ON COMPLEX LOADING WITH A PRESERVED DIRECTION OF THE AXES OF THE STRESS TENSOR

V. I. Maksak

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

At present, the processes of loading a body during plastic deformation are usually divided into two types: simple loading and complex loading. Under simple loading, the stresses increase proportionally to one parameter and the principal axes of stress retain their direction. Under complex loading, the stresses change nonproportionally, while the principal axes of stress may remain constant or rotate in space. In what follows, the case of complex loading in which the principal axes retain a constant direction is considered.

The loading process of an element of a body can be described by means of invariants of the stress tensor. As such, here the so-called auxiliary invariants of the stress tensor [1] are adopted:

\[ \sigma = \frac{1}{3}(\sigma_1+\sigma_2+\sigma_3), \]

\[ \overline{\sigma}_i = \frac{1}{\sqrt{6}} \sqrt{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}, \]

\[ \psi = \operatorname{arc\,tg} \frac{2\hat{\sigma}_2-\hat{\sigma}_1-\hat{\sigma}_3} {\sqrt{3}(\hat{\sigma}_1-\hat{\sigma}_3)}, \]

where \(\sigma\) is the mean normal stress;
\(\overline{\sigma}\) is the intensity of shear stresses;
\(\psi\) is the angle of the type of the stress state.

Since, for moderate values of stress, the mean stress does not have a large influence on the magnitude of plastic deformation, the loading process can be described by means of the intensity of shear stresses and the angle of the type of the stress state.

In the space of principal stresses, the quantity \(\overline{\sigma}\) represents the shear stress acting on a plane equally inclined to the axes of the principal stresses, and its direction in this plane is characterized by the angle \(\Psi\). Under complex loading, when the principal axes retain their direction, the quantity \(\overline{\sigma}\) changes its magnitude and direction, but remains at all times in the octahedral plane. Thus, \(\overline{\sigma}\) is the shear-stress vector, which

lies on the octahedral plane, and its direction is specified by the angle of the type of stress state \(\Psi\).

Numerous experiments show that under simple loading the stress intensity is a definite function of the strain intensity, independent of the type of stress state, i.e., independent of \(\Psi\),

\[ \sigma_i=f(e_i), \]

where

\[ \sigma_i=\sqrt{3}\,\bar{\sigma} =\frac{1}{\sqrt{2}}\sqrt{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}, \]

\[ e_i=\frac{\sqrt{2}}{3}\sqrt{(e_1-e_2)^2+(e_2-e_3)^2+(e_3-e_1)^2} \]

This dependence is determined from a tensile test. Under complex loading, as experiments show, this dependence is not satisfied, and the strain intensity must be a function of the stress intensity and of the angle of the type of stress state,

\[ e_i=\varphi(\sigma_i,\psi). \]

It is necessary to determine how a change in \(\Psi\) affects the change in strain intensity. Probably the most suitable loading paths for this purpose will be those in which, during the process of complex loading, only \(\Psi\) changes, while the value \(\sigma_i\) remains constant and equal to its value at the beginning of complex loading. Such experiments have been carried out. Thus, in [2] the results are described of tests with such loading of two specimens made of steel 30KhN3A. The specimens were loaded into the plastic range so that \(\sigma_1=\sigma_2\), and then loading proceeded along the curve \(\sigma_i=\mathrm{const}\) in the coordinates \(\sigma_1-\sigma_2\), with the loading paths of the specimens going in different directions.

In Fig. 1 the experimental points are shown for the portion of complex loading in the coordinates \(e_i\)—\(\Psi\). It is seen from the graph that the experimental points lie rather close to straight lines, which have the same slope. The straight lines do not pass through the origin, which can probably be attributed to errors that may have arisen when obtaining the data from the graphs in [2]. In another work [3], a specimen of aluminum alloy 14S—T6 was subjected to such a loading path. The experimental points for this specimen are also plotted in Fig. 1. For this specimen as well, the points lie rather close to a straight line; however, with a considerable increase in \(\Psi\) they depart from this straight line.

In order to determine how the magnitude of the stress intensity affects plastic deformation for loading paths \(\sigma_i=\mathrm{const}\), experiments on complex loading were carried out for these paths.

Tubular specimens of steel 40, 30 mm in diameter and with a wall thickness of 1 mm, were used. The length of the working part was 130 mm. The uniform wall thickness of the specimens was checked on a device generally similar to the device described in [4]. The deviations in wall thickness for one specimen were \(\pm 0.05\) mm. The specimens were annealed at a temperature of \(600^\circ\) for 6 hours with furnace cooling. After annealing, the material of the specimens had a yield strength \(\sigma_s=3600\ \mathrm{kg/cm^2}\), modulus of elasticity \(E=2.1\cdot10^6\ \mathrm{kg/cm^2}\). The tests were carried out on a ZDMU-30 machine by loading the specimen with a tensile force and internal pressure. Longitudinal strains were measured by a specially manufactured device using two indicators with a scale division of 0.01 mm over a 100 mm gauge length. Transverse strains were measured using a micron indicator with a scale divi-

tion of 0.001 mm on a diameter of 30 mm. Four specimens were tested under complex loading at constant stress intensity. The tests were carried out as follows: the specimen was stretched into the plastic range up to a prescribed value of \(\sigma_i\), and then complex loading was performed by varying the axial force and the internal pressure

Fig. 1.

in such a way that the stress intensity remained constant during the process of complex loading. The axial force and the internal pressure were changed in small steps. The readings of the instruments measuring the strain were taken one to two minutes after loading to a given step. For the different specimens the following values of stress intensity were prescribed: \(4000\ \text{kg}/\text{cm}^2\), \(5000\ \text{kg}/\text{cm}^2\), \(5500\ \text{kg}/\text{cm}^2\), and \(6000\ \text{kg}/\text{cm}^2\). The experimental points corresponding to complex loading are plotted in Fig. 1 in the coordinates \(e_i\)—\(\Psi\). It is seen from the graph that, for each value of the stress intensity, the experimental points lie close to a straight line. As the stress intensity increases, the tangent of the angle of inclination of the straight line increases. At a stress intensity equal to the elastic limit, such complex loading will not produce additional plastic deformations, and the tangent of the angle of inclination will be equal to zero. Figure 2 shows the dependence of the tangent of the angle of inclination on the magnitude of the stress intensity in the coordinates \(n\)—\(\sigma_i\), where

\[ n=\frac{e_i}{\psi}. \]

Let us consider an arbitrary loading path, which may be represented as the hodograph of the stress-intensity vector in octa-

rical plot (Fig. 3), and take two infinitely close points \(a\) and \(b\). In going from point \(a\) to point \(b\), the quantities \(\sigma_i\) and \(\psi\) will acquire increments \(\Delta\sigma_i\) and \(\Delta\psi\). Let us split the loading path into two paths. On loading path 1 the quantity \(\sigma_i\) remains constant, while the quantity \(\psi\) changes by \(\Delta\psi\). On loading path 2, \(\sigma_i\) changes by the amount \(\Delta\sigma_i\), while \(\psi\) remains constant. In other words, it is proposed to decompose an arbitrary loading into a complex loading, in which the stress intensity remains constant, and a simple loading, in which the ratio between the stresses remains constant and equal to the ratio that existed at the end of the complex loading.

Fig. 2.

Fig. 3.

It is further assumed that the increment of plastic strain on the segment of complex loading \(a\)—\(b\) is equal to the sum of the increments of plastic strains on the elementary loading paths 1 and 2.

\[ \Delta e_i=\Delta e_{i1}+\Delta e_{i2}. \]

Of course, this assumption to some extent simplifies the actual situation; however, similar attempts to split the loading path have been used earlier [5].

As a first approximation, it may be assumed that, under complex loading, when the stress intensity is constant, the increment of strain intensity is proportional to the increment of \(\psi\),

\[ \Delta e_{i1}=n\psi, \]

and the coefficient \(n\) depends linearly on \(\sigma_i\) (Fig. 1)

\[ n=\kappa(\sigma_i-\sigma_{is}), \]

where \(\sigma_{is}\) is the yield limit of the material, and \(\kappa\) is a proportionality coefficient which, under the assumptions made, is considered constant for the given material. Under simple loading, the increment of strain intensity is determined by

\[ \Delta e_{i2}=\varphi'(\sigma_i)\Delta\sigma_i, \]

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

The total increment of plastic strain is expressed as

\[ \Delta e_i=\varphi'(\sigma_i)\Delta\sigma_i+\kappa(\sigma_i-\sigma_{is})\Delta\psi. \]

By reducing the elementary loading path \(a\)—\(b\), this expression can be written in differentials as

\[ de_i=\varphi'(\sigma_i)d\sigma_i+\kappa(\sigma_i-\sigma_{is})d\psi. \]

Fig. 4.

Labels in the figure: \(M0\), \(M1\), \(M11\), \(M14\); \(p_\theta\); \(\sigma_\theta,\ \mathrm{kg/mm^2}\); \(\sigma_z,\ \mathrm{kg/mm^2}\); \(\sigma_\theta/\sigma_z\); “Loading path.”

Visible scale markings: \(-0.02\), \(-0.01\), \(0\), \(0.01\), \(0.02\), \(0.03\), \(0.04\), \(0.05\); \(0\), \(10\), \(20\), \(30\); \(40\), \(47.5\), \(55\), \(62.5\).

Under the assumptions made, this expression is the hardening condition in differential form for \(\sigma_i > \sigma_{is}\) and for such loading paths when \(d\sigma_i \geq 0\).

The expressions for the increments of the components of the total strain in the theory of plastic flow, under the given hardening condition, are written as:

\[ de_1 = \frac{3(\sigma_1-\sigma)}{2\sigma_i} \left[\varphi'(\sigma_i)\,d\sigma_i+\kappa(\sigma_i-\sigma_{is})\,d\psi\right] +\frac{1}{E}(d\sigma_1-\mu d\sigma_2-\mu d\sigma_3), \]

\[ de_2 = \frac{3(\sigma_2-\sigma)}{2\sigma_i} \left[\varphi'(\sigma_i)\,d\sigma_i+\kappa(\sigma_i-\sigma_{is})\,d\psi\right] +\frac{1}{E}(d\sigma_2-\mu d\sigma_3-\mu d\sigma_1), \]

\[ de_3 = \frac{3(\sigma_3-\sigma)}{2\sigma_i} \left[\varphi'(\sigma_i)\,d\sigma_i+\kappa(\sigma_i-\sigma_{is})\,d\psi\right] +\frac{1}{E}(d\sigma_3-\mu d\sigma_1-\mu d\sigma_2), \]

where \(E\) is the modulus of elasticity;
\(\mu\) is Poisson’s ratio.

Figure 4 presents the test results for two specimens (No. 10 and No. 11) and their loading paths. Both specimens were stretched until plastic strains appeared, and then complex loading was performed along different paths. In the graph, points denote the experimental data on the segment of the complex-loading path \(A—B\); the solid line denotes the results obtained from the proposed expressions, and the dashed line denotes those obtained from flow theory.

As can be seen from the graph, the proposed expressions describe the process of complex loading better than flow theory.

On the basis of the foregoing, the following conclusions may be drawn:

1. The experimental data on complex loading, when the stress intensity remains constant during loading and the principal axes of stress preserve their direction, show that the strain intensity increases approximately in proportion to the change in the angle specifying the form of the stressed state.

2. The coefficient of proportionality increases with increasing stress intensity.

3. The proposed expressions for the strain components may be used as a first approximation for describing the plastic deformation of isotropic metals under complex loading of this type.

REFERENCES

1. V. V. Novozhilov. Theory of Elasticity. Sudpromgiz, 1958.

2. A. M. Zhukov. On plastic strains of an isotropic metal under complex loading. Izv. AN SSSR, OTN, No. 8, 1955.

3. Marin J., Hu L. W., On the Validity of Assumptions Made in Theories of Plastic Flow for Metals, Trans. ASME, vol. 75, No. 6, 1953.

4. A. M. Zhukov. Investigation of plastic strains of steel under complex loading. Inzhenernyi sbornik, vol. 18, Publishing House of the Academy of Sciences of the USSR, 1954.

5. V. D. Kliushnikov. On the laws of plasticity for a particular class of loading paths. PMM, vol. XXI, issue 4, Publishing House of the Academy of Sciences of the USSR, 1957.