ELASTOPLASTIC DEFORMATION UNDER SIMPLE LOADING

G. A. Doshchinskii

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

The principal regularities of the theories of elastoplastic deformations are the relations between stresses and strains. The state of stress in the vicinity of a certain point of a body is completely determined by the three principal stresses: $\sigma_1$, $\sigma_2$, $\sigma_3$. Knowledge of the magnitudes and directions of these stresses makes it possible to determine the stresses on any area passing through the point. Similarly, knowledge of the three principal elongations: $e_1$, $e_2$, $e_3$—makes it possible to determine the deformation in any direction on the basis of the relations of the theory of deformations. The assumption that the principal directions for stresses and strains coincide in an isotropic body (which, in turn, can be justified as a necessary condition for the minimum work of the deformation process [3]) substantially simplifies the problem of establishing the connection between stresses and strains. In an orthogonal coordinate system coinciding with the principal directions for stresses and strains, this problem reduces, for an isotropic body, to establishing the magnitude of each of the principal elongations as a function of the three principal stresses and the properties of the material. Mathematically, each component of deformation may be considered as a certain continuous function of three independent variables—the principal stresses. Infinitely small increments of the stresses will also cause infinitely small increments of the strain components. Owing to the smallness of the increments of stresses and strains, a linear dependence may be assumed between them (since the concepts of derivatives in the theory of infinitesimals are based on taking into account only the linear parts of increments), in the form:

\[ de_1 = a_{11}\cdot d\sigma_1 + a_{12}\cdot d\sigma_2 + a_{13}\cdot d\sigma_3, \]

\[ de_2 = a_{21}\cdot d\sigma_2 + a_{22}\cdot d\sigma_3 + a_{23}\cdot d\sigma_1, \tag{1} \]

\[ de_3 = a_{31}\cdot d\sigma_3 + a_{32}\cdot d\sigma_1 + a_{33}\cdot d\sigma_2, \]

where $a_{ik}$ are angular coefficients, predetermined by the properties of the material, of the dependence of the strain increment on the increments of each of the principal stresses at the given stage of the deformation process.

If each of the principal stresses is placed in proportional dependence on some parameter $\lambda$ (for example, time),

\[ \sigma_1=\overline{\sigma}_1\cdot\lambda,\qquad \sigma_2=\overline{\sigma}_2\cdot\lambda,\qquad \sigma_3=\overline{\sigma}_3\cdot\lambda, \tag{2} \]

where \(\bar{\sigma}_1,\ \bar{\sigma}_2,\ \bar{\sigma}_3\) are certain constant quantities, then for this type of loading (called simple loading) we shall have:

\[ d\sigma_1=\bar{\sigma}_1\cdot d\lambda,\qquad d\sigma_2=\bar{\sigma}_2\cdot d\lambda,\qquad d\sigma_3=\bar{\sigma}_3\cdot d\lambda. \tag{3} \]

Substituting these values into equations (1) and integrating with respect to the loading parameter \(\lambda\), we obtain:

\[ \begin{aligned} e_1 &= \bar{\sigma}_1\int_0^\lambda a_{11}\cdot d\lambda +\bar{\sigma}_2\int_0^\lambda a_{12}\cdot d\lambda +\bar{\sigma}_3\int_0^\lambda a_{13}\cdot d\lambda,\\ e_2 &= \bar{\sigma}_2\int_0^\lambda a_{21}\cdot d\lambda +\bar{\sigma}_3\int_0^\lambda a_{22}\cdot d\lambda +\bar{\sigma}_1\int_0^\lambda a_{23}\cdot d\lambda,\\ e_3 &= \bar{\sigma}_3\int_0^\lambda a_{31}\cdot d\lambda +\bar{\sigma}_1\int_0^\lambda a_{32}\cdot d\lambda +\bar{\sigma}_2\int_0^\lambda a_{33}\cdot d\lambda . \end{aligned} \tag{4} \]

Taking (2) into account and denoting

\[ \frac{1}{\lambda}\int_0^\lambda a_{ik}\cdot d\lambda=A(\lambda)_{ik}, \tag{5} \]

where \(A(\lambda)_{ik}\) are material characteristics depending on the stage of deformation (the loading parameter), but invariant with respect to the type of stress state, we have:

\[ \begin{aligned} e_1 &= A(\lambda)_{11}\cdot\sigma_1+A(\lambda)_{12}\cdot\sigma_2+A(\lambda)_{13}\cdot\sigma_3,\\ e_2 &= A(\lambda)_{21}\cdot\sigma_2+A(\lambda)_{22}\cdot\sigma_3+A(\lambda)_{23}\cdot\sigma_1,\\ e_3 &= A(\lambda)_{31}\cdot\sigma_3+A(\lambda)_{32}\cdot\sigma_1+A(\lambda)_{33}\cdot\sigma_2 . \end{aligned} \tag{6} \]

The isotropy properties of the material impose a number of relations on the characteristics \(A(\lambda)_{ik}\). Thus, for tension in the first direction,

\[ \sigma_1=\sigma,\qquad \sigma_2=\sigma_3=0, \]

\[ e_1=e_{\mathrm{long}},\qquad e_2=e_3=e_{\mathrm{trans}},\qquad e_{\mathrm{long}}=A(\lambda)_{11}\cdot\sigma, \]

\[ e_2=A(\lambda)_{23}\cdot\sigma=e_{\mathrm{trans}},\qquad e_3=A(\lambda)_{32}\cdot\sigma=e_{\mathrm{trans}},\qquad A(\lambda)_{23}=A(\lambda)_{32}. \]

For the same tension in the second direction,

\[ \sigma_2=\sigma,\qquad \sigma_1=\sigma_3=0,\qquad e_2=e_{\mathrm{long}},\qquad e_1=e_3=e_{\mathrm{trans}}, \]

\[ e_{\mathrm{long}}=A(\lambda)_{21}\cdot\sigma,\qquad e_1=A(\lambda)_{12}\cdot\sigma=e_3=A(\lambda)_{33}\cdot\sigma=e_{\mathrm{trans}},\qquad A(\lambda)_{12}=A(\lambda)_{33}. \]

For the third direction,

\[ \sigma_3=\sigma,\qquad \sigma_1=\sigma_2=0,\qquad e_3=e_{\mathrm{long}},\qquad e_1=e_2=e_{\mathrm{trans}}, \]

\[ e_{\mathrm{long}}=A(\lambda)_{31}\cdot\sigma,\qquad e_1=A(\lambda)_{13}\cdot\sigma=e_2=A(\lambda)_{22}\cdot\sigma=e_{\mathrm{trans}},\qquad A(\lambda)_{13}=A(\lambda)_{22}. \]

The equivalence of all three directions for the longitudinal deformation leads to the condition

\[ A(\lambda)_{11}=A(\lambda)_{21}=A(\lambda)_{31} =\frac{e_{\mathrm{long}}}{\sigma}=\frac{1}{E(\lambda)}, \tag{7} \]

where \(E(\lambda)\) is the secant modulus of the tensile diagram (Fig. 1a). The equivalence of all directions for the transverse deformation, with (7) taken into account, gives:

\[ A(\lambda)_{23}=A(\lambda)_{32}=A(\lambda)_{12}=A(\lambda)_{33} =A(\lambda)_{13}=A(\lambda)_{22} =\frac{e_{\mathrm{trans}}}{\sigma} = \]

\[ =\frac{e_{\mathrm{trans}}}{e_{\mathrm{long}}E(\lambda)} =\frac{\mu(\lambda)}{E(\lambda)}, \tag{8} \]

where \(\mu(\lambda)\) is the transverse-strain coefficient, determining the ratio of transverse and longitudinal strains at various stages of the deformation process.

Consequently, for an isotropic body the properties of the material are characterized by two quantities, \(E(\lambda)\) and \(\mu(\lambda)\), which vary in the course of deformation. Finally, proceeding from (6), (7), (8), we obtain expressions for the principal strains in terms of the stresses and the characteristics of the isotropic material in the form:

\[ \begin{aligned} e_1 &= \frac{1}{E(\lambda)}\,[\sigma_1-\mu(\lambda)(\sigma_2+\sigma_3)],\\[6pt] e_2 &= \frac{1}{E(\lambda)}\,[\sigma_2-\mu(\lambda)(\sigma_3+\sigma_1)],\\[6pt] e_3 &= \frac{1}{E(\lambda)}\,[\sigma_3-\mu(\lambda)(\sigma_1+\sigma_2)]. \end{aligned} \tag{9} \]

Expressions (9) establish the magnitude of the principal strains under loading, without determining the character of the deformation—elastic (reversible) or plastic (irreversible). Elastic deformation will be characterized by the coincidence of the expressions for the strains under loading and unloading. The presence of residual strain and its components is determined by the difference between such dependencies in the processes of loading and unloading. In particular, for linearly elastic deformation under loading and unloading the material characteristics do not change in the deformation process, that is, they become independent of the loading parameter \(\lambda\) (Fig. 1b), and relations (9) pass into the expressions of the generalized Hooke’s law.

Fig. 1.

Equations (9) determine the values of individual components of the deformed state. Taken separately, they do not determine the deformed state of the body as a whole, which can be characterized only by the entire set of strains. Since expressions (9) have been obtained under assumptions on the independence of the material characteristics \(E(\lambda)\) and \(\mu(\lambda)\) from the type of stressed state, the set of strains adopted as a measure of the degree of deformation of the body must also be invariant with respect to the type of stressed state. As a measure of the degree of deformation of the body it is convenient to use the quantity

\[ \sqrt{\frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2}}, \]

taking into account the weight of the strain components-

Fig. 2

Diagram of one-sided deformation

Annotations in the drawing:

• unloading
• from the pressure synchronizer
• ring of the longitudinal-deformation strain gauge
• diagram of two-sided deformation
• from pump No. 2
• into the specimen piston
• to the specimen piston
• level
• into the cylinder of axial tension
• from pump No. 1

Top-left graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
“Torsion.” Curves: \(e_1\), \(\sigma\), \(e_3\).

Top-right graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.935 \]
Curves: \(e_z\), \(\sigma\), \(e_t\), \(e_m\).

Second row, left graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_m}{\sigma_t}=0.93 \]
Curves: \(e_t\), \(e_m\), \(\sigma\), \(e_z\).

Second row, right graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.85 \]
Curves: \(e_m\), \(e_t\), \(\sigma\), \(e_z\).

Third row, left graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_m}{\sigma_t}=0.5 \]
Curves: \(e_t\), \(\sigma\), \(e_m\), \(e_z\).

Third row, right graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.38 \]
Curves: \(e_m\), \(\sigma\), \(e_t\), \(e_z\).

Fourth row, left graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_m}{\sigma_t}=0.79 \]
Curves: \(e_t\), \(\sigma\), \(e_m\), \(e_z\).

Fourth row, right graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.55 \]
Curves: \(e_m\), \(\sigma\), \(e_t\), \(e_z\).

Fifth row, left graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.25 \]
Curves: \(e_m\), \(\sigma\), \(e_t\), \(e_z\).

Fifth row, right graph: \(\sigma\), kg/cm\(^2\); \(e \cdot 10^{-3}\); \(e_p \cdot 10^{-3}\).
\[ \frac{\sigma_t}{\sigma_m}=0.1 \]
Curves: \(e_m\), \(\sigma\), \(e_t\), \(e_z\).

Fig. 3

formed state corresponding to the longitudinal strain in tension and insensitive to the sign of the strain (elongation or shortening) for a material that behaves identically in tension and compression. Using equations (9), we arrive at an expression establishing the relation between the stressed and deformed states of the body as a whole, giving the basic characteristic of the material’s deformability.

\[ \sqrt{ \sigma_1^2+\sigma_2^2+\sigma_3^2 - \frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} (\sigma_1\cdot\sigma_2+\sigma_3\cdot\sigma_2+\sigma_1\cdot\sigma_3) } = \]

\[ =E(\lambda)\cdot \sqrt{ \frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2} } \]

\[ (\sigma_{\mathrm{gen}}=E(\lambda)\cdot e_{\mathrm{gen}}). \tag{10} \]

Expression (10) introduces no new relations into the connection between stresses and strains (since it is an identity and therefore does not contradict the system of equations (9)), but establishes the correspondence of various stressed states according to the loading parameter \(\lambda\). For simple tension it becomes the tensile diagram \(\sigma=E(\lambda)\cdot e\), that is, it makes it possible, from tensile-test data, to indicate the behavior of the body under any stressed state. For a volumetrically incompressible body, condition (10) becomes the relation, used in the theory of plasticity, between the intensities of stresses and strains.

For experimental verification of dependencies (9), (10), tests were carried out under conditions of simple loading on thin-walled specimens of steel St. 3. The specimens were prepared from a single bar and annealed at \(600^\circ\mathrm{C}\) in sealed ampoules with cast-iron powder. Isotropy was checked on Gagarin’s press on specimens cut with different orientations. The specimens had wall thickness

\[ t=1^{\pm 0.01}\ \mathrm{mm}, \]

mean diameter

\[ d=41^{\pm 0.02}, \]

and length of the thin-walled cylindrical part \(140\ \mathrm{mm}\). To ensure constancy of the ratio of pressures in the longitudinal-tension cylinder and in the specimen cavity, both feed pumps were connected by a pressure synchronizer, whose lever was maintained in a horizontal position by regulating the feed of one of the pumps. The longitudinal force was measured by means of a mercury barometer with a scale division value of \(10\ \mathrm{kg}\). The internal pressure in the specimen cavity was determined by a standard manometer (class \(0.35\), 1 division \(=0.5\ \mathrm{atm}\)). The longitudinal strain was measured over a base of \(10\ \mathrm{cm}\) by indicators with a scale division value of \(0.01\ \mathrm{mm}\). A micron indicator was used in the transverse-strain device. The test setup is shown in Fig. 2. Experiments on torsion of thin-walled specimens were also carried out.

Preliminarily, on the basis of simple tension and compression tests, the initial \(\sigma-e\) diagram was recorded, with determination of the current values of \(E(\lambda)\) and \(\mu(\lambda)\) as functions of the degree of strain. The experimental values of the stresses were determined from the measured pressure \(q\) in the specimen cavity and the longitudinal force \(P\):

\[ \sigma_t=\frac{qd}{2t},\qquad \sigma_m=\frac{P}{\pi dt}+\frac{qd}{4t},\qquad \sigma_r=-\frac{q}{2}. \]

In torsion:

\[ \sigma_1=-\sigma_3=\frac{2M_{\mathrm{tor}}}{\pi d^2t}, \qquad e_1=-e_3=\frac{\varphi\cdot d}{2L}. \]

For presentation of the test results, the $\sigma - e$ characteristic was transformed into a $\sigma - e_p$ curve $\left(e_p = \sqrt{e_1^2 + e_2^2 + e_3^2}\right)$. The theoretical values of the strain components corresponding to expressions (9) were also plotted with solid lines as functions of the value of $e_p$ in Fig. 3.

The experimental values of $\sigma_{\mathrm{gen}}$ and of the strain components are shown by points in Fig. 3.

The results of the experiments indicate the acceptability of expressions (9) and (10) as laws of elastic-plastic deformation.

REFERENCES

1. A. R. Rzhanitsyn. Calculation of Structures Taking Account of the Plastic Properties of Materials. Stroiizdat, Moscow, 1956.

2. Yu. N. Rabotnov. Strength of Materials. Moscow State University Press, 1962.

3. G. A. Doshinskii. On the principal directions of stresses and strains. In the present collection.