ON A METHOD FOR STUDYING PLASTIC DEFORMATION UNDER COMPLEX LOADING

V. I. Maksak, G. A. Doshchinskii

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

The main task of the general theory of plasticity is to establish the relation between stresses and irreversible deformations of material bodies. This problem may be considered satisfactorily solved only for a proportional change of stresses, or one close to it \[1, 2\]. Under loading with a change of stresses that differs greatly from proportional loading, and also when the axes of the stress tensor are rotated, none of the existing theories gives the necessary agreement with experimental data. Therefore, in recent years extensive experimental and theoretical studies have been carried out on the plastic behavior of a material under various laws of stress variation.

Of definite interest is complex loading in which the principal stresses remain constant in magnitude but change their direction relative to the element of the body. In the literature there are almost no experimental data on this type of complex loading, and the available information indicates the greatest discrepancy with existing theories. Systematic experimental studies under such loading will undoubtedly contribute to achieving the general goal of the theory of plasticity.

The most developed method for investigating plastic deformations under a complex stress state is the study of the deformation of a thin-walled tubular specimen under the simultaneous action of tension, internal pressure, and torsion. Existing machines for complex loading make it possible to apply all three types of loading simultaneously and independently of one another. The stresses arising in the specimen are calculated by the following formulas:

\[ \sigma_z = \frac{P}{2\pi Rt} + \frac{qP}{2t}, \qquad \sigma_\theta = \frac{qR}{t}, \qquad \sigma_r \approx 0, \qquad \tau_{\theta z} = \frac{M}{2\pi R^2 t}, \tag{1} \]

where \(P\), \(q\), and \(M\) are, respectively, the tensile force, internal pressure, and torque. \(R\) and \(t\) are the mean radius and wall thickness of the specimen. By specifying \(P\), \(q\), and \(M\), any plane stress state can be obtained; and by changing them according to a definite law, it is possible to achieve such loading that the principal stresses will rotate relative—

relative to the element of the specimen, while remaining constant in magnitude. The angle of rotation of the principal stresses is then determined from the formula

\[ \tg 2\varphi = -\frac{2\tau_{\theta z}}{\sigma_z-\sigma_\theta}. \tag{2} \]

On the faces of the element cut out from the wall of the specimen there act the normal stresses \(\sigma_z\) and \(\sigma_\theta\) and the shear stress \(\tau_{\theta z}\). The principal stresses are determined by the formula

\[ \sigma_{1,2}=\frac{\sigma_\theta+\sigma_z}{2}\pm \frac{1}{2}\sqrt{(\sigma_\theta-\sigma_z)^2+4\tau_{\theta z}^{\,2}}, \tag{3} \]

Considering \(\tau_{\theta z}\) independent, and \(\sigma_1\) and \(\sigma_2\) constant, we determine \(\sigma_\theta\) and \(\sigma_z\). Solving the system of equations (3), we obtain

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

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

Substituting in these expressions the values of \(\sigma_\theta\) and \(\sigma_z\) from equalities 1, we obtain expressions relating \(P\), \(q\), and \(M\),

\[ P=\pi Rt\left[\frac{\sigma_1+\sigma_2}{2}+\frac{3}{2}\sqrt{(\sigma_1-\sigma_2)^2-\frac{M^2}{\pi^2R^4t^2}}\right], \]

\[ q=\frac{t}{R}\left[\frac{\sigma_1+\sigma_2}{2}-\frac{1}{2}\sqrt{(\sigma_1-\sigma_2)^2-\frac{M^2}{\pi^2R^4t^2}}\right]. \tag{5} \]

In the right-hand sides of these expressions all quantities are constant, except the twisting moment.

The study of deformation during rotation of the axes and with an unchanged magnitude of the stresses can be carried out as follows. The specimen is brought into the plastic region with prescribed values of \(\sigma_1\) and \(\sigma_2\) by proportional loading with a longitudinal force and internal pressure. Then additional loading is carried out in small steps of twisting moment. To each value of the latter there correspond definite values of the axial force and internal pressure, which are calculated from formulas (5). In this case the loading path represents a broken line. However, if the loading is carried out in small steps, and taking into account the local stability of the deformation process [3], it may be assumed that the results obtained under stepwise loading will not differ greatly from the results that would be obtained under continuous variation of \(M\), \(P\), and \(q\).

The change of \(\sigma_z\), \(\sigma_\theta\), and \(\tau_{\theta z}\) during rotation of the principal stress axes is well represented by Mohr circles. Figure 1 shows the Mohr circle for arbitrary values of \(\sigma_1\) and \(\sigma_2\). The arrows indicate the directions of change of the stresses \(\sigma_z\), \(\sigma_\theta\), and \(\tau_{\theta z}\). At the beginning of complex loading the shear stress is absent and the principal stresses are equal, respectively, to \(\sigma_z\) and \(\sigma_\theta\). Under loading by a twisting moment, \(\sigma_\theta\) must increase, and \(\sigma_z\) decrease. In this case each loading step will correspond to a quite definite angle of rotation of the principal stresses and, when \(\tau_{\theta z}\) changes from zero to its maximum value \((\sigma_1-\sigma_2)/2\), the principal stress axes will rotate from zero to \(45^\circ\). If now \(\tau_{\theta z}\) is decreased in small steps, and the normal stresses are changed so that \(\sigma_\theta\) increases...

increased, while \(\sigma_z\) decreased, then the angle of rotation of the principal stresses will increase and, at \(\tau_{\theta z}=0\), will reach \(90^\circ\). When the specimen is twisted in the other direction and \(\sigma_z\) and \(\sigma_\theta\) are changed correspondingly, the angle of rotation will increase to \(180^\circ\). By periodically repeating such a loading procedure, one can obtain a rotation of the principal stresses through any angle.

The described method makes it possible to isolate, for independent study, the question of the influence of rotation of the axes of the stress tensor on the process of plastic deformation.

For measuring strains on thin-walled specimens, essentially similar instrument schemes are used, differing only in their structural design. In our investigations, instruments are used whose structural design is shown in Fig. 2. All strains are measured by means of dial indicators. The base for measuring longitudinal strain is equal to \(100\) mm.

Fig. 1.

Such instruments are usually installed directly on the specimen. The presence of appreciable transverse strain requires provision for mobility in the radial direction of the elements that fix the two transverse base sections of the specimen. For this purpose, elements sliding in guides along radial directions are often used. However, such designs require great care in manufacture and are inevitably associated with the influence of transverse strain on the results of measuring the longitudinal strain. This drawback can be eliminated by using backlash-free elastic systems that take up transverse displacements.

The basis of the instruments we used consists of two elastic ring springs, whose cross section is shown in Fig. 2. The high bending stiffness of the spring in the vertical direction (with insignificant radial stiffness), as well as the symmetry of the system, which eliminates torsional errors, make it possible to obtain very stable results in measuring longitudinal strain.

The ring springs \(1\) are fastened to specimen \(3\) by means of pointed screws \(2\). A ground ring \(5\) is attached to the upper spring by means of screws \(4\). On the lower spring, in exactly the same way, ring \(6\) is fastened, to which two posts \(7\) are attached; these serve for mounting two indicators \(8\) for measuring longitudinal strains (the second indicator is not shown).

The indicator feelers bear against the ground surface of ring \(5\).

To measure angular strains, a bracket \(9\) is attached to the lower ring \(6\); in this bracket an indicator is clamped by means of screw \(10\) (not shown in the figure). Another bracket \(11\), against which the indicator feeler bears, is attached to the upper ring \(5\).

For measuring transverse strain, ring \(12\) is used, in which indicator \(13\) is mounted. On the opposite side a stop screw \(14\) is screwed into the ring; on its end it has a flat hardened surface with which it bears against the wall of the specimen. To hold

the ring on the specimen are two prismatic knives 15, made in the form of a dovetail and connected to the ring by means of two flat springs 16, forming elastic parallelograms. The clamping force of the ring is adjusted by the stop screw 14. The prismatic knives 15 fix the central ring 12 relative to the specimen. The brackets for measuring angular deformation and the device for measuring transverse deformation are shown in the figure as conventionally rotated relative to the axis of the specimen by 90°.

Fig. 2.

For calibrating the instrument, a solid specimen of steel 3 was made, with a diameter of 30 mm and a length of 250 mm. The instrument was mounted in the middle part of the specimen.

Using micron indicators, the longitudinal and transverse strains were measured, and plots of \(\sigma=\sigma(\varepsilon_{\text{long}})\) and \(\sigma=\sigma(\varepsilon_{\text{trans}})\), shown in Fig. 3, were constructed. It is evident from the plot that the experimental points for the longitudinal and transverse strains lie satisfactorily on a straight line. From these data, Poisson’s ratio \(\mu \approx 0.32\) and the modulus of elasticity \(E \approx 2.18\cdot 10^6\ \text{kg}/\text{cm}^2\) were determined. Repeated calibration many times gave the same results. In addition, the modulus of elasticity was determined using two Guggenberger strain gauges, which gave the value

\[ E \approx 2.2\cdot 10^6\ \text{kg}/\text{cm}^2 . \]

The angle of twist was measured using an ordinary indicator with a scale division of \(0.01\ \text{mm}\). The shear modulus \(G\) was determined using the instrument and a mirror setup.

In both cases, \(G \approx 8.45\cdot 10^6\ \text{kg}/\text{cm}^2\). Figure 3 presents the plot \(\tau=\tau(\gamma)\). The experimental points lie satisfactorily on a straight line.

After loading and unloading in the elastic range, the indicator pointers for measuring the longitudinal and angular strains practically always returned to zero.

In order to refine the experimental procedure and test the instruments, the work included testing two tubular specimens under complex loading: one steel specimen made of steel 40 and one copper specimen. The tests were carried out on a ZDMU-30 machine for complex loading in the following manner. First, plastic deformation of the specimen was produced by simple tension, and then, by appropriately changing the axial force, internal pressure, and torque, loading was carried out such that the stressed state remained linear and constant in magnitude, while the principal stress rotated relative to the material particles of the specimen. For the steel specimen the principal stress was \(\sigma_1=5000\ \text{kg}/\text{cm}^2\), and for the copper specimen \(\sigma_1=1200\ \text{kg}/\text{cm}^2\).

Fig. 3.

For both specimens, an increase in the strain intensity was observed during the process of complex loading. Figure 4 shows the change in strain intensity as a function of the angle of rotation of the principal stress. It is evident from the plot that the increment of intensity over the segment of complex loading depends practically linearly on the angle of rotation of the principal stress, although for the copper specimen a small deviation can be observed.

For the steel specimen, after tension the strain intensity was \(3\%\); over the segment of complex loading it increased by a further \(1.57\%\).

For the copper specimen, respectively, 2.35 and 1.35%. Thus, in both cases, in the region of complex loading the increment in strain intensity amounted to more than 50% of the preliminary value under tension.

Earlier [4], a study was carried out of plastic strains under rotation of the principal axes of stresses during linear compression of specimens in the form of cubes.

The results of the study showed that, with rotation of the compressive stress, the strain intensity increases, and its increment is approximately proportional to the angle of rotation. However, shortcomings of the method (discontinuity of the loading process, not entirely homogeneous stress state) made it possible to draw only qualitative conclusions.

The tests carried out on tubular specimens according to the method set forth in the present work, and free from the shortcomings of the cube-testing method, confirm the results of work [4].

Fig. 4

Thus, under complex loading characterized by rotation of the constant tensile or compressive stress, the strain intensity increases, and its increment is proportional to the angle of rotation of the principal stress.

REFERENCES

1. A. A. Ilyushin. Plasticity, Gostekhizdat, 1948.

2. A. M. Zhukov. Complex loading and the theory of plasticity of isotropic metals. Izvestiya AN SSSR, OTN, No. 8, 1955.

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

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