RESISTANCE TO PLASTIC DEFORMATION UNDER TENSION AND TORSION OF STRAIN-HARDENING MATERIALS

K. K. Pakhotin

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

The resistance of strain-hardening materials to tension throughout the entire course of plastic deformation up to fracture is determined from the flow curve “maximum normal stress—maximum elongation.” The resistance of these same materials to shear can be characterized by the flow curve “maximum shear stress—true maximum shear,” which is constructed from the results of torsion tests of solid round specimens.

In [1] it was proposed that, as a criterion for comparing various processes of plastic change of shape, the value of the specific work of plastic deformation at equal shears be adopted.

Under tension, the stressed and deformed states of the material are regarded as homogeneous (at any rate, up to the formation of a neck). In this case the specific work of plastic deformation is determined from the area of the tensile diagram [2]:

\[ a=\int_{0}^{\varepsilon}\sigma\,d\varepsilon, \tag{1} \]

where

\(\sigma\) — the true normal stress at a given instant of plastic deformation,

\(\varepsilon\) — the conventional relative elongation.

Knowing the conventional relative elongation at each stage of deformation, it is easy to calculate the value of the true (logarithmic) shear at this stage

\[ g=(3/2)\ln(1+\varepsilon). \tag{2} \]

Under torsion of a cylindrical specimen, the stressed and deformed states are nonuniform over the cross section of the specimen, and

\[ \frac{\gamma}{\gamma_R}=\frac{\rho}{R}, \tag{3} \]

where

\(\gamma\) — the shear of a fiber located at a distance \(\rho\) from the axis of the bar.

The loading history of each elementary volume with cross section \(dF\) is such that it is described by a family of parallel “\(\tau-\gamma\)” curves with limiting curves \(BC\) (for \(\rho=R\)) and \(AD\) (for \(\rho=r\)), shown in Fig. 1.

The magnitude of the shear stresses at the given moment, when the shear of the outer fibers of the rod is equal to \(\gamma_R\), is described by the curve \(DC\).

Fig. 1. Stresses and strains inside a circular hollow specimen under torsion

Considering the stressed and deformed states to be homogeneous along the length of the specimen \(l\) [2], the work \(dA\) expended on plastic deformation of an elementary volume \(dF\) of length \(l\) can be calculated as

\[ dA = \tau \gamma l dF . \tag{4} \]

The specific work expended on plastic deformation of a unit volume of the specimen will be expressed as:

\[ a = \frac{1}{Fl}\int_F \tau \gamma l dF \]

or

\[ a = \frac{1}{F}\int_F \tau \gamma\, dF . \tag{5} \]

For a hollow specimen with radii \(r\) and \(R\), one may write:

\[ a = \frac{2R^2}{\gamma_R^2\left(R^2-r^2\right)} \int_{\gamma_r}^{\gamma_R} \tau \gamma^2\, d\gamma \]

or

\[ a = \frac{2\gamma_R}{R\left(R^2-r^2\right)} \int_r^R \tau \rho^2\, d\rho . \tag{6} \]

Idealizing the material, i.e., replacing the strain-hardening material by an ideally plastic one (plane \(Q\) in Fig. 1), the shear resistance \((\tau_{\mathrm{cp}})\) can be determined as

\[ \tau_{\mathrm{cp}} = \frac{3M}{2\pi R^3\left[1-\left(\frac{r}{R}\right)^3\right]} . \tag{7} \]

Considering, therefore, the stressed state to be homogeneous over the cross section of the specimen, instead of (6) we write:

\[ a=\tau_{\mathrm{cp}}\cdot \frac{2}{3}\gamma_R \left[1+\frac{r^2}{R(R+r)}\right] \]

or

\[ a=\tau_{\mathrm{cp}}\cdot \gamma_{\mathrm{cp}}, \tag{8} \]

where

\[ \gamma_{\mathrm{cp}}=\frac{2}{3}\gamma_R \left[1+\frac{r^2}{R(R+r)}\right]. \tag{9} \]

The true (logarithmic) shear in torsion \((g_{\mathrm{cp}})\) can be calculated by the formula of V. G. Osipov [3]

\[ g_{\mathrm{cp}}=2\ln\left(1+\frac{\gamma_{\mathrm{cp}}}{2}\right). \tag{10} \]

Table 1 gives the results of tensile tests of steels P18, P12, P6M3, 2X13, St3, brass, and copper. Table 2 gives the results of torsion tests of these same materials.

Table 1

The data given in Tables 1 and 2 make it possible to determine the specific work of plastic deformation in tension by formula (1) and in torsion by formula (8). The dependence of the specific works of plastic deformation in tension on the magnitude of the true (logarithmic) shear, determined by formula (2), is shown in Fig. 2 by open circles. The relation between the specific works of plastic deformation in torsion and the magnitude of the true (logarithmic) shear, determined by formula (10), is shown in Fig. 2 by oblique crosses.

The comparison made of the experimental results in tension and in torsion for a number of materials that differ very substantially in their mechanical properties confirms the reliability of the adopted comparison criterion at all stages of plastic deformation up to fracture for all the materials investigated. The greatest deviation of the experimental data in tension from the corresponding data in torsion occurs for brass, but even here this discrepancy does not exceed 3%.

Taking the flow curve of materials in the coordinates “specific work of plastic deformation—maximum logarithmic shear” as a single curve for a number of plastic forming processes, it is easy to calculate the resistance to torsion from the tensile diagram.

Table 2

Average shear stress over the cross-section under torsion, kg/mm².

Legend in the figure: ○ — tension; × — torsion.

Fig. 2. Comparison of experimental results for tension and torsion of various materials
(according to the author’s experiments)

REFERENCES

1. L. M. Sedokov. Resistance of Materials to Large Plastic Deformations. Publishing House of Tomsk State University, Tomsk, 1966.
2. N. A. Shaposhnikov. Mechanical Testing of Metals. Mashgiz, Moscow, 1954.
3. V. G. Osipov. On Nadai’s error... Vestnik inzhenerov i tekhnikov, No. 1, 1950.