PLASTIC DEFORMATION OF STEEL UNDER COMPLEX LOADING

V. I. Maksak

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

One of the fundamental questions in the theory of plasticity is the question of the relationship between stresses and strains. However, for arbitrary loading this question has not yet been solved and at present the accumulation of experimental and theoretical facts is continuing.

Below are presented the results of experimental data and their analysis in the investigation of plastic deformation under complex loading.

Thin-walled tubular specimens of steel 40, 30 mm in diameter and with a wall thickness of \(1^{\pm 0.05}\) mm, were tested. After annealing, the material of the specimens had a yield point \(\sigma_s = 36\ \text{kg/mm}^2\), and an elastic modulus \(E = 2.1 \cdot 10^4\ \text{kg/mm}^2\).

The experiments were carried out on a ZDMU-30 machine by loading the specimens with a tensile force, internal pressure, and torque. The strains were measured with a specially manufactured device.

Nine specimens were tested in such a way that, over the range of complex loading, the stress intensity \(\sigma_i\) remained constant. From the formulas for the plane stress state it follows that the stresses \(\sigma_z\) and \(\sigma_\theta\) must vary as functions of the principal stresses and the shear stress \(\tau_{\theta z}\) according to the expressions:

\[ \sigma_z=\frac{\sigma_1+\sigma_2}{2}+\sqrt{\frac{(\sigma_1-\sigma_2)^2}{4}-\tau_{\theta z}^2}, \tag{1} \]

\[ \sigma_\theta=\frac{\sigma_1+\sigma_2}{2}-\sqrt{\frac{(\sigma_1-\sigma_2)^2}{4}-\tau_{\theta z}^2}. \]

The first four specimens were tested so that the principal stresses remained constant in magnitude, but rotated during the course of the experiment. The ratios \(\sigma_2/\sigma_1\) for each specimen remained constant and were respectively: \(0;\ 0.25;\ 0.5;\ 0.75\). The stress intensity was \(\sigma_i = 50\ \text{kg/mm}^2\). In all cases, over the range of complex loading, \(e_i\) increased, and did so differently for different ratios between the stresses.

The fifth specimen was tested only in tension with internal pressure at the same stress intensity. The principal stresses retained their direction, while the ratio between them changed.

The sixth specimen was tested in tension and torsion. Here the principal stresses rotated and the ratio between them changed. For these two specimens the strain intensity also increased.

Thus, in the process of complex loading with constant stress intensity, the strain intensity increased.

Let us consider the geometrical picture of the loading process. For this purpose, instead of $\sigma_z$, $\sigma_\theta$, and $\tau_{z\theta}$ we introduce the quantities [1]

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

They are the projections of the stress vector in rectangular coordinates; moreover, its modulus coincides, up to a constant factor, with the stress intensity, and its direction coincides with the direction of the octahedral shear stress.

\[ |\overline{P}|=\sqrt{\frac{2}{3}}\,\sigma_i, \tag{3} \]

Any loading process can be represented by a curve described by the end of this vector. A sphere with its center at the origin of coordinates and with radius equal to $|\overline{P}|$ represents a surface of equal stress intensity.

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

\[ dS=\sqrt{dS_1^2+dS_2^2+dS_3^2}, \tag{4} \]

and, according to (1), (2), (3), the arc length is

\[ S=\frac{\sigma_1-\sigma_2}{\sqrt{2}}\arcsin \frac{\tau_{\theta z}\sqrt{2}}{\sigma_1-\sigma_2}. \tag{5} \]

For loading with preservation of the direction of the stress axes and for loading in tension with torsion, analogous arguments lead to the expressions

\[ S=\sigma_i\sqrt{\frac{2}{3}}\arcsin \frac{\sigma_\theta\sqrt{3}}{2\sigma_i}, \tag{6} \]

\[ S=\frac{\sigma_i}{\sqrt{3}}\arcsin \frac{\tau_{\theta z}\sqrt{3}}{\sigma_i}. \tag{7} \]

The experiments cited above are characterized by the fact that the loading curves for all specimens are located on a sphere, but have different forms. If the arc length is taken as the parameter, then the change in strain intensity can be represented as a function of this parameter. Figure 1 presents a graph of this dependence. It is seen that all the experimental points, obtained under different conditions, are located close to one straight line.

Taking this circumstance into account, it may be assumed that, for this class of complex loadings, the increment of strain intensity is a function of the arc length of the loading path and does not depend on the type of loading.

Taking this dependence, in the first approximation, to be linear, one may write

\[ de_i=n\,dS, \tag{8} \]

where $n$ is the proportionality coefficient.

This assumption makes it possible to reduce the required number of experiments and to bring them down to the simplest ones, for example, to a tensile test with internal pressure.

Fig. 1.

The coefficient \(n\) in (8) must depend on the value of \(\sigma_i\) at which the experiment is carried out. To determine this dependence, three more specimens, Nos. 7, 8, 9, were tested in tension and internal pressure. The stress intensity was respectively 40, 55, and 60 kg/mm\(^2\). Figure 1 presents the results of these tests. With increasing \(\sigma_i\), the coefficient \(n\) increases. This dependence is shown in Fig. 2.

Fig. 2.

Consider an arbitrary loading path—Fig. 3—and take on it two close points \(a\) and \(b\). Let us divide it into two paths. The first represents loading along the surface of a sphere, and the second—simple loading.

Next, it is assumed that the increment of plastic strain on \(ab\) is equal to the sum of the increments on segments 1 and 2. Of course, this simplifies the real picture, but similar attempts at subdivision have been made earlier [2].

Fig. 3.

As a first approximation one may assume that on segment 1 the strain increment is determined by expression (8), while \(n\) depends linearly on \(\sigma_i\), Fig. 2:

\[ n = k(\sigma_i - \sigma_s), \tag{9} \]

where \(k\), under the assumptions made, is regarded as constant for the given material.

According to (8) and (9), the total strain increment is

\[ de_i = \varphi'(\sigma_i)\,d\sigma_i + k(\sigma_i - \sigma_s)\,dS, \tag{10} \]

where \(\varphi'(\sigma_i)\) is the reciprocal of the plastic modulus of the stress–strain curve in tension.

This expression is the hardening condition for loading paths \(d\sigma_i > 0\).

Using the hypothesis of similarity of the deviators of stresses and increments of plastic strain, the components of the strain increments may be represented in the form:

\[ de_x=\frac{3(\sigma_x-\sigma)}{2\sigma_i}\left[\varphi'(\sigma_i)d\sigma_i+k(\sigma_i-\sigma_s)dS\right] +\frac{1}{E}\left[d\sigma_x-\mu d(\sigma_y-\sigma_z)\right]. \]

\[ d\gamma_{xy}=\frac{3\tau_{xy}}{\sigma_i}\left[\varphi'(\sigma_i)d\sigma_i+k(\sigma_i-\sigma_s)dS\right] +\frac{d\tau_{xy}}{G}. \]

Here \(de_x \ldots d\gamma_{xy}\ldots\) are the components of the strain increments,
\(\sigma\) is the mean normal stress,
\(\sigma_x \ldots \tau_{xy}\ldots\) are the components of the stress tensor,
\(E, G, \mu\) are elastic constants.

REFERENCES

1. A. A. Ilyushin. Plasticity, Publishing House of the USSR Academy of Sciences, 1963.
2. V. D. Klyushnikov. On the laws of plasticity for a particular class of loading paths. PMM, vol. XXI, no. 4, 1957.