Proceedings of the Tomsk Order of the October Revolution
and Order of the Red Banner of Labor
Polytechnic Institute named after S. M. Kirov

Vol. 224　　　　　　　　　　　　　　　　　　　　　　　　　1976

INVESTIGATION OF PLASTIC FLOW UNDER COMPLEX LOADING

G. A. DOSHCHINSKY, A. M. KORENEVA

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

The theory of plastic flow, which is one of the directions in the theory of plasticity, finds broad application in a whole range of technological problems associated with the pressure working of metals. At the same time, it should be noted that at present the reliability of the basic propositions of this theory still cannot be regarded as fully experimentally proven. Despite the existence of a large number of experimental works relating to flow theory, verification of the basic regularities of this theory under various conditions of loading and deformation remains highly desirable and necessary for the development of the general theory of plasticity. Many experimental studies concern the verification of only particular aspects and consequences of flow theory: changes in the shape of the yield surface, changes in the instantaneous modulus under orthogonal loading, comparisons of calculated relationships of flow theory and of the theory of small elastic-plastic deformations with the results of particular experiments, etc. The search for methods of integrating the differential relations of the theory in order to determine finite values of stresses and strains also remains desirable.

The task of the present study was the direct investigation of the process of plastic flow under complex loading by means of a finite-difference representation of the differential relations of the theory. On the basis of experimental data, the possibility of an approximate calculation of strains was considered for conditions analogous to those realized in the experiment. The investigations were limited to the class of loadings with a preserved direction of the principal axes of the stress tensor.

The initial proposition of the theory of plastic flow assumes coaxiality and similarity of the deviators of stresses and of strain increments (strain rates): total (the Saint-Venant–Levy–Mises form) or plastic (the Prandtl–Reuss form).

\[ d\mathcal{E}'_{jk}=\sigma'_{jk}\cdot d\Phi . \tag{1} \]

It is assumed that the multiplier \(d\Phi\) does not depend on the type of loading. In the class of loadings under consideration, for the principal directions we have:

\[ d\mathcal{E}_j-d\varepsilon=d\Phi(\sigma_j-\sigma),\quad (j=1,2,3);\qquad d\varepsilon=\frac{1}{3}\sum_{j=1}^{j=3} d e_j;\qquad \sigma=\frac{1}{3}\sum_{j=1}^{j=3}\sigma_j . \]

Under the assumption of constancy of volume during plastic deformation, \(d\varepsilon=0\).

The basic system of equations:

\[ \begin{aligned} d\mathcal{E}_1&=d\Phi(\sigma_1-\sigma)=\frac{2}{3}\,[\sigma_1-0.5(\sigma_2+\sigma_3)];\\ d\mathcal{E}_2&=d\Phi(\sigma_2-\sigma)=\frac{2}{3}\,[\sigma_2-0.5(\sigma_3+\sigma_1)]; \tag{2}\\ d\mathcal{E}_3&=d\Phi(\sigma_3-\sigma)=\frac{2}{3}\,[\sigma_3-0.5(\sigma_1+\sigma_2)] \end{aligned} \]

presupposes a definite structure of the differential multiplier \(d\Phi\), namely, using the representation

\[ d\mathcal{E}_j=d\mathcal{E}_i-\frac{d\varepsilon}{3}\quad (j=1,2,3), \]

squaring both sides of equations (2) and summing, we obtain

\[ \sqrt{\frac{2}{3}(d\mathcal{E}_1^2+d\mathcal{E}_2^2+d\mathcal{E}_3^2-d\mathcal{E}_1\cdot d\mathcal{E}_2-d\mathcal{E}_2\cdot d\mathcal{E}_3-d\mathcal{E}_3\cdot d\mathcal{E}_1)} \]

\[ =\left|\frac{2}{3}d\Phi\right| \sqrt{\frac{3}{2}(\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\cdot\sigma_2-\sigma_2\cdot\sigma_3-\sigma_3\cdot\sigma_1)}, \]

whence it follows that

\[ d\Phi=\frac{3}{2}\frac{d\mathcal{E}_i}{\sigma_i};\qquad d\mathcal{E}_j=\frac{3}{2}\frac{d\mathcal{E}_i}{\sigma_i}(\sigma_j-\sigma)\quad (j=1,2,3), \tag{3} \]

where \(d\mathcal{E}_i\) is the intensity of the strain increments;
\(\sigma_i\) is the stress intensity.

Equations (2) for the plane stress state \((\sigma_3=0)\) take the form:

\[ \begin{aligned} d\mathcal{E}_1&=d\Phi\,[\sigma_1-0.5\sigma_2];\\ d\mathcal{E}_2&=d\Phi\,[\sigma_2-0.5\sigma_1]; \tag{4}\\ d\mathcal{E}_3&=d\Phi\,[-0.5(\sigma_1+\sigma_2)], \end{aligned} \]

where

\[ d\Phi=\frac{d\mathcal{E}_i}{\sigma_i}. \]

A direct verification of these equations was the purpose of the present experiment.

The tests were carried out on copper annealed tubular specimens with variation of the longitudinal load and internal pressure according to a prescribed program. The wall thickness \(t\) and the mean diameter \(2R\) were respectively equal to \(1.5\ \mathrm{mm}\) and \(33.3\ \mathrm{mm}\). The tests were performed on a TsDMU-30 machine. The design of the instruments for measuring strains is described in [1].

In the series of experiments under consideration, a parametric specification of the loading path \(\sigma_1=f_1(\lambda)\), \(\sigma_2=f_2(\lambda)\) was used, specifically of a power-law character,

\[ \sigma_1=A\lambda^n,\qquad \sigma_2=B\lambda^m. \tag{5} \]

The axial stress \(\sigma_1\) and circumferential stress \(\sigma_2\) were determined from the expressions:

\[ \sigma_1=\frac{P}{2\pi Rt}+\frac{qR}{2t}; \qquad \sigma_2=\frac{qR}{t}. \]

The prescribed program values of the axial load \(P\) and internal pressure \(q\) were then:

\[ P=2\pi Rt\left(A\lambda^n-\frac{B\lambda^m}{2}\right), \qquad q=\frac{B\lambda^m\cdot t}{R}. \]

The values \(A, B, n, m\) were selected so that the loading paths converged at one point. The implemented loading programs are shown in Fig. 1. To check the isotropy of the material of the specimens, simple loading \(\sigma_1=\sigma_2\) was used. The axial \(\mathcal{E}_1\) and circumferential \(\mathcal{E}_2\) strains, as is seen from Fig. 2, differed little from each other. The same purpose was served by duplicating some programs \((2, 3)\) (Fig. 1) with the stresses \(\sigma_1\) and \(\sigma_2\) interchanged in the axial and circumferential directions.

Fig. 1

At each loading stage, for the corresponding value of the parameter \(\lambda\), the axial \(e_1\) and circumferential \(e_2\) strains were measured. The plastic parts of the strain were determined by excluding the elastic components, calculated on the basis of the generalized Hooke’s law, with the values:

\[ E=10^6\ \text{kg}/\text{cm}^2, \qquad \mu=0.32. \]

\[ \mathcal{E}_j=e_j-\varepsilon_j=e_j-\frac{1}{E}\left[\sigma_j-\mu(\sigma_e+\sigma_k)\right], \qquad \begin{gathered} j\\[-0.4ex] e\ k \end{gathered} \ (j,e,k=1,2,3). \tag{6} \]

The radial plastic strain was determined from the condition of constancy of volume under plastic deformation. The dependence of the obtained plastic strains \(\mathcal{E}_1\) and \(\mathcal{E}_2\) on the loading parameter is shown in Fig. 2 (the multiplier \(A\) was introduced for convenience in comparing the curves).

First of all, it may be noted that the experimental results clearly indicate the absence of a plastic strain potential. If, in elastic deformation, different loading paths give the same values of the strain components at a common point, then in the present case the values of \(\mathcal{E}_1\) and \(\mathcal{E}_2\) for the state corresponding to the common end point (marked by a cross) depend very strongly on the loading history.

To check the condition of similarity of the deviators of the stresses and of the increments of plastic strain, we exclude the factor \(d\Phi\) in expressions (4):

\[ \frac{d\mathcal{E}_2}{d\mathcal{E}_1} = \frac{\sigma_2-0.5\sigma_1}{\sigma_1-0.5\sigma_2}, \qquad \frac{d\mathcal{E}_3}{d\mathcal{E}_1} = \frac{-0.5(\sigma_1+\sigma_2)}{\sigma_1-0.5\sigma_2}. \tag{7} \]

Fig. 2

From the experimentally measured values of stresses and strains, plots of \(\sigma_1\), \(\sigma_2\), \(Э_1\), \(Э_2=\psi_j(\lambda)\) were constructed; from these, for each stage \(\xi\) of differentiation, with step \(\Delta=\dfrac{\lambda}{21}\), the increments of plastic strains and stresses were determined.

\[ dЭ_j=Э_j^{\xi+1}-Э_j^{\xi}, \qquad \sigma_j=\frac{\sigma_j^{\xi+1}+\sigma_j^{\xi}}{2}. \]

The results obtained for several programs are presented in Fig. 3. The solid lines correspond to the ratio of stresses, and the points to the ratio of strain increments. These results make it possible to note that the condition of similarity of the deviators of stresses and strain increments is satisfied for the conditions of simple loading (program 1) and, in a first approximation, also corresponds to the conditions of complex loading for the trajectories of moderate curvature under consideration. In the latter case, however, a picture similar to that shown in Fig. 3 for program 3 is predominantly observed, with some intersection even of curves similar in character that correspond to the ratios of stresses and increments of plastic strains.

Fig. 3

To what extent, under complex loading, the conditions of similarity of the deviators of stresses and strain increments are fulfilled better than those of the deviators of the strains themselves is shown in Fig. 4, where the corresponding values of the Lode parameters, calculated for program 5, are given. The values of the Lode parameters for strains deviate significantly from the diagonal that determines the similarity of the deviators, as compared with the corresponding parameter $\nu$ for increments of plastic strains.

According to the results of tests of specimens of this series, a slight deviation from the dependence $\sigma_i—e_i$, obtained under simple loading, was noted, although in other experiments on complex loading under various programs and materials, carried out by the authors, these discrepancies were substantially more noticeable.

The assumption of the independence of the function $\Phi=\Phi(\sigma_i)$ from the type of loading (more precisely, of the derivative function or differential multiplier $d\Phi$) was also subjected to experimental verification.

Starting from equations (4), the value of $d\Phi$ for each of the principal directions is determined, respectively, by

Fig. 4

\[ d\Phi_1=\frac{dЭ_1}{\sigma_1-0.5\sigma_2};\qquad d\Phi_2=\frac{dЭ_2}{\sigma_2-0.5\sigma_1};\qquad d\Phi_3=\frac{dЭ_3}{-0.5(\sigma_1+\sigma_2)}. \tag{8} \]

Using the experimentally observed values of the increments of strains and stresses for one and the same loading stage, characterized by the same stress intensity \(\sigma_i\), in different programs the mean value was calculated,

\[ d\Phi=\frac{d\Phi_1+d\Phi_2+d\Phi_3}{3}, \]

which was plotted on the graph \(d\Phi—\sigma_i\) (Fig. 5). Despite the natural and, with the given method, inevitable large scatter (since the measurement of increments is considerably less accurate than the measurement of the strains themselves), it is seen from graph 5 that, to a first approximation, the assumption of the independence of \(d\Phi\) from the type of loading is fulfilled in the class of loadings considered.

Fig. 5

If this assumption is accepted, then the quantity \(d\Phi\) can be established in a simple-loading test. For simple loading, the intensity of the strain increments is equal to the increment of the intensity (the intensity of the strain rates is equal to the time derivative of the strain intensity). This, in turn, makes it possible to outline the following possibility for an approximate calculation of strains.

Let the tensile diagram be approximated by the relation

\[ \sigma_i = C \cdot e_i^k, \tag{9} \]

where \(C\) and \(k\) are material constants (their values for the present material were determined as: \(C = 4850\ \text{kg}/\text{cm}^2,\ k = 0.45\)).

Then

\[ e_i=\left(\frac{\sigma_i}{C}\right)^{\frac{1}{k}}, \qquad de_i=\frac{1}{k\cdot C^{\frac{1}{k}}}\sigma_i^{\frac{1}{k}-1}\cdot d\sigma_i, \]

\[ d\Phi=\frac{d\mathrm{Э}_i}{\sigma_i}\approx\frac{de_i}{\sigma_i} =\frac{1}{k\cdot C^{\frac{1}{k}}}\sigma_i^{\frac{1}{k}-2}\cdot d\sigma_i. \tag{10} \]

The value of \(d\Phi\) found under simple loading is then used for arbitrary loading of the class under consideration. Expression (10) may be refined by excluding the elastic strain

\[ \mathrm{Э}_i=e_i-\varepsilon_i= \left(\frac{\sigma_i}{C}\right)^{\frac{1}{k}}-\frac{\sigma_i}{E}; \qquad d\Phi=\frac{d\mathrm{Э}_i}{\sigma_i} =\left(\frac{1}{k\cdot C^{\frac{1}{k}}}\sigma_i^{\frac{1}{k}-2} -\frac{1}{\sigma_i\cdot E}\right)d\sigma_i. \]

However, such a refinement substantially complicates the calculation of strains, and in the subsequent approximate calculation form (10) is used. A refinement of this kind is also possible by transforming the diagram \(\sigma_i-e_i\) into the diagram \(\sigma_i-\mathrm{Э}_i\), followed by an analytical approximation of the relation between the tensile stress and the longitudinal plastic strain.

In order to be able to integrate equations (4), we express all the quantities entering them, for example, through \(\sigma_1\).

For the considered type of complex loading, defined by conditions (5),

\[ \sigma_2=B\left(\frac{\sigma_1}{A}\right)^{\frac{m}{n}}, \qquad \sigma_i=\sqrt{\sigma_1^2+\sigma_2^2-\sigma_1\sigma_2} = \sqrt{ \sigma_1^2+ \frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}} - \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}+1} }. \]

Differentiating the latter with respect to \(\sigma_1\), we obtain

\[ d\sigma_i= \frac{1}{2\sigma_i} \left[ 2\sigma_1+ 2\frac{m}{n}\frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}-1} - \left(\frac{m}{n}+1\right) \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}} \right]d\sigma_1. \]

Therefore \(d\Phi\) takes the form

\[ d\Phi= \frac{1}{2k\cdot C^{\frac{1}{k}}} \left[ \sigma_1^2+ \frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}} - \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}+1} \right]^{\frac{1}{2}\left(\frac{1}{k}-3\right)} \]

\[ {}\cdot \left[ 2\sigma_1+ 2\frac{m}{n}\frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}-1} - \left(\frac{m}{n}+1\right) \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}} \right]d\sigma_1. \]

For integration, we expand the power factor by Newton’s binomial formula, restricting ourselves to the first two terms of the expansion,

\[ \left[ \sigma_1^2+\frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}} -\frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}+1} \right]^{\frac{1}{2}\left(\frac{1}{k}-3\right)} \simeq \]

\[ \simeq \sigma_1^{\frac{1}{k}-3} +\frac{1}{2}\left(\frac{1}{k}-3\right) \frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}+\frac{1}{k}-5} -\frac{1}{2}\left(\frac{1}{k}-3\right) \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}+\frac{1}{k}-4}. \]

The final expressions for calculating the strains take the form:

\[ \mathcal{E}_1= \int_0^{\sigma_1} \left( \sigma_1-\frac{B}{2A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}} \right)d\Phi; \qquad \mathcal{E}_2= \int_0^{\sigma_1} \left( \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}}-\frac{1}{2}\sigma_1 \right)d\Phi; \]

\[ d\Phi= \frac{1}{2k\cdot C^{\frac{1}{k}}} \left[ \sigma_1^{\frac{1}{k}-3} +\frac{1}{2}\left(\frac{1}{k}-3\right) \frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}+\frac{1}{k}-5} \right. \]

\[ \left. -\frac{1}{2}\left(\frac{1}{k}-3\right) \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}+\frac{1}{k}-4} \right] \]

\[ \cdot \left[ 2\sigma_1+ 2\frac{m}{n}\frac{B^2}{A^{2\frac{m}{n}}}\sigma_1^{2\frac{m}{n}-1} -\left(\frac{m}{n}+1\right) \frac{B}{A^{\frac{m}{n}}}\sigma_1^{\frac{m}{n}} \right]d\sigma_1. \tag{11} \]

The third component of the plastic strain can be found from the condition of constancy of volume

\[ \mathcal{E}_3=-(\mathcal{E}_1+\mathcal{E}_2). \tag{12} \]

To check expressions (11), consider simple loading of the form \(\sigma_1=\sigma_2=\sigma\). In this case \(A=B=m=n=1\). Substituting into (11), we obtain

\[ \mathcal{E}_1,\mathcal{E}_2= \frac{1}{2k\cdot C^{\frac{1}{k}}} \int_0^\sigma \left\{ \left(\sigma-\frac{\sigma}{2}\right) \left[ \sigma^{\frac{1}{k}-3} +\frac{1}{2}\left(\frac{1}{k}-3\right)\sigma^{\frac{1}{k}-3} \right.\right. \]

\[ \left.\left. -\frac{1}{2}\left(\frac{1}{k}-3\right)\sigma^{\frac{1}{k}-3} \right] (2\sigma+2\sigma-2\sigma) \right\}d\sigma = \frac{\sigma^{\frac{1}{k}}}{2C^{\frac{1}{k}}} = \frac{e_i}{2}. \]

On the other hand, from the expression for the strain intensity for such a case, bearing in mind the obvious relation for an isotropic material \(\mathcal{E}_1=\mathcal{E}_2=\mathcal{E}\), together with condition (12) we have

\[ \mathcal{E}_i= \frac{2}{\sqrt{3}}\sqrt{\mathcal{E}_1^2+\mathcal{E}_2^2+\mathcal{E}_1\cdot\mathcal{E}_2} = \frac{2}{\sqrt{3}}\sqrt{3\mathcal{E}^2} = 2\mathcal{E}, \]

whence

\[ \mathcal{E}_1=\mathcal{E}_2=\frac{\mathcal{E}_i}{2}\simeq\frac{e_i}{2}. \]

Using expressions (11), an analytical calculation was carried out for the strains \(\mathcal{E}_1\) and \(\mathcal{E}_2\) at the end of the loading segment for some of the implemented programs, using values of \(C\) and \(k\) approxi-

…that formed the diagram \(\sigma_i—e_i\) given above. The calculated values are shown on the right-hand side of graph 2 (indicated by the symbols corresponding to the program).

The data presented show that, under conditions of complex loading (even in the considered class of trajectories of moderate curvature), the deformation theory is untenable; the relations of the simplest flow forms (Lévy—Mises—Reuss) are justified only approximately.

REFERENCES

1. V. I. Maksak, G. A. Doshchinskii, “On a method for investigating plastic deformation under complex loading,” Izvestiya TPI, vol. 147, published by Tomsk University, 1966.

2. V. V. Novozhilov, “On a class of complex loadings characterized by preservation of the directions of the principal axes,” Applied Mathematics and Mechanics, vol. XVIII, published by the Academy of Sciences of the USSR, 1954.

3. Khul, D. Merin, “Finding a theoretical relationship between stress and strain in the plastic region under a variable stress ratio,” Collection of translations, Mechanics, 5 (21), published by IL, 1953.