Experimental Study of Plastic Deformation under “Neutral Loading”

A. M. Koreneva, G. A. Doshchinskii

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

One of the fundamental concepts in the theory of plasticity is the concept of a loading surface separating the regions of elastic and plastic states. The change of this surface in the process of deformation corresponds to the laws of hardening of the material. It is assumed here that loading processes represented by the motion of the loading point along the indicated surface are not accompanied by an increment of plastic deformation (neutral loading). A number of experimental works [2–5] and others have in recent years been devoted to studying the shape of the loading surface; from these it follows that a surface analytically represented as a function only of the second invariant of the stress deviator is not a surface of neutral loading. However, the laws of additional plastic deformation during motion along a loading surface of the indicated type are still not clear at present. At the same time, loading with $\sigma=\mathrm{const}$ may be regarded as an alternative to simple loading, in which the increment of plastic deformation is determined not by the increment of stresses but only by the influence of the “complexity” of the loading process in the general case. Knowledge of the laws of this increment may contribute to the solution of the general problem of complex loading in the theory of plasticity.

In previously conducted studies [6, 7] it was established that the increment of the intensity of deformation proves to be approximately proportional to the length of the arc of the loading trajectory on the surface of a hypersphere ($\sigma_i=\mathrm{const}$) in the five-dimensional deviatoric stress space. The experiments described below were a continuation of these studies. The tests were carried out on thin-walled tubular specimens made of steel U8, produced from a solid bar by turning and grinding, and subjected to recrystallization annealing.

The diameter of the specimens was 30 mm, with a wall thickness of 1 mm. The test was carried out on a ZMDU-30 machine under various types of biaxial tension. To measure longitudinal and transverse deformation, indicators with scale divisions of 0.01 and 0.001 mm were used.

The tests were programmed according to stresses specified in the plane of the two-dimensional stress vector $\bar S$

\[ \bar S = S_1 \bar i + S_2 \bar j \]

with components

\[ S_1=\sqrt{\frac{2}{3}}\left(\sigma_z-\frac{1}{2}\sigma_\theta\right),\qquad S_2=\frac{\sqrt{2}}{2}\sigma_\theta . \]

From the values of the axial and circumferential strains obtained in the experiment, the components of the strain vector were calculated:

\[ \vec{\Э}=\Э_1\vec{i}+\Э_2\vec{j},\qquad \Э_1=\sqrt{\frac{3}{2}}\,e_z,\quad \Э_2=\sqrt{2}\left(e_\theta+\frac{1}{2}e_z\right). \]

The moduli of these vectors, accurate to a constant factor, correspond to the intensities of stresses and strains:

\[ |S|=\sqrt{\frac{2}{3}}\,\sigma_i,\qquad |\Э|=\sqrt{\frac{3}{2}}\,e_i . \]

Fig. 1

Four specimens were tested in tension in the longitudinal and circumferential directions in order to verify the properties of isotropy; from the data of these experiments, the mean values of the modulus of elasticity, Poisson’s ratio, and yield stress were determined.

In the main tests, each specimen was brought by tension into the plastic region up to a specified value of the intensity

stresses of 30, 35, 40, 45, 50, 55, 57, and 60 kg/mm², and then the longitudinal force and internal pressure were varied so that the stress intensity remained constant.

For the loading segment under consideration, with constant intensity, the curvature of the trajectory is constant in an experiment, but differs for each specimen. This makes it possible to establish the dependence of the increments of plastic strains on the curvature of the loading trajectory. Another possible parameter for these conditions—the length of the arc of the loading path—can be determined by the expression [6]

\[ S = \sigma_i \sqrt{\frac{2}{3}} \arcsin \frac{\sigma_\theta \sqrt{3}}{2\sigma_i}. \]

The test results are presented in Figs. 1 and 2. Figure 1 shows the vectors of plastic strain at individual points of the loading trajectories (the elastic components of strain are excluded). The experiments

Fig. 2

show that, when moving along a surface of constant intensity, there occurs a slight but systematic increase in plastic strains.

The increment of the intensity of plastic strains, as is seen from Fig. 2, depends both on the length of the loading path and on its curvature.

As in the preceding experiments [6], an approximately proportional increase in the increment as a function of the length of the loading path was observed at first; however, subsequently the increase in strains decreased noticeably.

It is likely that in the preceding experiments, where the motion was carried out along more restricted paths, an analogous state was not reached.

As the radius of curvature of the path increases, the intensity of the increase in plastic strains increases; moreover, this increase proves to be inversely dependent on the degree of hardening of the material.

At the same time, the lag properties were studied in the work. The experiments basically confirm the proposition that the principle of lag will not be satisfied on paths of the given type. From the proposition that under neutral loading the strain remains elastic, there follow the collinearity of the stress and elastic-strain vectors on this segment and, correspondingly, the invariance in modulus and the collinearity of the plastic-strain vectors at different points of one path.

The magnitude of the lag trace was determined on the loading path of one of the specimens, consisting of three segments: tension, a straight segment with a bend of \(30^\circ\), and a segment of constant curvature. The lag trace approximately corresponds to the length of the second segment of this test. The length of the path of constant curvature was incomparably greater than the lag trace. From Fig. 1 it is seen that the plastic-strain vectors tend toward the direction of the active loading process and definitely do not coincide with the tangents to the path of neutral loading over a length several times greater than the lag trace. Nevertheless, a weak but systematic manifestation of lag properties is also observed for this type of loading.

Consequently, during motions along paths of constant stress intensity, a certain change is observed both in the scalar and in the vector properties of the material.

REFERENCES

1. A. A. Ilyushin. On the relation between stresses and small strains in continuum mechanics. PMM, issue 6, No. 18, 1954.

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

3. Hu L. W., Marin J. Anisotropic loading functions for combined stress in the plastic range. J. Appl. Mech. v. 22, No. 1, 1955.

4. S. S. Gill, I. Parker. Plastic stress-strain relations hips-some experiments on the effect of loading path and loading history. J. Appl. Mech. v. 2, No. 1, 1959.

5. G. B. Talyshov. The yield and fracture limit of low-carbon steel in the case of simple and complex loading. Izv. AN SSSR, “Mechanics and Mechanical Engineering,” No. 6, 1961.

6. G. A. Doshchinsky, V. I. Maksak. Plastic deformation of steel under complex loading with constant stress intensity. “Mechanics of Solids,” No. 3, Moscow, 1966.

7. G. A. Doshchinsky, V. I. Maksak. Experimental study of plastic strains under complex loading. “Mechanics of Solids,” No. 3, Moscow, 1966.