the stress state under which they were obtained. Therefore, the use of the flow curve in tension or compression for other stress–strain states is associated with the adoption of a definite law of plastic flow, i.e., with the introduction of a correction factor \(\beta\) into formula (1)

\[ \tau_{\max}=\beta F(g). \tag{4} \]

The coefficient \(\beta\) depends on the type of stress–strain state of the metal and on the adopted flow law.

Two flow laws are widely used:

a) for equal maximum shears, the maximum shear stresses are equal;

b) for equal octahedral shears, the octahedral shear stresses are equal. According to the first law, the coefficient \(\beta\) in formula (4) is always equal to unity. According to the second law, the coefficient \(\beta\) is taken equal to unity for stress–strain states close to axisymmetric tensile or compressive deformation, and equal to \(2:\sqrt{3}=1.15\) for stress–strain states close to plane deformation. The second flow law is often expressed through the stress intensity \((\sigma_i)\) and the strain intensity \((\varepsilon_i)\), which are proportional to the corresponding octahedral quantities,

\[ \sigma_i=F(\varepsilon_i). \tag{5} \]

Experimental studies show that some materials obey the first law, while others, including ordinary structural steels, obey the second. Systematic deviations from the second flow law are noted by Yu. I. Yagn and N. I. Mitrokhin [4], A. A. Lebedev and G. S. Pisarenko [5], V. I. Sinelnikov [6], S. I. Ratner [7], and other investigators. G. A. Smirnov-Alyaev [8] points out that relation (5), obtained from tensile tests, cannot be extended to torsion, since in torsion the requirements of simple loading and monotonic deformation are violated, which cannot but cause additional difficulties in calculating resistance to plastic deformation.

Guided by the foregoing, as well as by the studies of P. O. Pashkov [9], A. M. Rozenberg and A. N. Eremin [10], one may formulate yet another law of plastic flow of metals: the specific work of plastic deformation, referred to a unit of maximum shear, does not depend on the type of stress–strain state,

\[ \frac{a(g)}{g}=f(g), \tag{6} \]

As applied to the particular case of a plane stress state, all three laws of plastic flow may be written as follows:

\[ \left(\frac{\sigma}{\sigma_s}\right)^2+\left(\frac{c\tau}{\sigma_s}\right)^2=1, \tag{7} \]

where \(\sigma\) and \(\tau\) are the normal and shear stresses on a given plane;

\(\sigma_s\) is the yield stress in the process of plastic tension or compression;

$c$ is the coefficient under shear stress, which according to the first law of plastic flow is equal to 2, according to the second to $\sqrt{3}$, and according to the third to 1.5.

Figure 2 gives a comparison of (7) with the results of the experiments of Taylor and Quinney, taken from the work of A. Nadai [11]. If a mean line is drawn through all the experiments, it will lie between the lines $c=\sqrt{3}$ and $c=1.50$. Consequently, the third law of flow is approximately as reliable as the second. A certain systematic understatement of the results of the experiments in comparison with the third law has its own fundamental explanation. The transition from a linear stress state to a plane one, according to Ludwik [11], is accompanied by a considerable decrease in the principal elongation at the moment of fracture.

Fig. 2. Comparison of the results of Taylor and Quinney’s experiments with the plasticity conditions for a plane stress state.

Legend in the figure: ○ copper; ● aluminum; × decarburized steel; ▽ mild steel.

It can also be shown that the shear process is more energy-intensive than tension. Therefore, the developing plastic deformation, while changing the deformed state, will tend to pass closer to equiaxial deformation in tension or compression.

G. A. Dyoshinsky [12] showed that for different materials under identical stress states the deformed states will differ not only in the magnitude of the components but also in their character. Such a difference will also occur for one material, but at different stages of deformation, owing to a change in the value of the transverse-deformation coefficient. In this connection G. P. Zaitsev proposed a new elastic constant [13].

It is well known [8] that under large plastic deformations the correspondence between the stressed and deformed states, which underlies condition (5), is violated. If condition (6) is adopted, then the transition coefficient in the general case will be determined by the dependence

\[ \beta=\frac{2}{3}\left[2+\frac{\hat{\delta}_{2}}{\delta_{1(3)}}\right]. \tag{8} \]

Here \(\delta_{1(3)}\) is the greatest, in absolute value, principal elongation; \(\delta_2\) is the mean principal elongation.

In the case of upsetting rectangular prisms between parallel plates, the ratio of deformations may be replaced by the ratio of the transverse dimensions of the prism. If \(b\) is the width and \(l\) is the length of its base, then

\[ \beta = \frac{2}{3}\cdot\frac{2l+b}{l+b}. \tag{9} \]

Then the upsetting stress for a rectangular prism of any cross section, \(q\), can be determined through the normal stress for a square prism \((l=b)\) of the same material and the same height under the same temperature–strain-rate deformation conditions, \(\sigma\):

\[ q=\beta\sigma=\frac{2}{3}\cdot\frac{2l+b}{l+b}\,\sigma. \tag{10} \]

Figure 3 shows flow curves during compression of a square prism, 1, and two rectangular prisms with \(b_0:l_0=0.2\) and \(0.1\) (curves 2 and 3).

In Fig. 3: ordinate \(q\), kg/mm\(^2\); dashed line—calculation; solid line—experiment; material—aluminum; \(1\)—\(b_0:l_0=1.0\), \(2\)—\(b_0:l_0=0.2\), \(3\)—\(b_0:l_0=0.1\); abscissa \(\ln(h_0/h)\).

Fig. 3. Verification of equation (8) from the results of compression of rectangular prisms

The dashed lines indicate the flow curves according to equation (10) for these same prisms. The agreement between calculated and actual values is quite satisfactory.

The mean value of the maximum stresses under ordinary tension over the entire deformation period, from the beginning to its final ultimate stage,

\[ \bar{\tau}_{\mathrm{cp}}=\frac{1}{g}\int_{0}^{g}F(g)\,dg. \tag{11} \]

Guided by relations (2) and (3), we obtain

\[ \bar{\tau}_{\mathrm{cp}}=\frac{A}{m+1}g^m, \tag{12} \]

\[ \bar{\tau}_{\mathrm{cp}}=\tau_0(1+ag). \tag{13} \]

On the basis of (6), the resistance to plastic deformation under a complex stress state will be determined by the relations

\[ \tau'_{\mathrm{av}}=\beta\tau_0(1+ag). \tag{14} \]

\[ \tau'_{\mathrm{av}}=\beta\frac{A}{m+1}g^m. \tag{15} \]

In the cutting process, as a rule, plane deformation takes place (there is no widening of the chip). Then the resistance to cutting can be calculated by the formulas:

\[ \tau_{\mathrm{cut}}=\frac{4}{3}\frac{A}{m+1}g^m, \tag{16} \]

\[ \tau_{\mathrm{cut}}=\frac{4}{3}\tau_0(1+ag). \tag{17} \]

A. M. Rozenberg and A. N. Eremin [10], comparing the processes of compression and cutting, take as the measure of deformation the conventional relative shear \((\varepsilon)\), which, as is known, is related to the logarithmic shear by the following dependence

\[ \varepsilon=2\,\mathrm{sh}\,\frac{g}{2}. \tag{18} \]

On the basis of (17), the resistance to cutting can be determined by the formula

\[ \tau_{\mathrm{cut}}=\frac{4}{3}\tau_0(1+c\varepsilon). \tag{19} \]

Table 3 gives the characteristics of resistance to compression \((\tau_0\) and \(c)\) for a number of steels. It also indicates the actual resistance to cutting of these steels for the limiting values of the range of variation of the degree of plastic deformation in cutting. On the basis of the data presented in Fig. 4, a comparison is given between the actual resistance to cutting and the calculation by formula (19).

Fig. 4. Results of A. M. Rozenberg’s experiments on metal cutting in dimensionless coordinates

In N. N. Zorev’s experiments [2] on metal cutting, a decrease in the actual tangential stresses with increasing deformation is often observed, which contradicts any law of hardening. This fact is explained

by the fact that, with an increase in the degree of plastic deformation (chip compression), the broadening of the chip increases, which indicates an ever greater deviation of the deformation scheme from plane strain. In this case the coefficient \(\beta\), determined by formula (8), will be less than its limiting value. The decrease in the coefficient \(\beta\) will have a stronger effect than the increase in the average shear stress due to the increase in the degree of deformation. The ratio of deformations in formula (8) can be replaced by the ratio of chip broadening to chip compression, which will make it possible to calculate more accurately the resistance of the metal being machined to cutting.

Table 3

Dependence of the ratio of cutting resistance to the conventional yield strength on the magnitude of the relative shear according to the experiments of A. M. Rozenberg and A. N. Eremin [10]

The foregoing makes it possible to draw two conclusions:

1. Plastic deformation in metalworking by pressure and by cutting obeys the same laws.

2. The plastic flow of real metals over a large range of degrees of deformation is described quite satisfactorily by equation (6).

References

1. S. A. Bocharova. The stressed state of a tube under the action of uniform internal pressure and longitudinal force at large plastic deformations. In: Strength Calculations, issue 9, Mashgiz, 1963.

2. N. N. Zorev. Calculation of the Projection of the Cutting Force. Mashgiz, 1958.

3. Yu. E. Bondarev. Compression curves in dimensionless coordinates. Factory Laboratory, No. 2, 1962.

4. Yu. I. Yagn and N. M. Mitrokhin. On the systematic nature of deviations from the laws of plasticity. Doklady of the Academy of Sciences of the USSR, vol. 135, No. 4, 1960.

5. A. A. Lebedev, G. S. Pisarenko. Investigation of the strength of steel IX18N9T under conditions of a complex stressed state at temperatures up to 820°C. Proceedings of the Second Scientific and Technical Conference. Problems of High-Temperature Strength in Mechanical Engineering. Kiev, 1963.

6. Yu. I. Sinelnikov. Application of the energy condition of plasticity in the theory of metal forming. Proceedings of Higher Educational Institutions, Mechanical Engineering, No. 8, 1963.

7. S. I. Ratner. Strength and Plasticity of Metals. Oborongiz, 1949.

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

1. P. O. Pashkov. Plasticity and Fracture of Metals. Sudpromgiz, Leningrad, 1949.

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

3. A. Nadai. Plasticity and Fracture of Solids. Foreign Literature Publishing House, 1954.

4. G. A. Dolinskii. Theory of the limiting elastic state. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.

5. G. P. Zaitsev. A new dimensionless elastic constant, the transverse-deformation coefficient, and formulas for calculating large deformations. Transactions of the Siberian Physicotechnical Institute, issue 34, 1955.

6. L. M. Sedokov. Dimensionless parameters of the resistance of metals to plastic deformation. Zavodskaya laboratoriya, 1964, No. 10.