Torsion of Rods with Rectangular Cross-Sections Made of Hardening Material

G. D. Del’

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

Studies of torsion by the methods of the theory of plasticity have been carried out under serious assumptions, connected mainly with the idealization of the flow curve \[1, 2\]. Such idealization is not always acceptable for a real material, especially in the region of large plastic deformations, which accompany the technological operation of plastic torsion. This accounts for the need for further experimental investigations in this direction.

The aim of the present work is:

1. Experimental investigation of the stress-strain state in plastic torsion of rods of rectangular cross-section.

2. Establishment, on this basis, of the relation between the torsion diagrams of rods of rectangular cross-section and of a circular rod made of the same material.

The investigation was performed by the method of hardness measurement, which in the present case makes it possible to obtain a complete picture of the stress-strain state \[3\].

The stress-strain state was investigated mainly in high-strength stainless steel 1Kh18N9T.

A calibration graph relating hardness to the intensities of stress and strain was constructed from the results of a torsion test of a circular rod.

Figure 1 presents the distribution of the stress intensity $\sigma_i$ over the cross-section of a specimen twisted to a relative angle of twist $\Theta = 32\ \text{m}^{-1}$. The stress intensity reaches its greatest value at the points where the axes of symmetry emerge on the lateral surface. The law of variation of the stress intensity along the minor semiaxis of symmetry and along the segment $AB$ of the major axis, equal in length to half the minor axis, is practically the same. The same figure presents the distribution of shear stresses and logarithmic shears along the axes of symmetry (along the axes of symmetry there is pure shear; therefore the shear stress is

$$ \tau = \frac{1}{\sqrt{3}} \sigma_i, $$

and the shear is determined from the shear diagram).

Investigation of the stress state of a number of specimens of various cross-sections at various angles of twist showed:

1. The most highly loaded points are the points where the axes of symmetry emerge on the lateral surface.

1. The character of the shear-stress contours remains the same in the process of plastic torsion. The greatest shear stress occurs at the point where the axes of symmetry meet the lateral surface; the smallest is at the center of the specimen. The outlines of the contours are close to the shear diagram. As the angle of twist increases, the shear stresses increase.

2. The shears along the axes of symmetry are distributed according to a law close to linear,

\[ g=\frac{2g_0}{b}\,x, \tag{1} \]

where \(g_0\) is the shear on the surface at points \(A\) and \(C\), \(b\) is the smaller side of the rectangle, and \(x\) is the coordinate measured from the center of the specimen for the minor axis and from point \(B\) for the major axis.

From the shear-stress contours \(\tau\), the contours of residual stresses were determined. For this purpose the unloading stresses, determined by the formulas of the theory of elasticity [4], were subtracted from \(\tau\). It turned out

Fig. 1

that, in character, the contours of residual stresses are close to the contours in the absence of hardening, although deviations from the linear law of distribution are also observed. As the angle of twist increases, the residual stresses increase.

As is known, the twisting moment is equal to twice the volume enclosed under the so-called stress surface \(z=F(x,y)\). In the absence of hardening this surface has the form of a roof. Let us consider the deformation of the stress surface under the influence of hardening.

Methods for constructing it from the results of hardness measurements are described in article [3].

In Fig. 1 the stress surface is represented by stress lines (lines along which \(F=\mathrm{const}\)). Fig. 2 gives longitudinal and transverse sections of the surface for the same cross-section at relative angles of twist \(\Theta\) of 16 and 32 \(1/\mathrm{m}\). The contours of the sections of the stress surface, unlike the “roof,” are no longer rectilinear; the height of the surface increases with increasing \(\Theta\). However, to determine the volume enclosed under the surface, it is possible with sufficient

with sufficient accuracy, replace it by its “roof.” Such a replacement is equivalent to the assumption that the yield point is constant over the cross-section. The volume enclosed under the “roof” is readily calculated. We have

\[ M=b\left(a-\frac{b}{3}\right)F_0, \tag{2} \]

where \(a\) is the larger side of the rectangular cross-section, and \(F_0\) is the height of the “roof,” or the value of the stress function at the center of the cross-section. By definition of \(F\),

\[ F_0=\int_0^{b/2}\tau\,dx. \tag{3} \]

Thus, \(F_0\) is equal to the area of the shear-stress diagram along the smaller semiaxis of symmetry of the section.

From (1) we have

\[ dx=\frac{b}{2g_0}\,dg. \tag{4} \]

Fig. 2. Sections of the stress surface

Therefore

\[ F_0=\frac{b}{2g_0}\int_0^{g_0}\tau\,dg=\frac{b}{2}\,\bar{\tau}, \tag{5} \]

where \(\bar{\tau}\) denotes the mean shear stress. Thus,

\[ M=\frac{b^2}{2}\left(a-\frac{b}{3}\right)\bar{\tau}. \tag{6} \]

The multiplier before \(\bar{\tau}\) is equal to the plastic section modulus \(W_{\mathrm{p}}\); therefore

\[ M=W_{\mathrm{p}}\bar{\tau}. \tag{7} \]

In the absence of hardening (\(\tau=\mathrm{const}\)), (7) reduces to the well-known formula for the plastic torque.

The quantity \(\bar{\tau}\) of a hardening body depends on \(\Theta\). Expanding this dependence will make it possible to relate \(M\) to \(\Theta\), i.e., to obtain the torsion diagram of a rod of the given cross-section.

The relation between \(\bar{\tau}\) and \(g_0\) is a material characteristic and can be constructed from the shear diagram. Having specified \(g_0\), one deter-

divide the area enclosed under the shear diagram from its origin to \(g_0\). Dividing this area by \(g_0\), we obtain the corresponding \(\bar{\tau}\).

From (7) we have

\[ \bar{\tau}=\frac{M}{W_n}. \tag{8} \]

Having torsion diagrams \(M(\Theta)\) for rods of different sizes made of different materials, one can, at different angles of twist, determine from (8) the value of \(\bar{\tau}\), and from the last quantity on the diagram \(\bar{\tau}(g_0)\), the value of the corresponding \(g_0\). In this way it is possible to determine how \(g_0\) depends on the angle of twist, the dimensions, and the material of the specimens.

Fig. 3. Relation between shear on the surface and the relative angle of twist

The investigation showed that, irrespective of the dimensions of the cross section and of the material, \(g_0\) is related to \(\Theta\) by a linear dependence of the form

\[ g=\alpha\Theta \tag{9} \]

In Fig. 3 this dependence is given for specimens of square cross section \(18\times18\) mm made of 1Kh18N9T steel, steel 15, and D1 duralumin. As is clear from this figure, the angular coefficient \(\alpha\) is practically the same for different materials. The dependence of \(\alpha\) on the cross-sectional dimensions, established experimentally, is presented in Fig. 4.

Thus, having the shear diagram, one can construct the diagram \(\bar{\tau}(g_0)\). From the cross-sectional dimensions, \(W_n\) is determined, and from Fig. 4, \(\alpha\). Then, assigning \(\Theta\), from (9) we find \(g_0\), and from \(g_0\)—\(\bar{\tau}\). From (7) we determine the torque. In this way one can construct the torsion diagram of a specimen of rectangular cross section from the shear diagram.

Fig. 4

Figure 5 presents the results of checking this method—the actual torsion diagrams are compared with the calculated ones. All

The specimens, with the exception of the specimen with an \(18 \times 18\) cross section made of steel 15, were brought to failure. The agreement of the diagrams is satisfactory.

Fig. 5. Comparison of actual torsion diagrams with calculated ones.

REFERENCES

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

2. V. V. Dudukalenko, D. D. Ivlev. Izvestiya AN SSSR, OTN, No. 3, 1964.

3. G. D. Del. Investigation of plastic deformation by measuring hardness, Izvestiya TPI, vol. 138, No. 1, 1965.

4. N. M. Belyaev. Strength of Materials. Moscow, 1965.