Resistance to Plastic Torsion, Cutting, and Tension

L. M. Sedokov

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

In torsion of rods of circular profile, plane deformation takes place. In machining plastic metals by cutting, plane deformation is also very often realized. This allowed S. I. Gubkin [1] to conclude that the same mechanical characteristics that describe the torsion process should be taken as the basis for describing the cutting process. Consequently, such a complex process of plastic deformation as cutting can be directly compared with an elementary process of plastic deformation—torsion [1].

However, the process of plastic torsion of solid round rods has been studied poorly. Therefore it is necessary to develop a method for determining the mechanical characteristics of a metal from the results of its torsion test. Such a method will be of particular interest if the mechanical characteristics obtained from it can be checked numerically by a special calculation from the tensile diagram of this metal.

The magnitude of the shear stresses in plastic torsion of rods made of an ideally plastic material remains constant over the entire cross section. It is equal to the yield strength of the material in shear. In this case the torque required to produce plastic deformation of the entire volume of the rod is determined by the equation

\[ M_{\mathrm{cr}} = W_{\mathrm{pl}} \tau_s^{*}, \tag{1} \]

where \(\tau_s^{*}\) is the resistance of the material to plastic shear (the yield strength in shear),

\[ W_{\mathrm{pl}} \]

is the plastic section modulus of the cross section. For a round rod \((D = 2R)\)

\[ W_{\mathrm{pl}} = \frac{2}{3}\pi R^3. \tag{2} \]

The relation between the resistance of the material in shear—\(\tau_s^{*}\)—and in tension—\(\tau_s\)—is determined by the adopted condition of plastic flow

\[ \tau_s^{*} = \beta \tau_s. \tag{3} \]

The coefficient \(\beta\), according to the theory of maximum shear stresses, is equal to unity; according to the theory of octahedral stresses, it is \(2:\sqrt{3} = 1.15\); and according to the theory of greatest elongations (if the coefficient of transverse

deformation has its limiting value \(\mu=0.5\)—\(4:3=1.33\) [2—5].

In the case of plastic torsion of bars made of a strain-hardening material, the required torque will increase with increasing angle of twist.

For engineering calculations, the shears and shear stresses that vary over the radius of the bar may be replaced by average values, which permits the calculation to be carried out using equation (1).

Adopting the usual assumptions that plane sections remain plane also during plastic deformation, and that straight radii in them remain straight [2], we obtain [5]:

\[ \gamma_{\mathrm{av}}=\frac{1}{\pi R^2}\int_0^R 2\pi\rho\gamma\,d\rho =\frac{2}{3}\gamma_R, \tag{4} \]

\[ {}^{*}\tau_{\mathrm{av}}(\gamma_R)=\frac{1}{\gamma_R}\int_0^{\gamma_R}{}^{*}\tau_s\,d\gamma =\alpha\tau_R(\gamma_R). \tag{5} \]

Here \(\gamma_R\) and \(\tau_R\) are, respectively, the maximum shear and the maximum shear stresses at the periphery of the twisted bar; \(\alpha\) is a coefficient depending on the strain-hardening of the metal.

With a linear approximation of the tensile curve

\[ \sigma(\delta)=2A\delta^m. \tag{6} \]

we obtain

\[ \sigma_{\mathrm{av}}(\delta)=\frac{2A}{1+m}\delta^m, \]

\[ \tau_{\mathrm{av}}(\gamma)=\frac{A}{1+m}\left(\frac{2}{3}\gamma\right)^m. \]

Whence

\[ \alpha=\frac{\tau_{\mathrm{av}}(\gamma)}{\tau_s(\gamma)} =\frac{1}{1+m}. \tag{7} \]

Then the value of the reduced shear stresses in torsion of solid round bars, on the basis of (3), (5), and (7), may be determined by the formula

\[ \tau_{\mathrm{tor}}=\alpha\beta\tau_s(\gamma). \tag{8} \]

The calculated value of the resistance to torsion, determined by formula (8), taking into account the power-law approximation of the flow curve in tension (6) at each stage of plastic deformation characterized by the magnitude of the maximum shear (for torsion, \(\gamma_{\mathrm{av}}\)), may be computed according to the theory of maximum shear stresses

\[ \tau_{\mathrm{tor}}=A\left(\frac{4}{9}\gamma_R\right)^m, \tag{9} \]

according to the theory of octahedral stresses

\[ \tau_{\mathrm{tor}}=\frac{2}{\sqrt{3}}A\left(\frac{4}{9}\gamma_R\right)^m \tag{10} \]

and according to the theory of greatest elongations

\[ \tau_{\mathrm{tor}}=\frac{4}{3}\frac{A}{1+m}\left(\frac{4}{9}\gamma_R\right)^m. \tag{11} \]

The actual resistance to torsion is easily determined from the known torque (1).

Table 1 and Fig. 1 present the corresponding calculations and the actual data of G. A. Smirnov-Alyaev [6]. The mechanical characteristics of the materials ($A$ and $m$) were taken on the basis of tensile diagrams. The shear curves ($\tau_R-\gamma_R$), obtained as a result of processing the experimental data ($M_{\mathrm{tor}}-\gamma_R$) by the method of Ludwik and Shaw, practically coincided with the shear curves ($\tau_{\mathrm{tor}}-\gamma_{\mathrm{av}}$) obtained by the proposed method [5].

Table 1

Comparison of actual and calculated values of resistance to torsion

Note:
I — $\tau_{\mathrm{tor}}:\tau_{\mathrm{tens}}=1.00$,
II — $\tau_{\mathrm{tor}}:\tau_{\mathrm{tens}}=1.15$,

III — $\tau_{\mathrm{tor}}:\tau_{\mathrm{tens}}=1.33\,\dfrac{1}{1+m}$.

Although all three calculation methods give results close to the actual values, preference should be given to calculations based on the theory of maximum elongations. The advantages of the latter are clearly visible in Fig. 2, where the dependence of the ratio of the resistances to torsion and tension on the hardening exponent ($m$) is given. The actual values of the indicated ratio are very close to the calculated ones (curve III).

The calculated values of resistance to torsion by the proposed method are somewhat lower than the actual ones (see Table 1 and Fig. 1). This is explained by the fact that the center of the actual epure of tangential stresses is located somewhat farther from the center of gravity of the section than was assumed in the calculation scheme (one-half of the radius). Considering the data shown in Fig. 2, one can establish a correction coefficient that takes into account the location of the center of gravity of the actual stress epure. This coefficient must be associated with the hardening of the metal. It may be taken as equal to $0.2m$.

\[ \tau_{\mathrm{tor}}= \frac{1+0.2m}{1+m}\cdot\frac{4}{3}\, A\left(\frac{4}{9}\gamma_R\right)^m . \tag{12} \]

Fig. 1. Actual (solid lines) and calculated (dashed) torsion curves for solid circular rods.
1 — alloy steel, 2 — low-alloy steel, 3 — mild steel, 4 — brass, 5 — copper, and 6 — aluminum.

Fig. 2. Dependence of the ratio of torsional and tensile resistances on the strain-hardening exponent. Calculation: I — according to the theory of maximum shear stresses, II — according to the theory of octahedral stresses, III — according to the theory of the greatest elongations, IV — with a correction. Actual dependences: 1 — steel L., 2 — steel M., 3 — steel MP, 4 — aluminum, 5 — copper, and 6 — brass.

The curve according to equation (12) is shown in Fig. 2 by a dashed line. The mean stress in the tensile process can be determined by planimetering the tensile diagram.

Iosimara Iosimur [7] gives flow curves in tension and in torsion (solid lines in Fig. 3). Calculation of the flow curve in torsion by the proposed method gave satisfactory results (dashed line in Fig. 3).

As was indicated above, the resistance to plastic deformation in cutting metals can be compared with the resistance to torsion. Consequently,

\[ \tau_{\text{rez}}=\tau_{\text{kr}}= \frac{A}{1+m}\cdot \frac{4}{3}\left(\frac{2}{3}\gamma_p\right)^m, \tag{13} \]

where \(\gamma_p\) is the relative shear in cutting.

If the flow curve of the machined metal, obtained as a result of compression tests with the influence of friction excluded, is approximated by the function [8]

\[ \sigma = B\gamma^c, \tag{14} \]

then, in accordance with what has been said,

\[ \tau_{\text{rez}}=\frac{2}{3}\frac{B}{1+c}\gamma^c. \tag{15} \]

Fig. 3. Flow curves: 1 — in tension, 2 — in torsion, and 3 — calculated.

The theory of maximum shear stresses and the energy theory respectively give

\[ \tau_{\text{rez}}=\frac{1}{2}B\gamma^c, \tag{16} \]

\[ \tau_{\text{rez}}=\frac{1}{\sqrt{3}}B\gamma^c. \tag{17} \]

Table 2 gives the mechanical characteristics of the machined metals (\(B\) and \(c\)), the limiting values of the relative shear in cutting, the actual values of the resistance to cutting obtained by dividing the known specific work of plastic deformation by the value of the maximum shear, according to the experiments of A. M. Rozenberg and A. N. Eremin [9], and three calculation variants.

The data of Table 2 demonstrate the reliability of the calculation of the resistance to cutting by the proposed method, which completely coincides with the calculation of the resistance to torsion of solid round rods.

The foregoing confirms the well-known proposition that the process of cutting plastic metals is one of the types of plastic shaping. The dependences presented relate the resistance to plastic deformation in tension, torsion, and

cutting of metals make it possible, from the characteristics of one of the named deformations, to calculate the characteristics of another.

Table 2

Comparison of actual and calculated values of resistance to cutting

References

1. S. I. Gubkin. Plastic Deformation of Metals. Vol. III, Metallurgizdat, Moscow, 1961.

2. S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Vol. I, Mashgiz, Moscow, 1956.

3. P. O. Pashkov. Plasticity and Fracture of Metals. Sudpromizdat, Leningrad, 1949.

4. M. M. Filonenko-Borodich. Mechanical Theories of Strength. Moscow State University Press, 1961.

5. L. M. Sedov. Resistance of Materials to Large Plastic Deformations. Moscow State University Press, Moscow, 1966.

6. G. A. Smirnov-Alyaev. Processes of plastic tension and torsion of metals in their mutual comparison. In: Experimental Methods for Determining Stresses and Strains in Elastic and Plastic Zones. ONTI, NKTP, 1935.

7. Yosimaru Yosimura. Remarks on the sliding theory of Batdorf and Budiansky. Mechanics, No. 2 (60), 1960.

8. A. M. Rozenberg (ed.). Cutting of Metals and Tools. Mashinostroenie, Moscow, 1964.

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