CALCULATION OF THE ACTUAL SHEAR RESISTANCE FROM TORSION TEST RESULTS FOR STRAIN-HARDENING MATERIALS

K. K. PAKHOTIN, L. M. SEDOKOV

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

A very important characteristic of the mechanical properties of plastic materials is the shear resistance, which for strain-hardening materials depends on the degree of deformation. Determination of this characteristic from the results of torsion of thin-walled tubes, in whose walls the stress-strain state may be assumed homogeneous, is impossible, since such specimens rapidly lose stability [1].

In the torsion of solid circular rods, additional difficulties arise, associated with the nonuniformity of the stress-strain state and the presence of an elastic core at the center of the specimen. The use of thick-walled tubes makes it possible to eliminate the influence of the elastic core, but requires refinement of the calculation relationships.

At present there are three methods for processing experimental data on torsion of cylindrical rods:

a) P. Ludwik’s method, which forms the basis of GOST 3565-58.
b) The two-specimen method.
c) The method of mean values.

All three methods are based on two hypotheses:

1. Plane sections remain plane during deformation, i.e., the axial strain \((\varepsilon_z)\) does not vary along the radius of the specimen \((\rho)\):

\[ \varepsilon_z(\rho)=\text{const}. \tag{1} \]

1. Straight radii remain straight during deformation. This means that the maximum conventional relative shear \((\gamma)\) varies proportionally to the radius over the cross section

\[ \gamma = \gamma_R \frac{\rho}{R}, \tag{2} \]

where

\(\gamma_R, R\) are the maximum conventional relative shear at the periphery and the outer radius of the section.

In Ludwik’s method, the use of graphical differentiation of the experimental curve “torque—angle of twist” is mandatory.

In the torsion of thick-walled tubes, the shear resistance is determined [2]

\[ \tau_R - \tau_r \left( \frac{r}{R} \right)^3 = \frac{3}{2\pi R^3}\left( M + \frac{\gamma_R}{3}\frac{dM}{d\gamma_R} \right), \tag{3} \]

where

\[ \tau_R,\ \tau_r \]

are, respectively, the shear stresses in the outer and inner fibers of the tube,

\[ r \]

is the inner radius of the tube cross section,

\[ M \]

is the torque.

For \(r=0\), formula (3) is transformed into Ludwik’s formula [3]; for \(r \approx R\), formula (3) may be rewritten as

\[ \tau_R = \frac{3M}{2\pi R^3\left[1-\left(\dfrac{r}{R}\right)^3\right]} . \]

In the two-specimen method, the test results for two solid specimens of somewhat differing diameters are used [4].

The difference between the torsion diagrams of two specimens corresponds to the torsion diagram of a thin-walled “tube” with radii \(r\) and \(R\). In this case, the shear resistance is determined by formula (4).

The method of mean values [5] is based on replacing the strain-hardening material by an ideally plastic one, and also on reducing the deformed state to a homogeneous one along the radius of the specimen cross section.

In torsion of thick-walled tubes, the mean shear stresses \(\tau_{\mathrm{av}}\) in the section are determined as

\[ \tau_{\mathrm{av}} = \frac{3M}{2\pi R^3\left[1-\left(\dfrac{r}{R}\right)^3\right]}, \tag{5} \]

and the maximum conventional relative shear averaged over the volume, \(\gamma_{\mathrm{av}}\), on the basis of (2), is equal to

\[ \gamma_{\mathrm{av}} = \frac{2\gamma_R}{R(R^2-r^2)} \int_r^R \rho^2\,d\rho = \frac{2}{3}\gamma_R \left[ 1+\frac{r^2}{R(R+r)} \right]. \tag{6} \]

For \(r=0\), the formulas obtained are:

\[ \left. \begin{aligned} \tau_{\mathrm{av}} &= \frac{3M}{2\pi R^3},\\ \gamma_{\mathrm{av}} &= \frac{2}{3}\gamma_R \end{aligned} \right\}. \tag{7} \]

Below a comparison and analysis are given of the above-described methods for torsion of solid round bars made of materials with different strain-hardening capacity \(m\). Here \(m\) is the strain-hardening exponent in approximating the shear diagram by a power function (see Table 1).

Table 1

The torsion diagrams of these materials were constructed in the coordinates “\(M-\gamma\),” and from them the shear resistance was calculated by the described

above-mentioned methods. The shear curves are given in Figs. 1 and 2. As can be seen from the figures, the shear curves (III), constructed by the method of mean values, run somewhat above the corresponding curves (I), constructed by Ludwik’s method, while the curves (II), constructed by the two-specimen method, pass below curves (I), with the underestimation varying over wide limits—from 1% (P18) to 15% (copper).

Fig. 1. Shear curves from torsion results for solid round rods

The correctness of calculating the actual shear resistance in torsion by the two-specimen method depends on the wall thickness of the “tube,” the hardenability \(m\), the inhomogeneity of the material, and experimental errors. Approximating the shear diagram by a power function, the deviation \(\Delta\) of the shear diagram by the two-specimen method from the shear curve by Ludwik’s method, taking into account the influence of the factors listed above, is obtained as

Fig. 2. Shear curves from torsion results for solid round rods

\[ \Delta = 1 - \frac{3}{3+m}\, \frac{1-\left(\frac{r}{R}\right)^{3+m}} {1-\left(\frac{r}{R}\right)^3} \pm \frac{\Delta_l}{1-\left(\frac{r}{R}\right)^{3+m}}, \tag{8} \]

where

\[ \Delta_l \]

is a characteristic of material inhomogeneity and experimental error.

As the ratio \(r/R\) increases, the third term on the right-hand side of expression (8), reflecting the inhomogeneity of the material, increases substantially.

therefore, in the case of a small wall thickness of the “tube,” it is necessary that the material of the initial specimens be as homogeneous as possible. In the experiments described, the wall thickness of the “tube” was varied so that the ratio \(r/R\) was not less than 0.7.

Let us consider the reason for the divergence of curves (I) and (III). From formulas (3) and (7), for a solid bar we have

\[ \tau_R(\gamma_R)-\tau_{\mathrm{av}} = \frac{3}{2\pi R^3}\frac{dM}{d\gamma_R} (\gamma_R-\gamma_{\mathrm{av}}). \tag{9} \]

It follows from expression (9) that the values of \(\tau_{\mathrm{av}}\) on the shear curve are obtained by the intersection of the tangent \(\left(\frac{3}{2\pi R^3}\frac{dM}{d\gamma_R}\right)\) to the shear curve and the perpendicular to the shear axis drawn from the point \(\gamma_{\mathrm{av}}\) (Fig. 3). The greater the curvature of the curve “\(\tau\)—\(\gamma\),” the greater the difference between \(\tau_{\mathrm{av}}\) and \(\tau_R(\gamma_{\mathrm{av}})\); in turn, the curvature of the shear curve increases with increasing hardening \(m\). The theoretical value of the deviation of \(\tau_{\mathrm{av}}\) from \(\tau_R(\gamma_{\mathrm{av}})\) at \(m=0.5\) is 4.9%.

In the experiments shown in Figs. 1 and 2, the greatest divergence of curves (I) and (III) was found for brass—3.6% (\(m=0.44\)). For the remaining materials, curves (I) and (III) practically coincided.

Fig. 3. Comparison of the method of average values with Ludwik’s method

Fig. 4. Shear curves from torsion results for thick-walled tubes

For the experimental results presented (Fig. 4) on torsion of thick-walled tubes made of materials 2Kh13 (\(m=0.25\)), St3 (\(m=0.23\)), and an-

duralumin ($m = 0.28$), it was not possible to compare all three methods for the following reasons:

a) formula (3) cannot be used to determine the actual shear resistance, since two quantities in it are unknown—$\tau_R$ and $\tau_r$—and it requires knowledge of an approximation of the material shear diagram;

b) the two-specimen method is not applicable to thick-walled tubes with holes of different diameters and, moreover, it has no advantages over the other methods (there is no elastic core).

Therefore, the shear curves were constructed by the method of mean values and were compared with the shear curves constructed by the method of mean values and by Ludwik’s method for solid specimens, which practically coincided (Fig. 4). In Fig. 4, the shear curves for solid specimens are shown by solid lines, and the shear curves for tubular specimens by individual points. The scatter of the shear curves is such that it was not possible to find a dependence of the shear resistance on the tube-wall thickness, the ratio $r/R$, and so forth.

For the tubular specimens, the wall thickness varied from 4 mm to 8.6 mm, and the ratio $r/R$ from 0.45 to 0.80.

The root-mean-square deviation of the shear resistance from its mean value averaged 5% for duralumin, 4% for St3, and 3% for 2Kh13.

Conclusions

1. Ludwik’s method is not applicable for calculating the actual shear resistance in torsion of thick-walled tubes, and for solid rods it requires cumbersome graphical differentiation.

2. The two-specimen method gives significant errors, since in this method torques that are close in magnitude are compared.

3. The method of mean values is simple in calculating the actual shear resistance and the shear itself, does not require laborious layouts, and is applicable for processing torsion results for solid and hollow (thick-walled and thin-walled) specimens.

4. The good agreement between the shear curves obtained by Ludwik’s method and the shear curves obtained by the method of mean values makes it possible to recommend the latter for introduction into broad practice.

References

1. S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Vol. 1, Mashgiz, Moscow, 1956.

2. I. A. Noritsyn and P. E. Kislyi. Free plastic torsion of round rods. Izv. vuzov “Mashinostroenie”, No. 11, pp. 34–48, 1960.

3. P. Ludwik und R. Scheu. Stahl und Eisen, Bd. 45, S. 373, 1925.

4. V. I. Maksak. Author’s abstract of a Candidate dissertation. Tomsk, 1966.

5. L. M. Sedokov. Resistance of Materials to Large Plastic Deformations. Tomsk, 1966.