On the Dislocation Hypothesis of Plasticity

M. V. Klassen-Neklyudova and T. A. Kontorova

In the February issue of UFN for 1952, a translation was published of a review article by the American physicist Cottrell, “The Theory of Pinning in the Crystal Lattice”*).

Cottrell’s article sets forth the so-called “dislocation” concepts of the mechanism of plastic deformation of crystalline bodies. As is known, these concepts were first formulated by Taylor as early as 1934[^1]. In the two decades that have passed since then, the dislocation hypothesis of the plasticity of crystals has been intensively developed by foreign scientists. The number of published works devoted to these questions is very large.

It is essential to note that at the present time the adherents of the dislocation theory include all foreign investigators of plasticity, both theoreticians and experimentalists.

In the foreign literature, moreover, there is a tendency to explain from the “dislocation” point of view also a number of properties of crystals that have no direct relation to the phenomenon of plasticity. Thus, for example, in a number of works a dislocation model of grain boundaries in polycrystals is considered, the question of a “dislocation” mechanism of crystal growth is discussed, and so on.

The translation of Cottrell’s article is not accompanied by editorial comments. The reader may therefore get the impression that the correctness of the dislocation theory of plasticity is beyond doubt, and also that this theory is adhered to by all Soviet scientists as well.

In our article “The Development of Modern Theoretical Conceptions of the Nature of Plastic Deformation,” published in UFN in 1944[^2], it was noted that the “dislocation” conceptions of the mechanism of plasticity of crystals encounter serious fundamental objections.

*) UFN, XLVI, issue 2, pp. 179–230 (1952).

Relying on the more modern exposition of dislocation theory given by Cottrell, let us try to determine whether these ideas have undergone any substantial changes, whether the basic shortcomings of the theory have been eliminated, and whether any successes have been achieved on the path toward explaining the experimental regularities observed in the plastic deformation of crystals.

On the first page of his review Cottrell writes:

“It has already become evident that a special type of defect in the crystal lattice, called a dislocation, is an important link connecting the atomic structure of metals with their crystallographic and plastic properties...”

What, then, is a “dislocation” (or “catching”), i.e., precisely what defects are assumed to be responsible for the plastic properties of crystals?

On p. 180 we read:

“All dislocations reduce to a few standard types having quite definite forms.”

As the basic type of dislocation Cottrell further considers the linear dislocation, whose model was proposed by Taylor in 1934.^1 Let us recall that, according to Taylor, an individual dislocation is such a disturbance of the regular arrangement of lattice particles in which the number of atoms in one of the atomic rows is greater by one (condensation) or less (rarefaction) than the number of atoms in a neighboring atomic row; such a “nonus”-type distortion embraces a considerable region of the crystal. Taylor believes that in crystals there exist regular “dislocation lattices,” consisting of alternating rows of “positive” and “negative” dislocations (condensations and rarefactions).

To the second basic type of dislocation Cottrell assigns the so-called “screw” dislocations, introduced for consideration by Burgers in 1939.^3

The basic propositions of the dislocation hypothesis of plasticity reduce to the following:

1) It is assumed that in real crystals dislocations of the type mentioned above exist before the beginning of plastic deformation.

2) It is assumed that plastic deformation is the result of the displacement of dislocations along slip planes.

Speaking of the basic properties of dislocations, Cottrell writes:

“The basic properties of dislocations are their mobility along the slip plane and, as a result of this motion, the relative displacement of parts of the crystal situated on opposite sides of the slip plane in the direction of slip. These properties explain the important significance of dislocations in the theory of plasticity and sharply distinguish dislocations from other types of lattice defects” (p. 183).

What are the reasons for the appearance of dislocations in a crystalline lattice? Let us recall that Taylor once assumed that both individual dislocations and “dislocation lattices” arise in crystals as a result of thermal fluctuations. We have pointed out ^2 that distortions arising in a crystal lattice due to thermal fluctuations cannot exist in it for any appreciable length of time; they must gradually dissipate under the influence of the very same causes that produced their appearance. All the more implausible is Taylor’s hypothesis concerning the possibility of the formation of dislocations in crystals of a perfect lattice. Over this period a number of authors have become convinced that thermal fluctuations indeed cannot be the cause of the occurrence of dislocations. Calculations have shown that the energy of thermal fluctuations is insufficient for the formation of distortions of the Taylor-dislocation type extending over a considerable region of the crystal lattice. Still less is it sufficient for the formation of more complex “screw” dislocations.

Discussing the question of possible reasons for the appearance of dislocations in crystals, Cottrell is now compelled to admit:

“A large energy of formation leads to the conviction that dislocations cannot exist as stable details of the structure of a crystal in a state of equilibrium and can hardly be formed by thermal fluctuations, even in a strongly stressed, but in other respects perfect, crystal” (p. 227).

It should be noted that, on the question of the mechanism of dislocation formation, among the supporters of the dislocation hypothesis of plasticity there is still no unanimity.

Zaitz ^4 believes that dislocations arise in an overstressed zone at the ends of cracks present in real crystals. Cottrell criticizes this point of view, noting that sufficiently large stress concentrations corresponding to the energy of dislocation formation could occur only in the presence of cracks of very great depth (of the order of 1 mm).

To this, it seems to us, the following must be added. If cracks really could serve as a source of the formation of “dislocations,” then dissolution of the surface layer of a crystal, leading to the elimination of surface cracks, should entail not only an increase in tensile strength (the well-known “Joffe effect” observed experimentally), but also a noticeable increase in the elastic limit and the yield point.

In particular, the tensile curve of rock-salt crystals in water should then lie above the tensile curve taken in air.

Special experimental investigations ^5, however, have shown that dissolution of the surface of NaCl crystals is not accompanied by an increase in their elastic limit or yield point.

Another point of view is also very widespread, according to which grain boundaries may serve as a source of dislocations.

If this were so, then, other conditions being equal, fine-grained polycrystals, characterized by the greatest extent of grain boundaries, should have shown the greatest tendency toward plastic deformation.

Numerous experimental data, however, testify quite convincingly to the fact that in reality polycrystals are less plastic than single crystals.

Incidentally, the proponents of this point of view leave unexplained the very mechanism by which dislocations arise at boundaries, i.e., those physical causes by which the intercrystalline layer could be regarded as a supplier of dislocations determining the plastic properties of a crystalline grain.

Discussing the possible causes of the presence of dislocations in crystals, Cottrell inclines toward Frank’s hypothesis, according to which the growth of a nucleus of the crystalline phase in a melt or vapor is possible only on the condition that this nucleus contains dislocations and, consequently, dislocations appear already in the process of crystallization and “must naturally be encountered as structural details of real crystals” (p. 228).

It is indisputable that real crystals contain various structural defects that arose during crystallization, including, probably, a certain number of distortions of the “dislocation” type.

Let us recall, however, that according to Cottrell the principal and special property of dislocations that determines the plasticity of crystals is their high mobility. If such dislocations really arose during crystallization, then a fresh, newly grown crystal would have to deform under the smallest stresses, while at the same time undergoing hardening and gradually losing its plastic properties.

In order to explain the appearance of new dislocations that ensure the preservation of the crystals’ capacity for plastic deformation, it would then inevitably be necessary to resort to an additional assumption concerning the presence of some other mechanism for their formation. As for Frank’s hypothesis itself, it too encounters serious objections. Thus, for example, Buckley, in his article “On the Question of the Difficulties of Modern Theories of Crystal Growth”⁶, discussing Frank’s hypothesis, points out that it is highly doubtful that the occurrence of screw dislocations plays an essential role in the process of crystal growth.

Analyzing a very extensive body of experimental material on crystal growth, Buckley comes to the following conclusion:

“One can find few such examples that would indisputably indicate the applicability of the theory of screw dislocations, and there are ...”

hundreds of different types of crystals and tens of thousands of such individual crystals with respect to which there cannot even be any suspicion that screw dislocations play any substantial role for them.

At the beginning of the paragraph “The origin of dislocations,” which is the very last paragraph of the review, Cottrell writes:

“The justification for discussing the question of the origin of dislocations only at the end is that this question is still the least clear part of the theory of dislocations” (p. 227).

Such a statement by the author of an extensive review, written in 1949, seems to us very indicative for characterizing the present state of this theory.

Already after the publication of Cottrell’s review, Fisher and Kochendörfer[^7], on the basis of new, more accurate quantitative calculations, again came to the conclusion that the energy of formation of an individual dislocation in a perfect lattice is very large—of the order of \(1 \div 2\) eV. The authors suggest that the source of the origin of dislocations may be stresses caused by defects in the structure of the crystal. In this way they align themselves with Seitz’s hypothesis, the criticism of which was given by us above.

Let us now turn to a discussion of the second basic proposition of the dislocation hypothesis, according to which plastic deformation is effected by the displacement of dislocations along slip planes and, consequently, is impossible if the crystal does not contain dislocations. It follows from this that, according to the dislocation hypothesis, the capacity for plastic deformation is inherent only in a damaged crystal lattice containing distortions of a quite definite type. According to Taylor–Cottrell, a perfect crystal lattice would not possess the property of plasticity.

Is such a point of view consistent with experimental facts? Experimental observations show quite convincingly that the more perfect the structure of a crystal, the more clearly expressed is its capacity for plastic flow.

It is well known, in particular, that the slower the growth of a single crystal from the melt takes place, the more perfect its structure and the lower its elastic limit[^8].

Annealing and resting of crystals[^9,^10], as we know, lead to a sharp decrease in the elastic limit. It is also known that the purer a single crystal is, the fewer impurities it contains, the more plastic it is[^11]. It is further well known that the presence in crystals of any distortions whatsoever—in particular, a transition zone of one type or another (grain boundaries in polycrystals, or boundaries of mosaic blocks in single crystals)—is always the cause of the inhibition of plastic shifts. All these facts, it seems to us, are located

in obvious contradiction with the initial propositions of the dislocation theory of plasticity.

Does the dislocation hypothesis explain other experimental regularities observed in the study of plastic deformation?

Let us recall that the concept of dislocations was introduced at one time by Taylor and other authors above all in the hope of explaining the low practical value of the elastic limit of crystals. Attempts at a theoretical calculation of the elastic limit on the basis of dislocation concepts, as Cottrell himself notes (p. 195), have so far not led to satisfactory agreement with experiment. Cottrell’s conclusion that “a dislocation proves capable of moving under the action of the very smallest external shear stresses” (p. 195) is therefore not convincing.

Let us note further that such well-known experimental regularities as the crystallographic directionality of plastic flow and the presence of hardening in shear likewise have not received a sufficiently rigorous explanation within the framework of dislocation theory. Existing attempts to explain the phenomenon of hardening are based on so large a number of unfounded and inconsistent assumptions that, in discussing the hardening theory of Burgers, Kochendörfer, and Laurent, Cottrell is forced to acknowledge: “although the flow equations derived on the basis of these considerations can be brought into agreement with certain experimental facts, the rather large number of assumptions introduced here makes this theory not entirely convincing” (p. 211).

Let us pose the question as follows: is an explanation of the plastic properties of crystals really possible only on the basis of ideas about the existence of defects of a special type, called “dislocations,” present in crystals before the onset of deformation?

The principal characteristic feature of crystals is the regular, lawful arrangement of the atoms or ions that form them, and not at all the presence in them of defects of one or another type.

At the same time, the capacity for plastic deformation belongs among the specific properties of crystalline bodies. In constructing a physical theory of the plasticity of crystals, it would therefore be natural to try to explain the phenomenon of plasticity first of all from the properties of the regular crystalline lattice.

Ya. I. Frenkel and T. A. Kontorova\(^ {12}\) showed that, in a regular crystalline lattice, under certain conditions, a special type of atomic displacement is possible, representing a consecutive, mutually coordinated transition of particles of a given atomic row from one equilibrium position to another and leading to a gradual shift of this row relative to neighboring atomic

One may think that precisely this type of coordinated “collective” displacement of atoms, inherent above all in a regular crystal lattice, underlies the phenomenon of plastic deformation.

In 1937 the authors mentioned obtained a number of equations describing motion of this type; in particular, they determined the minimum energy required for the onset of shear formation, the rate of shear propagation, etc. Ten years later analogous equations were derived by Frank\(^{13}\) and Eshelby\(^{14}\), proceeding from macroscopic considerations of the theory of elasticity. It is curious to note that proponents of the dislocation hypothesis of plasticity try to interpret the Frenkel–Kontorova theory in the spirit of dislocation concepts. Thus, Cottrell gives the formulas of Frenkel and Kontorova, asserting that they describe the motion of a special variety of “fast” dislocations (p. 203). We also encounter an attempt at a similar use of these formulas in Kochendörfer’s monograph\(^{15}\).

In reality, the initial premises of the Frenkel–Kontorova theory are essentially different from the initial premises of the theory of dislocations. According to Frenkel and Kontorova, the capacity for plastic flow is one of the fundamental properties of a regular crystal lattice; whereas, according to the views of dislocation theory, a crystal not containing dislocations would in general display no plastic properties.

Of course, the process of gradual propagation of shear in a regular crystal lattice is inevitably accompanied by the formation of condensations and rarefactions of atoms in those regions of the crystal which at the given moment are encompassed by the shear. These condensations and rarefactions exist, however, only temporarily, only in the process of shear propagation, being a consequence, not the cause, of shear formation. They have a purely dynamic character, thereby differing essentially from those static dislocations whose presence, according to Taylor–Cottrell, is a necessary prerequisite for the beginning of plastic deformation of a crystal*).

In the theory of Frenkel and Kontorova, ways are indicated for explaining such specific features of plastic deformation of crystals as its crystallographic directionality and the presence of strengthening under shear. The crystallographic directionality of plastic properties can be explained by the fact that the minimum energy of shear formation is different for different crystallographic directions.

) To avoid misunderstandings it is necessary to note that Ya. I. Frenkel, in his book Introduction to the Theory of Metals*, also calls the dynamic distortions arising during shear in a regular crystal lattice “dislocations.” The physical content of this term, however, is different from the generally accepted one.

The gradual outflow of energy into the adjacent layers of the crystal, which inevitably accompanies the process of shear propagation, may cause its gradual damping, thereby being one of the causes of hardening[^12].

A significant shortcoming of the theory of Frenkel and Kontorova is the circumstance that it has not been brought to a state in which its direct quantitative comparison with experimental data would be possible. It shows, however, that in order to explain the phenomenon of plastic deformation of crystals there is no necessity whatever to resort to the concept of static dislocations.

The theory of the plasticity of crystals, developed over a number of years by another Soviet scientist, A. V. Stepanov[^16], also proceeds from the idea that the capacity for plastic deformation is a property of a regular crystal lattice. According to A. V. Stepanov, the onset of plastic flow is a consequence of the loss of elastic stability of the crystal lattice under the action of external forces. It is known that such a loss of elastic stability in a macroscopic system is observed upon reaching a certain critical stress, which is considerably smaller than the strength of this system. According to A. V. Stepanov’s conception, the loss of elastic stability of the crystal lattice occurs at a certain critical stress lying below the theoretical strength of this lattice. The loss of elastic stability entails a rearrangement of the crystal lattice, as a result of which the entire system passes into a more equilibrium state. A very important feature of A. V. Stepanov’s theory is its account of the character of the elastic anisotropy of crystals. A major achievement of this theory is the fact that it was the first to attempt to connect the plastic properties of crystals with their elastic properties. This makes it possible to bring in the theory of the elasticity of anisotropic bodies for elucidating the possible conditions and character of the fracture of crystals.

CONCLUSION

In summary, the following may be said. In the 19 years that have passed since the introduction of the concept of a “dislocation,” the dislocation theory of plasticity, vigorously developed by a number of foreign scientists, has not freed itself from the internal contradictions that were inherent in it from the very beginning, and has likewise not achieved any substantial successes in explaining experimental facts.

When reading modern voluminous works devoted to the development of the dislocation hypothesis and containing very complex mathematical relations, one may get the impression that we are dealing with a rigorous, well-developed microscopic

theory of the plasticity of crystals. This impression, however, is mistaken. The authors of these works have concentrated all their attention on the external formulation of ideas devoid of a real physical foundation.

For the correct construction of a physical theory of plastic deformation, it is first of all necessary to understand the mechanism of this phenomenon. The modern dislocation theory of plasticity is flawed methodologically, since its initial assumptions about the special role and special properties of static dislocations remain physically unsubstantiated to this day.

In the process of crystallization in real crystals, distortions of the “vernier” type as well as distortions of the screw-dislocation type may probably form.

It does not follow from anywhere, however, that such distortions must possess a special property of mobility that substantially distinguishes them from all other defects of the lattice. On the contrary, there is every reason to think that these distortions have no direct relation to the occurrence of plastic shear, and that their presence is merely an obstacle to the propagation of plastic deformation.

In conclusion, we consider it necessary to note that an uncritical presentation of the dislocation theory of plasticity gives the Soviet reader an incorrect idea of the present state of the question and may hinder the development of other theories, which may subsequently prove more fruitful.

References Cited

1. G. Taylor, Proc. Roy. Soc. A 145, 362, 388 (1934).
2. M. V. Klassen-Neklyudova and T. A. Kontorova, UFN 26, issue 2, 217 (1944).
3. J. M. Burgers, Proc. Roy. Acad. Sci. Amsterdam 42, 293 (1939); Proc. Phys. Soc. London 52, 23 (1940).
4. F. Seitz, Physics of Metals, Gostekhizdat, pp. 117–118 (1947).
5. M. V. Klassen-Neklyudova, ZhETF 6, 6, 598 (1936).
6. Buckley, Zeits. f. Elektrochemie. 56, 4, 275 (1952).
7. P. Fischer und A. Kochendörfer, Zeits. f. Naturforschung 7a, 11, 735 (1952).
8. W. Boas und E. Schmid, Zeits. f. Phys. 54, 16 (1929).
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14. Eshelby, Proc. Phys. Soc. 62, 307 (1949).
15. A. Kochendörfer, Plastische Eigenschaften von Kristallen und metallischen Werkstoffen. Berlin, 1941–1944.
16. A. V. Stepanov, ZhETF 19, 4, 492 (1949); ZhETF 20, 10, 1194 (1950).