METHODOLOGY AND STUDY OF LARGE PLASTIC DEFORMATIONS UNDER SIMPLE LOADING

V. I. Maksak, G. A. Doshchinskii

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

The experimental study of large plastic deformations under conditions of a complex stressed state is of known importance for solving many technological problems in the theory of plasticity. The most reliable method for creating complex stressed states is the testing of thin-walled tubular specimens by loading them with an axial force, internal pressure, and a torque. In this case, a plane stressed state is realized in the wall of the specimen, which may, with a sufficient degree of accuracy, be regarded as homogeneous. However, the rapid loss of stability of the specimen shape often does not allow large deformations to be achieved in the experiment.

Therefore, attempts have repeatedly been made to develop an experimental method using other specimen shapes. Thus, in [1] it was proposed to create a biaxial compressive stressed state in a cube-shaped specimen by means of a special reverser that decomposes the force of the press into two components. Subsequently this method was improved, which made it possible to study deformations in a cube with an edge of 5 mm under triaxial compression [2].

A very valuable feature of this method is the possibility of studying large deformations under a volumetric stressed state. Its disadvantages include the inevitable extrusion of material between the punches, the inhomogeneity of the stressed state due to friction, and the small dimensions of the specimen, which reduce the accuracy of deformation measurement. The need to measure deformations and to change punches requires complete unloading, which does not allow continuous loading to be carried out. Other methods [3], based on stretching rods of rectangular cross section in the longitudinal direction and compressing them in the transverse direction, have the same disadvantages.

The search for a methodology free from the noted shortcomings inherent in the testing of cubes, and allowing the study of large deformations under a complex stressed state, appears necessary. A possible variant of such a methodology is presented below.

Let us suppose that, under simple loading of a solid circular cylinder and a tubular specimen by an axial force \(P\) and a torque \(M\), the following conditions are satisfied:

1. The deformation is carried out in such a way that, on the base \(l\) of the specimen, the elongation \(\Delta l\) is proportional to the angle of twist \(\varphi\) (or to the circumferential displacements at a distance from the axis equal to the specimen radius \(R\)).

\[ \Delta l = k\varphi R, \tag{1} \]

where \(k\) is the proportionality coefficient.

1. The sections remain plane, and the radii remain straight. Accordingly, in a section there will act only axial normal stresses \(\sigma_\rho\) and shear stresses \(\tau_\rho\), depending only on the radius.

Approximating the flow curve \(\sigma_i - e_i\) by the power function \(\sigma_i = A e_i^n\), we determine the value of the force \(P_1\) and of the moment \(M_1\) for a certain state characterized by the prescribed values \(\Delta l\) and \(\varphi\). From the theory of deformation it follows that

\[ \sigma_\rho = \frac{\sigma_i}{e_i} e_\rho = A e_i^{\,n-1} e_\rho; \qquad \tau_\rho = \frac{\sigma_i}{3e_i}\gamma_\rho = \frac{1}{3} A e_i^{\,n-1}\gamma_\rho, \tag{2} \]

where

\[ e_\rho = \frac{\Delta l}{l} = \frac{k\varphi R}{l} = k\Theta R, \qquad \gamma_\rho = \frac{\varphi\rho}{l} = \Theta\rho, \]

\(\Theta\) is the angle of twist per unit length,

\[ e_i = \sqrt{e_\rho^2 + \frac{1}{3}\gamma_\rho^2}. \]

The force and moment of a specimen of radius \(R\) are determined by

\[ P_1 = 2\pi \int_0^R \sigma_\rho \rho\,d\rho; \qquad M_1 = 2\pi \int_0^R \tau_\rho \rho^2\,d\rho. \tag{3} \]

Expanding expressions (2), substituting into (3), and integrating, we obtain

\[ P_1 = \frac{6\pi A k\Theta^n R^{n+2}}{n+1} \left[ \left(k^2+\frac{1}{3}\right)^{\frac{n+1}{2}} - k^{n+1} \right], \]

\[ M_1 = \frac{6\pi A\Theta^n R^{n+3}}{(n+1)(n+3)} \left[ \left(k^2+\frac{1}{3}\right)^{\frac{n+1}{2}} \left(\frac{1}{3}n+\frac{1}{3}-2k^2\right) +2k^{n+3} \right]. \tag{4} \]

Let us take another cylindrical specimen with radius \(r=\alpha R\), where \(\alpha\) is less than unity. If it is loaded so that condition (1) is satisfied, analogous calculations give the value of the force \(P_2\) and moment \(M_2\) for the same \(\Delta l\) and \(\varphi\).

\[ P_2 = \frac{6\pi A k\Theta^n R^{n+2}}{n+1} \left[ \left(k^2+\frac{1}{3}\alpha^2\right)^{\frac{n+1}{2}} - k^{n+1} \right], \]

\[ M_2 = \frac{6\pi A\Theta^n R^{n+3}}{(n+1)(n+3)} \left[ \left(k^2+\frac{1}{3}\alpha^2\right)^{\frac{n+1}{2}} \left(\frac{1}{3}n\alpha^2+\frac{1}{3}\alpha^2-2k^2\right) +2k^{n+3} \right]. \tag{5} \]

For a tubular specimen with outer radius \(R\) and inner radius \(r=\alpha R\), for the same values \(\Delta l\) and \(\varphi\) and under condition (1), calculation of the force \(P\) and moment \(M\) leads to the expressions

\[ P = \frac{6\pi A k\Theta^n R^{n+2}}{n+1} \left[ \left(k^2+\frac{1}{3}\right)^{\frac{n+1}{2}} - \left(k^2+\frac{1}{3}\alpha^2\right)^{\frac{n+1}{2}} \right], \tag{6} \]

\[ M=\frac{6\pi A\Theta^n R^{n+3}}{(n+1)(n+3)} \left[ \left(k^2+\frac{1}{3}\right)^{\frac{n+1}{2}} \left(\frac{1}{3}n+\frac{1}{3}-2k^2\right) - \left(k^2+\frac{1}{3}\alpha^2\right)^{\frac{n+1}{2}} \left(\frac{1}{3}n\alpha^2+\frac{1}{3}\alpha^2-2k^2\right) \right]. \]

If, for identical values of \(\Delta l\) and \(\varphi\), for two solid specimens and for a thin-walled tubular specimen, the differences \((P_1-P_2)\) and \((M_1-M_2)\) are determined for the solid specimens, then they will give values of the force and moment that fully coincide with (6), i.e., with the force and moment in the tubular specimen.

This permits the following supposition. Since in solid specimens, at points with equal coordinates \(\rho\), the deformed state is identical, then, according to (2), the stressed state will also be identical. This means that if the stress-distribution diagrams \(\sigma_\rho\) and \(\tau_\rho\) over the radius for the large and small specimens are superposed, then in their common part they will coincide at every point. Consequently, the axial force and moment in the small specimen and the parts of the force and moment of the large specimen produced by the stresses \(\sigma_\rho\) and \(\tau_\rho\) in the circle of radius \(r\) are equal. The other parts of the force and moment of the large specimen, due to the stresses in the remaining part in the “tube” with wall thickness \(t=R-r\), are determined as the differences \((P_1-P_2)\) and \((M_1-M_2)\). And these differences coincide with the force and moment in the tubular specimen.

Thus, the essence of the method is as follows. Two cylindrical specimens of different diameters are loaded by an axial force and a twisting moment in such a way that they vary according to the same law for both specimens. The diagrams \(P-\Delta l\) and \(M-\varphi\) are constructed. For identical angles of twist \(\varphi_i\), the values of the moments are subtracted and the \(M-\varphi\) diagram of the “tube” is constructed. From the loading law relating force and moment, to each value \(M_i\) there corresponds a definite value of the force \(P_i\), both for the large and for the small specimen. For identical values \(\Delta l_i\), the values of the forces \(P_i\) are subtracted and the \(P-\Delta l\) diagram of the “tube” is constructed. Further processing of the experimental results is carried out by the usual methods.

Since the force and moment in the tube vary in the same way as for solid specimens, the program is calculated in the usual manner, but in doing so it is necessary to take into account the change in the specimen dimensions. From the condition of constancy of volume under plastic deformation, the current radius \(R\) and the wall thickness of the “tube” \(t\) are determined:

\[ R=R_0\sqrt{\frac{l_0}{l}};\qquad t=t_0\sqrt{\frac{l_0}{l}}, \tag{7} \]

where \(l\) is the current gauge length of the specimen, initially equal to \(l_0\);
\(R_0\) is the initial mean radius of the “tube.”

The known formulas for thin-walled tubular specimens, with allowance for (7), give expressions for the true stresses:

\[ \bar{\sigma}_z=\frac{P}{2\pi Rt} =\frac{P}{2\pi R_0t_0}\cdot\frac{l}{l_0}; \qquad \bar{\tau}_{\theta z}=\frac{M}{2\pi R^2t} =\frac{M}{2\pi R_0^2t_0} \left(\frac{l}{l_0}\right)^{\frac{3}{2}}. \tag{8} \]

The true longitudinal strain and the shear angle are

\[ \bar{e}_z=\ln(1+e_z); \qquad \gamma=\frac{\varphi R}{l} =\frac{\varphi R_0}{l_0(1+e_z)^{\frac{3}{2}}} =\frac{\gamma_0}{(1+e_z)^{\frac{3}{2}}}. \tag{9} \]

True shear (according to Nadai)

\[ \bar{g}=\ln \left[1+\frac{\gamma}{2}\left(\gamma+\sqrt{4+\gamma^{2}}\right)\right] =\ln \left[1+\frac{\gamma_0^{2}+\gamma_0\sqrt{4(1+e_z)^3+\gamma_0^2}}{2(1+e_z)^3}\right]. \tag{10} \]

Let it be required to realize the loading

\[ \sigma_1=m\sigma_2. \tag{11} \]

From the formulas for the plane stress state

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

we obtain, taking into account that \(\sigma_\theta=0\),

\[ \sigma_z=\sigma_2(1-m);\qquad \tau_{\theta z}=\sigma_2\sqrt{m}. \tag{12} \]

The force and moment in the “tube,” according to (8) and (12), are expressed as

\[ P=2\pi R_0 t_0\sigma_2(1-m)\frac{l_0}{l};\qquad M=2\pi R_0^2 t_0\sigma_2\sqrt{m}\left(\frac{l_0}{l}\right)^{\frac{3}{2}}. \tag{13} \]

Comparing expressions (13), we obtain

\[ P=\frac{1-m}{R_0\sqrt{m}}\sqrt{\frac{l}{l_0}}\cdot M. \tag{14} \]

Thus, the force and moment under simple loading and large deformations vary nonproportionally; therefore the loading program cannot be calculated in advance. In the experiments described below, loading was carried out in small steps, and each subsequent step was calculated in the course of the test. This was done as follows. For the next loading step, the value of the moment was prescribed.

The current length \(l\) and the increment in length \(\Delta l\) at the preceding step are known. Assuming that, at the step being calculated, the increment \(\Delta l\) will be the same as at the preceding one, the estimated value of \(l\) is determined. Then, by formula (14), the magnitude of the force \(P\) was calculated. If by the end of the loading step it turned out that \(\Delta l\) was much greater than assumed, the loads were corrected.

On the basis of the procedure set forth above, tests were carried out on steel specimens under conditions of simple loading at various ratios of the principal stresses. The loading program provided for tests for six types of stress states with ratios between the principal stresses \(\sigma_2/\sigma_1\) equal to:

\[ -1;\quad 0;\quad -0.2;\quad -0.4;\quad -0.6;\quad -0.8. \]

The loading paths obtained after processing the experimental results, in the coordinates \(\tau_{\theta z}\)—\(\sigma_z\), are shown in Fig. 1. The solid lines correspond to the planned loading paths, and the points to those obtained.

For all specimens and for each loading step, the values of the intensities of the true stresses and strains were calculated. From these data, flow curves were plotted, which in the coordinates \(\sigma_i\)—\(e_i\) are shown in Fig. 2. All experimental points lie fairly close to the curve obtained under pure torsion \((\sigma_2/\sigma_1=-1)\). The greatest discrepancy is shown by the curves obtained at ratios \(\sigma_2/\sigma_1\) equal to \(-0.2\) and \(-0.4\), at a strain \(e_i\approx 12\%\).

This discrepancy is approximately \(5.5\%\). However, both curves are located on different sides of the curve corresponding to tors-

...direction, at approximately the same distance, while all the others are located between them.

The curves for the stress and strain components are presented in Figs. 3 and 4.

On the basis of the foregoing, certain conclusions may be drawn concerning the methodology and the results of the study of large deformations.

The loading paths obtained as a result of processing the experimental data coincided completely with the loading paths that it was intended to realize. This indicates that the assumptions taken as the basis of the methodology being developed are satisfied with a sufficient degree of accuracy.

The slight scatter of the points on all the graphs presented indicates the high stability of the results of experiments carried out by means of this methodology.

Fig. 1

Since for one loading path it is necessary to test two specimens, the time spent on the experiment and its processing increases. The calculation of the loading program during testing is inconvenient. However, this is compensated for by the simple design of the specimens, the incomparably simpler manufacturing technology, the low cost, and the possibility of realizing loading under large shear strains, which the other methodology does not allow.

Fig. 2

Unlike the cube-testing method, the use of this methodology makes it possible to measure deformations with a necessary, predetermined degree of accuracy, since the length of the specimen can be made to a value corresponding to the required accuracy of deformation measurement.

Fig. 3

Fig. 4

This methodology can also be used to study creep phenomena at large deformations.

The results of the experiments carried out show that, with the degree of accuracy provided by the methodology and which occurred in the experiments, one may assume the existence of a single flow curve in the coordinates $\sigma_i — e_i$ also for large plastic deformations.

A criterion of the reliability of the methodology is the obtaining of a single flow curve at small deformations under simple loading, as is also shown by tests of thin-walled tubular specimens.

The methodology presented makes it possible, for mixed (tensile–compressive) stress states, to achieve deformations of considerably greater magnitude than those limited by the loss of stability of the deformation shape of thin-walled tubular specimens. The limiting values here are governed by the onset of nonuniform local deformation during the formation of a neck on the specimen. The latter was observed the earlier, the closer the stress state under study was to linear tension.

All of the above makes it possible to regard this methodology as suitable also for studying plasticity under complex loading, and makes it a useful supplement to existing methods.

REFERENCES

1. I. N. Vinogradov, Yu. I. Yagn. A cruciform reversor for testing materials under biaxial compression. Factory Laboratory, No. 8. Metallurgizdat, 1955.

2. G. A. Solntseva. Improvement of methods for testing small specimens under triaxial and biaxial compression under conditions of simple and complex loading. Proceedings of LPI, No. 211, Publishing House of the Leningrad Polytechnic Institute, 1960.

3. I. A. Chalitskii. Resistance of Metals to Plastic Deformation. Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1962.