STUDY OF LARGE PLASTIC DEFORMATIONS UNDER COMPLEX LOADING

V. I. MAKSAK, G. A. DOSHCHINSKII

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

At present, experimental and theoretical studies of the plastic deformation of metals under various conditions are being carried out intensively.

However, the available data give only an approximate idea of the character of plastic deformation under conditions of arbitrary loading. Whereas for small deformations there exists a fairly considerable amount of experimental data, this cannot be said of large plastic deformations. Experimental data on the study of large deformations under conditions of complex loading are practically absent. At the same time, such investigations are necessary for solving many technological problems of the theory of plasticity.

Analysis of the experimental data shows that the greatest discrepancies between existing theories and experiment occur under complex loading with rotation of the axes of the stress tensor. Below, on the basis of a method proposed by the authors, which, in comparison with ordinary investigations, makes it possible to study the plastic behavior of metals at substantially increased values of deformation [1], the results are presented of an experimental investigation of plastic deformations under complex loading with constant stress intensity.

The loading was carried out in the following manner. First, large and small specimens were twisted through the same angle. For this angle, from the measured moments $M_1$ and $M_2$, the moment for the “tube” was calculated as $M_0 = M_1 - M_2$, and the corresponding stress intensity was determined. Then the tensile force and the moment in each of the two solid specimens were changed in such a way that loading $\sigma_i = \mathrm{Const}$ was ensured in the “tube.”

The stresses $\bar{\sigma}_z$ and $\bar{\tau}_{\theta z}$ are related to the stress intensity by the following dependence

$$ \sigma_i = \sqrt{\bar{\sigma}_z^2 + 3\bar{\tau}^2}. $$

Substituting the values of the true stresses $\bar{\sigma}_z$ and $\bar{\tau}_{\theta z}$ [1], we obtain

$$ \sigma_i = \sqrt{ \frac{P^2}{4\pi^2 R_0^2 t_0^2}\left(\frac{l}{l_0}\right)^2 + 3\frac{M^2}{4\pi^2 R_0^4 t_0^2}\left(\frac{l}{l_0}\right)^3 }. \tag{1} $$

At the beginning of complex loading \(P=0\), therefore

\[ \sigma_i=\frac{M}{2\pi R_0^2 t_0}\sqrt{3\left(\frac{l}{l_0}\right)^3}. \tag{2} \]

Substituting (2) into (1) and simplifying, we obtain the relation between \(M_0\), \(M\), and \(P\) that ensures complex loading with \(\sigma_i=\mathrm{Const}\):

\[ P=\frac{\sqrt{3}}{R_0}\sqrt{\frac{l}{l_0}}\sqrt{M_0^2-M^2}. \tag{3} \]

Under complex loading, the program was also calculated during the test process, i.e., as in simple loading [1].

Fig. 1

Five pairs of specimens were tested. In each experiment the stress intensity remained constant and was equal, respectively, to 83, 55, 69, 73, and 78 kg/mm\(^2\). The loading paths obtained as a result of processing the experimental data, in the coordinates \(\tau_z\)—\(\sigma_z\), are presented in Fig. 1. The solid lines correspond to the planned loading paths. The agreement between the planned and obtained loading paths indicates good stability of the results under complex loading as well.

In these experiments, just as for small deformations [2, 3, 4], a systematic increase in plastic deformations was observed in the complex-loading segment. If, as a measure of the change in the stress state for loading with \(\sigma_i=\mathrm{Const}\), one takes, as for small deformations [2—4], the arc length \(S\) of the loading path in the coordinates of the stress vector, then the change in the intensity of plastic deformations in the complex-loading segment can be related to the arc length.

Fig. 2

This dependence is presented in Fig. 2 for all the tested pairs of specimens. For the first “tube,” tested at \(\sigma_i=83\) kg/mm\(^2\), the experimental points are located higher than for the other “tubes,” and

are grouped rather close to a single straight line, which qualitatively corresponds to the results of experiments at small strains. For other “tubes,” at the beginning of the complex-loading segment the additional plastic strain increased by approximately up to 0.1% more intensely than over the remaining part of the segment. However, in the graph one can observe that after this the experimental points for “tubes” No. 2 ($\sigma_i = 55\ \text{kg}/\text{mm}^2$) and No. 5 ($\sigma_i = 78\ \text{kg}/\text{mm}^2$) lie on the corresponding straight lines.

A characteristic feature of these experiments is the insignificant increase in plastic strains during the period of complex loading in comparison with the preliminary strain. Thus, for “tube” No. 1 ($\sigma_i = 83\ \text{kg}/\text{mm}^2$), complex loading, which changed the stress state from shear to pure tension, caused an increase in the intensity of plastic strains by 0.4%, while the preliminary strain was about 47%.

Thus, the increment of strain amounts to less than one percent of the preliminary strain. Naturally, on the flow curve in the coordinates $\sigma_i - e_i$, complex loading with $\sigma_i = \mathrm{Const}$ will correspond to a horizontal plateau whose length is small in comparison with the total strain.

In the literature the opinion is expressed [5] that under complex loading there is a tendency for the resistance of the material to plastic deformation to decrease. Therefore one may assume that under simple loading the resistance of the material will be greatest, and under complex loading of the type $\sigma_i = \mathrm{Const}$ it will be smallest. Consequently, for other types of complex loading, when $\sigma_i$ increases, the curve $\sigma_i - e_i$ will be located between the horizontal plateau and the flow curve under simple loading. And since complex loading with constant stress intensity at large strains causes only an insignificant increase in plastic strains, and the curve $\sigma_i - e_i$ with respect to stresses differs little from the curve under simple loading, other loading paths with increasing $\sigma$ will also give a flow curve that differs little from the curve for simple loading.

This makes it possible to conclude that, at large strains, complex loading has little effect on the flow curve in the coordinates $\sigma_i - e_i$.

The qualitative agreement of the experimental results at large strains with the data obtained at small strains, and the correspondence of the planned loading paths to those obtained in the experiments, testify to the possibility of applying the method to investigations of plasticity also under complex loading.

LITERATURE

1. V. I. Maksak, G. A. Doshchinskii. Methodology and investigation of large plastic strains under simple loading, this collection.

2. V. I. Maksak, G. A. Doshchinskii. Experimental investigation of complex loading with rotation of the axes of the stress tensor under uniaxial compression. Izv. TPI, vol. 133, 1965.

3. V. I. Maksak, G. A. Doshchinskii. Plastic deformation of brass under complex loading. Izv. TPI, vol. 147, 1966.

4. V. I. Maksak. Plastic deformation of steel under complex loading. Izv. TPI, vol. 139, 1966.

5. V. S. Lenskii. Experimental verification of the basic postulates of the general theory of elastic-plastic deformations. In: Problems of the Theory of Plasticity. Publishing House of the Academy of Sciences of the USSR, 1961.