THEORY OF THE LIMITING ELASTIC STATE

G. A. DOSHCHINSKY

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

The task of the theory of the limiting elastic state is to determine the magnitudes of stresses for a complex state of stress at which the elastic behavior of a material is disrupted. In a number of cases, the normal operating conditions of structural elements can be ensured only with insignificant residual deformations, or even with their absence. For a theoretical solution of this question, the condition of plasticity may be used. However, the condition of plasticity determines the ratio of stresses already in the process of plastic deformation. The transition, when strict requirements are imposed on calculation by the elastic limit rather than by the yield limit while retaining the former design expressions—as is sometimes recommended [1]—can hardly be regarded as sufficiently justified. The replacement of the laws of elastic deformation by the prevailing laws of plastic deformation occurs within a certain transition region; therefore the design condition, strictly speaking, may be a function of the tolerance for the magnitude of residual deformation.

The problem of excluding residual deformation may also be approached from another side: on the basis of an analysis of elastic deformation and the use of a justified criterion determining the magnitudes of stresses after which the predominance of elastic properties over the remaining ones is violated. Such a theory—the theory of the limiting elastic state—may be constructed on the basis of a linear relation between stresses and deformations. We note that most “theories of strength” are constructed on a linear relation between stresses and deformations and therefore are theories of the limiting elastic state.

Let us consider some premises that must be taken into account by a theory evaluating the possibility that a material preserves its elastic properties under a complex state of stress.

A reliable determination of the limits of elastic relation is possible only in the case when the theory reflects sufficiently fully and correctly the behavior of the material in the process of the preceding elastic deformation. A static and geometric investigation of the behavior of a material body leads to the concepts of stress and deformation. Mathematically, the stressed state of the neighborhood of a point may be characterized by the stress tensor—\(T_{\sigma}\), and the deformed state by the strain tensor—\(T_{\varepsilon}\). As experiments with various loadings of bodies show, for a large number of materials within the limits of elasticity a linear relation may be adopted between stresses and deformations, giving the simplest solutions with accuracy sufficient for a number of practical problems. Since the question considered here is that of determining the limiting elastic state, all the propositions refer to the idealized linear theory of elasticity of a material body, one that operates identically in tension and compression.

In the most general mathematical form, the dependence between stresses and strains may be represented by the equation:

\[ T_{\sigma}=f(T_e). \tag{1} \]

Under the assumption of proportionality of stress to strain, the dependence (1) may be represented in the simplest form as follows:

\[ T_{\sigma}=C\cdot T_e \tag{2} \]

i.e., it is assumed that any component of the stress tensor is directly proportional to the corresponding component of the strain tensor. The coefficient of proportionality \(C\) in this case would be the sole elastic constant of the material. However, materials described by this equation are far from the majority of real materials. This is evident from the example of simple tension, for which, as is known from experiment, longitudinal strain is accompanied by transverse contraction

\[ e_{\text{trans}}=\mu\cdot e_{\text{long}} \]

as a result of which the stress tensor is not proportional to the strain tensor, and only for materials with \(\mu=0\) is condition (2) satisfied. Thus the dependence (2) can be regarded only as a particular case. Linear elasticity theory establishes a more general relation between stresses and strains in the form of a system of equations in which each component of the stress tensor has a linear dependence on all components of the strain tensor. Analysis of these dependences shows that for an isotropic elastic body there are only two independent elastic constants. This corresponds to the decomposition of the stress and strain tensors into two parts, determining the change of volume and of shape (the spherical and deviatoric parts).

\[ T_{\sigma}=T_{\sigma}^{\circ}+D_{\sigma} \qquad T_e=T_e^{\circ}+D_e, \tag{3} \]

where

\[ T_{\sigma}^{\circ}=3K\,T_e^{\circ} \qquad D_{\sigma}=2G\,D_e . \]

The numerical coefficients here are the result of the corresponding method of decomposition; therefore one may write:

\[ T_{\sigma}^{\circ}=\alpha\cdot T_e^{\circ} \qquad D_{\sigma}=\beta\cdot D_e \tag{4} \]

and, instead of two distinct linear dependences between the corresponding parts of the tensors, on the basis of synthesizing equations (3) and (4), represent the generalized law of elasticity in the form of a linear tensor function of two independent variables:

\[ T_{\sigma}=\alpha\cdot T_e^{\circ}+\beta\cdot D_e \qquad \text{for stresses} \tag{5} \]

or

\[ T_e=\frac{1}{\alpha}\cdot T_{\sigma}^{\circ}+\frac{1}{\beta}D_{\sigma} \qquad \text{for strains.} \tag{6} \]

The possibility of such a representation of the generalized law of elasticity follows from the fact that it makes it possible to obtain from this initial form all the basic relations characteristic of elastic deformation. In matrix representation, equation (5) takes the form:

\[ \begin{vmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \cdot & \sigma_y & \tau_{yz} \\ \cdot & \cdot & \sigma_z \end{vmatrix} = \alpha \begin{vmatrix} e_{\mathrm{cp}} & 0 & 0 \\ \cdot & e_{\mathrm{cp}} & 0 \\ \cdot & \cdot & e_{\mathrm{cp}} \end{vmatrix} + \beta \begin{vmatrix} e_x-e_{\mathrm{cp}} & \dfrac{1}{2}\gamma_{xy} & \dfrac{1}{2}\gamma_{xz} \\ \cdot & e_y-e_{\mathrm{cp}} & \dfrac{1}{2}\gamma_{yz} \\ \cdot & \cdot & e_z-e_{\mathrm{cp}} \end{vmatrix}, \tag{7} \]

where

\[ e_{cp}=\frac{e_x+e_y+e_z}{3}=\frac{e_1+e_2+e_3}{3}. \]

From (7), the stress components can be expressed in terms of the strain components:

\[ \begin{aligned} \sigma_x&=\alpha\cdot e_{cp}+\beta\,(e_x-e_{cp})& \tau_{xy}&=\tau_{yx}=\frac{\beta}{2}\gamma_{xy}=\frac{\beta}{2}\gamma_{yx}\\ \sigma_y&=\alpha\cdot e_{cp}+\beta\,(e_y-e_{cp})& \tau_{yz}&=\tau_{zy}=\frac{\beta}{2}\gamma_{yz}=\frac{\beta}{2}\gamma_{zy}\\ \sigma_z&=\alpha\cdot e_{cp}+\beta\,(e_z-e_{cp})& \tau_{zx}&=\tau_{xz}=\frac{\beta}{2}\gamma_{zx}=\frac{\beta}{2}\gamma_{xz}. \end{aligned} \]

Similarly, expressions can also be obtained for the strain components. The proportionality coefficients determine the two principal constants characterizing the material. Any relation between these quantities may be regarded as a derived constant.

For comparison of the elastic behavior of different materials, let us write the generalized law of elasticity in the indicated form for two materials:

\[ T_{\sigma 1}=\alpha_1\cdot T^\circ_{e1}+\beta_1\cdot D_{e1}, \qquad T_{\sigma 2}=\alpha_2\cdot T^\circ_{e2}+\beta_2\cdot D_{e2}. \]

For identical stressed states

\[ T_{\sigma 1}=T_{\sigma 2} \quad \text{and, consequently} \quad \alpha_1\cdot T^\circ_{e1}+\beta_1\cdot D_{e1} = \alpha_2\cdot T^\circ_{e2}+\beta_2\cdot D_{e2}, \]

but in the general case for different materials

\[ \alpha_1\ne \alpha_2,\qquad \beta_1\ne \beta_2 \]

and equality becomes possible under the condition:

\[ T^\circ_{e1}\ne T^\circ_{e2} \quad \text{and} \quad D_{e1}\ne D_{e2}. \]

Consequently, for different materials under identical stressed states, the deformed states may differ not only in the magnitude of the components, but also in character. Conversely, it is obvious that for different materials under identical deformed states, the stressed states may be different. The features of volumetric deformation and change of shape manifest themselves differently in different materials.

For one and the same material under different stressed states:

\[ T_{\sigma 1}=T^\circ_{\sigma 1}+D_{\sigma 1}, \qquad T_{\sigma 2}=T^\circ_{\sigma 2}+D_{\sigma 2}, \]

where

\[ T^\circ_{\sigma 1}\ne T^\circ_{\sigma 2}, \qquad D_{\sigma 1}\ne D_{\sigma 2} \]

the deformed states, on the basis of (6), are determined by:

\[ T_{e1}=\frac{1}{\alpha}T^\circ_{\sigma 1}+\frac{1}{\beta}D_{\sigma 1}, \qquad T_{e2}=\frac{1}{\alpha}T^\circ_{\sigma 2}+\frac{1}{\beta}D_{\sigma 2}. \]

From this it can be seen that, in the general case, owing to the different relationship between the components of the stress and strain tensors, the stressed and deformed states will not be identical in character. This is possible only in special cases: \(\alpha=\beta\); \(\alpha=\infty\); \(\beta=\infty\), or when all the components \(T^\circ_{\sigma}\) or \(D_{\sigma}\) vanish. In this case condition (6) passes into (2).

Consequently, both for different materials under identical stressed states and for one material under different stress—

... states by means of two elastic constants determines the influence of the type of stressed state on the character of the deformed state. This circumstance must not be overlooked when formulating a criterion of equivalence of states. The parameters adopted as defining the stressed or deformed state can enter into the initial condition of equivalence only together with the two elastic constants of the materials. Otherwise such a condition cannot reflect the features of the elastic behavior of different materials. When equivalent states are compared, both constants cannot simultaneously drop out of the equation, which would be equivalent to denying the different tendency of materials under different conditions toward volumetric deformation and deformation without change of density. However, if the generalized law of elasticity for two stressed states of one material is written in the form:

\[ T_{\sigma 1}=\alpha T_{\sigma 1}^{\circ}+\beta D_{e1} =\alpha\left[T_{e1}^{\circ}+\frac{\beta}{\alpha}D_{e1}\right], \]

\[ T_{\sigma 2}=\alpha T_{e2}^{\circ}+\beta D_{e2} =\alpha\left[T_{e2}^{\circ}+\frac{\beta}{\alpha}D_{e2}\right], \]

then one may assume the possibility of canceling one of the elastic constants in the equation of equivalence. But when one of the constants is canceled, in order to reflect the features of the elastic behavior of materials there must remain some ratio of the characteristics of the materials, i.e., some derived constant. One can also arrive, from energetic considerations, at the conclusion that it is necessary to retain in the equivalence condition some characteristic reflecting the features of the elastic behavior of different materials.

From the standpoint of the expenditure of work on deformation, it may be said that work is expended on changing the density of the material and on changing the configuration of the given material body. The value of the specific work of deformation (or of the numerically equal specific potential energy) for an elastic isotropic body is determined by the following relations:

\[ A=A_{\mathrm{vol}}+A_{\mathrm{f}}, \]

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

\[ A_{\mathrm{vol}}=\frac{1-2\mu}{6E}(\sigma_1+\sigma_2+\sigma_3)^2, \]

\[ A_{\mathrm{f}}=\frac{1+\mu}{3E}\left[ \sigma_1^2+\sigma_2^2+\sigma_3^2 -(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1) \right], \]

where \(A\) is the total work of deformation,

\(A_{\mathrm{vol}}\) is the work expended on changing the volume of the element,

\(A_{\mathrm{f}}\) is the work of change of shape,

\(E\) and \(\mu\) are the elastic constants of the material.

As an extreme case, it may be assumed that all the work is expended only on changing the volume,

\[ A=A_{\mathrm{vol}}, \]

whence, after transformations, we obtain:

\[ \frac{1}{E}(1+\mu)(\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1)=0. \]

The latter equality is possible when \(\sigma_1^2+\sigma_2^2+\sigma_3^2=\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1\), i.e., for a definite stressed state (such is hydrostatic pressure), and for any stressed state when \(\mu=-1\). Thus a material with \(\mu=-1\) would be insensitive to the character of the stressed state; under any stressed state its deformation would proceed only in a definite way, namely, it would be purely volumetric. Another limiting value can be obtained by assuming that all the work of deformation goes into change of shape (deformation without change of density):

\[ A=A_{\phi}. \]

After transformations we obtain

\[ \frac{1}{E}(1-2\mu)(\sigma_1+\sigma_2+\sigma_3)^2=0. \]

The equality is possible when \(\sigma_1+\sigma_2+\sigma_3=0\) or \(\mu=0.5\). Thus such deformation will occur under any force action for a material with \(\mu=0.5\), i.e., such a material is indifferent to the character of the stressed state. The possibility \(E=\infty\), which follows from both conditions, corresponds to an absolutely rigid body. For materials with \(\mu\ne0.5\) and \(\mu\ne-1\), the deformed state will differ not only with the magnitudes of the stresses, but also with their ratio, i.e., with the type of stressed state. For real materials \(\mu=0\div0.5\).

The quantity \(\mu\) determines the ratio of the works spent on volumetric deformation and change of shape for identical stressed states, depending on the type of material, i.e., it reflects the deformation characteristics of the material. Therefore the role of this constant may be defined as an indicator of the deformation capacity of the material. For two materials with different \(\mu\), under identical stressed states, the character of the deformed state will be different, and conversely. Moreover, volumetric deformation and change of shape are conditioned by a definite set of the three principal stresses. A theory that does not take into account even one of the components of the stressed or deformed state permits considerable arbitrariness for this component, thereby introducing a certain uncertainty into the characterization of the behavior of the material body. A theory that does not fully reflect the picture of elastic behavior cannot take into account the peculiarities of volumetric deformation and change of shape of different materials, and therefore cannot reliably predetermine the limiting elastic state. Therefore, all those theories based on analysis of the stressed state, the deformed state, or their combination, which do not take into account the deformation characteristics of the material, must be recognized as not entirely satisfactory for determining the limiting elastic state.

Let us see to what extent this can be attributed to the basic “theories of strength” [2].

The theory of the greatest normal stresses \(\sigma_{\max}\le[\sigma]\) is very limited; the initial condition does not fully reflect the stressed state and the elastic properties of the material, and therefore is a crude approximation.

The theory of greatest strain \(e_{\max}\le[e]\) introduces into the initial condition only one component of the deformed state, and thus the deformed state is characterized incompletely.

The theory of maximum shear stress \(\tau_{\max}\le[\tau]\) (and likewise Mohr’s theory \(\sigma_1-\sigma_3\le[\sigma]\)) does not completely cover the stressed state and does not introduce the elastic characteristics of materials into the condition.

The theory of specific potential energy of change of shape

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1}\le[\sigma], \]

while completely characterizing the stressed...

… state does not take into account the deformation characteristics of the material and asserts the impossibility of residual volumetric deformations, which does not correspond to certain experimental facts.

The theory of total potential energy, \(\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-2\mu(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}\leq[\sigma]\), seems to correspond best to what has been noted; it fully reflects the stressed and deformed state, as well as the strength and deformation characteristics of materials (\([\sigma]\) and \(\mu\)). The most substantial objection to this theory is the fact that plastic materials have an increased capacity to retain elastic properties under hydrostatic pressure, whereas according to the theory this state is almost as dangerous as simple tension—compression.

Here we shall confine ourselves only to an analysis of the principal theories; the others are characterized by still more arbitrary assumptions and by a less definite physical meaning. It should also be noted that experimental verification often does not permit a definite assessment to be made, since it was carried out under allowances corresponding to an already sufficiently developed plastic deformation, and sometimes under incorrect ones. For verifying the theory of the limiting elastic state these allowances must be maximally stringent. Therefore it may be considered that the question of the theoretical prediction of the limits of elastic relation under a complex state of stress has not been finally resolved in favor of any one theory. Hence the necessity of theoretical and experimental investigations in this direction.

Below, one possible theoretical determination of the limiting elasticity of materials under a complex state of stress is considered.

Since within the limits of elasticity there exists a definite relation between stresses and strains, the choice of one or another side for formulating the initial condition of the theory of the limiting elastic state is immaterial and is determined by the adoption of the corresponding criterion. In order to reflect completely the statical, geometrical, and physical aspects of the elastic behavior of a material, it is necessary that the initial condition contain all components of the stressed \((\sigma_1;\sigma_2;\sigma_3)\) or deformed state \((e_1;e_2;e_3)\), together with the two characteristics of the elastic properties of the material in either case. Under this condition, one of the aspects taken separately completely reflects the other as well. Since the issue is the difference in force action and the state of stress is an arbitrary aspect, the deformed state may be adopted as the determining factor. The physical state of a material subjected to different force actions may be assumed to be identical if, in the process of these actions, the material has undergone the same deforming action, i.e., is in an equally deformed state. The same may be said of the state of the material preceding the occurrence of plastic deformations, i.e., the limiting elastic state. If this state has been established from some test, for example tension, then the magnitude of the stresses under any other loading will be determined from the condition that this loading brings the material into the same deformed state as that to which tension brought it at the moment residual deformations appeared. For a continuous, linearly elastic, and completely isotropic material (i.e., one that also works equally in tension and compression), we shall assume that the measure for comparing deformed states is the magnitude of the mean strain, whose value for limiting states we shall assume to be constant. The assumption of identical behavior of the material in tension and compression makes essential only the consideration of the absolute values of the strain components. In such cases, as is known from the theory of means, the more convenient…

and the root-mean-square value is preferable in comparison with the arithmetic mean,

\[ e_{cp}=\sqrt{\frac{e_1^2+e_2^2+e_3^2}{2}}=C . \tag{8} \]

The total relative strains in the directions of the principal elongations are expressed, respectively, as:

\[ e_1=\frac{1}{E}(\sigma_1-\mu\sigma_2-\mu\sigma_3) \qquad e_2=\frac{1}{E}(\sigma_2-\mu\sigma_3-\mu\sigma_1) \qquad e_3=\frac{1}{E}(\sigma_3-\mu\sigma_1-\mu\sigma_2). \]

Substituting into the initial condition, we have:

\[ \frac{1}{E\sqrt{2}}\sqrt{(\sigma_1-\mu\sigma_2-\mu\sigma_3)^2+(\sigma_2-\mu\sigma_3-\mu\sigma_1)^2+(\sigma_3-\mu\sigma_1-\mu\sigma_2)^2}=C \]

whence, after transformations,

\[ \frac{1}{E\sqrt{2}}\sqrt{(1+2\mu^2)(\sigma_1^2+\sigma_2^2+\sigma_3^2)+(2\mu^2-4\mu)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} . \]

For equivalent tension \(\sigma_2=\sigma_3=0;\ \sigma_1=\sigma_0\), where \(\sigma_0\) is the limiting tensile stress at which the material still retains elastic properties,

\[ \frac{1}{E\sqrt{2}}\sqrt{(1+2\mu^2)\sigma_0^2}=C. \]

Equating and reducing, we obtain:

\[ \sigma_1^2+\sigma_2^2+\sigma_3^2+\frac{2\mu^2-4\mu}{1+2\mu^2}(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)=\sigma_0^2. \tag{9} \]

The final expression of the condition ensuring the preservation of the elastic properties of a material under a complex stress state may be represented in the following form:

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-k_e(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}\leq k_\sigma, \tag{10} \]

where: \(\displaystyle k_e=\frac{4\mu-2\mu^2}{1+2\mu^2}\) is the deformation characteristic of the material,

\(k_\sigma\) is the strength characteristic of the material (elastic limit, proportionality limit).

\(\mu\) is Poisson’s ratio.

The expressions of the conditions of total energy

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-2\mu(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}\leq[\sigma] \tag{11} \]

and of the energy of change of form

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}\leq[\sigma] \tag{12} \]

may be regarded as special cases of equation (10), for which respectively

\[ k_e=2\mu \quad \text{and} \quad k_e=1. \]

Equating \(\displaystyle \frac{4\mu-2\mu^2}{1+2\mu^2}=2\mu\), we obtain \(2\mu^3+\mu^2-\mu=0\).

The last equation gives three roots: $\mu = 0$; $\mu = -1$; $\mu = 0.5$. Thus, for materials whose bodies can change under deformation only in volume or only in shape, and for a material in which the longitudinal deformation of a specimen is not accompanied by a transverse one, the condition of total potential energy reflects equally deformed (in the indicated sense) states.

For

\[ \frac{4\mu - 2\mu^2}{1 + 2\mu^2} = 1 \]

we have as the only root of this equation

\[ \mu = 0.5. \]

Consequently, the condition of energy of form change considers equally deformed states only for a material that does not change density under deformation.

Let us compare the values of $k_e$ for different materials according to conditions (9, 11, 12), (Table 1).

Table 1

It is evident from the table that the influence of the type of material (its deformation properties) on the design condition in the group of metals with $\mu = 0.3 \div 0.4$, denied entirely by condition (12), will, according to condition (9), be considerably smaller than is assumed by theory (11).

In order to note the difference in the limiting values of stresses according to conditions (9), (11), (12), depending on the type of stressed state, let us give a graphical interpretation of these conditions for a plane stressed state ($\sigma_3 = 0$).

We have:

\[ \sigma_1^2 + \sigma_2^2 - k_e \cdot \sigma_1 \sigma_2 \leq k_s^2. \]

Passing to the notation $\sigma_1 = x$, $\sigma_2 = y$ and replacing the inequality by equality, we obtain the equation of the limiting curve

\[ x^2 + y^2 - k_e \cdot xy = k_s^2. \]

Let us rotate the coordinate system through the angle $\alpha = 45^\circ$, carrying out the coordinate transformation by the formulas:

\[ x = \xi \cdot \cos \alpha - \eta \cdot \sin \alpha,\qquad y = \xi \cdot \sin \alpha + \eta \cdot \cos \alpha. \]

Substituting

\[ \sin \alpha = \cos \alpha = \frac{\sqrt{2}}{2} \]

\[ x = (\xi - \eta)\frac{\sqrt{2}}{2};\qquad y = (\xi + \eta)\frac{\sqrt{2}}{2} \]

and replacing $x$ and $y$ in the original equation of the limiting curve, we obtain:

\[ \frac{(\xi - \eta)^2}{2} + \frac{(\xi + \eta)^2}{2} - \frac{k_e}{2}(\xi - \eta)(\xi + \eta) = k_s^2, \]

whence, after transformations, it follows that:

\[ \frac{\xi^2}{\dfrac{2k_\sigma}{2-k_e}}+ \frac{\eta^2}{\dfrac{2k_\sigma}{2+k_e}}=1. \]

Introducing the notation:

\[ a=\frac{k_\sigma}{\sqrt{1-\dfrac{k_e}{2}}},\qquad b=\frac{k_\sigma}{\sqrt{1+\dfrac{k_e}{2}}}, \]

we see that the limiting curves of theories (9), (11), and (12) in the coordinate system \(\xi\) and \(\eta\) are ellipses with different semiaxes depending on the type of material

\[ \frac{\xi^2}{a^2}+\frac{\eta^2}{b^2}=1. \]

For comparison, we give the values of \(a\) and \(b\) for several materials (Table 2).

Table 2

A visual representation is given in Fig. 1, where the condition of maximum tangential stress is also shown.

In the stress space \(\sigma_1,\sigma_2,\sigma_3\), condition (9), in comparison with condition (11), represents an ellipsoid considerably more elongated in the direction of the isotropy axis, transforming into the Huber–Mises–Hencky cylinder for a material with \(\mu=0.5\). This to some extent eliminates the main shortcomings of theories (11) and (12) for stressed states close to triaxial tension-compression.

The specific nature of hydrostatic stresses excludes the possibility of shear deformation, but does not exclude the possibility of residual change of volume, which is also observed experimentally [6].

For hydrostatic compression-tension, theories (9), (11), (12) give:

\[ \sqrt{3p^2-k_e\cdot 3p^2}\leq k_\sigma, \]

whence the pressure, exceeding which leads to residual volumetric deformation, is

\[ p=\frac{k_\sigma}{\sqrt{3(1-k_e)}}. \]

According to the distortion-energy theory, for any material \(p=\infty\), which denies the possibility of the existence of residual change of volume. According to

according to the total-energy theory this pressure, for most materials, is approximately of the same magnitude as the elastic limit, whereas experiments establish it as considerably greater. According to the theory of uniformly deformed states, the pressure at which a residual change of volume begins to appear, for metals with \(\mu = 0.3 \div 0.45\), is

\[ p = (1.6 \div 6)\,k_\sigma . \]

Fig. 1

The limiting shear stress in pure shear, according to the theory of uniformly deformed states, is expressed by the following value:

\[ \sqrt{\tau^2+\tau^2+k_e \tau^2}\le k_\sigma, \qquad \text{whence} \qquad \tau_0=\frac{k_\sigma}{\sqrt{2+k_e}} . \]

For various materials,

\[ \tau_0=(0.577\div 0.71)\,k_\sigma , \]

whereas according to the theory of distortion energy it is, for all materials, \(0.577\,k_\sigma\).

Condition (9) may be brought to a somewhat different form:

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-k_e(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1) +2(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)-2(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} \]

\[ =\sqrt{(\sigma_1+\sigma_2+\sigma_3)^2-(2+k_e)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}\le k_\sigma \]

or

\[ \sqrt{(\sigma_1+\sigma_2+\sigma_3)^2 -\frac{2(1+\mu)^2}{1+2\mu^2} (\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} \le k_\sigma . \]

Whence it is evident that the condition is expressed through constants and the first and second invariants of the stress tensor; consequently, it satisfies one of the necessary requirements—invariance. Using the values of the invariants, this condition may be represented as a continuous function of the coordinates, which is convenient in the investigation of nonuniform stressed states:

\[ \sqrt{(\sigma_x+\sigma_y+\sigma_z)^2 -(2+k_e)(\sigma_x\sigma_y+\sigma_y\sigma_z+\sigma_z\sigma_x -\tau_{xy}^2-\tau_{yz}^2-\tau_{zx}^2)} \le k_\sigma . \]

In this form it makes it possible to estimate the possibility of violation of elastic properties under a complex state of stress without a preliminary calculation of the principal stresses.

Of decisive importance in evaluating theories is their agreement with experimental data over a wide range of stress states. Let us turn to certain experiments on testing theories of plasticity, which may give an approximate idea of the agreement of the theory of the limiting elastic state considered here with experiment. We shall use, in particular, the experimental data of Lode and of Taylor—Quinney, which in the theory of plasticity are usually regarded as fundamental.

The main difference between the results given by condition (9) for metals with \(\mu=0.3\div 0.4\), as compared with other theories for the plane state of stress, reduces to the following. In the region of biaxial tension the theory gives values of the limiting stresses lying between the results of the Saint-Venant and Huber–Mises–Hencky conditions, closer to the latter and analogous to it in character. Therefore the validity of the condition under consideration could serve as some explanation of the often contradictory opinions concerning the better agreement with experiment of one or another of the indicated theories, since these are the states most frequently realized in experiments. In Fig. 2 the data of Lode’s experiments under biaxial tensions are shown. The experimental points, while showing the advantage of the Huber–Mises–Hencky condition as compared with the Saint-Venant condition, nevertheless have systematic deviations from it and come close to the limiting curve of the hypothesis of uniformly deformed states (dashed line). (The data are taken from A. Nadai, Plasticity, ONTI, NKTP, 1936.)

In the region of mixed states (tension–compression), condition (9) is characterized by somewhat increased values of the limiting stresses.

In Fig. 3, representing the data of the Taylor–Quinney experiments [3], the curve corresponding to this condition is shown by a dashed line. In particular, for shear at \(\mu=0.3\) we have \(\tau_s=0.6\sigma_s\), which is very often observed in experiments with steel [4].

Analogous deviations from the Huber–Mises–Hencky condition are noted by A. M. Zhukov [5] and by other experimenters.

Fig. 2

The use of the resulting strain as an equivalence criterion occurs in Zandel’s theory (1919). However, Zandel’s condition in the form

\[ \sigma_{\mathrm{priv}}=E\cdot e_p \le [\sigma] \tag{7} \]

gives somewhat different results. At the same time, the condition (9), constructed on the basis of the resulting strain, as is evident from the above, is not inferior, on the whole, in a number of cases to the results given by the most commonly used theories. Therefore condition (9) may be regarded as a condition worthy of attention.

Fig. 3

References

1. Encyclopedic Handbook “Mechanical Engineering.” Vol. 1, Book 2. 1948.
2. S. D. Ponomarev et al., Fundamentals of Modern Methods of Strength Calculation in Mechanical Engineering. Vol. I. Mashgiz, 1950.
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