On the Theory of Elastic-Plastic Deformation

G. A. Doshchinsky

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

The objects of analysis of the mathematical theory of elastic-plastic deformation are bodies endowed with certain generalized and sometimes substantially simplified mechanical characteristics and properties. The theory establishes dependences between the stress and strain tensors at various points of a body, and on this basis makes it possible to solve a number of practically important problems. In doing so, it is usual in the theory to decompose the stress and strain tensors into spherical and deviatoric parts, expressing the relation between the corresponding parts of these tensors

\[ \begin{aligned} T_\sigma &= T_\sigma^\circ + D_\sigma, \qquad &T_\varepsilon &= T_\varepsilon^\circ + D_\varepsilon,\\ T_\sigma &= f_1(T_\varepsilon^\circ), \qquad &D_\sigma &= f_2(D_\varepsilon). \end{aligned} \tag{1} \]

The decomposition of tensors into parts in the form (1) has a definite meaning, making it possible to characterize the features of volumetric deformation and of shape change in various materials. The boundedness of the volumetric deformation of materials, in particular metals, makes it possible in a number of practical problems to formulate certain assumptions concerning volume deformation. Therefore the dependences between the indicated parts of the tensors are usually considered separately. Useful assumptions in solving problems are those of incompressibility of materials, constancy of volume under plastic deformation, linearly elastic volumetric deformability, etc.

The magnitude of the volumetric deformation by itself often has no substantial significance. However, of definite interest is the analysis of changes of quantitative and qualitative character that are introduced into the theory together with the indicated assumptions. These changes may be taken into account when studying experimental facts that do not fit within the framework of the theory and when analyzing deviations revealed in experimental verification of the laws of elastic-plastic deformation. On this basis some refinement of the theory is also possible. Modern experimental technique permits verification also of the conclusions that may be obtained in such an investigation.

The experimental basis for posing such a problem consists of experiments revealing residual change of volume (the investigations of Davidenkov [1], Meyer [1], Andrew and Lee [9], and others), numerous Bridgman experiments showing that volumetric deformation is not linearly elastic [7], certain systematic deviations in experiments testing plasticity conditions, etc.

This problem, in particular, was pointed out by Yu. N. Rabotnov in a report at the joint session of the Division of Technical Sciences and the Division of Physical and Mathematical Sciences of the Academy of Sciences of the USSR on the problem of strength and its physical foundations in 1954:

“Of all the laws of plasticity, perhaps the greatest doubt is aroused—

... gives the first, namely, the condition of incompressibility of volume under plastic deformation” [4].

Below, one of the possibilities for analyzing elastic-plastic deformation is considered without any preliminary assumptions concerning volumetric deformation.

Basic relations between stresses and strains in elastic-plastic deformation

Consider a body in the stage of elastic-plastic deformation. Suppose that all components of the stress state have received infinitely small increments. These stress increments will correspond to certain infinitely small increments of the strain components. Owing to the smallness of the increments of stresses and strains, the dependence between them may be taken as linear. Assuming also the preservation of isotropy, one may represent the relations between infinitely small increments of stresses and strains in the form used for elastic deformation. Using the expression of the generalized law of elasticity for strains [13], we write it in differential form

\[ dT_{e}=\frac{1}{\alpha}\,dT_{\sigma}^{\circ}+\frac{1}{\beta}\,dD_{\sigma}. \tag{2} \]

For simple loading (according to A. A. Ilyushin [3]), taking the proportionality of all stress components to one parameter, we have:

\[ \begin{aligned} \sigma_x &= \bar{\sigma}_x\cdot\lambda, & \tau_{xy} &= \bar{\tau}_{xy}\cdot\lambda,\\ \sigma_y &= \bar{\sigma}_y\cdot\lambda, & \tau_{yz} &= \bar{\tau}_{yz}\cdot\lambda,\\ \sigma_z &= \bar{\sigma}_z\cdot\lambda, & \tau_{zx} &= \bar{\tau}_{zx}\cdot\lambda, \end{aligned} \]

where \(\bar{\sigma}_x, \bar{\sigma}_y, \bar{\sigma}_z, \bar{\tau}_{xy}, \bar{\tau}_{yz}, \bar{\tau}_{zx}\) are certain constant quantities, and \(\lambda\) is a variable parameter.

In generalized form one may write:

\[ T_{\sigma}=\bar{T}_{\sigma}\cdot\lambda = \left(\bar{T}_{\sigma}^{\circ}+\bar{D}_{\sigma}\right)\cdot\lambda = \bar{T}_{\sigma}^{\circ}\cdot\lambda+\bar{D}_{\sigma}\cdot\lambda. \]

Differentiating with respect to the parameter \(\lambda\), we have

\[ dT_{\sigma}^{\circ}=\bar{T}_{\sigma}^{\circ}\cdot d\lambda, \qquad dD_{\sigma}=\bar{D}_{\sigma}\cdot d\lambda, \]

and, consequently, equation (2) may be written in the form:

\[ d\bar{T}_{e} = \frac{1}{\alpha}\,\bar{T}_{\sigma}^{\circ}\cdot d\lambda + \frac{1}{\beta}\,\bar{D}_{\sigma}\cdot d\lambda. \]

Integrating (2) with respect to the parameter \(\lambda\), we obtain

\[ T_{e} = \bar{T}_{\sigma}^{\circ}\int_{0}^{\lambda}\frac{1}{\alpha}\,d\lambda + \bar{D}_{\sigma}\int_{0}^{\lambda}\frac{1}{\beta}\,d\lambda, \]

since

\[ \bar{T}_{\sigma}^{\circ}=\frac{T_{\sigma}^{\circ}}{\lambda}, \qquad \bar{D}_{\sigma}=\frac{D_{\sigma}}{\lambda}, \]

then

\[ T_e = T^\circ_\sigma \cdot \frac{1}{\lambda}\int_0^\lambda \frac{1}{\alpha}\,d\lambda + D_\sigma \cdot \frac{1}{\lambda}\int_0^\lambda \frac{1}{\beta}\,d\lambda . \]

or

\[ T_e = \frac{T^\circ_\sigma \cdot \frac{1}{\lambda}} {\displaystyle\int_0^\lambda \frac{1}{\alpha}\,d\lambda} + \frac{D_\sigma \cdot \frac{1}{\lambda}} {\displaystyle\int_0^\lambda \frac{1}{\beta}\,d\lambda}. \]

Introducing the notation:

\[ \alpha(\lambda)= \frac{\lambda} {\displaystyle\int_0^\lambda \frac{1}{\alpha}\,d\lambda} \quad \text{and} \quad \beta(\lambda)= \frac{\lambda} {\displaystyle\int_0^\lambda \frac{1}{\beta}\,d\lambda}, \]

we can represent the generalized law of elastic-plastic deformation in the form of a generalized law of elasticity. Here \(\alpha(\lambda)\) and \(\beta(\lambda)\) are no longer constants, but certain functions of the parameter \(\lambda\):

\[ T_e = \frac{1}{\alpha(\lambda)}T^\circ_\sigma + \frac{1}{\beta(\lambda)}D_\sigma . \tag{3} \]

From equation (3), in projections onto the coordinate axes \(x, y, z\), one can obtain expressions for the deformation components in terms of the stresses:

\[ \begin{aligned} a)\quad e_x &= \frac{\sigma_{\mathrm{cp}}}{\alpha(\lambda)} + \frac{\sigma_x-\sigma_{\mathrm{cp}}}{\beta(\lambda)}, & d)\quad \gamma_{xy}=\gamma_{yx} &= \frac{2\tau_{xy}}{\beta(\lambda)} = \frac{2\tau_{yx}}{\beta(\lambda)}, \\[1.2em] b)\quad e_y &= \frac{\sigma_{\mathrm{cp}}}{\alpha(\lambda)} + \frac{\sigma_y-\sigma_{\mathrm{cp}}}{\beta(\lambda)}, & e)\quad \gamma_{yz}=\gamma_{zy} &= \frac{2\tau_{yz}}{\beta(\lambda)} = \frac{2\tau_{zy}}{\beta(\lambda)}, \\[1.2em] c)\quad e_z &= \frac{\sigma_{\mathrm{cp}}}{\alpha(\lambda)} = \frac{\sigma_z-\sigma_{\mathrm{cp}}}{\beta(\lambda)}, & f)\quad \gamma_{zx}=\gamma_{xz} &= \frac{2\tau_{zx}}{\beta(\lambda)} = \frac{2\tau_{xz}}{\beta(\lambda)} , \end{aligned} \tag{4} \]

where

\[ \sigma_{\mathrm{cp}}=\frac{\sigma_x+\sigma_y+\sigma_z}{3}, \]

whence

\[ \beta(\lambda)=2\left(\frac{\tau_{xy}}{\gamma_{xy}}\right)=2G(\lambda), \tag{5} \]

where \(G(\lambda)\) is the secant shear modulus on the shear diagram \(\tau-\gamma\) (Fig. 1a)—a variable quantity determined by the properties of the material,

\[ G(\lambda)=\frac{\tau}{\gamma_{\mathrm{el.}}+\gamma_{\mathrm{res.}}}. \]

Fig. 1

Summing equations (4 a, b, c), we obtain

\[ e_x+e_y+e_z = 3\frac{\sigma_{cp}}{\alpha(\lambda)} + \frac{(\sigma_x+\sigma_y+\sigma_z)}{\beta(\lambda)} - \frac{3\sigma_{cp}}{\beta(\lambda)} = 3\frac{\sigma_{cp}}{\alpha(\lambda)} . \]

Denoting the relative change of volume by \(\Theta\),

\[ \Theta=(e_x+e_y+e_z)=(e_1+e_2+e_3) \]

and introducing the hydrostatic pressure \(p=\sigma_{cp}\), we have:

\[ \Theta=\frac{3p}{\alpha(\lambda)}, \qquad \text{whence} \qquad \alpha(\lambda)=3\left(\frac{p}{\Theta}\right)=3K(\lambda). \tag{6} \]

\(K(\lambda)\) is the secant modulus of volumetric elasticity under elastic-plastic deformation (Fig. 1b)

\[ K(\lambda)=\frac{p}{\Theta_{\text{el.}}+\Theta_{\text{res.}}}. \]

Usually, in the theory of small elastic-plastic deformations, the volumetric deformation is assumed to be linearly elastic,

\[ K(\lambda)=\mathrm{const}=K, \]

as a result of which the behavior of the material under elastic-plastic deformation is determined by the single variable characteristic \(G(\lambda)\) and the constant \(K\). However, numerous investigations by Bridgman [7] show that the volumetric deformation of metals obeys the law

\[ \Theta=a\cdot p-b\cdot p^2, \tag{7} \]

where \(a\) and \(b\) are experimental constants of the material. Therefore, the behavior of materials under elastic-plastic deformation, without simplifying assumptions, must be described by two variables \(G(\lambda)\) and \(K(\lambda)\), or \(\alpha(\lambda)\) and \(\beta(\lambda)\).

The expressions (4), after substitution of \(\sigma_{cp}\) and transformations, are reduced to the following form:

\[ e_x= \frac{1}{ \left[ \dfrac{3\alpha(\lambda)\cdot\beta(\lambda)} {\beta(\lambda)+2\alpha(\lambda)} \right] } \cdot \left[ \sigma_x - \frac{\alpha(\lambda)-\beta(\lambda)} {\beta(\lambda)+2\alpha(\lambda)} \cdot (\sigma_y+\sigma_z) \right], \qquad \gamma_{xy}=\frac{2}{\beta(\lambda)}\tau_{xy}. \]

\[ e_y = \frac{1}{\left[\dfrac{3\alpha(\lambda)\cdot \beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)}\right]} \cdot \left[ \sigma_y - \frac{\alpha(\lambda)-\beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)} \cdot (\sigma_z+\sigma_x) \right], \qquad \gamma_{yz}=\frac{2}{\beta(\lambda)}\tau_{yz} \tag{8} \]

\[ e_z = \frac{1}{\left[\dfrac{3\alpha(\lambda)\cdot \beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)}\right]} \cdot \left[ \sigma_z - \frac{\alpha(\lambda)-\beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)} \cdot (\sigma_x+\sigma_y) \right], \qquad \gamma_{zx}=\frac{2}{\beta(\lambda)}\tau_{zx}. \]

For tension, \(\sigma_z=\sigma,\ \sigma_x=\sigma_y=0,\ \tau_{ik}=0\), and the strain components are expressed as:

\[ e_z= \frac{\sigma_z}{\left[\dfrac{3\alpha(\lambda)\cdot \beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)}\right]}, \qquad e_y=e_x= - \left[ \frac{\alpha(\lambda)-\beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)} \right] \cdot \frac{\sigma_z}{\left[\dfrac{3\alpha(\lambda)\cdot \beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)}\right]}. \]

Whence it follows:

\[ \frac{3\alpha(\lambda)\cdot \beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)} = \frac{\sigma_z}{e_z} = \frac{\sigma_1}{e_1} = E(\lambda), \qquad E(\lambda)=\frac{\sigma}{e_{\text{el.}}+e_{\text{res.}}} \tag{9} \]

\(E(\lambda)\) is the secant modulus on the tensile diagram,

\[ \text{and}\qquad \frac{\alpha(\lambda)-\beta(\lambda)}{\beta(\lambda)+2\alpha(\lambda)} = \frac{-e_x}{e_z} = \frac{-e_y}{e_z} = \mu(\lambda) \tag{10} \]

plays the role of the coefficient of transverse deformation under elastic-plastic deformation with simple loading (Fig. 2). Using these characteristics, equations (8) for the principal directions can be represented in the form:

\[ e_1= \frac{1}{E(\lambda)} \left[ \sigma_1-\mu(\lambda)\cdot(\sigma_2+\sigma_3) \right], \]

\[ e_2= \frac{1}{E(\lambda)} \left[ \sigma_2-\mu(\lambda)\cdot(\sigma_3+\sigma_1) \right], \tag{11} \]

\[ e_3= \frac{1}{E(\lambda)} \left[ \sigma_3-\mu(\lambda)\cdot(\sigma_1+\sigma_2) \right]. \]

Fig. 2

From expressions (5), (6), (9), (10) there follow relations between the characteristics \(E(\lambda)\), \(\mu(\lambda)\), \(K(\lambda)\), \(G(\lambda)\):

\[ E(\lambda)=\frac{9K(\lambda)\cdot G(\lambda)}{G(\lambda)+3K(\lambda)} =3K(\lambda)\,[1-2\mu(\lambda)]; \qquad G(\lambda)=\frac{E(\lambda)}{2[1+\mu(\lambda)]}; \]

\[ \mu(\lambda)=\frac{3K(\lambda)-2G(\lambda)}{6K(\lambda)+2G(\lambda)} =\frac{E(\lambda)}{2G(\lambda)}-1; \qquad K(\lambda)=\frac{E(\lambda)}{3[1-2\mu(\lambda)]}. \tag{12} \]

Solving equations (4) with respect to \(\sigma_x;\sigma_y;\sigma_z;\tau_{xy};\tau_{yz};\tau_{xz}\), one can obtain expressions for the stress components in terms of the strain components:

\[ \begin{aligned} \sigma_x&=\beta(\lambda)\cdot e_x+[\alpha(\lambda)-\beta(\lambda)]\cdot\frac{\Theta}{3}\\ \sigma_y&=\beta(\lambda)\cdot e_y+[\alpha(\lambda)-\beta(\lambda)]\cdot\frac{\Theta}{3}\\ \sigma_z&=\beta(\lambda)\cdot e_z+[\alpha(\lambda)-\beta(\lambda)]\cdot\frac{\Theta}{3} \end{aligned} \qquad\left\|\begin{aligned} \tau_{xy}&=\tau_{yx}=\beta(\lambda)\cdot\frac{\gamma_{xy}}{2} =\beta(\lambda)\cdot\frac{\gamma_{yx}}{2}\\ \tau_{yz}&=\tau_{zy}=\beta(\lambda)\cdot\frac{\gamma_{yz}}{2} =\beta(\lambda)\cdot\frac{\gamma_{zy}}{2}\\ \tau_{zx}&=\tau_{xz}=\beta(\lambda)\cdot\frac{\gamma_{zx}}{2} =\beta(\lambda)\cdot\frac{\gamma_{xz}}{2} \end{aligned}\right. \tag{13} \]

Subtracting from equations (13) the value \(\sigma_{cp}\), expressed for the right-hand side in terms of \(e_{cp}\), we obtain:

\[ \sigma_x-\sigma_{cp}=2G(\lambda)(e_x-e_{cp}) \]

\[ \sigma_y-\sigma_{cp}=2G(\lambda)(e_y-e_{cp}) \tag{14} \]

\[ \sigma_z-\sigma_{cp}=2G(\lambda)(e_z-e_{cp}), \]

whence follow proportions that determine the similarity of Mohr circles for stresses and strains under elastic-plastic deformation in conditions of simple loading. Assumptions concerning the coincidence of the principal directions for stresses and strains may be considered justified only within the limits in which the material retains the properties of isotropy.

Consequently, under the indicated conditions the basic relations between stresses and strains for elastic-plastic deformation can be represented in a form outwardly similar to the analogous relations of the theory of elasticity. Such a representation of equations (11), etc., is not new [10, 11, 12]. However, it should be noted that equations (11) have a certain quantitative and qualitative peculiarity in that \(\mu(\lambda)\) here is an independent characteristic. In the theory of elastic-plastic deformation, \(\mu(\lambda)\) is a quantity either constant and equal to \(0.5\), or determined from the strength characteristic \(\sigma=f(e)\) and the elastic constants

\[ \mu(\lambda)=\frac{1}{2}-\frac{1}{6K}E(\lambda), \qquad \text{where } K=\text{const}. \]

The behavior of an isotropic material under “purely elastic” deformation is determined by two constants \(K\) and \(G\). If one assumes the preservation of the properties of isotropy under elastic-plastic deformation, then the relations between stresses and strains (as shown above) are also determined by means of two quantities \(K(\lambda)\) and \(G(\lambda)\) \((\alpha(\lambda)\) and \(\beta(\lambda)\),

which are already variable quantities depending on the degree of deformation. Instead of \(K(\lambda)\) and \(G(\lambda)\), derived characteristics \(E(\lambda)\) and \(\mu(\lambda)\) may be used to express the relation between stresses and strains. With any choice of the form in which the expressions for the components of stress and strain are written, the use of two material characteristics is necessary. Any assumption that makes it possible to represent the dependences between stresses and strains through a single characteristic will ascribe to the material certain properties following from such an assumption. Thus, for an exact description of the behavior of a material under elastic-plastic deformation, it is necessary to obtain experimentally two curves as functions of the degree of deformation. Taking tension, as the simplest experiment, as such curves one may take

\[ E(\lambda)=f_1(e) \quad \text{and} \quad \mu(\lambda)=f_2(e), \]

where \(e\) is the longitudinal strain of the specimen.

\(E(\lambda)=f_1(e)\) characterizes the increase in resistance (stress) in connection with the growth of strain, and therefore may be replaced by the curve of change of the strength properties of the material \(\sigma=f(e)\).

\(\mu(\lambda)=f_2(e)\) shows the change in the ratios between the strain components, which ultimately determines the distribution of the work of deformation between deformation of volume and change of shape; it may therefore be called the curve of change of the deformation properties of the material.

The curves of change of the strength and deformation properties of a material can be obtained from a tensile test only on the condition that, in the course of the experiment, both the longitudinal and the transverse strains of the specimen are measured. However, measurement of the latter somewhat complicates the experiment and is usually not performed. The information about the behavior of the material during deformation that is not obtained for this reason is ascribed to the material on the basis of various assumptions, such as invariance of volume under plastic deformation, linear elasticity of volumetric deformation, and so on. This predetermines the approximate character of theories of elastic-plastic deformation and, probably, may even lead to the fact that certain experimental facts prove not to fit within the framework of a theory constructed on incomplete information about the material.

Obtaining the curves \(E(\lambda)=f_1(e)\) and \(\mu(\lambda)=f_2(e)\) from a tensile test appears in practice to be more convenient than obtaining \(K(\lambda)=f_3(\theta)\) and \(G(\lambda)=f_4(\gamma)\) from separate experiments. The experimental determination of \(K(\lambda)=f_3(\theta)\) is distinguished by the extraordinary complexity of the experiment. Moreover, the relation between the results of different tests can be established only on the basis of some assumptions about the equivalence of degrees of deformation in two separate experiments. These assumptions, however, in turn require additional justification.

Condition of equivalence of different stress states under elastic-plastic deformation. The law of hardening

The basis for constructing numerous computational expressions in the theory of elastic-plastic deformation is the assumption of the existence of a generalized yield curve independent of the type of stress state. The possibility of comparing different stress states for the stage of elastic-plastic deformation has been analyzed by a number of investigators and has repeatedly been subjected to experimental verification. Some experimental facts have sometimes served as grounds for expressing doubts about the possibility of determining equivalent states of a material under different loadings.

The generalized flow curve represents a characteristic of the behavior of a material subjected to mechanical action. The reduction of various forces to a mechanical equivalent makes it possible to assume the possibility of determining this characteristic. Of course, a theory constructed only on the categories of mechanics will reflect the actual behavior of the material only to the extent that the physical, chemical, etc., phenomena not taken into account by it in the process of the material’s operation, and their influence on the mechanical aspect of the deformation process, are insignificant.

The existence of a generalized curve makes it possible to characterize the behavior of a material under any loading on the basis of an analysis of its behavior in the simplest test. The tensile test is the most commonly used for this purpose. The accuracy of characterizing various stressed states from tensile-test data will, naturally, depend on the completeness of the information on the behavior of the material obtained from the tensile test.

The possibility of characterizing the behavior of a material under any type of loading by a single dependence was apparently first studied in detail by Ludwik [8]. He regards such a generalized curve as a basic technological characteristic. In Ludwik’s opinion, the generalized curve is the diagram \(\tau_{max}—\gamma_{max}\). The method he proposed for constructing the generalized curve from tensile-test data, based on measuring only the longitudinal strain, has in principle been retained up to the present time.

The generalized curve \(\tau_{max}—\gamma_{max}\), corresponding to the Saint-Venant plasticity condition, is analyzed in a number of theoretical and experimental works, from which one may conclude that this dependence is approximate and reflects only a plane scheme. Therefore, in recent times the indicated dependence has been displaced by the dependence \(\sigma_i—e_i(\tau_{okm}—\gamma_{okm})\), corresponding to the Mises–Hencky plasticity condition. Numerous experiments confirm the possibility of characterizing the behavior of a material by a generalized curve, and for metals and alloys with a stable structure this is regarded as almost beyond doubt, although in experiments some divergence of the curves is also found. These divergences are not only the result of experimental inaccuracies. Exhibiting a certain systematic character, they make it possible to suppose as one of the causes the approximate nature of equivalence conditions constructed without obtaining from experiment the actual curve of change of deformation properties, with the missing information compensated by assumptions of various kinds. The deformation characteristic of the material is not taken from experiment and is absent both in the Saint-Venant plasticity condition \(\tau_{max}=\tau_s\) and in the Mises–Hencky condition \(\sigma_i=\sigma_s\). Nor is it introduced into the corresponding expressions of the hardening law. For example, in accordance with the commonly used expression of the hardening law: the stress intensity is a function of the strain intensity, independent of the stressed state—the coincidence with the tensile diagram takes place only under the assumption of incompressibility of the material, namely, the dependence \(\sigma_i=\Phi\left(\dfrac{3}{2}\dfrac{1}{1+\mu(\lambda)}\cdot e_i\right)\) becomes \(\sigma_i=\Phi(e_i)\) when \(\mu(\lambda)=0.5\) is adopted [3]. On this basis the physical equations of the theory of small elastic-plastic deformations are also constructed.

The condition determining the equivalence of different stressed states in the process of elastic-plastic deformation, represented in the form of a dependence of a certain reduced stress on a reduced strain, must take into account the change in the properties of the material, charac-

characterized by the two indicated curves. This is necessary for ensuring the compatibility of the equations describing the process of elastic-plastic deformation, since the relation between the components of stresses and strains is expressed by means of two independent characteristics.

Above, the possibility was considered of representing the basic relations in elastic-plastic deformation under simple loading in the form characteristic of elastic deformation. Continuing this analogy, it is necessary to try to represent also the equivalence condition in elastic-plastic deformation in the same form as the equivalence condition of different stressed states for elastic deformation.

For this purpose let us turn to the condition considered earlier, used to determine the limiting elastic state and found to be in good agreement with experimental data [13]. Assuming that equivalent states under different loadings are determined by equality of the mean strain, we write the initial condition in the form:

\[ e_{cp}=\sqrt{\frac{e_1^2+e_2^2+e_3^2}{3}}=C, \]

using the values of the principal elongations represented in the form (11) and equating the expressions for the mean strains for the general case and for equivalent tension, after transformations we arrive at the equation of equivalence in stresses:

\[ \sigma_1^2+\sigma_2^2+\sigma_3^2-k(\lambda)_e\cdot(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)=\sigma_s^2, \tag{15} \]

where \(\sigma_s\) is the strength characteristic of the material at the given stage of deformation, corresponding to the equivalent stress in tension.

\[ k(\lambda)_e=\frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} \]

is a characteristic taking into account the deformation properties at this stage.

A change of \(\mu(\lambda)\) within the limits \(0.25 \div 0.5\) corresponds to values

\[ k(\lambda)_e=0.777 \div 1. \]

From relations (12) there follows an expression for the coefficient of transverse strain in elastic-plastic deformation

\[ \mu(\lambda)=\frac{1}{2}-\frac{1}{6K(\lambda)}\cdot\frac{\sigma}{e_{upr}+e_{ost}}, \tag{16} \]

which, under the assumption of “purely plastic” deformation occurring without change of volume, gives \(\mu(\lambda)=0.5\); in this case condition (15) passes into the Mises-Hencky plasticity condition.

The hardening law may be expressed as follows: the reduced stress for each material is a definite function, independent of the nature of the stressed state, of the reduced strain

\[ \sigma_{priv}=\Phi(e_{priv}), \]

where

\[ \sigma_{priv}=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-k(\lambda)_e(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}, \]

\[ e_{priv}=\frac{e_p}{\sqrt{1+2\mu(\lambda)^2}} =\sqrt{\frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2}}. \]

The function \(\sigma_{\mathrm{priv}}=\Phi(e_{\mathrm{priv}})\) coincides completely with the tensile diagram \(\sigma-e\) and may be taken from a tensile test\({}^{1}\).

Under the plane interpretation of this condition, the equivalent states of the material in the process of elastic-plastic deformation for various stress states are characterized by points located between the Mises–Hencky ellipse and the ellipse corresponding to the equation

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-\frac{4\mu-2\mu^2}{1+2\mu^2}\left(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1\right)}=\sigma_e, \]

where \(\mu\) is Poisson’s ratio.

Analysis of experiments checking the plasticity conditions shows that precisely in these regions (hatched in Fig. 3) the overwhelming majority of experimental points are located.

Fig. 3

Thus, on the basis of the use of two experimental functions, it appears possible to describe the process of elastic-plastic deformation by equations similar in form to the equations of the theory of elasticity. On this basis it is also possible to estimate the changes introduced into the theory of small elastic-plastic deformations by various assumptions concerning volumetric deformation.

REFERENCES

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

3. Il’yushin A. A. Plasticity. Gostekhizdat, 1948.

4. Rabotnov Yu. N. Small plastic deformations as a problem of mechanics. Izvestiya AN SSSR, OTN, No. 7, 1954.

5. Collection: Theory of Plasticity. Edited by Rabotnov Yu. N. GIIL, 1948.

\({}^{1}\) In Sandel’s theory (1919), \(e_{\mathrm{priv}}=e_p-e\), which also leads to a different expression for \(\sigma_{\mathrm{priv}}\) [6].

1. Rivosh O. A. Strength of Materials. Methodological Manual No. 9. Publication of the Central Correspondence Mechanical-Engineering Institute, 1933.

2. Bridgman P. W. Recent work in the field of high pressures. GIIL, 1948.

3. P. Ludwik. Elemente der Technologischen Mechanik. Berlin, 1909.

4. Andrew, Lee A. O. Iron and Steel Inst. Vol. 165, p. 2. 1950.

5. Rabotnov Yu. N. “Strength of Materials.” Moscow State University Press, 1950.

6. Rzhanitsyn A. R. “Calculation of Structures with Allowance for the Plastic Properties of Materials.” Moscow, 1954.

7. Bezukhov N. I. “Theory of Elasticity and Plasticity.” GITTL, 1953.

8. Doshchinskii G. A. Theory of the limiting elastic state. Izv. TPI, vol. 85. Tomsk, 1957.