GENERAL LAW OF THE RELATIONSHIP BETWEEN STRESSES AND STRAINS¹

G. A. Doshchinskii

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

Strength of materials, in solving engineering problems, is based on a set of regularities establishing the relationship between the stressed and deformed state of a material body. Among these fundamental regularities are such laws as Hooke’s law for tension—compression, Hooke’s law for shear, the volumetric Hooke’s law, and the dependence between the intensity of stresses and strains in plastic deformation. Each of these dependences is often regarded as an independent law determining the behavior of a material under various deformation conditions. It is shown below that all these regularities can be generalized, for practical calculations, by a single expression determining the relationship between the stressed and deformed state in an isotropic body.

The stressed state in the neighborhood of any point of a loaded body is completely determined if the three principal stresses, \(\sigma_1;\ \sigma_2;\ \sigma_3\), are known (in magnitude and direction). Knowledge of these stresses makes it possible, by the known relations of the theory of the stressed state, to determine the stresses on any plane passing through this point.

The deformed state can, analogously, be characterized by the three principal linear strains: \(e_1;\ e_2;\ e_3\), from which, on the basis of the relations of the theory of strains, the linear and angular changes in any direction are determined.

In elastic behavior of a material, the principal strains coincide in direction with the principal stresses. A similar proposition is usually assumed to be valid also in plastic deformation. Therefore, the physical dependence between the stressed and deformed state can be determined by an equation relating the magnitudes of the three principal stresses to the three principal strains.

¹ Report at a meeting of the Scientific Council of the Academy of Sciences of the USSR on the problem “Scientific Foundations of Strength and Plasticity,” held jointly with Moscow State University and the Institute of Mechanics of the Academy of Sciences of the USSR on June 15–16, 1961, in Moscow.

The relationship between stresses and strains within the limits of elasticity is established in elementary form by the well-known relations of the generalized Hooke’s law, which for the principal directions have the form

\[ \varepsilon_1=\frac{1}{E}\,[\sigma_1-\mu\sigma_2-\mu\sigma_3]; \qquad \varepsilon_2=\frac{1}{E}\,[\sigma_2-\mu\sigma_3-\mu\sigma_1]; \tag{1} \]

\[ \varepsilon_3=\frac{1}{E}\,[\sigma_3-\mu\sigma_2-\mu\sigma_1], \]

where \(E\) is the modulus of normal elasticity, \(\mu\) is Poisson’s ratio. Each of these expressions by itself determines the value of an individual strain component for a given state of stress along one of the principal directions.

As the most general equation relating the stressed and deformed state as a whole, the following dependence may be proposed:

\[ \sqrt{ \sigma_1^2+\sigma_2^2+\sigma_3^2 -\frac{4\mu-2\mu^2}{1+2\mu^2} (\sigma_1\cdot\sigma_2+\sigma_2\cdot\sigma_3+\sigma_3\cdot\sigma_1) } = E\cdot \sqrt{\frac{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2}{1+2\mu^2}} . \tag{2} \]

From this expression follow the dependences between stress and elastic strain that are characteristic for any particular type of deformation.

For example, for tension

\[ \sigma_1=\sigma; \quad \sigma_2=\sigma_3=0; \quad \varepsilon_1=\varepsilon; \quad \varepsilon_2=\varepsilon_3=-\mu\varepsilon . \]

Substituting these values into (2), we obtain the expression of Hooke’s law for tension

\[ \sigma=E\cdot\varepsilon, \]

for shear \(\sigma_1=\tau;\ \sigma_2=0;\ \sigma_3=-\tau;\ \varepsilon_1=\dfrac{\gamma}{2};\ \varepsilon_2=0;\ \varepsilon_3=-\dfrac{\gamma}{2}\)

and from (2) we obtain the expression of Hooke’s law for shear

\[ \tau=\frac{E}{2(1+\mu)}\cdot\gamma, \qquad (\tau=G\cdot\gamma) \]

For purely volumetric deformation under hydrostatic pressure

\[ \sigma_1=\sigma_2=\sigma_3=p; \qquad \varepsilon_1=\varepsilon_2=\varepsilon_3=\varepsilon_{\mathrm{av}}=\frac{\Theta}{3}. \]

Equation (2) gives

\[ p=\frac{E}{3(1-2\mu)}\cdot\Theta, \quad \text{or} \quad p=K\cdot\Theta . \]

Consequently, elastic deformation of any kind is characterized by dependence (2), while its individual components are determined by expressions (1), and in characterizing the elastic behavior of a material in practical calculations there is no need to proceed from the independent, separate existence of the laws of elastic shear, linear, and volumetric deformation.

Dependence (2), which may be called the generalized formula of Hooke’s law, is not merely a simple construction identical to the system of equations (1), but is a regularity estab-

…establishing the correspondence of the material characteristics \((E, G, K\), etc.) for different types of deformation. This is revealed more clearly in elastic-plastic deformation, when the corresponding characteristics depend on the state of the material and may be regarded as functions of the degree of deformation.

If, beyond the elastic limits, the relation between infinitesimal increments of stresses and strains is taken in the form of a linearized dependence characteristic of elastic deformation, then integration with respect to the loading parameter (under simple loading) shows that the total (consisting of elastic and plastic parts) components of the principal strains can be determined from expressions analogous to equations (1) [4]:

\[ e_1=\frac{1}{E(\lambda)}\,[\sigma_1-\mu(\lambda)(\sigma_2+\sigma_3)];\qquad e_2=\frac{1}{E(\lambda)}\,[\sigma_2-\mu(\lambda)(\sigma_3+\sigma_1)]; \]

\[ e_3=\frac{1}{E(\lambda)}\,[\sigma_3-\mu(\lambda)(\sigma_1+\sigma_2)], \tag{3} \]

where \(E(\lambda)=\dfrac{\sigma}{e}\) is the secant modulus in the original tensile diagram, and \(\mu(\lambda)=\dfrac{e_{\text{trans.}}}{e_{\text{long.}}}\) is the coefficient of transverse elastic-plastic deformation—material characteristics that vary with increasing degree of deformation under isotropic hardening.

It should be emphasized that by simple loading we mean here such loading under which all total stress components \(\sigma_1;\ \sigma_2;\ \sigma_3\) vary proportionally to one parameter, while preserving their direction unchanged. Let us note that usually in the theory of plasticity simple loading is understood as loading with the components of the stress deviator varying proportionally to one parameter, although under practical loading conditions the components of the spherical part of the stress tensor also usually vary proportionally to this parameter (this is evident in the example of even tensile deformation). Such a usual approach in the theory of plasticity is associated with a certain approximation, ignoring a number of features of the elastic behavior of the material beyond the elastic limit and accepting the initial hypothesis of elastic volumetric deformability of materials. However, the latter, as noted below, may also not be an absolutely necessary proposition.

Let us note at the same time that the elastic-plastic state here denotes the state of a material in which each elementary volume of the body, under deformation, is characterized by distortions that partially disappear after unloading and are partially irreversible (for example, a bar stretched beyond the elastic limit in the unloaded state). Particular cases of such elastic-plastic deformability, as a general property of real materials, when one of the aspects is neglected, may yield “purely elastic” and “purely plastic” states.

Sometimes in practical problems a state is called “elastic-plastic” when, in a loaded part, there exist two zones: an “elastic” and a “plastic” one. Such a division can have only a conditional character, presupposing in advance an approximation, since upon removal of the load, disappearing (“elastic”) deformations will also occur in the “plastic” zone. More consistent with the character of the deformations occurring in the part would be a subdivision into a zone of “elastic” and a zone of “elastic-plastic” deformations (if the material exhibits a clearly expressed elastic limit).

For the type of simple loading indicated above, the relation between the stressed state and the strain under elastic-plastic deformation can be determined by a dependence similar in form to expression (2).

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2- \frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} (\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} = E(\lambda)\cdot \sqrt{\frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2}} . \tag{4} \]

Since \(\mu(\lambda)\) is a relative dimensionless characteristic, dependence (4) may be regarded as a relation between generalized stress and generalized strain

\[ \sigma_{\text{gen.}}=E(\lambda)\cdot e_{\text{gen.}} \tag{5} \]

The admissibility of using this law in practical calculations can be justified by the fact that deformation curves for various stressed states, constructed from the data of many investigators, as well as from the author’s experiments with steel, in the coordinates \(\sigma_{\text{gen.}}—e_{\text{gen.}}\), almost merge into a single curve characterizing the elastic-plastic behavior of materials. Quantitatively, moreover, expression (4) gives results close to the practically used dependence between the intensities of stresses and strains.

Analogously to what was presented above, from one and the same dependence (4) we can obtain, for tension,

\[ \sigma=E(\lambda)\cdot e, \]

for shear

\[ \tau=G(\lambda)\cdot\gamma, \]

and for hydrostatic pressure

\[ p=K(\lambda)\cdot\Theta . \]

Dependence (4) completely coincides with the tension diagram; this makes it possible to determine from this experiment the material characteristics \(E(\lambda)\) and \(\mu(\lambda)\) for any degree of strain.

From the transition of dependence (4) to the expression \(p=K(\lambda)\cdot\Theta\) under hydrostatic compression, there follows the possibility of another interpretation of the behavior of materials under conditions of all-round uniform pressure. This type of loading need not be regarded as something special, for which materials possessing elastic-plastic deformability under various stressed states, with behavior characterized by the characteristics \(E(\lambda)\) and \(\mu(\lambda)\) varying with increasing strain, are as if reborn and become only elastic under the particular type of loading—hydrostatic pressure. From the experimental fact of the approximate equality of \(K(\lambda)\) to the value of the bulk modulus of elasticity \(K\),

\[ K(\lambda)=\frac{E(\lambda)}{3[1-2\mu(\lambda)]}\simeq K \]

there follows not only the unambiguous, usually used solution characteristic of elastic deformation,

\[ E(\lambda)=E;\quad \mu(\lambda)=\mu, \]

but \(E(\lambda)\) and \(\mu(\lambda)\) may also vary in such a way that the ratio

\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]} \]

may remain weakly varying and even constant. Therefore elastic-plastic deformation, identical in character for all stressed states, under conditions of hydrostatic pressure will only appear to be elastic, owing to the equality of this ratio—

tion during loading and unloading. In an experiment with all-round uniform pressure, the measured quantities are pressure and volume. Therefore the only reliable experimental fact may be considered to be that

\[ K(\lambda)=\frac{p}{\Theta}\simeq K, \]

but the constancy of \(\nu(\lambda)=\nu\) in this case is certainly not confirmed by experiment, since measurements of \(\nu(\lambda)\) are possible only in the presence of a difference between the strain components \(e_1; e_2; e_3\) (this is also evident from (3)), but such a difference is not observed under the given conditions. A similar solution, when \(K(\lambda)\) is determined by an independent variation of \(E(\lambda)\) and \(\nu(\lambda)\), also permits a nonlinearity of the relation between hydrostatic pressure and volumetric strain, which corresponds to Bridgman’s data for high pressures [3].

The equality \(E(\lambda)=E\) and \(\nu(\lambda)=\nu\) for \(K(\lambda)=K\), obviously, may be accepted for all stressed states, including hydrostatic pressure, during unloading.

The components of plastic strain may be determined as the difference between the total (elastic-plastic) and elastic strains in the form

\[ \varepsilon_1=\frac{1}{D(\lambda)}[\sigma_1-\nu_0(\sigma_2+\sigma_3)];\qquad \varepsilon_2=\frac{1}{D(\lambda)}[\sigma_2-\nu_0(\sigma_3+\sigma_1)]; \]

\[ \varepsilon_3=\frac{1}{D(\lambda)}[\sigma_3-\nu_0(\sigma_1+\sigma_2)], \tag{6} \]

where

\[ D(\lambda)=\frac{E(\lambda)\cdot E}{E-E(\lambda)} \]

is the secant modulus of plasticity, and

\[ \nu_0=\frac{E\cdot\nu(\lambda)-E(\lambda)\cdot\nu}{E-E(\lambda)} \]

is the coefficient of transverse plastic strain. Under the assumption of constancy of volume for plastic strain,

\[ \nu_0=0.5 \quad\text{and}\quad \varepsilon_1+\varepsilon_2+\varepsilon_3=0. \]

Since

\[ \varepsilon_n=\varepsilon_n-\frac{\varepsilon_1+\varepsilon_2+\varepsilon_3}{3} \qquad (n=1,2,3), \]

the generalized dependence between stresses and strain in the form (4), under plastic deformation, passes into the relation—known in the theory of plasticity—between the intensities of stresses and strains

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1} = D(\lambda)\cdot \sqrt{\frac{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2}{1+2\nu_0^2}} \]

\[ = \frac{2}{3}D(\lambda)\cdot \sqrt{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2-\varepsilon_1\varepsilon_2-\varepsilon_2\varepsilon_3-\varepsilon_3\varepsilon_1}. \tag{7} \]

Thus, dependence (4) may be regarded as a single generalized expression of the relation between the stressed and deformed states under elastic, plastic, and elastic-plastic strains.

The adoption, from the initial experiment (for example, tension), of the additional material characteristic \(\nu(\lambda)\) makes it possible to take into account a number of features of the volumetric deformability of materials, such as, for example, the noted

the above-mentioned nonlinearity, residual volume deformation in materials with defects of continuity (real metals, foam materials, etc.). If these features are neglected, then in practical calculations one may confine oneself to a single experimental function

\[ E(\lambda) = \frac{\sigma}{e} \]

with the additional determination of \(\mu(\lambda)\) from the expression

\[ \mu(\lambda) \approx 0.5 - \frac{1 - 2\mu}{2E}\cdot \frac{\delta}{e}, \]

which, according to the experiments of A. M. Zhukov [2] and others, gives values of \(\mu(\lambda)\) that are fairly close to the actual characteristics of the material.

Dependence (4), by taking into account the change in the characteristics of the material in the course of the development of deformation, gradually and automatically transforms from the general formula of Hooke’s law (2) into relation (7), corresponding to the principal regularity of plastic behavior. Such universality can undoubtedly be convenient for practical calculations.

LITERATURE

1. Ilyushin A. A. Plasticity. Gostekhizdat, 1948.
2. Zhukov A. M. On Poisson’s ratio in the plastic region. Izvestiya AN SSSR, OTN, No. 12, 1954.
3. Bridgman P. W. Recent Works in the Field of High Pressures. GIIL, 1948.
4. Rzhanitsyn A. R. Analysis of Structures with Allowance for the Plastic Properties of Materials. Moscow, 1954.