On the Law of Volumetric Deformation
Abstract Generated abstract
The paper reexamines the common assumption in plasticity theory that volumetric deformation remains linearly elastic even in the plastic range. Using relations for simple loading with deformation-dependent elastic-plastic characteristics, it argues that hydrostatic compression experiments determine only the ratio between pressure and volume change, not the separate constancy of the modulus and Poisson-type coefficient. The author proposes that these characteristics may vary while their ratio remains nearly constant, allowing residual volumetric deformation and nonlinear pressure-volume behavior consistent with high-pressure observations. The paper further suggests that the law of volumetric deformation need not be an independent postulate, but may follow as a special case of more general stress-strain relations.
Full Text
ON THE LAW OF VOLUMETRIC DEFORMATION
G. A. Doshchinskii
(Presented by the scientific seminar of the Department of Strength of Materials)
The reliability of any theory in explaining the various aspects of the phenomenon being described depends on the soundness of its initial assumptions. The approximate character, limitations, and particular conditions underlying these assumptions cannot but affect the possibilities and completeness of the theory as a whole. This proposition hardly requires special proof. In this connection, let us dwell on one of the initial laws of the theory of elastic-plastic deformation—the law of volumetric deformation.
Common to all existing theories of plasticity is the assertion of the linear elasticity of volumetric deformation. Accordingly, a proportionality is established between the spherical parts of the stress and strain tensors. The separation of this dependence from the general interrelation between the stress and strain tensors is sometimes called the first law of the theory of plasticity; the remaining regularities of plastic deformation are formulated as relations between the stress and strain deviators, their components, and invariants.
The justification usually accepted for the indicated law of volumetric elasticity is the results of observations of the behavior of materials under conditions of hydrostatic compression, in which even quite high hydrostatic pressure supposedly does not lead to the transition of materials into a plastic state. The assertion that “the change in volume is always elastic, both in the elastic and in the plastic region,” has firmly entered textbooks on strength of materials [6].
Leaving aside the not entirely clear content of the concept of “transition into a plastic state” for a polycrystalline material in general, we shall show that the indicated assertion is not an unambiguous conclusion following from experimental investigations of hydrostatic compression; nevertheless, it has become so generally accepted that other possibilities in explaining the behavior of materials at high pressures are thereby overlooked.
As was shown by A. A. Ilyushin [5], under conditions of simple loading, i.e., when stresses increase in proportion to a single parameter, various theories of plastic deformation coincide with the simplest of them—the theory of small elastic-plastic deformations; moreover, it proves possible to represent the dependences between the principal components of the stress and strain tensors as-
deformations in a form similar to the expressions of the generalized Hooke law, but with variables depending on the degree of deformation of the characteristics of elastic-plastic behavior \(E(\lambda)\) and \(\mu(\lambda)\) [6], [7], [3],
\[ e_1=\frac{1}{E(\lambda)}[\sigma_1-\mu(\lambda)(\sigma_2+\sigma_3)], \]
\[ e_2=\frac{1}{E(\lambda)}[\sigma_2-\mu(\lambda)(\sigma_3+\sigma_1)], \tag{1} \]
\[ e_3=\frac{1}{E(\lambda)}[\sigma_3-\mu(\lambda)(\sigma_1+\sigma_2)]. \]
Such a representation presupposes the coincidence of the principal directions of stresses and strains, as well as proportionality between the maximum shears and the tangential stresses. These propositions, in turn, are rather well confirmed by experimental data for various stressed states and various materials up to the development of considerable anisotropy; therefore, we shall consider the further use of equations (1) acceptable.
The relative change in volume, proceeding from elementary geometrical constructions and neglecting small quantities of higher order, is known to be determined by the sum of the principal elongations
\[ \Theta=e_1+e_2+e_3. \tag{2} \]
Using expressions (1), one may obtain
\[ \Theta=\frac{3[1-2\mu(\lambda)]}{E(\lambda)} \left(\frac{\sigma_1+\sigma_2+\sigma_3}{3}\right). \tag{3} \]
Denoting
\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]}=K(\lambda) \]
for hydrostatic pressure, we arrive at the expression
\[ p=K(\lambda)\cdot\Theta, \tag{4} \]
which determines the change in volume as a function of pressure.
The adoption of a law of linear elasticity for volumetric deformation corresponds to the assertion that the quantity \(K(\lambda)\) is equal to the bulk modulus of elasticity
\[ K(\lambda)=K=\frac{E}{3(1-2\mu)} \tag{5} \]
with the simultaneous equality characteristic of elastic deformation
\[ \mu(\lambda)=\mu=\mathrm{const}\quad \text{and}\quad E(\lambda)=E=\mathrm{const}, \]
where \(E\) and \(\mu\) are the elastic constants of the material—the modulus of normal elasticity and Poisson’s ratio. This is a possible solution, usually also used as the basis of most theories of plasticity. However, such a solution is not the only possible one.
The quantities \(E(\lambda)\) and \(\mu(\lambda)\), characterizing elastic-plastic behavior, may be regarded as varying in a limited manner, so that in the given particular type of loading the ratio
\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]} \]
as a whole may remain either constant or only weakly varying. From one equation
\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]}=K \]
the two quantities \(E(\lambda)\) and \(\mu(\lambda)\) cannot be determined uniquely.
Experiment also does not permit an unambiguous conclusion. In an experiment with uniform all-round pressure, the only measurable quantities can be pressure and volume. Therefore the only reliable (more precisely, approximate) experimental fact can be considered to be that
\(K(\lambda)=\dfrac{p}{\Theta}=K\), and nothing more. At the same time, constancy of \(\mu(\lambda)=\mu\)
and \(E(\lambda)=E\) cannot in any way be asserted by this experiment, since measurements of \(\mu(\lambda)\) are possible only when there is a difference between the components of relative deformation: \(e_1; e_2; e_3\), which, however, is not observed under the given conditions (this is also seen from equations (1), substituting into which \(e_1=e_2=e_3=e\) and equating the right-hand sides, we arrive only at the identity \(\mu(\lambda)\equiv\mu(\lambda)\) (6).
One must also reject any indirect method of measuring the quantity \(\mu(\lambda)\) under these conditions (X-ray and the like), since any indirect method is based on correspondence between the quantity being studied and the indirect one. If the quantity under study (as, for example, the coefficient of transverse deformation, determined by the ratio of directly measured macrodeformations) in certain conditions loses the fundamental possibility of being determined, then the same must also hold for the indirect characteristic; otherwise the complete correspondence would be doubtful.
Leaving aside other formally possible variants, let us note some possibilities of the proposed solution
\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]}\simeq K \quad \text{with } \mu(\lambda)\ne \mathrm{const} \text{ and } E(\lambda)\ne \mathrm{const}. \]
From such a solution there follows, first of all, the possibility of a different (in our view, more natural) explanation of the behavior of materials under conditions of uniform all-round pressure. This type of loading need not be regarded as something exceptionally special, for which materials possessing definite elastic and plastic properties as it were “degenerate” and become only elastic under hydrostatic pressure. Elastic-plastic deformation, identical in character and determined by the quantities \(E(\lambda)\) and \(\mu(\lambda)\) for any stress states, under conditions of hydrostatic pressure, as a particular loading possibility with equality of stresses, owing to the limited variability of the characteristics \(E(\lambda)\) and \(\mu(\lambda)\), will only beyond the limits of elasticity show a relation between pressure and volume differing little from this relation under elastic deformation. This relation, like any mathematical relation without additional explanations, can define only the quantitative side of the phenomenon.
The equality \(E(\lambda)=E=\mathrm{const}\) and \(\mu(\lambda)=\mu=\mathrm{const}\) for \(K(\lambda)=K=\mathrm{const}\), obviously, may be adopted equally for all stress states (including hydrostatic pressure) during unloading, if the latter is regarded as obeying the laws of elasticity. Moreover, if one takes into account that experiment indicates only the approximate equality
\[ \frac{E(\lambda)}{3[1-2\mu(\lambda)]}\simeq K, \tag{8} \]
then this makes possible the determination of residual volumetric deformation, observed experimentally in real materials, in particular in foam materials and others, and even in metals, although in insignificant amounts [1], [2]. Such deformation is possible in actual-
...ity, but if one proceeds from the law of volumetric linearly elastic deformability, it should not exist.
The quantities \(E(\lambda)\) and \(\mu(\lambda)\), measured in the initial tension (compression) experiment and corresponding to the current values of the secant modulus
\(E(\lambda)=\dfrac{\sigma}{e}\)
and of the coefficient of total transverse deformation
\(\mu(\lambda)=\dfrac{e_{\text{transv.}}}{e_{\text{long.}}}\),
make it possible to predict the behavior of the material at high pressures, whose experimental realization is associated with difficulties. At the same time, the independent variation of the characteristics \(E(\lambda)\) and \(\mu(\lambda)\) makes the quantity \(K(\lambda)\) somewhat different from the constant value \(K\). This, in turn, allows for a certain nonlinearity of the law of volumetric deformation, which agrees with the numerous investigations of P. Bridgman [1] at high pressures, who established that the dependence between the relative change in volume and pressure is more accurately described by the equation
\[ \Theta = ap - bp^{2}, \tag{9} \]
where \(a\) and \(b\) are experimental constants of the material, than by the linear dependence
\[ \Theta = \frac{1}{K}\cdot p. \tag{10} \]
It follows from identity (6) that the law of volumetric deformation proves to be identical to the system of equations (1) written for hydrostatic pressure. This points to the possibility of excluding the law of volumetric deformation as an independent law of the theory of elastic-plastic deformation. Such an exclusion, of course, will be legitimate only in the case where the derivation of equations (1) itself is not obtained on the basis of an initial postulation of the law of volumetric linear elasticity. The majority of theories of plasticity (including the deformation theory of Hencky—Nadai—Ilyushin, etc.) arrive at the form of equations (1) by means of the preliminary assumption of volumetric linearly elastic deformability.
However, in principle, in order to obtain expressions for the relation between principal stresses and strains in the form (1), it is apparently sufficient to assume the coincidence of the principal directions for stresses and strains and the existence of an isotropic effect of transverse deformation. One may arrive at the form of equations (1) without a preliminary postulation of the linear elasticity of volumetric deformation, in particular, in the following way.
If the dependence between infinitesimal increments of stresses and strains is represented in the form of linear functions (which is entirely admissible in the theory of infinitesimals), then integration with respect to the loading parameter for simple loading of an isotropic material leads to equations (1). In a similar way, these equations were obtained in works [7], [3].
In the author’s theory of elastic-plastic deformation [3], [4], which proceeds from the assumption of the existence of a general law relating any stressed and deformed state of an isotropic body by the dependence (11),
\[ \sqrt{ \sigma_1^2+\sigma_2^2+\sigma_3^2 - \frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} (\sigma_1\cdot\sigma_2+\sigma_2\cdot\sigma_3+\sigma_3\cdot\sigma_1) } = \]
\[ = E(\lambda)\cdot \sqrt{ \frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2} }, \]
the law of volumetric deformation follows as a particular value of expression (11), written for the case of hydrostatic pressure [4], and is not a separate independent regularity.
REFERENCES
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