Problems of Inelastic Deformation of Metals*

K. Zener and J. H. Hollomon

In all fields of research it is useful from time to time to assess the possibilities presented by various directions for further work. Such an assessment is of particular importance in the field of plastic deformation and fracture of metals, in view of the large number of variable factors in experiments of the kind used in studying these problems, and also in view of the variety of materials investigated. In this and the following article an attempt is made to outline the problems of research on inelastic deformation and fracture of metals, both in those directions in which work is already being conducted and in possible new directions.

I. Introduction

In an idealized metal, the dependence between stress and deformation is divided into two parts—elastic and plastic.

Fig. 1. Idealized relation between stress and deformation.

An essential feature of the first part is its reversibility, i.e., the single-valuedness of the dependence. This dependence is not necessarily linear, although it is usually taken to be so. The distinguishing feature of the second part consists in the presence of residual deformation. In Fig. 1 the curve \(ABB'\) corresponds to an increase, \(B'A'\) to a decrease, of the load; \(AA'\) represents the residual deformation. The slope of the stress–deformation curve to the axis of abscissas in the plastic region usually has, but not necessarily, a positive value.

No real metal possesses ideal elasticity. Even in the case where there exists a definite value of the stress below which no residual deformation is obtained, the deforma—

* Journal of Applied Physics, 17, 69 (1946). Translated by N. Fuks.

...formation is nevertheless a function of the value of the stress not only at the given instant, but also at previous instants of time. The deformation diagram in the preplastic region may, therefore, have the form shown in Fig. 2: after removal of the load the metal gradually returns to its initial state. If the experimenter were to stop his observations before the hysteresis loop had closed, he would conclude that there is residual deformation in the metal, i.e. that the latter is plastic under the given conditions. That property of a solid body by virtue of which deformation and stress in the preplastic region are related ambiguously has been called “anelasticity.” Consideration of the phenomena of “anelasticity” is therefore of essential importance for a complete understanding of plastic deformation.

The plastic part of the deformation diagram in Fig. 1 has been idealized in two respects. First, in the case of alloys, serrations are observed on the curve; second, the curve for a real metal approaches the curve shown in Fig. 1 only in the case where, throughout the entire test, certain parameters remain constant, namely the rate of deformation and the temperature, which is not always the case in ordinary tests. It is not substantially difficult to derive, from a simple tensile test, the elastic part of the idealized deformation diagram for any character of the stress distribution, provided only that Poisson’s ratio has been measured. Similarly, taking the value \(1/2\) for this ratio, one can, from data obtained in a simple tensile test, predict the course of the plastic part of the idealized deformation diagram for an arbitrary stress distribution. Such predictions, however, are not as successful as in the elastic region.

Fig. 2. Typical dependence between stress and deformation in a real metal in the preplastic region.

It follows from this brief introduction that the “anelastic” properties of metals can be described in a simple way only in the presence of very special conditions. In order to understand the variety of reactions of a metal to forces applied to it, it is first necessary to understand the microscopic details of deformation.

The physical causes of “anelasticity” have recently been analyzed by one of the authors of the present article[^1]. These causes may be divided into two groups. The first includes diffusion processes caused by externally applied forces—for example, thermal diffusion, atomic diffusion, and magnetic diffusion. The second group is connected with the presence of isolated regions possessing viscous properties. Whatever the origin of “anelasticity,” it is possible, as Boltzmann showed[^2], to predict the behavior of a specimen under any conditions of increasing load...

according to observations made under the sudden application of a constant load.

A detailed description of one microscopic mechanism of plastic deformation in crystal lattices was first given by Orowan³ in developing Polanyi’s ideas. According to this description, plastic deformation proceeds in two stages. In the first, a special type of lattice distortion occurs near regions of low resistance to shear. These distortions are known as “displacements.” Their formation is illustrated in Fig. 3. In the second stage, the displacements propagate through the whole lattice. Their propagation is impeded by any disturbance of the lattice structure—other displacements (this type of obstacle causes the phenomenon of work hardening during deformation), precipitates, grain boundaries, etc.

Fig. 3. Illustration of the formation of displacements.

A. Undistorted lattice. The shaded regions represent areas of weak bonds.
B. Distorted lattice in which a pair of displacements has formed. The shaded region has increased in its extent.

Another possible mechanism of plastic deformation is the formation of twins. However, up to the present time there is still no satisfactory dynamical theory of twinning.

If these four mechanisms—diffusion, viscous flow in isolated regions, displacements, and twinning—exhaust all possible mechanisms of the “anelastic” properties of metals, then for all known types of these “anelastic” properties, as well as for their dependence on the microstructure of the metal, there must exist an explanation based on one or another of the mechanisms indicated.

II. “ANELASTICITY”

Viscous flow in isolated regions can cause deviations from ideal elasticity exceeding 100%. Viscous flow may be expected in regions in which the arrangement of atoms is amorphous, noncrystalline in character. Thus, atoms situated on different sides of the boundaries between crystalline grains are arranged irregularly with respect to one another, and therefore at these boundaries one may expect the manifestation of viscous properties. The strong influence of viscous

sliding on the relation between stress and deformation is seen from Fig. 4, constructed from data recently obtained in our laboratory by West\(^4\). A very important practical problem in which viscous sliding along the boundaries between crystalline grains can play the chief role is the relaxation of stresses at elevated temperature. Thus, in the example illustrated in Fig. 4, the stress decreased by 60% owing to relaxation caused by reversible inelastic deformation associated with sliding along the boundary between grains, and by no more than 6% owing to plastic deformation. Another problem, still more important for practice, in which viscous sliding along the boundaries between grains can play the principal role, is creep (creep strain) at elevated temperature, which is of special importance in those cases where preservation of the dimensions of a product within very narrow tolerances is required. Thus, in the iron specimen from which the results shown in Fig. 4 were obtained, 10-minute heating at 500°C increased the deformation under constant load by more than a factor of two. At least 95% of this additional deformation belongs to the reversible inelastic type associated with sliding along the boundaries between grains.

Fig. 4. Inelastic behavior of iron wire caused by viscous flow along the boundaries of crystalline grains, according to West’s experiments (reference 4).

Rosenhain\(^5\) pointed out in 1910 the important significance of viscous sliding along the boundaries between grains and carried out a series of experiments proving the existence of such sliding. Despite the above-mentioned practical importance of this phenomenon, during the 30 years that have elapsed since then not a single attempt has been made to determine, in any metal, the relation between the rate of sliding along the boundary between grains and the force causing this sliding. This complete absence of experimental data is due to the fact that almost all physical tests of metals, especially in the U.S.A., have been and are being carried out with complete disregard for the microstructure of metals. Such a mode of action is caused by a lack of faith in the ability of physics to predict the macroscopic mechanical properties of metals on the basis of the mechanical properties of individual structural elements. The authors of the present article do have this faith.

Until an atomic interpretation of all inelastic phenomena has appeared, it is advisable to use certain phenomenological concepts concerning the mechanical properties of microscopic structural elements with dimensions exceeding atomic dimensions. Such, for example, is the concept of viscous boundaries between crystalline grains. Another concept that may prove useful is that of viscous slip bands. This concept can be studied by measuring “inelasticity.”

As early as 1900 it was established that plastic deformation occurs only along slip lines^6, that this deformation causes phenomena of “inelasticity,” for example elastic hysteresis, and that “inelasticity” can be eliminated by heating the metal even to \(100^\circ\mathrm{C}\).^7 Rosenhain^8 pointed out in 1905 that these phenomena during plastic deformation indicate that, in the newly formed slip planes, the metal possesses the properties of a disordered body, i.e. a viscous character, but that upon ageing of the metal it gradually assumes the ordered structure of the surrounding main mass of the metal.

During the subsequent 40 years no one carried out experiments aimed at further elucidating the properties of the newly formed slip planes. Many observations were made on special questions of “inelasticity” caused by plastic deformation. A survey of these works is given in the article cited above.^1 However, one can hardly expect these observations to help in understanding the mechanism of deformation until they are interpreted on the basis of the mechanical properties of the slip planes.

III. MECHANISM OF PLASTIC DEFORMATION

Plastic, i.e. permanent, deformation of metals may proceed by one of two different mechanisms—by twinning and by slip. In each particular case the deformation may represent a combination of these two types.

Two neighboring parts of a crystal, separated by a plane, form a twin if their crystallographic axes are different but are directed so that each part is the mirror image of the other with respect to their common plane. The crystallographic relations in the twins studied were described by Shmid and Barrett.^9 Twinning^10 is called deformation with the formation of a twin. It may be assumed that the final configuration is obtained by a homogeneous simple shear parallel to the common plane. Twinning is not accompanied by any distortion of the crystal lattice.

Deformation by slip is thought to occur in those cases in which it is confined to narrow bands that do not form twins with the adjoining zone. As observations show, the orientations

slip bands, and also the direction of slip, are connected by definite crystallographic relations with the parent lattice. These relations are also given in the monographs mentioned above.^{11,12} Within a slip band the orientation of the substance is, generally speaking, the same as in the parent lattice, but undergoes considerable fluctuations in individual regions.

Temperature and strain rate are the principal factors determining the type of deformation experienced by a crystalline substance. With an increase in temperature or with a decrease in the rate of deformation, the resistance to deformation by slip falls rapidly; the resistance to twinning, however, does not fall so rapidly, or does not change at all. Another factor that has some influence on the type of deformation is the stress gradient in the metal. Macroscopic deformation may be represented as the superposition of many small deformations. The smallest possible deformation is called elementary. Each such elementary deformation is localized in a limited region of the substance. As a first approximation, only the average value of the stress over the whole region determines the presence or absence of an elementary deformation. In deformation by twinning or by slip, the width of the region is comparable with the size of the crystal. It has not yet been clarified whether localization of stress favors twinning or slip. Still another factor that exerts a certain influence on the type of deformation is the chemical composition of the metal. Thus, for example, the addition of silicon to iron increases the resistance to slip to a greater degree than the resistance to twinning.^{13} Whether this action of silicon is general for all solid solutions has not yet been established. The influence of insoluble impurities on the type of deformation is also unknown.

A. Twinning

In his experiments with tin, Chalmers^{14} was apparently the first to point out that twinning is a macroscopically discontinuous process, or, in other words, that the region of elementary deformation in this case has macroscopic dimensions. One might have expected that, under conditions of uniform stress, there would exist a critical value of the shearing stress necessary for the formation of a twin. In reality, however, the conditions necessary for the formation of a twin are so structurally sensitive that in no work on single crystals has it been possible to obtain reproducible critical values of the shearing stress in twinning. In contrast to deformation by slip, twinning can begin in a limited region with a large value of the stress and spread over a great distance through regions in which, before the onset of deformation, the stress was equal to zero (see Fig. 47 in Schmid and Boas). It may be supposed that, owing to the sp—

properties of twins to arise under the action of a local high stress, the critical value of the shear stress required for the formation of twins must be sensitive to irregularities on the surface of the specimen; however, the possible effect of surface defects has not yet been investigated.

Once a twinning band has formed, it can grow continuously further by extension of the twinning plane. Thus, in recrystallized α-brass, twinning bands expand as the crystalline grains grow. The dependence of the rate of propagation of twinning planes on various parameters, for example on the magnitude of the stress and on the temperature, has not yet been studied.

Phase transformations caused by shear in the lattice have much in common with twinning. Examples are the formation of martensite in steel and martensite-like structures in β-brass. Any success in understanding one phenomenon will contribute to the understanding of the other.

B. Slip

In a first approximation, deformation by slip may be regarded simply as the sliding of one group of planes over another. This picture is sufficiently accurate to predict the planes and directions of slip. If it is assumed that one crystal plane slides uniformly over another, the potential energy of such a system must undergo periodic fluctuations. The amplitude of the fluctuations will be smallest for planes with maximum close packing, since in such planes the distance between neighboring atoms is minimal and the distance between planes is maximal. Similarly, the fluctuations will be smallest in those directions along which the atoms are arranged most closely. Thus, the picture of simple slip satisfactorily explains the experimentally observed facts—that the planes of easiest slip are the planes with the closest packing of atoms, and that the direction of slip is the direction in which the atoms are arranged most closely.

However, the idea of the slip of an atomic plane as a whole over another plane cannot be correct. For such slip in an unalloyed crystal, a shear stress several orders of magnitude greater than that found experimentally would be required. Polanyi¹⁵ was apparently the first to discover that this difficulty can be resolved by introducing into the consideration lattice distortions of a definite type, known as “dislocations.” Such dislocations move through the lattice and thus lead to the resultant macroscopic deformation at a stress of considerably smaller magnitude than that which would be required for the slip of one plane over another as a single whole. To estimate the full effect of dislocations in reducing the resistance

for slip one must also take into account the thermal energy of the lattice. Fluctuations of thermal energy may, over a certain interval of time, produce an elementary slip process under conditions in which the stress by itself is insufficient for slip. Such fluctuations of thermal energy cannot, however, produce slip of one atomic plane, as a whole, over another, since the probability of the fluctuation required for this, a very considerable one, is too small. Let \(E\) be the magnitude of the directed kinetic energy which the system must possess in order, together with the shearing stress, to produce elementary slip. The probability that, at a given instant of time, the system will possess a store of directed kinetic energy equal to or exceeding \(E\) is proportional to \(e^{-E/kT}\). Thermal fluctuations may thus be important only in the case where \(E\) does not exceed \(kT\) by several orders of magnitude, and consequently when the elementary slip process occurs in a comparatively small volume, as is the case in displacements.

If the influence of temperature on plastic deformation is due exclusively to energy fluctuations, it follows that, at constant stress, the rate of deformation must increase with temperature; in other words, the stress required to attain a definite rate of deformation is the smaller the higher the temperature. At sufficiently high temperatures additional effects appear, such as, for example, the growth of crystalline grains, spheroidization of precipitates, etc., and finally, if the initial deformation was sufficiently large, the phenomenon of recovery is observed, which may be accompanied (not necessarily) by recrystallization.

At lower temperatures, when such additional processes do not occur, one may expect, as Ludwig first indicated\(^{16}\), the existence of a mechanical equation of state relating the stress \(S\), the deformation \(\varepsilon\), the rate of deformation \(\dot{\varepsilon}\), and the temperature \(T\). This equation may be written symbolically as:

\[ S = S(\varepsilon, \dot{\varepsilon}, T), \tag{1} \]

or

\[ \dot{\varepsilon} = \dot{\varepsilon}(\varepsilon, S, T). \tag{2} \]

The existence of a mechanical equation of state is equivalent to the assumption that the stress necessary to obtain a definite rate of deformation at a given temperature depends only on the magnitude of the deformation at the given moment and does not depend on the temperature and deformation at previous moments of time. According to Taylor’s ideas\(^{17}\), a discontinuous change in temperature does not produce an equally discontinuous change in stress, but only a discontinuous change in the rate of increase of deformation with increasing stress. In order to determine whether a mechanical equation exists

states, experiments were carried out in our laboratory in which, during the recording of the stress–strain diagram, the temperature was suddenly changed. The results of these experiments, presented in Fig. 5, show that the stress does not depend on the temperature at which the deformation had previously been carried out and that, consequently, a mechanical equation of state really does exist (at least at low temperature). It follows from this that, at the temperatures of our experiments, the influence of temperature consists exclusively in the occurrence of fluctuations of thermal energy.

Fig. 5. Example of the applicability of the mechanical equation of state to unalloyed carbon steel at low temperatures (tensile tests).

Twinning and slip are processes characteristic of a crystal. Therefore it may be expected that any disturbance of the crystalline structure increases the resistance of the material to deformation of these two types. This is indeed the case in reality. Such a disturbance may be caused by prior plastic deformation, by the boundaries of crystal grains, by soluble and insoluble impurities, etc. The increase in resistance to plastic deformation caused by prior deformation was investigated theoretically by Taylor^17, who used the concept of displacements. Although Taylor’s theory should be regarded as a serious step toward understanding the phenomenon of strain hardening during deformation, it is based on overly naive assumptions and cannot withstand the test obligatory for any physical theory—the ability to predict new phenomena. Up to the present time there has not appeared a single theoretical work in which these unfounded assumptions have been eliminated, for example, concerning the existence of impenetrable barriers and parallel displacements of indefinite length. The increase in resistance to plastic deformation caused by crystal boundaries—

...of grains was also treated theoretically by Taylor^18. Although his theory gives the correct magnitude of the discrepancy between the deformation diagrams of aluminum single crystals and polycrystalline specimens of this metal, the theory does not explain the frequently observed dependence of the resistance to deformation on the grain size in a polycrystalline material^19,^20. Essential in Taylor’s theory is the assumption that the restraining action of neighboring grains is distributed uniformly throughout the grain. Meanwhile, the observations mentioned above indicate that this action is limited to a more or less narrow zone adjoining the grain boundaries. To date no modification of Taylor’s theory has been proposed that takes into account the indicated localization of the restraining action. Gensamer^21 found a relation between the increase in the resistance to plastic deformation caused by dissolved atoms and the difference in size between dissolved atoms and solvent atoms. Further, Gensamer and his co-workers^22 found a relation between the increase in resistance to deformation caused by insoluble precipitates and the mean free path between these precipitates. A satisfactory explanation of these dependences likewise does not yet exist.

Although deformation previously undergone by a metal generally increases its resistance to further deformation, this cannot be said of every element of deformation. A small deformation may, in fact, lower for an instant the resistance to subsequent deformation. Thus, a smaller stress is required for the propagation of a slip band than for the formation of a band. Further, in many cases a lower stress is required for the formation of a slip band adjoining an already existing band than for the formation of the latter. Such an instantaneous decrease in the resistance to further deformation, caused by a small deformation, leads to a whole series of new phenomena. Thus, for example, when a constant stress is applied to a specimen, elongation may occur in steps. Conversely, if the total rate of deformation is kept constant, the stress may change in separate jumps. Whether each jump is associated with an individual slip band or with a whole series of bands, as in the case of duralumin, depends to a considerable extent on the elasticity of the system. Despite the large number of experiments in which discontinuous deformation has been observed, there is no satisfactory theory that would explain the instantaneous drop in resistance at each jump and the accompanying phenomena. The ideas of Orowan^3 concerning the origin of displacements could serve as the basis for such a theory. According to Orowan, displacements arise in certain weak regions of the lattice, shaded in Fig. 3. As is seen from Fig. 3 B, after one pair of displacements has formed and has separated by a certain distance, the weak region expands and now, for the formation of a new pair of displacements, ...

a lower stress is already required. The occurrence of slip is thus an autocatalytic-type process.

Discontinuous deformation is especially pronounced in mild steel and in duralumin. In both cases the phenomenon is apparently connected with a state of supersaturation: in iron—with respect to dissolved carbon or nitrogen, or both together; in aluminum—with respect to copper or zinc. Indeed, the phenomenon of discontinuity disappears if all the carbon and nitrogen are removed from the iron, and becomes considerably less noticeable during the aging of aluminum. It seems probable that in these cases the phenomenon is connected with the presence of regions in which the dissolved atoms have reached an especially high concentration or have separated out in the form of flakes. Such regions usually appear before the visible precipitation of flakes; their formation is probably caused by the fact that such a distribution reduces the deformation energy of the system \(^{23}\). The flakes are a restraining factor with respect to the propagation of slips. As soon as a slip has crossed a flake, thereby dividing it into two parts, the flake no longer offers resistance to further slips in the same plane. Many problems relating to this phenomenon still remain to be solved. For example, it is not observed in single crystals of iron, but does occur in single crystals of aluminum alloys. The theory of the formation of concentration flakes has also not yet been developed.

IV. GENERAL PICTURE OF PLASTIC DEFORMATION

A. Mechanical equation of state

The assumption of the existence of a mechanical equation of state leads to a number of interesting conclusions of a qualitative character.

Fig. 6. Example of the relation between deformation and stress in the case when the stress is held constant for some time.

Consider, as an example, a specimen deformed isothermally and at constant rate up to some final amount of deformation \(E_0\); then the stress is maintained constant for a long time, after which deformation is resumed at the previous rate. Figure 6 shows the deformation diagram obtained under these conditions. While the stress is maintained constant, the deformation increases at a rate determined by equation (2), and therefore the stress now required in order to return to the initial rate of deformation also increases according to equation (1). Deformation diagrams of this type were in fact obtained by Hanson and Wheeler \(^{24}\). As another

as an example, let us consider the case of a constant load (as distinct from the case of a constant stress), analyzed in detail by Ludwik. As can be seen from Fig. 7, the sudden application of the load leads to a high initial rate of deformation. Then the rate of deformation gradually decreases, reaches a minimum at the deformation value \(E_0\), and thereafter increases again. The curve of deformation versus time thus has the form shown in Fig. 7, with the inflection point of the curve corresponding to the onset of neck formation.

The relation between deformation and time, presented in Fig. 7, is typical for creep tests. Although Ludwik’s work was published more than 40 years ago, the generally accepted interpretation of the portion with a constant creep rate on creep diagrams was that, in this portion, there is equality of the rates of strengthening caused by deformation and of softening due to recovery or recrystallization of the metal. A solution of the question of the correct interpretation of creep diagrams might be achieved by means of experiments in which the temperature would be changed during the test, for example according to the following program: deformation of \(1\%\) at room temperature; then heating to the temperature at which the creep test is carried out, cooling to room temperature and, finally, determining whether the strengthening caused by the initial small deformation has decreased. It is especially necessary to emphasize that Nadai’s\({}^{25}\) determination of the stress function \(S_T(\dot{\varepsilon}, \varepsilon)\) for a number of temperatures by means of measurements performed at constant deformation rates cannot establish the existence of a mechanical equation of state. In order for such a function to represent a mechanical equation of state, it is additionally necessary to discover that the stress required for further increasing the deformation of a specimen held for some time at

Fig. 7. Deformation as a function of time according to the concept of a mechanical equation of state.

with a constant magnitude of the strain, does not decrease with increasing holding time.

Two attempts were made to establish the form of a mechanical equation of state for metals. Since neither of them was entirely successful, it is useful to examine them in detail in order to determine the direction of further research that might lead to the goal.

The first attempt was undertaken by Becker in 1926[^26] and analyzed by Orowan[^3] in 1936. In it, central attention is given to the magnitude of the shear stress that an elementary region must possess in order to undergo plastic deformation. Let \(S_0\) be the actual magnitude of the shear stress required for deformation, and \(S\) the magnitude of the macroscopic shear stress. Thermal fluctuations supply an additional stress component \(S_0 - S\). If \(V\) is the minimum volume in which the stress \(S_0\) must exist in order for deformation to occur, then the rate of deformation is expressed by the equation

\[ \dot{\varepsilon}=f_0 \exp[-V(S_0-S)^2/2GkT]. \tag{3} \]

The second factor expresses the probability that the given region will at some moment possess a thermal fluctuation shear stress exceeding \(S_0-S\) (\(G\) is the shear modulus). The first factor may be put equal to

\[ f_0=n\varepsilon_0\tau, \tag{4} \]

where \(n\) is the number of effective regions of volume \(V\) contained in a unit volume, \(\varepsilon_0\) is the average strain of the specimen caused by elementary plastic deformation in one region per unit volume, and \(\tau\) is the average time during which the shear stress \(S_0\) must exist in the elementary region before slip occurs in it. In his analysis of Becker’s mechanical equation of state, Orowan pointed out that \(S_0\) is of the same order of magnitude as the macroscopic shear stress required to produce deformation at room temperature. From this Orowan concluded that, at any moment of time, deformation is localized in regions with a large concentration of stress.

The second attempt, undertaken by Condon[^27] in 1938, was further developed by Kauzmann[^28] and by Dushman[^29] and his collaborators. The basic point of this theory is the activation energy necessary for the elementary process of slip. If \(Q_0\) is the magnitude of the activation energy in the absence of applied external stress, then upon application of a stress \(S\) the activation energy is lowered, as is assumed—

make the authors of the theory, by an amount proportional to \(S\); thus

\[ Q(S)=Q_0-VS. \]

The corresponding mechanical equation of state will therefore be

\[ \dot{\varepsilon}=f_0 \exp\left[-(Q_0-VS)/kT\right], \tag{5} \]

where both factors have the same physical meaning as in Becker’s equation.

Equations (3) and (5) are equivalent only if \(S \ll S_0\), i.e., if the stress causing deformation is considerably smaller than the stress necessary to produce deformation at absolute zero temperature. However, as Orowan indicated, \(S\) and \(S_0\) are of the same order of magnitude and, consequently, these two equations cannot be regarded as equivalent. A more detailed analysis of the elementary process of slip shows that both equations are approximations to the more general equation

\[ \dot{\varepsilon}=f_0 \exp\left[-Q(S)/RT\right]. \tag{6} \]

In the Becker approximation an implicit assumption is made that the probability of attaining, by thermal stress, a certain magnitude in a given elementary region does not depend on the stress caused by external forces. In view of Orowan’s representation of the very large stresses arising under the action of external forces in the regions in which slip occurs, it is unlikely that this assumption is fully correct. In the Kauzmann–Dushman approximation an explicit assumption is made that the activation energy may be taken as a linear function of the stress. From the standpoint of Orowan’s concepts, this assumption too is debatable. In particular, the linear correction in the expression for the heat of activation does not take into account the circumstance that stress changes the positions of potential peaks and wells, as well as their height.

Applying equation (5) to experimental data, Kauzmann and Dushman found it necessary to assume that \(V\) varies with temperature, although they were unable to give a satisfactory explanation for this variation. As an illustration of the applicability of equation (6), we shall present Figs. 8 and 9, based on experimental data obtained by Dushman.

In our recent investigations\({}^{30}\) on the question of the influence of changes in deformation rate and temperature on the strength of steel, we came to the conclusion that the stress at a given magnitude of deformation depends on the deformation rate and temperature through only one parameter \(P\), defined by the formula

\[ P=(\dot{\varepsilon}/f_0)e^{-Q/RT}. \tag{7} \]

It is of interest to ascertain under what conditions this conclusion is compatible with the general mechanical equation of state. Suppose that we

we shall be interested in that range of strain rates and temperatures in which the stress is close to \(S_0\). Denoting by \(\Delta S\) the difference between \(S\) and \(S_0\), we obtain from equation (6):

\[ \Delta S=\{\ln(f_0 \dot e)-Q(S_0)/RT\}\,RT/Q'(S_0). \tag{8} \]

In the range of strain rates and temperatures in which the stress is close to \(S_0\), the first factor contains two large numbers that almost compensate one another. A definite change in temperature will therefore be reflected much more strongly in the first factor than in the second. The influence of changes in strain rate and temperature is thus manifested chiefly through the parameter \(P\), determined by equation (7), in which \(Q\) is taken at \(S=S_0\).

Despite the large number of experimental investigations on creep that have been carried out previously and are being carried out at the present time, especially in the USA, no one except Dorn has set himself the aim of establishing a mechanical equation of state. Therefore there are no experimental data on the dependence of the value of the constant \(f_0\) and of the function expressing the magnitude of the activation energy \(Q(S)\) on various metallurgical variables, such as, for example, dissolved elements, precipitates, grain size, etc. Since Orowan’s old work, there has not appeared

Fig. 8. Application of the mechanical equation of state to pure aluminum, \(f_0=2.2\cdot10^5\) (according to the data of Dorn et al.\(^{28}\)).

...not a single theoretical work in which the mechanism of plastic deformation and the dependence of the latter on thermal energy were examined in greater detail.

Mechanical equilibrium can exist only under such conditions in which the metal does not change appreciably with time at zero rate of deformation. Thus, the metal must not soften owing to recovery or recrystallization. The smaller the preceding deformation was, the higher is the temperature required to eliminate work hardening. The agreement of the experimental results presented in Figs. 8 and 9 with the predictions derived from the equation of state indicates that, in the metals used, the work hardening under the experimental conditions did not decrease appreciably owing to recovery or recrystallization. In order to determine the conditions under which the mechanical equation of state is applicable, it is necessary to develop a theory of recovery and recrystallization.

Fig. 9. Application of the mechanical equation of state to an aluminum alloy with 2% magnesium.
$f_0 = 2.5 \cdot 10^9$ (according to the data of Dupman et al.^28).

B. Strengthening during deformation (work hardening)

As has already been indicated, the resistance to deformation usually increases with the increase of the latter. Thus, in equation (1) the stress $S$ increases with the deformation $\varepsilon$. In a recently published article^31 the question of the exact form of the dependence of the tensile stress on elongation at constant rate of deformation and temperature is analyzed.

If discontinuous deformation does not occur, the stress in this case is a power function of the deformation up to the value of the latter equal to 0.4. Thus,

\[ S=S_0(\varepsilon/\varepsilon_0)^m, \]

where \(S_0\) is the stress at some arbitrary deformation \(\varepsilon_0\). The dependence of the constants \(S_0\) and \(m\) on metallurgical factors has been studied in considerable detail in the case of iron and steel. Thus, in the absence of precipitates, the addition of soluble alloys increases \(S_0\), without noticeably changing the value of the exponent \(m^{32}\). Conversely, in steels whose strength is due primarily to insoluble carbides, the exponent \(m\) is inversely proportional to \(S_0\) at any carbon content. Whether analogous relations exist in other metals is still unknown at present.

C. Generalized stress and deformation

In all that has been set forth above it was implied that the stress and deformation referred to uniaxial tension. We shall now examine the problems that arise in passing to stresses of a more complicated type.

The route by which the laws of uniaxial tension can be generalized so that they encompass a stress system of the most general type was indicated by Mises\(^{33}\). A certain invariant function of the stresses, \(S_1\), plays the same role in determining the rate of deformation as the tensile stress plays in the case of uniaxial tension. This function is defined in such a way that \(S_1^2\) is proportional to the energy of elastic shear. Thus,

\[ S_1^2=\frac{1}{2}\{(Y_y-Z_z)^2+(Z_z-X_x)^2+(X_x-Y_y)^2\}+3(Y_z^2+Z_x^2+X_y^2). \tag{9} \]

The type of deformation corresponding to the generalized system of stresses is identical with the type of deformation that an amorphous specimen would possess. Thus

\[ \dot{\varepsilon}_{xx}=\lambda\{X_x-(Y_y+Z_z)/2\}. \]

Analogous equations are obtained for \(\dot{\varepsilon}_{yy}\) and \(\dot{\varepsilon}_{zz}\) with the same value of the factor \(\lambda\). Further,

\[ \dot{\varepsilon}_{yz}=3\lambda Y_z \]

and analogous equations for \(\dot{\varepsilon}_{zx}\) and \(\dot{\varepsilon}_{xy}\). The generalized mechanical equation of state can therefore be written as

\[ \dot{\varepsilon}_{xx}=\{X_x-(Y_y+Z_z)/2\}\,S_1^{-1}f_0\exp[-Q(S_1)/RT] \tag{10} \]

and analogous equations for $\dot{\varepsilon}_{yy}$ and $\dot{\varepsilon}_{zz}$, and also:

\[ \dot{\varepsilon}_{yz}=3X_{yz}S_1^{-1}f_0\exp[-Q(S_1)/RT] \tag{10'} \]

and analogous equations for $\dot{\varepsilon}_{zx}$ and $\dot{\varepsilon}_{xy}$.

In the case of uniaxial tension we saw that strain hardening during deformation could be taken into account by regarding $f_0$ and $Q$ as functions of stress. In the case of a stress of general type one can proceed analogously. First, however, it is necessary to establish a measure for relative deformation of general type. To find such a measure one must first find a method of summing the increments of relative deformation corresponding to given increments of absolute deformation. Then the increments of relative deformation must be determined as functions of the increments of absolute deformation. The first problem was solved by following the proposal of Ludwik1: to determine the increment of relative deformation associated with a given increment of absolute deformation by referring the latter to the configuration of the system immediately before this increment. Thus, if the length of a specimen increases under uniaxial tension from $l$ to $l+dl$, the corresponding increment of relative deformation is equal to $dl/l$, independently of the magnitude of the deformation experienced by the specimen before this. As Ludwik indicated, the total relative deformation in this case is simply equal to the sum of all increments of relative deformation. Thus, if the length of a specimen changes under uniaxial tension from $l_2$ to $l_1$, the corresponding total relative deformation is equal to $\ln(l_2/l_1)$. A satisfactory measure of the increment of relative deformation is an invariant function of the components of the relative deformation, proportional to $S_1$ in the case of an isotropic elastic medium. Such a function $d\varepsilon_1$ is defined by the equation

\[ d\varepsilon_1^2=\frac{1}{2}\{(d\varepsilon_{yy}-d\varepsilon_{zz})^2+(d\varepsilon_{zz}-d\varepsilon_{xx})^2+(d\varepsilon_{xx}-d\varepsilon_{yy})^2\}+ \]
\[ +\frac{3}{4}(d\varepsilon_{yz}^{\,2}+d\varepsilon_{zx}^{\,2}+d\varepsilon_{xy}^{\,2}). \tag{11} \]

Using the preceding equations of this section, it can be proved that the deformation so defined satisfies the following equation:

\[ \dot{\varepsilon}_1=f_0\exp[-Q(S_1)/RT]. \tag{12} \]

The quantities $f_0$ and $Q$ should be regarded as functions of the deformation $\varepsilon_1$. Equations (10), (11), and (12) form a consistent scheme of a general mechanical equation of state. The main questions now are: how correctly does this scheme reflect the properties of real metals, and what causes the corresponding deviations.

Let us note one consequence of the general equation of state: at constant temperature and rate of deformation there must exist a general hardening function for deformation of the type:

\[ S_1 = S_1(\varepsilon_1). \tag{13} \]

Nadai and Davis\({}^{34}\) performed an experiment in order to verify this equation. They measured the stress \(S_1\) as a function of the deformation \(\varepsilon_1\) for different types of stress distribution and found that, at small relative deformations not exceeding 0.2, the equation is valid, but that at larger deformations considerable deviations are observed.

Fig. 10. Comparison of the results of tensile and torsion tests by means of the concept of generalized stresses and deformations (tempered martensitic steel).

Fig. 11. Comparison of the results of tensile and torsion tests by means of the concept of generalized stresses and deformations (pearlitic steel).

In deriving the general mechanical equation of state by means of the invariant functions \(\varepsilon_1\) and \(S_1\), it was assumed that the substance initially was and remained isotropic. It should be expected that plastic deformation will cause the appearance of anisotropy, the more noticeable the greater the deformation. The degree of anisotropy caused by a given deformation quite probably also depends on the microstructure of the metal. To illustrate this proposition, in Figs. 10 and 11

the hardening curves (on a logarithmic scale) under tension and torsion in two types of microstructure are presented. As was to be expected, steel possessing the greater anisotropy of structure, namely pearlitic steel, exhibits greater deformation anisotropy. For the further development of the theory of plastic deformation under stresses of a general type, it is necessary to determine the magnitude of the anisotropy caused by deformation and to introduce into the mechanical equation of state a correction for this anisotropy.

Up to now we have assumed that the strain and its components in the equation of state referred only to plastic, i.e. permanent, strain. The magnitude of the components of elastic strain can be found from the equation of elastic deformation:

\[ (\dot{\varepsilon}_{xx})_{\text{elastic}} = E^{-1}\{ \dot{X}_{x} - \sigma(\dot{Y}_{y} + \dot{Z}_{z})\}. \tag{14} \]

The total components of strain are thus equal to the sum of the elastic and plastic components:

\[ (\dot{\varepsilon}_{xx})_{\text{total}} = (\dot{\varepsilon}_{xx})_{\text{elastic}} + \dot{\varepsilon}_{xx}. \tag{15} \]

Using equations analogous to (14) for the elastic part and equation (10) for the plastic part of the strain, one can thus derive a system of equations for the time derivatives of the components of the total strain. This was done by Prandtl^35 and Reuss^36. If one passes from one type of stress distribution to another at comparable magnitudes of elastic and plastic strain, the validity of these equations is violated for two reasons. First, the “inelasticity” of real metals does not allow a sharp boundary to be drawn between elastic and plastic deformation. Second, the Bauschinger effect in real metals is very significant for plastic strains comparable with the elastic ones, but becomes imperceptible for considerably larger plastic strains, as was clearly shown by Sachs and Shoen^37. Prager^38 applied the Prandtl–Reuss equation to the case of relaxation of residual stresses caused by deformation, at a total magnitude of strain comparable with elastic strains. Instead of attributing the discrepancy between theory and experiment to “inelasticity” and the Bauschinger effect, Prager applied an empirical rule (derived, according to the author’s assertion, from the “Hencky–Nadai theory”) relating plastic and total strain, and obtained agreement with the experimental data. Analysis of this rule shows, however, that it leads to different results in cases of continuous and interrupted plastic deformation; yet all experiments indicate the contrary. In opposition to Prager’s point of view, we believe that a correct understanding of the mechanism of small plastic deformations is attainable only by studying the causes of the phenomena of “inelasticity” and the Bauschinger effect, and not by attempts to find empirical formal relations between the type of stress and elastic and plastic deformation.

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1. Ludwik. ↩