Development of Modern Theoretical Concepts of the Nature of Plastic Deformation

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

Introduction

In the last twenty years a number of original works have appeared in the scientific literature devoted to the theoretical consideration of the phenomenon of residual deformation of crystalline bodies. In the Russian literature, until very recently these questions have been treated very scantily. In this connection it seems of interest to give a brief survey of modern theoretical conceptions of the plastic properties and the mechanism of plastic flow of crystals.

I. Becker’s Theory

One of the first attempts to construct a consistent theory of plastic deformation and to establish the quantitative laws governing this phenomenon belongs to Becker¹˒².

The capacity of solid bodies for plastic deformation, from Becker’s point of view, can be explained only by taking into account the thermal motion of the particles of the solid body.

Becker distinguishes two types of plasticity: the “amorphous” type and the “crystalline” type. He notes that the first variety of plasticity, observed in amorphous bodies, is due to the displacements of individual atoms or molecules—their exchange of places. Becker’s works contain no more detailed ideas about the mechanism of this phenomenon.

In order for the exchange of places to take place, the atoms that change places must acquire the additional energy necessary to overcome the potential barrier separating two neighboring positions of equilibrium. This additional energy may be obtained at the expense of the random fluctuations accompanying the thermal motion of the particles.

Becker describes the behavior of an amorphous body under the action of an external force by means of Maxwell’s relaxation equation³

\[ \frac{d\sigma}{dt} = G\frac{ds}{dt} - \frac{\sigma}{\vartheta}, \tag{1} \]

where \(\sigma\) is the shear stress, \(s\) is the strain, \(\vartheta\) is the relaxation time, and \(G\) is the shear modulus. The first term on the right-hand side of the equation characterizes the elastic part of the deformation; the second term, the relaxation of stress.

At a constant value of \(\sigma\), the rate of flow is determined by the relation

\[ \frac{ds}{dt} = \frac{\sigma}{G\vartheta}. \tag{2} \]

It is thus proportional to the magnitude of the acting force. Becker notes that in this case a definite viscosity, in the usual sense of the word, can be attributed to the material. The coefficient of viscosity \(\eta\), according to equation (2), will be equal to

\[ \eta = G \cdot \theta . \tag{3} \]

Becker’s “amorphous” plasticity is therefore nothing other than ordinary viscous flow.

Becker notes that, although the phenomenon of the exchange of positions of individual atoms also occurs in crystalline bodies, it is not this, however, that determines the plastic properties of these bodies.

Becker considers spontaneous fluctuations of energy accompanying the thermal vibrations of the particles of the crystal lattice to be responsible for “crystalline” plasticity as well. In contrast to the case of “amorphous” plasticity, the plasticity of crystalline bodies, according to Becker, is determined by fluctuation phenomena embracing not individual atoms, but entire regions of the crystal, containing a sufficiently large number of atoms and situated in the region of slip planes.

Becker assumes that in an ideally regular crystal lattice, at the temperature of absolute zero, slip in some definite crystallographic direction cannot occur until the component of the externally applied force, taken in this direction, reaches the value of the theoretical elastic limit \(\sigma_0\). In the case where the stress \(\sigma_0\) is reached, slip becomes possible. However, in Becker’s opinion, it must inevitably lead to complete separation of the material along the slip plane: “at the temperature of absolute zero we cannot imagine slow slip along a given plane that is not accompanied by rupture.”

At temperatures different from zero, slip in a given crystallographic plane also becomes possible at values of the external force lying considerably below the theoretical value of the elastic limit \(\sigma_0\).

Indeed, at some moment of time, as a result of fluctuations of the energy accompanying the thermal motion of the lattice particles, the component of the externally applied stress, taken in the given crystallographic direction, may by chance be increased up to the value \(\sigma_0\).

The probability \(P\) of the occurrence of a state characterized by some excess energy \(\Delta U\) is determined, as is known, by the relation

\[ P = \mathrm{const.}\cdot e^{-\frac{\Delta U}{kT}}, \tag{4} \]

where \(k\) is Boltzmann’s constant.

The additional energy \(\Delta U\) necessary for the onset of slip can be written as

\[ \Delta U = \frac{V(\sigma_0-\sigma)^2}{2G}, \tag{5} \]

where $\sigma_0$ is the theoretical elastic limit, $\sigma$ is the component of the external forces acting in the slip plane and taken in the direction of slip, $V$ is the volume encompassed by the fluctuation, and $G$ is the shear modulus.

Hence

$$ P=\operatorname{const.}\cdot e^{-\frac{V(\sigma_0-\sigma)^2}{2GkT}} . \tag{6} $$

Every local increase of the stress $\sigma$ up to the value $\sigma_0$ must, according to Becker, lead to a local shear along the given crystallographic plane over some small segment $\lambda$. Becker assumes that the magnitude $\lambda$ is independent, on the average, both of the temperature of the experiment and of the value of $\sigma$.

In this case the rate of plastic flow $w$ will be determined solely by the probability $P$ of a spontaneous increase of $\sigma$ to the value $\sigma_0$. Becker writes:

$$ w=C\cdot e^{-\frac{V(\sigma_0-\sigma)^2}{2GkT}}, \tag{7} $$

where $C=\lambda n$, with $\lambda$ the elongation of the specimen corresponding to a separate local shear, and $n$ the number of regions of the crystal simultaneously participating in slip (i.e. the number of regions in which stress fluctuation occurs).

According to relation (7), the rate of plastic deformation of crystalline bodies should depend very sharply on the temperature of the experiment.

Becker notes that, according to the available experimental data, the plasticity of crystals does not display such a sharp temperature dependence. He considers the cause of this circumstance to be the presence of hardening of the material, which inevitably accompanies plastic deformation and masks the influence of temperature.

In Fig. 1 are shown the schematic curves given by Becker, illustrating the dependence of the elongation of a specimen on time for two different experimental temperatures. Becker notes that under ordinary conditions of testing materials the intervals of time $t_1$ and $t_2$, during which plastic deformation proceeds at an appreciable rate, are very small in comparison with the time required to attain the elongation $s$. As a result, usually only the final value of the elongation $\Delta s$ is measured, corresponding to that stage of deformation at which the flow of the material has practically already ceased and which, according to Fig. 1, in fact does not depend on temperature. In Becker’s opinion, in order to reveal the temperature dependence of the plastic properties of a material, one should study the process of plastic deformation at its initial stage.

Fig. 1

For a direct verification of the validity of relation (7), Becker1 carried out special experiments to study the plastic deformation of wires of monocrystalline tungsten at various temperatures—from room temperature to $100^\circ$. Curves of the dependence of deformation on time were recorded. The elongation of the wires was registered with

with the aid of a mirror device giving a magnification of 60 times. The wire was heated by an electric current; the temperature was measured from the change in its resistance.

As a measure of the temperature dependence of the flow rate \(w\) and its dependence on the stress \(\sigma\), Becker introduces the following quantities, convenient for direct measurement:

\[ \begin{aligned} f_T &= T\frac{\partial \ln w}{\partial T},\\ f_\sigma &= \sigma\frac{\partial \ln w}{\partial \sigma}. \end{aligned} \tag{8} \]

Knowing \(f_T\) and \(f_\sigma\), with the aid of (7) it is easy to determine the relation between the theoretical value of the elastic limit \(\sigma_0\) and that value of the external shearing stress \(\sigma\) which in practice causes the appearance of the first residual deformations, and also to calculate the volume \(V\) in which the stress \(\sigma\), owing to thermal fluctuations, must be raised to the value \(\sigma_0\) in order that an individual local slip may occur.

Indeed, according to (7),

\[ \begin{aligned} f_T &= \frac{V(\sigma_0-\sigma)^2}{2GkT},\\ f_\sigma &= \frac{V\sigma(\sigma_0-\sigma)}{GkT}, \end{aligned} \tag{9} \]

whence

\[ \begin{aligned} \frac{\sigma_0}{\sigma} &= 1+\frac{2f_T}{f_\sigma},\\ V &= \frac{f_\sigma^2GkT}{2\sigma^2 f_T}. \end{aligned} \tag{10} \]

The measurements showed that the dependence of the plastic-flow rate of tungsten single crystals on temperature and stress does indeed obey Becker’s formula (7).

In this way the following values of \(\frac{\sigma_0}{\sigma}\) and \(V\) were obtained:

\[ \begin{aligned} \frac{\sigma_0}{\sigma} &= 2.5,\\ V &= 5\cdot 10^{-20}\,\text{cm}^3. \end{aligned} \tag{11} \]

In the experiments described, \(\sigma\) was \(96\ \text{kg}/\text{mm}^2\), whence

\[ \sigma_0 = 250\ \text{kg}/\text{mm}^2. \tag{12} \]

Becker notes that the theoretical value of the elastic limit \(\sigma_0\) proved to be a quantity of the same order as the tensile strength of tungsten. As for the volume \(V\) encompassed by the stress fluctuations, it is very small in comparison with the region encompassed by the slip caused by this fluctuation—in it there are only several thousand atoms.

Becker considers that these results are in complete agreement with the theoretical views developed by him.

We have already noted that, in Becker’s opinion, the detection of the temperature dependence of the rate of plastic deformation usually proves impossible because of the effect of hardening superimposed on the plastic flow of the material. According to Becker, the cause of the difficulty of slip that leads to hardening of the material is defects in the structure of the lattice of real crystals. By structural defects he understands, in this connection, the deviation of real slip planes from ideal crystallographic planes, caused by the natural imperfections of the crystal and intensified by the thermal motion of the lattice particles. The removal of hardening—the healing of the lattice—can be effected by the relocation of individual atoms.

In this connection Becker emphasizes that, in the case of crystalline bodies, diffusion plays an essential role only in the processes of recovery and recrystallization, whereas in the case of amorphous bodies it also lies at the basis of the very phenomenon of plastic flow.

Becker’s works are devoted, as we see, to the investigation of one of the fundamental questions of the theory of plasticity, concerning the conditions for the occurrence of plastic deformation.

One of the main shortcomings of these works is, it seems to us, the circumstance that plastic deformation is thereby assigned to the category of purely thermal phenomena. Meanwhile, it is well known that in the case of crystalline bodies plastic flow is observed even in the range of temperatures close to absolute zero, when thermal fluctuations cannot play any significant role.

Becker attempts to distinguish between “crystalline” and “amorphous” types of residual deformation. Nevertheless, throughout the theory he developed, the discussion is essentially only of the deformation process characteristic of purely amorphous bodies. He notes, it is true, that individual displacements of atoms are responsible for the deformation of amorphous substances, whereas displacements of whole groups of atoms are responsible for the deformation of crystals. The mechanism of propagation of deformation in the latter case is nevertheless assumed to be purely diffusional. Relation (7), which, according to Becker, characterizes the rate of plastic deformation of crystals, in essence relates to the rate of viscous flow—the only possible type of deformation of amorphous bodies.

With such an interpretation of the question of the plastic deformation of crystalline substances, the crystallographic directionality of this process remains unmarked and completely incomprehensible.

One of the very substantial shortcomings of the theory under consideration is also the circumstance that, in giving it quantitative form, the author completely ignores hardening, which, it is true, is not observed in the residual deformation of amorphous bodies, but which constitutes one of the specific features of the plastic deformation of crystals.

The chief merit of Becker’s works is sometimes considered to be the circumstance that in them, allegedly for the first time, the discrepancy between the theoretical and practical values of the elastic limit was explained.

It is necessary to note that the quantity \(\sigma\) appearing in Becker’s theory has nothing in common with the so-called practical elastic limit.

The concept of an elastic limit within the framework of the given theory is not introduced at all, since, according to formula (7), at temperatures different from zero the rate of plastic flow \(\mathfrak{w}\) proves to be different from zero for any values of the external shearing force \(\sigma\), including also \(\sigma=0\).

We shall return to a more detailed consideration of this question in the next section, when discussing the basic relations of Orowan’s theory.

II. OROWAN’S WORKS

For a number of years Orowan has supplemented and improved Becker’s theory. In one of his first works\(^4\) he uses the basic relation of Becker’s theory (7), which determines the dependence of the rate of plastic flow on temperature, in order to find the temperature dependence of the practical elastic limit.

We have already noted above that, according to formula (7), the rate of plastic deformation \(\mathfrak{w}\) proves to be different from zero for arbitrarily small values of the stress \(\sigma\). It becomes accessible to experimental observation, however, only beginning with some definite minimum value of \(\sigma\).

In this connection Orowan introduces the concept of the practical elastic limit, understanding by it that smallest value of the shearing stress \(\sigma_{\min}\) acting in the slip plane, beginning with which the rate of plastic flow \(\mathfrak{w}\) becomes measurable.

Putting in Becker’s formula (7)

\[ \sigma=\sigma_{\min} \]

and, correspondingly,

\[ \mathfrak{w}=\mathfrak{w}_{\min}, \]

we obtain:

\[ \mathfrak{w}_{\min}=C\cdot e^{-\frac{V(\sigma_0-\sigma_{\min})}{2GkT}} . \tag{13} \]

Whence

\[ \sigma_{\min}=\sigma_0-B\sqrt{T}, \tag{14} \]

where

\[ B=\sqrt{\frac{2Gk}{V}\ln\frac{C}{\mathfrak{w}_{\min}}}. \]

Relation (14) indicates that the practical elastic limit of a material must depend substantially on the temperature of the experiment, decreasing as it increases.

This result is, as is known, in qualitative agreement with experimental data: the higher the temperature of the crystal, the smaller those

values of the external force at which it is possible to record the presence of plastic flow.

Studying the dependence of the critical shear stress of zinc and cadmium crystals on the temperature of the experiment, Orowan comes to the conclusion that relation (14) is well justified, starting from the melting temperature down to the region of low temperatures. The numerical values of the ratio \(\dfrac{\sigma_0}{\sigma_{\min}}\) prove, in this case, to be very close to the value 2.5 previously obtained by Becker for tungsten single crystals [see (11)]:

\[ \left(\frac{\sigma_0}{\sigma_{\min}}\right)_{\mathrm{Zn}} = 2.3;\qquad \left(\frac{\sigma_0}{\sigma_{\min}}\right)_{\mathrm{Cd}} = 2.4. \]

Orowan fully accepts Becker’s view that thermal fluctuations of the energy of the particles forming the crystal lattice can lead to an increase of the stress in individual elements of the crystal, thereby promoting the onset of plastic flow.

Already in the first part of his paper\(^4\), however, he notes that in the case of real crystals, when considering the conditions for the occurrence of plastic deformation, alongside the thermal motion of the particles a very substantial role must be played by taking into account the various defects of structure of the crystal lattice, whose presence must also, in his opinion, give rise to local overstress in the crystal.

Among such defects Orowan includes not only cracks, holes, and all other possible flaws of the material, but also various kinds of microscopic disturbances of the regular arrangement of the particles of the crystal lattice. The latter arise, in his opinion, chiefly as a result of plastic deformation, but may also be present in the crystal in advance, before its onset. To all these flaws of the crystal he applies the general term “Kerbstelle.”

To characterize the degree of concentration of additional stresses caused by the presence of each of the defects, Orowan introduces a special coefficient \(q\) (Kerbwirkungsfaktor), equal to the ratio of the true value of the shear stress \(\sigma'\), acting in the region of the “defective site,” to the mean value of the shear stress \(\sigma\) caused by the action of external forces:

\[ q = \frac{\sigma'}{\sigma}. \tag{15} \]

According to Orowan, slip should occur in regions of the crystal characterized by the largest values of the quantity \(q\), under the condition that the stress acting there, \(\sigma' = q\sigma\), proves to be supplemented up to the theoretical value of the elastic limit \(\sigma_0\) by thermal fluctuations.

The rate of plastic flow \(w\) is then determined no longer by formula (7), but by the relation

\[ w = C \cdot e^{-\frac{V(\sigma_0-\sigma')^2}{2GkT}} = C \cdot e^{-\frac{V(\sigma_0-q\sigma)^2}{2GkT}} . \tag{16} \]

Accordingly, in formula (10), instead of \(\sigma\) one should now write \(\sigma'\) or \(q\sigma\), and formula (11) is rewritten in the form

\[ \frac{\sigma_0}{\sigma_*}=\frac{\sigma_0}{q\sigma_{\min}}=2.5 . \tag{17} \]

Like many investigators, Orowan asserts that between the theoretical elastic limit \(\sigma_0\) and its practical value there must exist a discrepancy of the same order of magnitude as between the theoretical and practical values of strength, i.e. that the ratio \(\dfrac{\sigma_0}{\sigma_{\min}}\) must be \(10^2\)—\(10^4\).

Since thermal fluctuations are capable of raising the stress \(\sigma'\) by only a few times [see (17)], in order to obtain the required ratio between \(\sigma_0\) and \(\sigma_{\min}\), Orowan has to assume that the numerical values of the multiplier \(q\) may be very large—of the order of \(10^2\)—\(10^4\).

The temperature dependence of the practical elastic limit \(\sigma_{\min}\), when allowance is made for overstresses associated with defects in the structure of crystals, is therefore determined by the relation

\[ \sigma_{\min}=\frac{\sigma_0}{q}-B'\sqrt{T}, \tag{18} \]

where

\[ B'=\frac{B}{q}. \]

Comparing this relation with formula (16), corresponding to Becker’s theory in its original form, it is interesting to note that if, according to Becker, at \(T=0\) plastic deformation is possible only under the condition \(\sigma_{\min}=\sigma_0\), then according to Orowan, in this case \(\sigma_{\min}\) is equal to

\[ \sigma_{\min}=\frac{\sigma_0}{q}. \tag{19} \]

Thus, even at the temperature of absolute zero, the concept is retained of a practical elastic limit distinct from \(\sigma_0\), owing to the presence in the crystal of defective regions that are sources of overstresses and thereby promote the occurrence of plastic flow.

At temperatures different from absolute zero, the occurrence of plastic deformation is already determined by the combined action both of these defective regions and of the stress fluctuations that accompany the thermal motion of the particles of the crystal. The source of shear, in Orowan’s opinion, must in each case be some defect of the crystal lattice.

In the preceding section we have already noted that, according to formula (7), the rate of plastic deformation proves to be different from zero even at \(\sigma=0\), i.e. in the complete absence of external forces. The inadequacy of Becker’s basic formula in the limiting case \(\sigma=0\) was first pointed out by Orowan.^5

It is necessary, however, to mention that in Becker’s first and principal work^1 this difficulty is absent: Becker assumes in it

that the dimensions of an individual local shear are proportional to the magnitude of the stress $\sigma$ itself and, accordingly, determines the rate of plastic deformation by the formula

$$ w=\mathrm{const.}\cdot \sigma e^{-\frac{V(\sigma_0-\sigma)^2}{2GkT}}, $$

according to which, for $\sigma=0$, $w$ in turn becomes zero.

In another, later work, however, Becker$^2$ determines the rate of flow by formula (7), which is generally known at the present time as “Becker’s formula.”

Considering that physical notions about the mechanism of plastic slip give no grounds for introducing the quantity $\sigma$ as a factor before the exponential function

$$ e^{-\frac{V(\sigma_0-\sigma)^2}{2GkT}}, $$

Orowan$^5$ attempts to improve Becker’s formula (7), proceeding from somewhat different premises.

In investigating the question of the role of thermal fluctuations, Becker limits himself to considering only those spontaneous fluctuations of the stress $\sigma$ which lead to its increase, supplementing it up to the value of the theoretical elastic limit, $\sigma_0$.

Orowan$^5$ notes that, in addition to such fluctuations, fluctuations of stress of the opposite sign are also possible, decreasing the magnitude of $\sigma$, as a result of which slip may ultimately arise in the direction diametrically opposite to $\sigma$.

If the stress fluctuation necessary for the onset of slip in the direction of $\sigma$ is equal to

$$ \sigma_0-q\sigma, $$

then, in order for slip in the opposite direction to become possible, it must correspondingly be equal to

$$ \Delta\sigma=\sigma_0-(-q\sigma)=\sigma_0+q\sigma. $$

The probability of occurrence of such a fluctuation is determined by the expression

$$ \mathrm{const.}\cdot e^{-\frac{V(\sigma_0+q\sigma)^2}{2GkT}}, \tag{20} $$

and the rate of plastic deformation $w$, taking into account the possibility of fluctuations of the stress $\sigma$ both in the direction of its increase and in the direction of its decrease, will accordingly be written as

$$ w=C\left[e^{-\frac{V(\sigma_0-q\sigma)^2}{2GkT}}-e^{-\frac{V(\sigma_0+q\sigma)^2}{2GkT}}\right]. \tag{21} $$

In contrast to formula (7), this relation, for $\sigma=0$, gives $w=0$. The introduction of the new term into formula (7), as Orowan himself notes, has only fundamental significance, since under ordinary experimental conditions it is so small in comparison with the first term

$$ e^{-\frac{V(\sigma_0-q\sigma)^2}{2GkT}} $$

that it may practically be neglected completely.

The difference between relations (21) and (7) could have been detected only at very small stresses and very high temperatures. Thus, for example, according to Orowan’s calculations, in the case of zinc and cadmium this difference could become noticeable only at temperatures lying considerably above the melting temperatures of these crystals.

Comparing Orowan’s additions to Becker’s works with Becker’s theory in its original form, it should, it seems to us, be noted that essentially only one of them is fundamentally new—the allowance for overstresses associated with the presence of defects in the crystal lattice. In Orowan’s view these overstresses must play a considerably more substantial role than purely thermal fluctuations of stress.

If the ideas developed earlier by Becker can equally well be referred both to ideal and to real crystals, then, according to Orowan, the plastic properties of a crystal must be expressed the more sharply, the more various kinds of defects it contains. This conclusion, as is well known, does not agree with experimental data.

As for Orowan’s ideas (shared, incidentally, by many other authors as well) concerning the large numerical value of the theoretical elastic limit \(\sigma_0\), they seem to us completely unfounded. The elastic limit of a crystalline material has never yet been calculated theoretically by anyone. In discussing the question of its possible theoretical value it is customary to refer to Born’s work\(^6\), which, however, concerns not the elastic limit but the strength of crystals. The elastic limit, meanwhile, should have nothing in common with the strength limit, since the mechanism of rupture and the mechanism of slip in crystals are essentially different from one another.

The remaining works of Orowan\(^7\), not cited by us, contain no fundamentally new propositions, and therefore we consider it possible not to dwell here on their discussion.

III. TAYLOR’S THEORY

In Taylor’s theory\(^8,9\) we encounter the first attempt to create definite ideas about the mechanism of propagation of plastic deformation in crystalline bodies.

Most contemporary theories of plasticity represent a further development of Taylor’s views.

In this connection we consider it necessary to dwell here on their detailed examination.

§ 1. Basic premises of the theory

Taylor’s theory is the first of the theories of plastic deformation that operates with a definite model of the crystal lattice. As such a model it considers a plane atomic net formed by linear chains of equidistant atoms of one and the same kind.

On the basis of experimental data, Taylor notes that in the crystal lattice plastic deformation can be carried out

appear only in strictly definite crystallographic planes and directions. In considering the causes of the crystallographic directionality of the shear-formation process he does not, however, go into this question.

Before Taylor’s work it was usually considered that the plastic deformation of crystals is effected by the displacement of one part of the crystal, as a whole, relative to another part of it; this displacement was assumed to occur simultaneously over the entire slip plane.

Discussing the question of the possible character of the propagation of plastic deformation, Taylor arrives, however, at the conclusion that “slip does not involve all the atoms situated in the slip plane simultaneously, but takes place in a limited region, which moves through the crystal in a finite interval of time.”

He notes that, for simultaneous slip along the whole plane, extraordinarily large external forces would have to be applied to the crystal, and also that under these conditions it would not be possible to explain the phenomenon of work-hardening in shear. In this connection he refers to experiments carried out at one time in the Leningrad Physico-Technical Institute[^10] on the study, in polarized light, of the plastic deformation of rock salt, which testify to the gradual propagation of shear.

Taylor’s ideas on the finite rate of propagation of the process of plastic deformation in a crystal are, it seems to us, very valuable. They are also shared by the authors of all later works on the theory of plasticity.

§ 2. The Concept of “Dislocation”

In considering the question of the mechanism of propagation of plastic deformation, Taylor from the very first lines uses the term “dislocation.”

Without going into explanations, he presents here a scheme of the arrangement of atoms in a crystal lattice: a) before the beginning of shear, b) for a certain intermediate stage of its propagation, and c) after its completion (Fig. 2).

From this scheme it follows that before shear the atoms were arranged in a completely regular manner, and that during the propagation of shear in the slip plane a rarefaction of atoms could be observed, while after the completion of shear the upper part of the crystal proved to have been displaced by one lattice constant with respect to the lower part of the crystal.

This means, according to Taylor, that “a dislocation has passed through the crystal.”

It is necessary to note that the concept of dislocation pertains first of all not to only one plane in which slip is effected, but to an entire region located near this plane and including layers of atoms lying above and below it.

Taylor distinguishes a “positive” dislocation, in which the chains of atoms lying above the slip plane turn out to be compressed in the direction of slip, while the atomic chains lying below the slip plane—

...—stretched (Fig. 2; a, b, c), and a “negative” dislocation, which is a mirror image of the positive dislocation (the formation of a “negative” dislocation is illustrated in Fig. 2, d, e, f).

Let us try to establish whether such dislocations were present in the crystal before the onset of slip, or whether they arose in the process of its propagation.

Consideration of the scheme given above shows that, both before the beginning of the shear and after its completion, there are no dislocations in the lattice; the atoms are arranged in a strictly regular manner. The conclusion suggests itself that the dislocation is born in the process of slip.

Fig. 2

Fig. 3

Taylor then gives, however, an atomic model of a dislocation in the absence of external forces (Fig. 3).

It follows from this that a dislocation is, as it were, a stable formation capable of existing in the lattice independently of the action of external forces.

Subsequently, throughout his entire work Taylor operates with the concept of dislocations, without saying a word about the causes and mechanism of their occurrence.

Only on one of the last pages do we find the phrase: “according to this theory the crystal is equally ideal in all directions; it may therefore be assumed that dislocations arise at the boundary or inside the crystal owing to thermal motion.”

Everything said above indicates that Taylor’s “dislocation” hypothesis refers not at all to the real crystal, as is usually assumed, but to an ideally regular crystalline lattice.

The “atomic model” of a positive dislocation shown in Fig. 3 consists of three linear chains of atoms. Here the upper chain \(AB\) is compressed, the lower \(CD\) is stretched, while slip is effected in the middle atomic chain \(EF\).

The number of atoms falling on a certain segment of length \(l\) in the upper and lower chains is not the same; \(N\) atoms of the lower chain corre-

comprises \((N+1)\) atoms of the upper chain. The dislocation is thus equivalent to an atomic “vernier.”

Under the scheme of the distribution of atoms in the presence of a dislocation in Fig. 3, the corresponding graph is given of the dependence of the potential energy of the atoms of the middle chain \(C_0, C_1, C_2, C_3\) on the distances between them.

The potential “wells” (states characterized by minimum values of the energy) correspond to the equilibrium positions of atoms in the crystal lattice.

This graph differs essentially from the graph of the potential energy of atoms in the absence of a dislocation (Fig. 4), namely: 1) not all atoms are here in equally stable equilibrium positions (the depths of the potential wells in the region of the dislocation differ from one another); 2) besides the potential wells occupied by atoms and corresponding to their stable equilibrium positions, there is in the lattice an additional potential well 0, unoccupied by an atom, called by Taylor the “center of the dislocation.”

Fig. 4

§ 3. Behavior of dislocations in a crystalline lattice

According to Taylor, plastic deformation is effected by the motion of dislocations along certain crystallographic planes and directions.

Taylor assumes that there exists a certain critical temperature \(T_0\), above which a dislocation can move quite freely along the slip plane even in the complete absence of external forces.

Taylor represents this motion of a dislocation in the following way: “in an ideal crystal the atoms must remain in the potential wells until a certain temperature is reached at which they can overcome the potential barriers. At this temperature the atom \(C_0\) can jump into the well occupied by \(C_1\). At lower temperatures the atom will not be able to rise over the potential barrier between \(C_0\) and \(C_1\), but will be able to jump into the well corresponding to the center of the dislocation... The temperature at which one of the two central atoms (\(C_1\) or \(C_2\)) can jump into the vacant central well 0 we shall call \(T_0\)...” (Fig. 3).

Taylor further considers that the potential well thus vacated in turn becomes the center of the dislocation, and “the neighboring atoms, of themselves, pass continuously into new equilibrium positions.”

Moving along the slip plane, the dislocation reaches the surface of the crystal, on which, as a result, a step is formed (Fig. 2). In this case one part of the crystal proves to be displaced relative to the other part by a distance equal to the lattice constant.

After the departure of the dislocation, the crystal lattice remains, as before, ideally regular.

From all that has been said above it follows that if there were only one dislocation in the crystal, then at temperatures lying above \(T_0\) the elastic limit would be equal to zero (at temperatures lying below \(T_0\), it would have to have a finite value).

However, when two or more dislocations are present in the crystal, the elastic limit at any temperature is different from zero.

In Taylor’s opinion, the presence of dislocations should give rise to elastic stresses in the crystal. In finding the law of distribution of the stresses, Taylor passes from consideration of the atomic model of the crystal lattice given above to consideration of a continuous elastic medium. Assuming that the elastic stresses near the “center of a dislocation” are very large, Taylor supposes that the presence of dislocations in a crystal is equivalent to the presence of cavities in a continuous elastic medium.

Fig. 5

In the case of the propagation of a positive dislocation, leading to a displacement over a segment \(d\) in the direction of the positive half-axis \(x\), this initially spherical cavity is deformed, acquiring the form shown in Fig. 5.

The problem of the distribution of stresses around a cavity of such a form was solved at one time by Timpe \(^{11}\). The latter showed that the stress components at a point taken at a sufficiently large distance \(r\) from the center of the plane \(O\) are determined by the relations:

\[ \left. \begin{aligned} F_{xx}=-F_{yy}&=-\frac{Gd}{\pi}\frac{y}{r^{2}},\\ F_{xy}&=\frac{Gd}{\pi}\frac{x}{r^{2}}, \end{aligned} \right\} \tag{22} \]

where \(G\) is the shear modulus. For a displacement through a distance \(d\), the components of the arising stress are directly proportional, thus, to the magnitude of the displacement itself.

Taylor uses the relations (22), found by Timpe, in considering the case when several dislocation centers are present simultaneously in a crystal.

If in the crystal there are two mutually parallel slip planes (situated perpendicular to the \(y\)-axis), and in one of them there is, for example, a positive dislocation center \(A\), while in the other there is a dislocation center \(B\), likewise positive and displaced relative to center \(A\) toward the positive half-axis \(x\), then, according to relations (22), center \(B\) must be displaced to the right under the influence of the shearing stress created by center \(A\) (\(F_{xy}\) in this case is positive).

Two positive dislocation centers must, therefore, repel each other.

As for two centers of opposite sign, they will attract each other ($F_{xy}$ is negative) until an equilibrium is established between them corresponding to the case $x=0$ (Fig. 6). The simultaneous existence of two dislocations of opposite sign in one and the same slip plane proves, consequently, to be impossible, since they mutually “neutralize” each other.

Fig. 6

The laws of interaction of “dislocation centers” are formally similar, as we see, to the laws of interaction of electric charges.

§ 4. Conditions for the occurrence of slip. Elastic limit

As we have already noted, according to Taylor, plastic deformation is effected by the displacement of dislocations along slip planes.

The relative displacement of two dislocations of opposite sign can, however, become possible only when we overcome the forces of their mutual attraction, i.e. only in the case when some finite stress is applied to the crystal. The elastic limit must, accordingly, be different from zero.

If a shearing stress $\sigma$, acting in the direction of the positive half-axis $x$, is applied to the crystal, then, according to relation (22), a positive dislocation must begin to move from left to right along the slip plane, while a negative dislocation will move in the opposite direction (Fig. 6).

If two dislocations of opposite sign are present in the crystal, for example, a positive $A$ and a negative $B$, then their relative displacement will continue until a new equilibrium state is established, determined by the condition

$$ \sigma=\frac{Gd x}{\pi\left(x^{2}+h^{2}\right)}, \tag{23} $$

where $h$ is the distance between the slip planes containing the dislocation (Fig. 6).

$\sigma$ acquires its maximum value under the condition $x=h$, i.e.

$$ \sigma_{\max}=\frac{Gd}{2\pi h}. \tag{24} $$

For values of the external stress $\sigma$ exceeding $\sigma_{\max}$, the equilibrium between the dislocations is disturbed and they begin to be displaced relative to one another. As a result, slip occurs along the slip plane.

The condition for the occurrence of plastic deformation in a crystal containing two unlike dislocations has, therefore, the following...

yielding the form:

\[ \sigma \geqslant \frac{Gd}{2\pi h}. \tag{25} \]

The quantity \(\dfrac{Gd}{2\pi h}\), therefore, is nothing other than the elastic limit of the given material.

The numerical value of the elastic limit turns out, as we see, to be inversely proportional to the distance between the slip planes \(h\). Taylor notes that, by virtue of this circumstance, simultaneous shear formation must take place in the slip planes that are farthest apart.

Taylor considers the simultaneous presence in a crystal of a large number of dislocation centers to be possible. Each such dislocation center arises as a result of thermal motion; therefore the sites at which dislocations arise in the crystal are arranged in a completely arbitrary manner. According to Taylor, however, the distribution of the dislocations themselves cannot remain chaotic because of the interaction between the dislocation centers. This interaction, in his opinion, must lead to the formation in the crystal of a regular “dislocation lattice,” which, in turn, consists of two dislocation lattices inserted into one another; one of these lattices contains only “positive,” and the other only “negative,” dislocation centers.

Fig. 7

Such a lattice is shown in Fig. 7. The application to the crystal of an external shearing stress causes a displacement of the “positive” dislocation lattice relative to the “negative” lattice. Just as was done above for the case of two dislocation centers, Taylor finds the distribution of stresses arising in the presence of a dislocation lattice, and also determines the elastic limit, which in this case is the maximum stress necessary to overcome the force of interaction of dislocation lattices of different sign.

The final expression determining the elastic limit of the crystal has in this case the following form:

\[ \sigma \geqslant \frac{GdF}{b}, \tag{26} \]

where \(d\) is the crystal-lattice constant, and \(a\) and \(b\) are the parameters of the dislocation lattice; \(F\) is a certain function of the ratio \(\dfrac{a}{b}\).

The displacement of the positive dislocation lattice relative to the negative lattice appears as plastic shear.

§ 5. Hardening

According to relation (26), the elastic limit of a crystal is inversely proportional to the parameter of the dislocation lattice \(b\). But the latter is determined by the number of dislocations, decreasing as it increases. Hence it follows,

that the elastic limit must be the greater, the greater the number of dislocations in the crystal.

Taylor assumes that, at the initial stage of deformation: “the first few dislocations can move through the crystal under the action of an infinitely small shearing stress. As deformation proceeds, the number of dislocations will increase and the mean value of \(b\) will decrease. As a result, the resistance to shear must increase.”

In Taylor’s theory we encounter for the first time an attempt to explain the phenomenon of hardening in shear, which is one of the most characteristic features of plastic deformation of crystalline materials.

§ 6. Relation between stress and deformation

In order to make it possible to compare the results he obtained with experimental data, Taylor next determines the theoretical dependence of the deformation of a specimen on the magnitude of the externally applied shearing stress.

The relative displacement of two parallel slip planes, situated at a distance \(h\) from one another, for the simplest case of a rectangular lattice of dislocations is determined as

\[ \delta=\frac{hLd}{ab}, \]

where \(L\) is the total distance over which displacement of a dislocation along the slip plane is possible, \(d\) is the crystal-lattice constant, \(a\) is the distance between neighboring dislocation centers along the slip plane, and \(b\) is the distance between two neighboring slip planes (Figs. 6 and 7). The crystallographic shear \(s\), defined as the ratio of the magnitude \(\delta\) to the distance between the slip planes \(h\), is therefore written as

\[ s=\frac{Ld}{ab}. \tag{27} \]

With the aid of relation (26), which determines the elastic limit of the material, Taylor finds the theoretical dependence of \(s\) on \(\sigma\):

\[ \frac{\sigma}{G\sqrt{s}}=\sqrt{\frac{ad}{bL}}\,F, \tag{28} \]

or

\[ \frac{\sigma}{G\sqrt{s}}=k\sqrt{\frac{d}{L}}, \tag{29} \]

where \(k\) is a constant equal to \(\sqrt{\frac{a}{b}}\,F^{1}\).

Taylor’s theory leads to a parabolic dependence between the magnitude of the deformation and the stress externally applied to the crystal.

\(^{1}\) The numerical value of \(k\) is determined by the character of the arrangement of the dislocation centers.

Experimental tensile curves for crystals of the cubic system (Al, Cu, Au, Fe) can indeed, as is known, be approximated by parabolas with sufficient accuracy.

Taylor notes that the regularity (29) applies only to temperatures exceeding the critical temperature \(T_0\). In the case of lower temperatures the elastic limit must be higher by a certain amount \(\sigma_T\), equal to the stress necessary for the displacement of the first dislocation (at temperatures above \(T_0\), as already noted, \(\sigma_T\) is assumed to be zero).

§ 7. Allowance for the Mosaic Structure of Crystals

It follows from relation (29) that the nature of the connection between deformation and stress depends on the value \(L\), which, according to what was said above, represents the distance over which a dislocation can move freely along the slip plane.

Taylor notes that \(L\) cannot be interpreted as the thickness of the crystalline specimen as a whole, since it is well known from experiment that the form of the tensile curve does not depend on the dimensions of the specimen. In this connection he uses the concept of the mosaic structure \(^{12}\) of real crystals, assuming that the free path length of a dislocation \(L\) is determined not by the spatial extent of the shear plane passing through the entire crystal, but by the linear dimensions of a mosaic block. The region of contact between two neighboring blocks, in Taylor’s view, in most cases proves to be impermeable to dislocations, as a result of which the latter must be retarded at the block boundaries.

Determining from the experimental tensile curves the ratio \(\sigma\) to \(\sqrt{s}\) and specifying the value \(k\) [see (29)], Taylor calculates the values of \(L\) for a number of metals (see the table). These values of \(L\) are of the same order of magnitude (\(10^{-4}\ \mathrm{cm}\)) as the linear dimensions of mosaic blocks according to the experimental data of Goetz \(^{12}\), Straumanis \(^{13}\), Belyaev \(^{14}\), and also Cvikl \(^{15}\).

Metal	\(L\) in cm \((k = 0.2)\)
Al	\(5.3\cdot 10^{-4}\)
Cu	\(2.7\cdot 10^{-4}\)
Au	\(4.4\cdot 10^{-4}\)
Fe	\(1.7\cdot 10^{-4}\)

Taylor notes that the free path length of a dislocation will not always coincide with the dimensions of the mosaic blocks, since the degree of impermeability of the block boundaries must depend on the temperature of the crystal: as the temperature rises, the boundaries should become more and more “transparent” to dislocations.

In this way he obtains the possibility of introducing into consideration the influence of temperature on the process of plastic flow, which he had not previously taken into account.

It should be expected that at high temperatures \(L\) will exceed the distance between the boundaries of neighboring blocks, gradually approaching it as the temperature is lowered. Using the experimental data of Boas and Schmid \(^{16}\), which give the relation between stress and deformation of aluminum crystals in the temperature interval from \(-185^\circ\) to \(600^\circ\), Taylor, by means of relation (29), calculates the values of \(L\) corresponding

different temperatures. It turns out that \(L\) does indeed decrease as the temperature is lowered, changing from \(4.0 \cdot 10^{-2}\ \mathrm{cm}\) at \(600^\circ\) to \(1.8 \cdot 10^{-4}\ \mathrm{cm}\) at \(-185^\circ\). Calculation shows that in the case of Al the boundary of the mosaic blocks should become almost completely transparent at \(480^\circ\). Taylor notes that this temperature is close to the recrystallization temperature of aluminum.

§ 8. Technical elastic limit

In considering the mechanism of plastic deformation in an ideally regular crystal lattice, Taylor, as was shown in § 4, defines the elastic limit as the force necessary to produce a relative displacement of two dislocation lattices of opposite sign.

Turning to consideration of the question of the elastic limit of real crystals, i.e., to the determination of the technical elastic limit, Taylor notes:

1. “If the observed elastic limit did indeed represent the shearing force necessary to displace an individual dislocation in an ideal crystal, then one would expect a sharp change in the elastic limit when the temperature is changed”...

Experimental data indicate, however, that at low temperatures the elastic limit depends only very weakly on temperature.

1. “The most careful investigations show that, in order for plastic deformation to begin, a finite shearing force is necessary... If the centers of dislocations can move freely even in the absence of shearing stresses (at absolute zero), then the existence of a finite elastic limit can be explained only on the assumption that the crystal, already in its initial state, is not free from internal stresses. The existence of surfaces separating mosaic blocks gives rise to internal stresses in the crystal.”

According to this theory, these stresses impede the free displacement of dislocations through the crystal. Plastic flow will be able to begin only when the externally applied shearing stress exceeds the stresses existing in the crystal owing to its mosaic structure.

In calculating the internal stresses acting at the boundaries between mosaic blocks, Taylor assumes that each of the blocks possesses an ideal crystal structure, but that the crystallographic axes of two neighboring blocks form some small angle with one another. Then, replacing each of the crystalline blocks by an elastic continuum, Taylor, with the aid of the relations of the theory of elasticity, calculates the shearing stresses \(\tau\) acting near the boundaries of the mosaic blocks.

The magnitude \(\tau\), in Taylor’s opinion, should represent the lower boundary of the elastic limit. The technical elastic limit is thus determined by him as the shearing stress acting along

slip plane and sufficient to overcome the “barrier of internal stresses” caused by the mosaic structure of the crystal.

For copper crystals \(\tau\) turns out to be equal to \(45\ \mathrm{g/mm^2}\), and for aluminum crystals—\(260\ \mathrm{g/mm^2}\).

Taylor notes that these figures are very close to the experimental values of the elastic limit of Cu and Al.

§ 9. Conclusion

Taylor’s theory, as we already noted at the very beginning of the present chapter, represents the first attempt to create definite ideas about the mechanism of propagation of residual deformation in crystalline bodies.

In fact, it consists of two almost mutually independent parts—the theory of plastic flow in an ideally regular crystal lattice and the theory of plastic deformation as applied to crystals possessing a mosaic structure.

It must be noted once again that in the first case the author succeeds in: 1) establishing a theoretical relation between deformation and stress that is in satisfactory agreement with experimental data for certain metals; 2) calculating the elastic limit of an ideally regular crystal lattice; 3) giving a formal description of the phenomenon of strain hardening.

Further, in considering plastic deformation in real crystals, the influence of temperature on the form of the deformation curves and on the elastic limit of the material is qualitatively taken into account, and the elastic limit is calculated with allowance for the mosaic structure. Its numerical values are in agreement with experimental data for copper and aluminum.

At the same time, in agreement with experimental data, the entire theory as a whole is constructed on the assumption of a finite velocity of propagation of plastic deformation in the crystal.

It is quite evident that Taylor’s theory represents a great step forward in comparison with Becker–Orowan’s theory.

A general, very substantial shortcoming of Taylor’s work, however, is its excessive artificiality and unwieldiness, as well as the insufficient development of certain theoretical propositions.

We shall attempt to analyze some of them.

1. One of the most characteristic features of the plastic deformation of crystals is, as is well known, the crystallographic orientation of the slip process. The anisotropy of plastic flow, however, does not follow from Taylor’s theory. Taylor merely postulates that dislocations can move along certain definite planes, and also in definite crystallographic directions.

2. The very important question of the velocity of plastic flow remains open. Taylor’s theory contains no attempts to calculate it, although this is entirely possible within the framework of the atomic model of dislocations proposed by the author.

3. Taylor’s notion of the existence of a critical temperature \(T_0\), below which a dislocation, in the absence of external forces, cannot freely

to move through the crystal, is extremely perplexing. The motion of a dislocation in the absence of external forces must be an ordinary diffusion process. For diffusion, as we know, there is no critical temperature; it proceeds with a finite velocity at any temperature values other than zero.

1. The specific feature of plastic flow—the hardening that accompanies it—essentially remains unexplained.

Taylor assumes that hardening is a consequence of a gradual increase in the number of dislocation centers in the process of deformation. The reason for the increase in the number of dislocations, however, remains entirely unclear, and the question of the physical nature of hardening therefore remains open.

1. The very method of describing the phenomenon of hardening leads to contradictions with one of the fundamental propositions of the theory advanced earlier. On the one hand, in the process of plastic deformation, dislocations, on reaching the surface of the crystal, must, as we know, leave it. On the other hand, Taylor asserts that the number of dislocations in the crystal must steadily increase and that precisely this circumstance leads to the hardening of the material.

2. The existence of slip traces, as Taylor himself also notes, is incompatible with the idea that lattices of dislocations are present in the crystal. Indeed, the relative displacement of two like-named dislocation lattices must occur throughout the entire volume of the crystal; it cannot lead to a localized deformation.

3. Taylor’s ideas concerning the elastic limit cannot be considered sufficiently clear and consistent.

In fact, in the case of an ideally regular crystalline lattice, the elastic limit in the presence of only a single dislocation and at temperatures lying below \(T_0\) has, as we recall, a finite value, whereas at temperatures exceeding \(T_0\) it is equal to zero. The temperature \(T_0\) itself, according to Taylor’s assertion, may coincide with absolute zero, whence it follows that the elastic limit at any temperatures may turn out to be equal to zero.

On the other hand, Taylor asserts that unlike lattices of dislocations are present in the crystal, the relative displacement of which is possible only under the action of a finite force. This proposition is equivalent to the assertion of a finite value of the elastic limit.

When considering the conditions of plastic flow in real crystals, Taylor then obtains a finite value of the elastic limit, of an order of magnitude close to the experimental data. But in doing so he completely ignores the definition of the elastic limit originally given by him and, in the process of calculation, no longer once makes use of the notions so elaborately developed in the “dislocation” theory.

1. Finally, the basic assumption of Taylor’s theory concerning the possibility of the existence in a crystal of a lattice of dislocations raises great doubts.

As a result of thermal fluctuations accompanying the oscillations of particles near the position of equilibrium, in individual regions of the lattice there may, co-

locally, of course, local distortions—“dislocations”—may arise. The prolonged existence of such distortions in the crystal lattice seems to us, however, impossible. They must inevitably disappear under the action of the same cause that produced their appearance, i.e. the thermal motion of the particles.

All the more implausible, to us, is Taylor’s assertion of the possibility of an entire “dislocation lattice” existing in a crystal. The presence of such a lattice in a crystal ought, incidentally, to impart to the crystal a number of quite specific properties, in particular to cause reduced electrical conductivity, an anomalous diffusion coefficient, etc.

Taylor’s dislocation hypothesis does not fit, in our opinion, within the framework of modern conceptions of the physical properties of crystalline solids.

IV. THE THEORY OF THE BURGERS BROTHERS

Each of the theories considered above, taken separately, does not provide an exhaustive description of the complex, many-sided process of plastic flow of crystalline bodies.

The Becker–Orowan theory contains, in essence, no definite ideas about the mechanism of plastic deformation in the crystal lattice. It operates, meanwhile, with the fundamental quantities that in practice characterize the process of flow—the rate of deformation and the elastic limit of the material.

Taylor’s theory, on the contrary, proposes a detailed microscopic mechanism for the process of propagation of plastic deformation, while leaving aside the question of the macroscopic rate of flow of crystals.

The next stage in the development of modern ideas on the plastic deformation of crystalline bodies is the attempt by the Burgers brothers (1935)[^17] to combine the points of view developed by Becker and Orowan, on the one hand, and by Taylor, on the other.

In doing so, the Burgers brothers wholly and completely accept Taylor’s ideas on the “dislocation” nature of plasticity, only partially supplementing and developing them.

1. They note, first of all, that Taylor’s work lacks clear ideas about the causes of the generation of dislocations in the crystal lattice.

In attempting to clarify this question, the Burgers brothers use the ideas on the structure and character of deformation of real crystals developed earlier by Orowan, and also by Smekal[^18]. Together with Smekal they suppose that “in every crystal there exists a system of structural defects (‘Lockerstellen’).”

Like Orowan, they consider that, owing to thermal fluctuations, at a corresponding degree of concentration of stresses in the region of these defects, local shears (“jumps”) may arise. They determine the frequency of occurrence of such jumps \(\nu\) by the Becker–Orowan formula:

\[ \nu = C \cdot e^{-\frac{V(\sigma_0 - q\sigma)^2}{2GkT}}, \tag{30} \]

where \(C\) is a constant, and all the other notation has the same meaning as before [see (16)].

It is precisely these local shears that, in the Burgers’ view, entail the formation of Taylor dislocations. Subsequently the dislocations behave in complete agreement with Taylor’s theory: “The dislocations that have formed move through the crystal lattice until they are stopped by some obstacle that proves impenetrable to them...”.

To determine the elastic limit, the Burgers use Orowan’s relation:

\[ \sigma_{\min}=\frac{\sigma_0}{q}-B\sqrt{T} \]

[see (18)], giving this quantity a somewhat different physical interpretation. They regard the elastic limit as the limiting stress below which “the formation of dislocations occurs at such a low rate that the deformation of the crystal remains inaccessible to practical observation. At stresses exceeding \(\sigma_{\min}\), the rate of plastic flow acquires an appreciable magnitude.”

The Burgers thereby develop the assertion, expressed by Taylor only in passing, according to which “dislocations arise owing to thermal motion,” resorting for this purpose to the aid of Becker–Orowan’s fluctuation theory.

1. According to Taylor’s theory, dislocations must, as we remember, be arranged in the crystal lattice in a more or less regular manner. In Taylor’s work this proposition remains, however, unsubstantiated. The Burgers point out that this difficulty can be resolved if one accepts, in agreement with SmeKalam, that structural obstacles (and, consequently, the dislocations arising near them) are distributed in the crystal lattice in a regular manner along certain definite crystallographic planes.

2. Strain hardening, which is one of the most characteristic features of the plastic flow of crystalline bodies, is, as we have already noted more than once, not taken into account quantitatively at all by the Becker–Orowan theory.

According to Taylor, strain hardening in real crystals is a consequence of the braking of dislocations at the boundaries of mosaic blocks.

The Burgers propose a new way of describing the phenomenon of hardening. They note that every stopped dislocation must (as follows directly from Taylor’s theory) create internal stresses around itself. In the presence of a sufficiently large number of dislocations that have been arrested in the lattice, an “internal” field of stresses is created. The authors show that this field will counteract the externally applied shearing force. As a result: “the probability of creating a sufficient concentration of stresses in the region of the structural obstacles will diminish; the formation of new dislocations at subsequent moments of time will occur at a reduced rate, as a consequence of which the rate of deformation itself will also decrease.”

Hardening from this new point of view is thus due not only to the braking of already existing dislocations, but also to a decrease in the rate of formation of new dislocations.

Burgers emphasize that it is precisely the latter circumstance that plays the most essential role. In the case when the number of arrested dislocations reaches such a value that the stress of the opposing internal field \(\tau\) almost completely compensates the external force \(\sigma\), the occurrence of new dislocations practically ceases almost entirely.

These considerations on the nature of hardening are then used by Burgers in the quantitative determination of the rate of plastic deformation, the elastic limit of the material, and the relation between stress and deformation.

They express the rate of deformation by Becker’s usual formula, replacing in it only the external shear stress \(\sigma\) by the difference \((\sigma-\tau)\), where \(\tau\) is the stress of the “internal” field. According to Taylor, the stress produced by dislocations is determined by a formula of the form [see (24)]:

\[ \sigma=\frac{C_0Gd}{h}, \]

or

\[ \sigma=C_0Gd\sqrt{N_0}, \tag{31} \]

where \(N\) is the number of dislocation pairs per unit area.\(^1\)

Burgers suppose that the stress of the internal field \(\tau\) can be determined in an analogous manner:

\[ \tau=C\cdot Gd\sqrt{N}, \tag{32} \]

where \(N\) is the number of arrested dislocations, \(C\) is a constant (in the general case different from the constant \(C_0\)).

The authors determine the rate of appearance of new dislocations by the expression

\[ \frac{\partial N}{\partial t} = \mathrm{const.}\cdot e^{-\frac{v[\omega-q(\sigma-CGd\sqrt{N})]^2}{2GkT}}. \tag{33} \]

The rate of plastic flow

\[ w=\frac{ds}{dt}, \]

where \(s\) is the crystallographic shear, is found by Burgers on the basis of the following simple considerations.

According to Taylor [see (27)]

\[ s=\frac{Ld}{ab}, \]

where \(d\) is the lattice constant, \(L\) is the length of the free path of a dislocation, and \(a\) and \(b\) are the parameters of the dislocation lattice.

\(^1\) The smallest distance between two neighboring dislocations is approximately inversely proportional to the square root of the number of dislocations.

If \(N\) is the number of dislocations per \(1\ \mathrm{cm}^{2}\), then the total crystallographic slip is

\[ s=dLN. \tag{34} \]

Hence the rate of plastic deformation

\[ \mathfrak{w}=Ld\frac{\partial N}{\partial t} =\mathrm{const.}\cdot d\cdot L\cdot e^{ -\frac{v\left[\sigma_0-q\left(\sigma-CGd\sqrt{N}\right)\right]^2}{2CGd} } \tag{35} \]

or, if we make secondary use of the proportionality between \(s\) and \(N\),

\[ \mathfrak{w}=\mathrm{const.}\cdot d\cdot L\cdot e^{ -\frac{v\left[\sigma_0-q\left(\sigma-CG\sqrt{\frac{ds}{L}}\right)\right]^2}{2GkT} }. \tag{36} \]

According to this relation, the rate of plastic flow decreases as the deformation \(s\) increases; it therefore determines the rate of plastic deformation with allowance for the hardening effect.

The elastic limit is found by the Bürgerses with the help of the same considerations as by Orowan, defining it as that minimum value of the stress \(\sigma_{\min}\) at which the rate of plastic flow \(\mathfrak{w}\) assumes the value \(\mathfrak{w}_{\min}\), accessible to measurement.

On the basis of (36),

\[ \mathfrak{w}_{\min}=\mathrm{const.}\cdot L\cdot d\cdot e^{ -\frac{v\left[\sigma_0-q\left(\sigma_{\min}-CG\sqrt{\frac{sd}{L}}\right)\right]^2}{2GkT} }, \tag{37} \]

whence

\[ \sigma_{\min}=\frac{1}{q}\left\{\sigma_0-\sqrt{\frac{2GkT}{v}\ln\frac{\mathrm{const.}\,Ld}{\mathfrak{w}_{\min}}}\right\} +CG\sqrt{\frac{d}{L}s}, \]

or else

\[ \sigma_{\min}=\frac{\sigma_0}{q}-B\sqrt{T}+CG\sqrt{\frac{d}{L}s}. \tag{38} \]

This expression for the elastic limit differs from the corresponding expression of Orowan’s theory in that, in addition to the first two terms, which characterize the influence of thresholds (the factor \(q\)) and the influence of temperature, it contains one further additional term, which takes account of the hardening of the crystal.

For a given constant value of the temperature, the relation between stress and deformation proves to be parabolic, as in Taylor’s theory:

\[ \sigma_{\min}=\mathrm{const.}\sqrt{s}+(\sigma_{\min})_0. \]

The axis of this parabola, however, is displaced by the amount \((\sigma_{\min})_0\) relative to the \(s\)-axis.

It cannot be denied that in the theory of the Bürgerses the most positive elements of all preceding theories of plastic deformation are combined in a very successful manner. But in essence it contains no fundamentally new ideas or concepts. The rate of plastic—

of viscous flow is determined, as we have seen, by Becker’s method, the elastic limit by Orowan’s method, while the microscopic picture of the propagation of the deformation process is borrowed entirely from Taylor.

As a result, the shortcomings inherent in each of these theories separately prove to be partially eliminated.

Burgers’ theory is the first theory of plasticity to determine simultaneously the rate of deformation, the elastic limit and its temperature dependence, as well as the character of the relation between stress and deformation. All these quantities are found here with allowance for the work-hardening effect.

In Burgers’ theory, however, the basic shortcomings of all preceding theories are automatically retained. Here too plastic deformation is assigned to the category of purely thermal phenomena, which, as we know, does not accord with the experimental data (low-temperature plasticity). As before, the specific feature of the plastic flow of crystals—its anisotropy and crystallographic directionality—is completely disregarded. Almost all the criticisms that we directed in due course at Taylor’s dislocation theory also remain in force.

As for the fluctuation mechanism of dislocation nucleation proposed by Burgers, it seems to us to be rather implausible. It is difficult to imagine that, as a result of a thermal fluctuation, a dislocation could be formed in the sense in which Taylor understands it.

A thermal fluctuation can cause only random displacements of individual atoms, practically without disturbing the regularity of the structure of the crystal lattice. Fluctuation phenomena cannot, it seems to us, lead to the simultaneous stretching or compression of several chains of atoms (see III, § 2).

Taylor’s dislocation theory, in its first part, referred, as we recall, to an ideally regular crystal lattice. In contrast to Taylor, Burgers assumes that dislocations can arise only near defects in the structure of the crystal. According to Burgers, plastic deformation is thereby assigned to the number of properties inherent only in real crystals. From this point of view, the most plastic crystals should be those whose structure deviates most sharply from the regular one. This conclusion, however, contradicts the experimental data.

On the other hand, the presence of structural defects is, according to Burgers, at the same time also the cause of the braking of dislocations, leading to the creation of an internal field and causing work hardening of the material. Hence, as Burgers themselves note, it follows that a crystal is the more plastic, the fewer structural defects it contains. This proposition contradicts the conclusion made above, but is, as we know, in agreement with the experimental data.

V. THE WORK OF KÖCHENDÖRFER

In a recently published paper by Köchendörfer (1938),^19 Burgers’ ideas on the mechanism of plastic deformation in real crystals received further development.

THE NATURE OF PLASTIC DEFORMATION

Together with his predecessors, Kochendörfer assumes that plastic flow is based on the displacement of dislocations along slip planes inside ideally regular blocks of the mosaic. At the same time, as in the Burgers theory, it is assumed that a dislocation arises at the boundary of a block and “travels” through the crystal lattice of the block until it is arrested upon encountering its new boundary.

In the preceding section we have already said that, according to Burgers’ idea, arrested dislocations create an internal stress field, the presence of which accounts for the hardening of the crystal.

In Kochendörfer we find an attempt to introduce into the basic relations of the Burgers theory new terms characterizing the phenomena of recovery (softening) of the material.

Taking account of the recovery effect in fact constitutes the main content of Kochendörfer’s work.

Taylor and Burgers, as we recall, introduced the idea that the boundaries of mosaic blocks, depending on the experimental conditions, may prove to be more or less “transparent” to dislocations.

The question of the causes and mechanism of the braking of dislocations at block boundaries, however, remained open.

Kochendörfer notes that the boundaries of mosaic blocks must differ from the interior of the block primarily in energetic respect, and that it is precisely from this point of view that one should approach the elucidation of the causes of dislocation braking. He assumes, moreover, that the mosaic blocks may be treated as completely independent of one another in the sense that in each of the blocks the displacement of dislocations and their emergence onto the surface of the block occur independently of the presence of neighboring blocks. In other words, according to Kochendörfer each mosaic block represents a single crystal possessing a free surface.

In considering the conditions for the formation of a dislocation on such a free surface, Kochendörfer uses the relation of the Burgers theory (35), which determines the rate of formation of dislocations. Writing it in the form

\[ \left. \begin{aligned} \frac{dN_1}{dt} &= A_1 e^{-\frac{U_1}{kT}\left(1-\frac{\sigma-\tau}{\sigma_1}\right)^2},\\[6pt] U_1 &= \frac{V c_0^{\,2}}{2G}, \qquad \sigma' = \frac{\sigma_0}{q}, \end{aligned} \right\} \tag{39} \]

where

he notes that \(U_1\) represents the activation energy that would be required for the formation of a dislocation in the absence of both external and internal stresses.

The product

\[ U_1\left(1-\frac{\sigma-\tau}{\sigma'}\right)^2 \]

determines the activation energy necessary for the formation of a dislocation in the presence of an external shear-

of the external stress \(\sigma\), as well as of the internal stress field \(\tau\). In all cases this activation energy is supplied by the thermal fluctuations accompanying the vibrations of the particles of the crystal.

Basing himself on the assumption of the mutual independence of the mosaic blocks, Kochendörfer writes the energy of formation of a dislocation \(U_1\) as the sum of two terms:

\[ U_1 = U'_1 + \Delta U, \tag{40} \]

where the term \(\Delta U\) is due to the existence of the free surface of the block (the surface energy of the dislocation).

The transition of the dislocation thus formed into the interior of the block must be accompanied by the liberation of part \(U'_2\) of the energy \(U'_1\).

As a result, inside the block the dislocation will possess an energy \(E\), equal to

\[ E = U'_1 - U'_2. \tag{41} \]

Further, under the action of the external shearing stress the dislocation “travels” through the block until it meets its next boundary. Here it may acquire the additional energy necessary for it to emerge onto the new surface of the block, equal to

\[ \left. \begin{aligned} &U_2 - U_2 \rho_2^2,\\ &\text{where}\\ &\rho_2^2 = \left(1 - \frac{\sigma - \tau}{\sigma''}\right)^2. \end{aligned} \right\} \tag{42} \]

The energy \(U_2\), according to Kochendörfer, is approximately equal to the energy \(U'_2\).

The factor \(\rho_2^2\) characterizes the presence of a boundary between the blocks (let us recall that \(\tau\) is the stress of the internal field created by dislocations that have already emerged onto the surface of the given block).

A dislocation that has penetrated to the surface of a block, from Kochendörfer’s point of view, proves to be “bound” (arrested). In this case it remains in the bound state until, by means of thermal fluctuations, it randomly receives the energy \(U_2 \rho_2^2\) lacking for it to reach the energy \(U_2\). Beginning from this moment, the dislocation ceases to be bound and acquires the ability for further motion.

\(U_2\) thus represents the activation energy necessary for the liberation of a dislocation arrested in a stressed crystal.

This process of “liberation,” caused by thermal fluctuations alone, is, in Kochendörfer’s opinion, what chiefly underlies the effect of softening, or recovery, of the material.

The essence of softening, according to Kochendörfer, consists in the fact that the “liberation” of dislocations leads to a decrease of the stress of the internal field \(\tau\), which characterizes the degree of hardening of the material.

The duration of a dislocation’s stay in the “bound” state, \(t_{\mathrm{bound}}\), is inversely proportional to the probability of its liberation:

\[ t_{\mathrm{bound}} \sim \frac{1}{e^{-\frac{U_2 b^2}{kT}}}. \]

The rate of “liberation” of dislocations, \(\frac{dN_2}{dt}\), i.e. the rate of softening, is defined by Kochendörfer by a relation analogous to formula (39), which, according to Burgers, characterizes the rate of birth of dislocations:

\[ \frac{dN_2}{dt} = A_2 e^{-\frac{U_2}{kT}\left(1-\frac{\sigma-\tau}{\sigma''}\right)^2}. \tag{43} \]

It is known from experiment that the hardening of a material at any given moment of time is determined not only by the rate of plastic flow and by the degree of deformation, but also by the magnitude of the preceding hardening.

On the basis of the ideas set forth above concerning the mechanism of hardening and softening, Kochendörfer determines the character of the dependence of the hardening \(\tau\) on all the parameters characterizing the experimental conditions. For this purpose he uses Taylor’s and Burgers’ assumption of direct proportionality between the number \(N\) of bound dislocations and the square of \(\tau\). For some instant of time \(t\),

\[ \tau = C \cdot Gd \sqrt{N}, \]

where \(d\) is the lattice constant and \(G\) is the shear modulus [see (32)].

If during the time \(dt\), \(dN_1\) dislocations are formed, and \(dN_2\) dislocations are “liberated” from the bound state, then the value of \(\tau\) at the time \(t+dt\) may be written in the form

\[ \tau + d\tau = C \cdot Gd \sqrt{N + dN_1 - dN_2}. \tag{44} \]

Substituting the values of \(\frac{dN_1}{dt}\) and \(\frac{dN_2}{dt}\) [see (39) and (43)], we obtain:

\[ 2\tau \frac{d\tau}{dt} = 2G^2 d^2 \left\{ A_1 e^{-\frac{U_1}{kT}\left(1-\frac{\sigma-\tau}{\sigma'}\right)^2} - A_2 e^{-\frac{U_2}{kT}\left(1-\frac{\sigma-\tau}{\sigma''}\right)^2} \right\}. \tag{45} \]

With the aid of relation (35) of Burgers’ theory, which determines the rate of plastic flow, Kochendörfer expresses \(\tau\) through the rate of plastic deformation \(\omega\) and the experimental temperature \(T\):

\[ \sigma-\tau = \sigma' \left\{ 1- \sqrt{ \frac{k}{U_1} \ln\left(\frac{A_1 L d}{\omega}\right) \sqrt{T} } \right\}. \tag{46} \]

Using this expression, he obtains the equation

\[ 2 \frac{d\tau}{dt} = \frac{C^2 G^2 d}{L} \left\{ \omega - A_2 dL e^{-\Phi(\omega,T)} \right\}, \tag{47} \]

where

\[ \varphi(w,T)=\left(1-\frac{\sigma'}{\sigma''}\right)\frac{U_2}{kT} +2\left(1-\frac{\sigma'}{\sigma''}\right) \sqrt{\ln\frac{A_1Ld}{w}\, \frac{\dfrac{U_2}{k}}{\dfrac{\sqrt{U_1}}{kT}}} + \frac{U_2}{U_1}\left(\frac{\sigma'}{\sigma''}\right)^2 \ln\frac{A_1Ld}{w}. \]

Equation (47) also contains a term corresponding to the softening of the material. The course of the hardening curves under the given experimental conditions is thus determined by the relation between the rates of hardening and recovery.

Integrating equation (47) with initial conditions corresponding to an undeformed and unhardened crystal, i.e., assuming that at

\[ t=0,\quad \tau=0 \quad \text{and} \quad s=0, \tag{48} \]

and using the definition of the strain rate \(w=\dfrac{ds}{dt}\) (where \(s\) is the crystallographic shear), Kochendörfer obtains:

\[ \frac{\tau^2}{s}=\frac{C^2G^2d}{L} \left(1-\frac{A_2Ld}{w}e^{-\omega(w,T)}\right). \tag{49} \]

The external shearing stress \(\sigma\) may be written as

\[ \sigma=\sigma_{\min}+\tau. \tag{50} \]

It follows from this that equation (49), for given constant values of \(w\) and \(T\), leads to a parabolic dependence of the deformation \(s\) on the stress \(\sigma\).

This result is not, however, unexpected, since the very derivation of equation (49) is based on Taylor’s assumption of direct proportionality between \(\tau\) and \(\sqrt{N}\), which in its time led Taylor to a parabolic relation between \(s\) and \(\sigma\).

Kochendörfer uses equation (49) to find the dependence of hardening on the temperature of the experiment and on the strain rate. For a constant value of the strain rate \(w\) we obtain:

\[ \frac{\tau^2}{s}=A\left[1-Be^{-\left(\frac{C}{T}+\frac{D}{\sqrt{T}}\right)}\right], \tag{51} \]

whereas for constant \(T\)

\[ \frac{\tau^2}{s} = A\left[ 1-\frac{B'}{w^{C'}}e^{-D'\sqrt{\ln\frac{A_1Ld}{w}}} \right]. \tag{52} \]

The theoretical value of the constants \(A, B, C, D, B', C'\) and \(D'\) entering into these equations can be found from equations (47) and (49).

Kochendörfer determines the elastic limit and the character of its dependence on temperature by the same method as Orowan and Burgers.

For an undeformed crystal, i.e., under conditions (48), relation (46) gives for the initial value of the shearing stress:

\[ \sigma_{\min} = \sigma' \left\{ 1-\sqrt{\frac{k}{U_1}\ln\frac{A_1Ld}{w_{\min}}\sqrt{T}} \right\}. \tag{53} \]

This equation determines the critical value of the shearing stress \(\sigma_{\min}\), necessary in order that already at the initial stages of deformation the rate of deformation \(w\) should have some definite value \(w_{\min}\). The magnitude of the external shearing stress for subsequent moments of time is determined by formula (50).

In conclusion, Kochendörfer compares relations (51) and (52) with experimental data on the course of strain-hardening curves of certain crystals under various experimental conditions.

For this purpose he first of all uses the results of the experiments of Boas and Schmid\({}^{16}\), who studied the deformation of Al and Cd at different temperatures. For Al, Boas and Schmid, as we recall, established a parabolic relation between deformation and stress. Kochendörfer proves that, according to the data of Boas and Schmid, the dependence of \(\frac{\tau^{2}}{s}\) on temperature can be satisfactorily described by a relation of the type (51).

Comparing the experimental and theoretical values of the coefficient \(A\) in formula (51), Kochendörfer calculates the length of the free path of a dislocation \(L\). The latter turns out in this case to be of the order of \(10^{-5}\) cm.

For crystals of the hexagonal system, in particular for cadmium, the relation between deformation and stress is, however, not parabolic but rectilinear; in connection with this, as Kochendörfer himself notes, equations (51) and (52) cannot be applied in this case.

Further, Kochendörfer attempts to subject relation (52), which gives the connection between the rate of deformation and strain-hardening, to experimental verification.

Assuming that the coefficients \(A, B, C\), and \(D\) in equation (51) have one and the same numerical value for all metals, the author uses the experimental value of these coefficients, obtained on the basis of the above-mentioned experiments with aluminum, to calculate the coefficients \(B'\), \(C'\), and \(D'\) in formula (52). Assigning definite values to the ratio \(\frac{U_{2}}{U_{1}}\), he then constructs theoretical curves of the dependence of \(\frac{\tau^{2}}{s}\) on the rate of deformation \(w\), corresponding to different numerical values of \(\frac{U_{2}}{U_{1}}\).

In view of the almost complete absence of experimental data on the dependence of the course of strain-hardening curves of metals on the rate of deformation, Kochendörfer compares these theoretical curves with the corresponding experimental curves obtained by him for naphthalene crystals. The experimental points in this case lie well on the theoretical curve for which \(\frac{U_{2}}{U_{1}} = 0.9\). From this Kochendörfer concludes that the activation energy \(U_{2}\), which determines the rate of the softening process, is less than the energy \(U_{1}\) necessary for the formation of a new dislocation.

Kochendörfer’s work is devoted, as we see, to an analysis of the conditions of strain-hardening and softening.

This analysis is based on consideration of the energetic state of a dislocation at the various stages of its existence—when it arises on the surface of a mosaic block, penetrates into the interior of the block, “trave-

“travel” along the block, upon emerging onto a new surface of the block, braking on this surface, and, finally, “release.”

Both the stage corresponding to hardening (braking of a dislocation) and the stage corresponding to softening (release of the braked dislocations) are thus treated from one and the same point of view—the evaluation of the activation energy required for the transition of a dislocation from the given stage to the next one and imparted to it by thermal fluctuations.

The mechanism of hardening and softening, according to Kochendörfer, thus proves to be completely identical.

Experimental data indicate, however, that the nature of these phenomena is entirely different.

The basis of the softening effect is indeed formed by processes of the diffusion type, in which individual atoms participate independently of one another and which lead to the gradual “healing” of the distorted lattice.

Hardening, meanwhile, is of an essentially different character, being associated with macroscopic displacements of whole regions of the crystal lattice that arise in the course of plastic deformation.

If softening is unquestionably a purely thermal phenomenon, then hardening should rather be assigned to the category of athermal phenomena. In this connection, it seems to us that there can be no question of identifying the mechanisms of hardening and softening.

However, even if one adopts Kochendörfer’s point of view, the meaning of the energy relations introduced by him remains completely obscure. Indeed, one may agree that dislocations should form primarily on the surfaces of mosaic blocks and that, having formed, they will tend to pass into the interior of the block, losing in the process part of their energy. One can also understand why, in order to emerge onto the surface of the block, upon encountering its new boundary, a dislocation must be supplied with some additional energy. It is entirely unclear, however, why as a result it will turn out to be “braked.” Why must some new energy be supplied to it for further motion, whereas a dislocation newly formed under exactly the same conditions, on the contrary, always itself tends to leave the boundaries of the block?

If one consistently adheres to Kochendörfer’s ideas about the mechanism of the phenomena under consideration, there should seemingly be no difference in the behavior of a dislocation newly arising at a given boundary and that of a dislocation released after braking at this same boundary.

It should also be noted that, according to Kochendörfer, the softening of the material ought to be accompanied not by restoration of the crystal lattice, the presence of which is attested by experimental data, but, on the contrary, by its deterioration as a consequence of the departure of the “released” dislocations into the interior of the block.

Further puzzling is the manner in which the author establishes the connection between hardening $\tau$, deformation $s$, deformation rate $w$, and the temperature of the experiment.

In integrating equation (45), Kochendörfer substitutes into the right-hand side of this equation the value of \(\tau\), expressed in terms of the quantities \(w\) and \(T\) by formula (46). But the latter determines the rate of plastic flow \(w\) without taking account of the effect of softening, whereas equation (45) itself was introduced by the author with the special aim of clarifying the conditions of equilibrium between hardening and softening. By omitting the term \(\dfrac{dN_2}{dt}\) in the definition of the strain rate \(w\), Kochendörfer thereby fails to take account of the participation of the “released” dislocations in plastic flow, for which, as a result, only the newly formed dislocations are responsible. The subsequent fate of the released dislocations remains completely unclear in this case.

It also remains unclear what considerations lead to the substitution of the function \(\tau=\tau(w,T)\) only into the right-hand side of this equation.

Finally, Kochendörfer’s assertion that the coefficients \(A, B, C, D\) and \(B', C', D'\) in equations (51) and (52) must have identical numerical values for all metals arouses great bewilderment. If this were so, then the tensile curves of all metals, obtained under identical experimental conditions, would have to coincide with one another.

All the more strange is the use of numerical values of these coefficients, obtained from experiments with aluminum, in considering experimental data relating to the study of the plastic properties of naphthalene.

VI. THE THEORY OF FRENKEL AND KONTOROVA

Somewhat apart from the theories considered above stands the theory of Frenkel and Kontorova (1938–1939).\(^{20}\) In view of the fact that it was published in Russian, we shall dwell here only on its most essential propositions.

This theory, unlike all the preceding ones, in considering the mechanism of plastic deformation, makes absolutely no appeal to the aid of “dislocations,” or, in general, to any defects in the structure of the crystal.

The authors propose that the capacity for crystallographically directed residual deformation is one of the characteristic features of an ideally regular crystal lattice. The work of Frenkel and Kontorova is devoted mainly to considering the microscopic picture of the process of propagation of an elementary shear in such a defect-free lattice.

It is assumed that plastic deformation is effected by the gradual transition of atoms of the crystal lattice from one equilibrium position to another.

At first the authors confine themselves to considering the one-dimensional case, investigating the process of displacement of atoms in an infinite atomic chain \(AB\), situated on a “substrate” of exactly the same atomic chains (Fig. 8).

In the first part of the theory the atoms of the chains that form the “substrate” are assumed to be fixed immovably at the lattice sites.

Without touching upon the question of the causes of the occurrence of the shift, the authors set up the equations of motion of the atoms participating in the process of

Fig. 8                Fig. 9

sliding. In doing so, the interaction of the atoms of the chain under consideration with one another is taken into account, as well as the action upon them from the atoms of the immobile “substrate.”

The forces of interaction of the atoms of the chain are, in the first approximation, assumed to be quasi-elastic. The total potential energy of the particles constituting the chain can in this case be written in the form:

\[ U=\sum_k A\left(1-\cos 2\pi\frac{\psi_k}{d}\right) +\frac{1}{2}a\sum_k(\psi_{k+1}-\psi_k)^2, \tag{54} \]

where \(d\) is the lattice constant, \(a\) is the coefficient of the quasi-elastic bond, and \(\psi_k\) and \(\psi_{k+1}\) are the displacements of two neighboring [the \(k\)-th and \((k+1)\)-th] particles from their equilibrium positions.

The first term of this expression characterizes the periodic force field of amplitude \(A\), created by the immobile atoms of the “substrate.” The state of the particles of the chain in such a field is similar to the state of heavy balls resting in equally spaced pits and connected with one another by elastic springs (Fig. 9).

The equation of motion of one of the particles, for example the \(k\)-th particle, is written in the form

\[ m\frac{d^2\psi_k}{dt^2} =-\frac{\partial U}{\partial \psi_k} =-2\pi\frac{A}{d}\sin 2\pi\frac{\psi_k}{d} +a(\psi_{k+1}+\psi_{k-1}-2\psi_k). \tag{55} \]

The equations of motion of all the other particles of the chain will have an entirely analogous form.

It is important to note that equation (55) contains not only the coordinate of the \(k\)-th particle, but also the coordinates of the neighboring \((k+1)\)-th and \((k-1)\)-th particles. The displacement of each of the particles from its equilibrium position therefore depends on the displacements of all the other particles of the chain; the transition of atoms to new equilibrium positions occurs, as it were, “collectively,” similarly to what takes place in the propagation of a sound wave in a crystal. This “collectivity,” however, by no means signifies that all atoms of the chain are simultaneously displaced by equal distances; on the contrary, the shift is not carried out at once throughout the entire chain, but propagates along it gradually.

Since the atoms of the “substrate” are assumed to be immobile, the total energy of the atoms of the chain under consideration must remain constant—

...throughout the entire passage of the shear, and the very process of shear formation must proceed with a constant velocity \(w\), equal to

\[ w=\frac{d}{\tau}, \tag{56} \]

where \(d\) is the lattice constant, \(\tau\) is the time of propagation of the shear over a distance \(d\), i.e., from atom to atom.

The displacement quantities \(\psi_{k+1}\) and \(\psi_{k-1}\), entering into the equation of motion (55), may therefore be expressed in terms of the displacement \(\psi_k\) and the time \(\tau\). Namely, if the direction of propagation of the shear coincides with the direction of numbering of the particles of the chain, then the \((k+1)\)-th particle will begin to move from its equilibrium position a time \(\tau\) later, and the \((k-1)\)-th particle a time \(\tau\) earlier than the \(k\)-th particle, whence

\[ \begin{aligned} \psi_{k+1}(t)&=\psi_k(t+\tau),\\ \psi_{k-1}(t)&=\psi_k(t-\tau). \end{aligned} \tag{57} \]

Substituting (57) into (55) and assuming that the change of the function \(\psi_k\) over the time \(\tau\) is small in comparison with the function \(\psi_k\) itself, the authors expand \(\psi_k(t+\tau)\) and \(\psi_k(t-\tau)\) in a series up to terms of the second order.

As a result, the equation of motion assumes the following final form:

\[ (m-\alpha\tau^2)\frac{d^2\psi_k}{dt^2}\cong -2\pi\frac{A}{d}\sin 2\pi\frac{\psi_k}{d}. \tag{58} \]

The right-hand side of this equation represents the force acting on the \(k\)-th atom of the chain from all the atoms of the substrate; the coefficient of \(\dfrac{d^2\psi_k}{dt^2}\) on the left-hand side of the equation plays the role of mass. The latter differs from the ordinary mass \(m\) by the term \(\alpha\tau^2\), the presence of which is due to taking into account the interaction of the atoms of the chain with one another.

The presence of a connection between the particles of the chain under consideration thus leads to the fact that their motion in the field created by the substrate occurs as though with a modified mass:

\[ m'=m-\alpha\tau^2. \tag{59} \]

Integration of equation (58) under the initial conditions

\[ \psi_k=0 \quad \text{and} \quad \frac{d\psi_k}{dt}=0 \]

gives the displacement velocity of the \(k\)-th particle of the chain:

\[ \frac{d\psi_k}{dt}=2\sqrt{-\frac{A}{m'}\sin 2\pi\frac{\psi_k}{dt}}, \tag{60} \]

and likewise the displacement itself \(\psi_k\) as a function of time \(t\),

\[ \psi_k=\frac{2d}{\pi}\operatorname{arctg}\left[C_k e^{\pm\frac{2\pi}{d}\sqrt{-\frac{A}{m'}}\,t}\right]. \tag{61} \]

The case \(m' > 0\) corresponds to small oscillations of particles near the equilibrium position, not accompanied by their displacement along the chain.

It is easy, however, to show that, under the condition

\[ m'' < 0 \tag{62} \]

the displacement \(\psi_k\), over the course of some interval of time, changes by an amount \(d\) (theoretically, when \(t\) changes from \(-\infty\) to \(+\infty\); practically, over a time of the order of \(T=\dfrac{d}{2\pi}\sqrt{\dfrac{|m'|}{A}}\)).

It is precisely this case that corresponds to the displacement of the particles of the chain by a distance equal to the lattice constant; this means that an elementary shift is effected in the chain.

The condition for shift formation (62) makes it possible to find the limits of variation of the velocity of propagation of the shift.

Introducing into consideration the speed of sound \(w_0\),

\[ w_0=\frac{d}{\tau_0}=d\sqrt{\frac{a}{m}} \tag{63} \]

(\(\tau_0=\sqrt{\dfrac{m}{a}}\) is the period of oscillation of an atom when it has only one “neighbor,” i.e., in the absence both of all the other particles of the chain and of the substrate), and taking (59) into account, the authors write inequality (62) in the form

\[ w < w_0 . \tag{64} \]

Thus, the velocity of propagation of the shift does not exceed the speed of sound in the given medium.

Further, the authors establish a relation between the velocity of propagation of the shift and the energy of the particles participating in the sliding process.

The total energy \(W\) of all particles of the chain under consideration is determined by the expression

\[ W=U+E=\sum_k A\left(1-\cos 2\pi\frac{\psi_k}{a}\right) +\frac{1}{2}a\sum_k(\psi_{k+1}-\psi_k)^2 +\frac{m}{2}\sum_k\left(\frac{d\psi_k}{dt}\right). \tag{65} \]

On the basis of (57) and (60) it can be written in the form

\[ W=4A\frac{a\tau^2}{|m'|}\sum \sin^2\pi\frac{\psi_k}{d}, \]

and, after carrying out the summation, in its final form

\[ W=\frac{4mw_0^2}{\pi}\sqrt{\frac{A}{m\left(w_0^2-w^2\right)}} . \tag{66} \]

From this, the velocity of propagation of a displacement \(w\) as a function of the total energy \(W\) is:

\[ w=w_0\sqrt{1-\frac{W_0^2}{W^2}}, \tag{67} \]

where

\[ W_0^2=\frac{16mw_0^2}{\pi^2}A^2. \tag{68} \]

It follows from relation (67) that the velocity of propagation of the displacement assumes real and nonzero values provided that the condition

\[ W>W_0 \tag{69} \]

is satisfied.

From this the authors conclude that the quantity \(W_0\) may be interpreted as that minimum value of the energy of the chain starting from which the propagation of a displacement in it becomes possible.

In Frenkel and Kontorova’s opinion, relations (68) and (69) make it possible to understand the physical nature of the anisotropy of residual deformation in crystalline bodies.

Let us recall that the constant \(A\), which determines the energy \(W\), is the amplitude of the tangential component of the force acting on each of the atoms of the chain under consideration from all the atoms of the “substrate.” For different crystallographic planes and directions this constant, and consequently also the energy \(W_0\), must assume different values.

Frenkel and Kontorova suppose that it is precisely this circumstance that explains the crystallographic directionality of plastic deformation.

The determination of the directions in which displacement can occur most easily and in which it will therefore most often be realized in actuality must, in their opinion, reduce to finding those chains of atoms for which the energy \(W_0\) and, correspondingly, the amplitude \(A\), assume the smallest of all possible values.

From the standpoint of these considerations one may assert that, first of all, displacement formation must take place in crystallographic planes and directions characterized by the densest packing of particles, since precisely these planes and directions must correspond to the smallest value of \(A\).

This conclusion is confirmed, as is known, by experimental data.

One may further suppose that, in comparing the plastic properties of different crystalline substances, the least “plastic” will turn out to be those among them for which the energy \(W_0\) has the greatest values.

\(W_0\) (or the amplitude \(A\)) may thus be interpreted as a measure of the “nonplasticity” of the crystal.

Investigating the limits of applicability of the approximate form of the equation of motion (58), the authors arrive at the conclusion that the sliding process of the type under consideration is possible only in such crystallographic …

directions for which the value of the amplitude \(A\) satisfies the inequality

\[ A < \frac{m}{4\pi^{2}} w_{0}^{2}, \tag{70} \]

i.e., for which

\[ W_{0} < \frac{2mw_{0}^{2}}{\pi^{2}}. \tag{71} \]

A comparison of the upper bound of the energy \(W_{0}\) (the quantity \(\frac{2mw_{0}^{2}}{\pi^{2}}\)), calculated for various metals, with experimental data on the character of the hardening curves, as well as on the shear moduli and hardness numbers of these metals, indicates that for crystals of the cubic system the quantity \(W_{0}\) may indeed be regarded as a criterion of “nonplasticity.”

In what follows the authors investigate the conditions for the propagation of elementary shear when allowance is made for the mobility of the atoms of the “substrate.”

They solve this problem by a method similar to that adopted in collision theory: the method of considering the question of the passage of an \(\alpha\)-particle through matter. The velocity of the \(\alpha\)-particle is usually assumed to be constant in this case, and the loss of energy experienced by it in collisions is determined from the energy acquired by the electrons of the atoms being smoothed over.

In the present case it is assumed that the mobility of the atoms of the substrate does not affect the character of the sliding process, but causes a gradual outflow of energy from the slip plane into the adjacent atomic layers, which entails a gradual damping of the shear in the chain under consideration.

The authors make an approximate calculation of the loss of energy experienced by the atomic chain as a result of “collisions” of the shear, propagating along it, with the vibrating particles of the substrate. The calculations show that at energies appreciably exceeding the minimum energy of shear formation, these losses are so insignificant that they should not exert any substantial influence on the rate of shear propagation.

The last part of Frenkel and Kontorova’s work contains a generalization of the preceding theory to the case of the twinning process.

Assuming that the mechanism of twinning is essentially no different from the mechanism of translation (a gradual rotation of atomic layers into new equilibrium positions, mirror-symmetric with respect to the initial ones), the authors calculate the rate of propagation of a twin and investigate the energetic conditions of twinning, guided by the same notions that underlay the theory of elementary shear in a linear atomic chain.

Turning to a discussion of this theory, it should first of all be noted that the model of atomic chains sliding relative to one another with which it operates is, of course, highly schematic.

Experimental data indicate that in reality the process of shear formation proceeds incomparably more complexly. In particular, before

until now it has not been possible to observe the phenomenon of translation in pure form. The latter usually turns out to be complicated by rotation and bending of the elements of the crystal located in the region of the slip planes. This circumstance, as well as a number of specific features of the structure of real crystals (the presence of mosaic blocks, various types of distortions and the overstresses associated with them), are completely not taken into account by the theory under consideration. In this connection it can hardly claim to give any complete description of the process of plastic flow of real crystals.

Nevertheless, the posing of the question of the possibility of residual shear formation in an ideally regular crystal lattice is of fundamental interest.

For a correct understanding of the physical nature of plastic deformation it is extremely important to ascertain whether the property of plasticity is inherent only in real crystals, or whether it could also occur in a crystal lattice possessing an ideally regular structure.

Let us recall that this theory for the first time gives a solution to this question, proving the possibility of shear formation in a crystal lattice devoid of any defects and distortions whatsoever. This conclusion is to a certain extent in agreement with experimental data indicating that the plastic properties of crystals are expressed the more sharply, the more regular their structure.

Returning to the basic propositions of the theory, let us note once more that, according to its authors, once the energy conditions for shear have been created, it will propagate with a finite velocity at any experimental temperatures, including at the temperature of absolute zero.

Within the framework of this theory, plastic deformation is thereby assigned to the category of athermal properties of crystals. It is known from experiment that in metallic crystals plastic deformation is indeed observed at all temperatures, down to temperatures very close to absolute zero.

Among the advantages of this theory is also the circumstance that it is the first to indicate the causes of the anisotropy of plastic flow.

It must be noted that many propositions of the Frenkel and Kontorova theory remain insufficiently developed.

First of all, it does not examine the causes and conditions of the occurrence of shear, i.e. the practical elastic limit of the material is not determined.

Furthermore, the authors do not touch upon one of the most essential questions in the theory of the plasticity of crystals—the question of hardening. Assuming that the cause of hardening is the gradual damage of the lattice in the process of deformation, they do not, however, give a quantitative estimate of this effect.

The question of the temperature dependence of the plastic properties of crystals also remains outside their field of view1.

In conclusion, mention should be made of Dellinger and Kochendörfer’s attempt \(^{23}\) to interpret the theory of Frenkel and Kontorova in the spirit of “dislocation” conceptions.

While agreeing that the Frenkel–Kontorova equation does indeed represent the process of plastic flow of crystals, Dellinger and Kochendörfer hold that these equations describe nothing other than the propagation of a Taylor dislocation, whose origin is due to the combined action of thermal fluctuations and local overstresses acting in the region of the “thresholds” of the crystal lattice.

In the process whereby atoms of a chain pass from one equilibrium position to another in a crystal, a distorted zone having the structure of a Nonius does indeed arise. It exists in the crystal only temporarily, however, and its occurrence, associated with the rearrangement of the lattice during shear, is a consequence, but not the cause, of shear formation.

Such a distorted zone, however, has nothing in common with Taylor’s statistical model of the lattice of dislocations, which exist for a long time in the crystal even before the onset of shear propagation.

As we have already noted, the basic idea of the Frenkel–Kontorova theory is that plasticity is above all a property of a regular crystalline lattice.

An attempt to unite this theory with “dislocation” theories of plastic deformation therefore contradicts the essence of each of the two directions in the development of physical conceptions concerning the nature of crystal plasticity that have now taken shape.

CONCLUSION

Fifteen years have passed since the appearance of the first theory of plastic deformation—the Becker theory.

During this period, physical conceptions of the nature of residual deformation of crystalline bodies have undergone a considerable evolution.

If in Becker (1925) we encounter an attempt to solve only one question—the causes of the onset of plastic flow—then Buerger (1935) and Kochendörfer (1938) develop a consistent quantitative theory of crystal plasticity, concerned with the principal specific features of this phenomenon. In these works we already find quite definite ideas both about the conditions for the occurrence of shears and about the mechanism of their propagation, as well as about the causes of the hardening and softening effects that usually accompany plastic deformation. They further establish a quantitative relation between the principal macroscopic quantities characterizing plastic flow in experiment—between stress, the magnitude of deformation, its rate, and the temperature of the experiment.

It must be noted that the appearance of Taylor’s work (1934) provided a sharp impetus to the further improvement of existing views on the nature of plastic deformation. The latter for a long time determined the direction and paths of development of modern theoretical conceptions of the plasticity of crystals.

It cannot be denied that the quantitative relations furnished by contemporary theories of plasticity are to a certain extent confirmed by experiment.

The physical concepts that have led to the establishment of these relations, however, can by no means be regarded as sufficiently consistent and convincing.

The dislocation hypothesis, which underlies the Taylor—Burgers—Kochendörfer theory, as we have already noted more than once, cannot, of course, be regarded as a sufficiently firm and unimpeachable foundation for a physical theory of plasticity.

As for the theory of Frenkel and Kontorova, according to which the occurrence of plastic shears is not connected with the presence of any distortions in the lattice, and the propagation of plastic deformation takes place by means of a gradual collective transition of atoms from some equilibrium positions to others, then, owing to its microscopic character, it is as yet not entirely clear by what path macroscopic equations characterizing the process of plastic flow of real crystals can be established on its basis.

LITERATURE

1. Becker, Physik. Z., 26, 919, 1925.
2. Becker, Z. techn. Physik, 7, 547, 1926.
3. Maxwell, Phil. Mag., 35, 139, 1868; Phil. Trans., London, 157, 49, 1867.
4. Orowan, Z. Physik, 89, 6, 1934.
5. Orowan, Z. Physik, 98, 382, 1936.
6. Born and Geppert-Mayer, Theory of the Solid State, ONTI, 1938.
7. Orowan, Z. Physik, 89, 619, 1934; 97, 573, 1935; 102, 112, 1936.
8. Taylor, Proc. Roy. Soc., 145, 362, 1934.
9. Taylor, Proc. Roy. Soc., 145, 388, 1934.
10. A. F. Ioffe, Physics of Crystals, 1929.
11. Timpe, Götting. Diss., Leipzig, 1905.
12. Goetz, Proc. Nat. Acad. Amer., 16, 99, 1930.
13. Straumanis, Z. Physik, 13, 316, 1931; 19, 63, 1932.
14. Belaiew, Proc. Roy. Soc., 103, 295, 1925.
15. Zwicky, Brown Boveri Review, 1929.
16. Boas und Schmid, Z. Physik, 100, 463, 1936.
17. W. Burgers a. G. Burgers, First report on viscosity and plasticity, p. 173, Amsterdam, 1935.
18. Smekal, Physik Z., 34, 633, 1933.
19. Kochendörfer, Z. Physik, 108, 244, 1938.
20. T. A. Kontorova and Ya. I. Frenkel, ZhETF, 8, 89, 1340, 1349, 1938.
21. Ya. I. Frenkel, ZhETF, 9, 1238, 1939.
22. T. A. Kontorova, Journal of Technical Physics, 9, 1086, 1939.
23. Dehlinger u. Kochendörfer, Z. Physik, 116, 576, 1940.

1. This question was, it is true, considered by the authors 21, 22, but its investigation was based on purely formal considerations and was carried out without any connection with the microscopic theory of plasticity set forth here. ↩↩