PLASTIC DEFORMATION OF BRASS UNDER COMPLEX LOADING

V. I. Maksak, G. A. Doshchinsky

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

It has been noted repeatedly [1, 2, 3] that the expanding Huber–Mises cylinders are unsuitable as a representation of the resistance of materials to plastic deformation. However, there are extremely few experimental data devoted to the study of plastic deformations under complex loading along the surface of this cylinder (i.e., at constant stress intensity).

Thus, in [1], two specimens of steel 30KhN3A were subjected to such loading; in the process of complex loading the principal axes of stress retained their direction, while the ratio between them changed. In [3], a specimen of aluminum alloy 14S—T6 was tested in the same way. In both cases, an increase in plastic deformations was observed in the interval of complex loading.

As for complex loading with rotation of the axes of the principal stresses, here one can note only work [4]: “...Of particular interest is the path in which the axes of the stress tensor rotate without a change in the invariant $\sigma_2$...,” i.e., at constant stress intensity. Specimens of “...carefully annealed fine-grained aluminum and $\alpha$-brass...” were used. Unfortunately, the number of specimens tested at constant stress intensity is not indicated. The authors also noted an increase in plastic deformations in the process of complex loading.

Earlier [5], experiments were carried out to clarify the qualitative picture of the change in deformations under complex loading with rotation of the linear compressive stress. Although the investigation method had certain shortcomings, analysis of the results makes it possible to conclude that, under complex loading with rotation of the axes of the principal stresses, plastic deformations increase.

Below are presented the results of an experimental study of the plastic deformations of brass under complex loading.

The specimens were sections of brass tube, each 200 mm long, with an outside diameter of 24 mm and a wall thickness of $1^{+0.18}_{\phantom{+}0}$ mm. All specimens were cut from one tube six meters long. To check the isotropy of the properties, two specimens were tested in tension in the longitudinal and transverse directions. In Fig. 1, the tensile diagram in the axial direction is shown by a solid line, in the transverse direction by a dashed line, and the points give the tensile data for thirteen other specimens whose test program provided for tension at the first stage of loading. It is evident from the graph that the mate-

a series of specimens is sufficiently isotropic and homogeneous. On the basis of these tests the following material characteristics were obtained:

\[ E = 8.45 \cdot 10^3 \ \text{kg/mm}^2,\quad \mu = 0.24,\quad \sigma_s = 15.6 \ \text{kg/mm}^2. \]

Sixteen specimens were tested under complex loading; the test results for two of them appeared doubtful and were repeated. The tests were carried out on a ZDMU-30 complex-loading machine. The specimens were fastened in the machine by means of a special fixture. The strains in the working section of the specimen, 100 mm long, were distributed uniformly. The testing procedure and the instruments for measuring strains are described in [6].

Four series of experiments were conducted. Under simple loading the specimen was loaded up to a certain value of the stress intensity, and then, by a corresponding change in the axial force, internal pressure, and torque, the loading was carried out so that the stress intensity remained constant and equal to its value at the end of simple loading.

Fig. 1

In each series four specimens were tested. For the first specimen, complex loading consisted of rotation of a linear constant tensile stress. For the second, it was rotation of a plane stress state, with the principal stresses not changing in magnitude, and their ratio \(\sigma_2/\sigma_1\) remaining constant and equal to 0.5. In the third specimen the principal axes of stress retained their direction, but the ratio between them changed, while the stress intensity remained constant. The fourth specimen was tested in tension and torsion; in this case the principal stresses rotated and the ratio between them changed; the stress intensity remained constant.

In each series the stress intensity remained constant and was equal, respectively, to \(18 \ \text{kg/mm}^2\), \(20 \ \text{kg/mm}^2\), \(22 \ \text{kg/mm}^2\), and \(24 \ \text{kg/mm}^2\).

Thus, in each series a fairly broad class of loading paths is covered.

Processing the experimental results for all specimens showed that, under complex loading, the strain intensity \(l_i\) increases, although the stress intensity \(\sigma_i\) remains constant.

Let us consider the geometrical picture of the loading process. Instead of the components of the stress tensor \(\sigma_z\), \(\sigma_\theta\), and \(\tau_{\theta z}\), introduce the quantities [7]

\[ S_1=\sqrt{\frac{2}{3}}\left(\sigma_z-\frac{1}{2}\sigma_\theta\right);\quad S_2=\frac{\sqrt{2}}{2}\sigma_\theta;\quad S_3=\sqrt{2}\tau_{\theta z}, \tag{1} \]

which are the projections of the stress vector in rectangular coordinates. Its modulus is proportional to the stress intensity

\[ |\bar P|=\sqrt{S_1^2+S_2^2+S_3^2}=\sqrt{\frac{2}{3}}\sigma_i, \]

and its direction coincides with the direction of the octahedral shear stress.

The loading process can be represented geometrically by the curve described by the end of this vector. The surface of equal stress intensity is a sphere with its center at the origin, and all the loading paths along which the tests were carried out lie on it; moreover, in each series of tests, for each of the four specimens, they have different forms.

The element of arc length of the loading-path curve is expressed as

\[ dS=\sqrt{dS_1^2+dS_2^2+dS_3^2}. \tag{2} \]

According to (1), (2), and the formulas for the plane stress state, the arc length for complex loading with rotation of the axes at constant principal stresses \(\sigma_1\) and \(\sigma_2\) is

\[ S=\frac{\sigma_1-\sigma_2}{\sqrt{2}}\arcsin \frac{\tau_{\theta z}\sqrt{2}}{\sigma_1-\sigma_2}. \tag{3} \]

For loading with preservation of the directions of the stress axes, and for loading in tension with torsion, analogous calculations lead to the expressions:

\[ S=\sigma_i\sqrt{\frac{2}{3}}\arcsin \frac{\sigma_\theta\sqrt{3}}{2\sigma_i}, \tag{4} \]

\[ S=\frac{\tau_i}{\sqrt{3}}\arcsin \frac{\tau_{\theta z}\sqrt{3}}{\sigma_i}. \tag{5} \]

Taking as the loading parameter the arc length according to (3), (4), (5), the change in strain intensity can be represented as a function of this parameter.

Figure 2 presents this dependence for each series of tests. It is seen that all the experimental points are located fairly close to a single straight line. The solid lines drawn on the graphs were obtained by processing the experimental data by the method of least squares.

Taking this circumstance into account, one may assume that, for such a class of complex loadings, the increment of strain intensity is a function of the arc length, independent (or only weakly dependent) on the type of loading at a given stress intensity.

For different values of $\sigma_i$ the slope of the straight line in the graphs of Fig. 2 is almost the same. Of course, it should depend on the magnitude of $\sigma_i$, but because of the large scatter of experimental points for nonferrous metals and the weak hardening of brass in the region of small deformations, this dependence probably lies within the limits of experimental error. Small deviations in the stresses from the loading program, under weak hardening, cause significant deviations in the strain components. It is likely that, in order to clarify this dependence, it is necessary to specify the force, torque, and internal pressure very, very precisely.

On the basis of what has been presented, the following conclusions may be drawn.

Under complex loading of brass, characterized by constancy of the stress intensity, the strain intensity increases.

The increment of the strain intensity as a function of the arc length of the loading path depends only weakly on the type of loading, and this dependence is close to linear.

For determining the dependence of the magnitude of the increment of the strain intensity on the magnitude of the stress intensity for brass and, probably, for other weakly hardening materials, more careful experiments are necessary.

Fig. 2

REFERENCES

1. A. M. Zhukov. On plastic deformations of an isotropic metal under complex loading. Izv. AN SSSR, OTN, No. 12, 1956.

2. A. M. Zhukov. Some features of the neutral-loading curve. Izv. AN SSSR, OTN, No. 8, 1958.

3. I. Marin, L. W. Hu. On the Validity of Assumption Made in Theories of Plasticity for Metals, Trans. ASME, v. 75, No. 6, 1953.

4. U. Dehlinger, I. Diehl, I. Meissner. Das Fließgesetz vielkristalliner Metalle, Z. Naturforschung B., 11-a, No. 1, 1956.

5. V. I. Maksak, G. A. Doshchinskii. Complex loading with rotation of the axes of the stress tensor under linear compression. Izv. TPI, vol. 133, 1965.

6. V. I. Maksak, G. A. Doshchinskii. On a method for investigating plastic deformation under complex loading. Izv. TPI, vol. 147, 1966.

7. A. A. Ilyushin. Plasticity. Publishing House of the Academy of Sciences of the USSR, 1963.