THEORY OF INTERLOCKS IN THE CRYSTAL LATTICE*)

A. H. Cottrell

A theoretical consideration of the plastic deformation of metals can be carried out in two different ways. One of them¹ consists in constructing a phenomenological theory of plasticity on the basis of empirical laws that idealize the observed plastic properties of metals. These laws are formulated mathematically, and then attempts are made to solve the resulting equations, which make it possible, under various conditions, to analyze the behavior of metals in such complex deformation processes as pressing, rolling, etc. The second path, which brings us closer to solving the problems considered here, has as its aim the creation of a theory of the mechanism of atomic motions accompanying plastic deformation. Although this molecular theory of plasticity is still at an early stage of its development, it has already become clear that a special type of crystal-lattice defect, called an interlock**), is an important link connecting the atomic structure of metals with their crystallographic plastic properties (with the exception of certain cases of noncrystalline plasticity, such as quasiviscous flow along grain boundaries). Taylor², Orowan³, and Polanyi⁴, almost simultaneously in 1934, expressed the idea that plastic slip is a consequence of the passage of interlocks through a crystal. Even earlier, Prandtl⁵ and Dehlinger⁶ introduced the concept of an interlock into the theory of mechanical hysteresis and crystal growth.

*) Progress in Metal Physics, Editor B. Chalmers, 1949, p. 77.
A. H. Cottrell, “The Theory of Dislocations.” Translated from the English by G. A. Gol’der, edited by Prof. G. S. Zhdanov.

**) Editor’s note to the translation. In the original, Cottrell’s article is entitled “The Theory of Dislocations.” The term “dislocation,” sometimes translated as “displacement,” and in most cases simply as “dislocation,” takes its origin from the theory of elasticity. In the theory of elasticity, the case usually considered is that of an elastic cylinder with a cut and a displacement of parts of the cylinder along the plane of the cut (Fig. 7). The use of the term “displacement” in the present case is apparently quite justified. In this model, not ...

A. H. COTTRELL

TYPES OF DISLOCATIONS

The concept of a dislocation is a natural consequence of the crystallographic nature of plastic flow. The observation of slip lines in stressed metals, revealing the most important crystal planes7, 8, the proof that the observed plastic deformations can be resolved into elementary shears along these planes9, and that plastic flow is determined by the shear stress acting on these planes10—all this clearly showed that flow occurs as a result of the sliding of certain atomic planes (slip or shear planes) relative to one another. Moreover, the structure in the slip planes remains crystalline during the process of flow, since slip takes place in the direction of closest packing of atoms in the slip planes, and not in the direction of maximum shear stress. Taking the picture described above as our starting point, let us consider a plane \(A\) of atoms sliding in some crystallographic direction over an adjacent plane \(B\). Because the atoms in a crystal are not rigidly fixed, but are only elastically bound to one another, and since thermal vibrations and other sources of distortions in the lattice may cause a nonuniform distribution of forces over the slip plane, different regions of the plane \(A\) may at some moment slide over the plane \(B\) in different ways. Dislocations are the boundaries between these different regions of the slip plane. The existence of a discrete atomic structure in the slip plane limits the difference in the magnitudes of shear between neighboring regions to the length of an interatomic distance or, in some lattices, to a small fraction of this distance. We call the difference in the magnitude of shear along the boundary the measure of the dislocation forming this boundary. A second consequence of the discreteness of the atomic structure is that all dislocations reduce to a few standard types having quite definite forms.

are reflected the forces holding the elastically displaced parts of the cylinder in positions of relative displacement. It is assumed that there are either external forces balancing the internal elastic forces in the deformed cylinder, or that there is a “glue” or “bonding” in the cut plane holding the displaced parts of the cylinder. The model of the cut cylinder (the theory of an elastic continuous medium) is often regarded as an analogue of the special kind of distortions arising in a crystal lattice during plastic deformation. These residual distortions, an initial idea of which can be obtained from considering the arrangement of bubbles in soap or oil foam (photographs I and II), possess a well-known stability and, unlike the cut elastic cylinder, are balanced by forces within the crystal lattice. The term “dislocation” applies to this type of distortion of the crystal lattice, arising during plastic deformation and, possibly, during crystal growth.

THEORY OF DISLOCATIONS IN A CRYSTAL LATTICE

The first type includes the linear dislocation described by Taylor, shown in Fig. 1. Here the slip plane \(ABCD\) is divided by the dislocation line \(EF\) into a region in which slip has occurred (\(ABEF\)) and a region without slip (\(FECD\)). The direction of slip \(A'F'\) is perpendicular to the dislocation line \(EF\). The atomic structure in a linear dislocation is shown in Fig. 2, which depicts a section perpendicular to the line \(EF\) (Fig. 1).

Fig. 1. Linear dislocation.

We see that the atoms in the upper half-space of the crystal \(P\) are compressed along the direction of shear, while the atoms in the lower half-space \(Q\) are stretched. There may also exist a dislocation opposite to that shown in this scheme, i.e. a dislocation in which the upper half-space of the crystal is expanded and the lower one compressed. Accordingly, dislocations of the type shown in Fig. 1 are called positive, and those opposite to them negative. Since, by turning over the scheme (the crystal), a positive dislocation can be transformed into a negative one and conversely, their difference would be inessential were it not for the important circumstance that the forces acting between dislocations depend on whether the dislocations have the same sign or different signs.

Fig. 2. Structure of a linear dislocation.

The second principal type of dislocation is the screw dislocation (Fig. 3), the idea of which was introduced by Burgers\(^{11, 12}\). In Fig. 3 the part \(ABEF\) of the slip plane is displaced in the direction \(EF\), while the remaining part \(FECD\) is immobile; the boundary \(EF\) forms a screw dislocation. It is necessary to note that, in contrast to the preceding case, the line of a screw dislocation lies in the direction of the shear. In both cases, further shear is caused by displacement of the dislocation region \(EF\) in the direction toward \(CD\), i.e. the slip region \(ABEF\) grows at the expense of the undeformed region \(FECD\). However, the dislocations differ

orientation of the direction of their motion relative to the direction of shear. The atomic arrangement in a screw dislocation is shown in Fig. 4, which presents a plan view of part of the atomic planes sliding one over the other; in this diagram the solid lines and black circles represent atoms in the upper plane, while the dashed lines and light circles represent atoms in the lower plane. As in the case of linear dislocations, screw dislocations of opposite signs, mutually inverse, may exist.

Fig. 3. Screw dislocation.

Fig. 4. Structure of a screw dislocation.

Linear and screw dislocations are linear ruptures; however, dislocations of an arbitrary nonlinear form can easily be produced by jointly combining segments of linear and screw dislocations, as shown in Fig. 5. It is probable that in real crystals dislocations belong to such a complex type. Burgers11, 12 noted that dislocation lines cannot end at some point inside a crystal; they must either form closed chains consisting of linear and screw segments, or end at the surface of the crystal. This becomes clear from a description of the picture of a dislocation terminating inside a crystal. Suppose, for example, that the linear dislocation of Fig. 1 extends only from point \(E\) to \(F'\); this means that the region of shear of the slip plane is only \(A'BEF'\), and not \(ABEF\). But because of this the line \(A'F'\) will inevitably be a screw dislocation. Consequently, we cannot con—

...obtain a linear dislocation inside the crystal without the occurrence of a screw dislocation, and conversely.

Metals, in reality, do not possess the simple cubic lattice that we have used here, and in the lattices of metals it is considerably more difficult to represent the structure of dislocations. However, many important properties of dislocations depend on the distribution of stresses in the region of the dislocation at distances from its center large in comparison with $\lambda$; in an elastically isotropic medium the distribution of stress at such distances depends only on the magnitude of the displacement, and not on the details of the shape of its center. In a real crystal the measure of a dislocation may be found by determining the distance, in the direction of slip, between neighboring equilibrium positions of atoms lying in the slip plane.

Although dislocations are too small for direct observation, as, for example, in metals, indirect evidence of their existence can nevertheless be given. It is interesting to note that in two-dimensional “crystals” made of bubbles, studied by Bragg and Nye[^13], plastic shear occurs as a result of the passage of dislocations along the directions of closest packing. Photograph I (see insert) shows an example of a dislocation in a cluster of soap bubbles.

Fig. 5. Complex dislocation formed by portions of a pure screw and a linear dislocation.
— linear dislocation.
|||||| — screw dislocation.

BASIC PROPERTIES OF DISLOCATIONS

The basic properties of dislocations are their mobility along the slip plane and, as a result of this motion, the relative displacement of the parts of the crystal situated on opposite sides of the slip plane in the direction of slip. These properties explain the important significance of dislocations in the theory of plasticity and sharply distinguish dislocations from other types of lattice defects. Thus, for example, although a vacancy is a mobile defect, this lattice defect, in the process of its displacement within the crystal, does not cause an external change in the shape of the latter. It is easy to show qualitatively that a dislocation is most mobile in the slip plane. In Fig. 6 the black circles depict a configuration of a dislocation whose center was initially on the line $AA'$. If the atoms in the upper layer move in the direction of the arrow to positions,

marked by white circles, the interlocking advances by one interatomic distance from \(AA'\) to \(BB'\). Each atom of the upper layer is attracted by the atoms of the lower layer toward the nearest node located vertically above an atom of the lower layer. As a result, when atoms that shift the interlocking from \(AA'\) to \(BB'\) move, the atoms situated to the left of \(AA'\) move under the action of the attractive forces of the lower plane, while the atoms situated to the right move against these forces. In the first approximation these forces balance each other; therefore an infinitesimally small external force is sufficient for the motion of the interlocking. In a higher approximation, balance does not occur, and a small force is necessary for the motion of the interlocking; the difficult problem of calculating this force will be discussed below.

If an external shear stress acts on the slip plane in the crystal in the direction of slip, a force arises which tends to move

Fig. 6. Motion of an interlocking.

the interlocking in such a direction that this leads to a reduction of the stresses in the crystal. This force is easily calculated by the method proposed by Mott and Nabarro\(^{14}\). Consider a crystal (Fig. 1 or 3) in the form of a cube with edge length \(L\), under a shear stress \(\sigma\) acting on the upper and lower faces in the direction of slip. The external forces are equal to \(\sigma L^2\). If the interlocking passes through the entire crystal, then the two halves of the crystal will be displaced by a distance \(\lambda\) relative to one another; in doing so the external forces will perform work equal to \(\lambda\sigma L^2\). If \(F\) is the force per unit length of the interlocking, then the total force is \(FL\), and the work done by this force in moving the interlocking through the crystal will be \(FL^2\). From the equality of these works we obtain the force acting on the interlocking,

\[ F=\lambda\sigma . \tag{1} \]

It is easy to find the displacement of the halves of the crystal and the shear caused by the motion of the interlocking. Consider a crystal of width \(L_1\) in the direction of slip and thickness \(L_2\) perpendicular to the slip planes. When a linear-type interlocking moves through the entire crystal, a displacement \(\lambda\) arises. For motion over a distance \(l\), the corresponding displacement will be \(\dfrac{\lambda l}{L_1}\), and the average shear deformation of the crystal

\[ \gamma=\frac{\lambda l}{L_1L_2}. \tag{2} \]

I. Interlocking in a two-dimensional cluster of bubbles.

II. Transitional boundaries between “crystals” in a cluster of bubbles.

To the article by A. Kh. Kottrell.

III. Boundaries revealed on polished and etched zinc crystals in the positions of the expected transition surfaces. The bending plane and the hexagonal axis lie in the plane of the drawing.

Similarly, if the velocity of motion of the dislocation is

\(v=\dfrac{dl}{dt}\), then the shear velocity of the crystal is

\[ \frac{d\gamma}{dt}=\frac{v\lambda}{L_1L_2}. \tag{3} \]

In many problems we are dealing with crystals containing a large number of dislocations; it is therefore convenient to introduce the quantity \(\rho\)—the dislocation density, as the number of dislocations per unit area of a plane perpendicular to these lines. Consequently, in the present case the number of dislocations will be \(\rho L_1L_2\). If each of them moves on average through a distance \(l\), then the shear deformation is

\[ \gamma=\rho l\lambda, \tag{4} \]

and if the average velocity is \(v\), then the shear velocity is

\[ \frac{d\gamma}{dt}=\rho v\lambda. \tag{5} \]

STRESSES AROUND A DISLOCATION

Each dislocation is a center of internal stresses in the crystal. Thus, in the case of a linear dislocation (see Fig. 2) the lower half-crystal acts on the boundary of division of the upper half, compressing it, while the upper half-crystal correspondingly stretches the lower half. Taylor\(^8\) showed that these stresses can be determined if the dislocation is regarded as a certain discontinuity, which had already been analyzed in the classical theory of elasticity by Timpe, Volterra, and others\(^ {15,16,17}\). In these works an isotropic\(^*\) elastic body of cylindrical form was considered, as shown in Fig. 7. A dislocation is formed by a radial cut \(OA\) followed by displacement of the cut surfaces relative to one another by a distance \(\lambda\) and their reverse joining together; thus, the points \(P\) and \(P'\), initially lying opposite one another in the plane \(AB\), undergo a relative displacement \(\lambda\) in the radial direction. In a continuous medium the stresses at the center become infinite, and therefore it is necessary to assume that inside the cylinder a cylindrical cavity \(H\) of radius

\(^*\) Here the influence of elastic anisotropy on the stresses around a dislocation is not considered. This question has not yet been studied to the extent it deserves, but there are some indications\(^ {11,16}\) that anisotropy changes the stress formulas only slightly.

of order \(\lambda\). It is assumed that both the inner and outer surfaces of the medium are free of stresses and that the outer radius of the cylinder is infinite. The stresses due to the presence of an interlocking can then be determined\(^{11,19,20}\) for points not too close to the center by considering the problem as a plane one, i.e., on the assumption that the interlocking is directed along the \(z\)-axis and that the displacements in this direction are zero.

Then the normal components of the stresses are:

\[ \sigma_{xx}=-\frac{G\lambda}{2\pi(1-\nu)}\,y\,\frac{3x^{2}+y^{2}}{(x^{2}+y^{2})^{2}}, \]

\[ \sigma_{yy}=\frac{G\lambda}{2\pi(1-\nu)}\,y\,\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}, \]

\[ \sigma_{zz}=\nu(\sigma_{xx}+\sigma_{yy}), \tag{6} \]

and the shear stresses are:

\[ \sigma_{xy}=\sigma_{yx}=\frac{G\lambda}{2\pi(1-\nu)}\,x\,\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}, \]

\[ \sigma_{yz}=\sigma_{zy}=\sigma_{xz}=\sigma_{zx}=0. \]

These formulas apply both to Fig. 2 and to Fig. 7; \(G\) is the shear modulus of the material, \(\nu\) is Poisson’s coefficient, and \(\lambda\) is the measure of the interlocking.

Fig. 7. Elastic cylinder with an interlocking.

The analysis for a screw interlocking was given by Burgers\(^{11}\). As is evident from equations (6), the stresses are inversely proportional to the distance from the interlocking; they are equal to zero at infinity and are infinite at the center. However, the region around the center that must be excluded because the stresses become too large for the application of the theory of elasticity is very small. Thus Keller\(^{19}\) excludes a cylindrical cavity of radius \(r_0\), determined from the condition that the maximum deformation on the surface of the cavity should not exceed 0.1, and finds that \(r_0\) reaches only \(6\) Å for an interlocking with measure \(\lambda=2.5\) Å. The indeterminacy of the stresses at the center for \(r_0=0\) is connected with the assumption of continuity of the medium and, of course, is absent for an interlocking in an atomic lattice. The stresses,

which are continuous functions of the coordinates, can represent the actual state in a discrete structure only when they vary slowly in the crystal over distances of the order of interatomic distances. Near the center, the continuous stress function loses its meaning, and therefore here it is necessary to consider the displacements of individual atoms. The atomic displacements at the center of the dislocation are finite, and it is meaningless to speak of a change of stress inside the region bounded by these atoms. Nabarro^18, following Peierls’ method^31, took into account the atomic structure on the slip plane and developed a more complete theory, which made it possible to analyze the structure of the dislocation core. In this theory the equilibrium of atoms in the plane \(P\) is considered (see Fig. 2). These atoms are acted upon by forces from the atoms of the upper half of the crystal, tending to distribute the compression uniformly along the direction of slip; at the same time they are held by forces coming from the atoms of the lower half of the crystal, in particular from atoms lying in the plane \(Q\).

Equilibrium is attained when these forces are equal. The atomic structure of the medium is taken into account by representing the forces caused by the atoms of the plane \(Q\) in the form of a periodic function of position with period \(\lambda\). Then the uncertainty in the stresses at the center disappears, since the factor \((x^2 + y^2)^{3/2}\) in the denominators of equations (6) is replaced by the expression \(\{x^2 + (y \pm \Gamma)^2\}^{3/2}\), where the positive sign is taken for the upper \((y > 0)\), and the negative sign \((y < 0)\) for the lower half of the crystal, and \(\Gamma = \nu \lambda / 2(1 - \nu)^*\).

The stresses caused by the dislocation at infinity decrease as \(\dfrac{1}{r}\), where \(r\) is the distance from the center; however, some quantities that are related to integration over the stress field diverge logarithmically. Thus, let us consider the stress energy associated with the dislocation field. The deformation energy in a volume element is determined by the product of the stress and the strain in this volume, and since the strain is proportional to the stress, the energy must decrease as \(\dfrac{1}{r^2}\). Integrating in polar coordinates over the whole field, we obtain:

\[ \int_{0}^{2\pi} \int_{r_0}^{\infty} \frac{1}{r^2}\, r\, d\theta, \]

*) This expression differs somewhat from the expression given in Nabarro’s paper.

which gives the total strain energy in the form of the term \(\lg \left( \dfrac{r_\infty}{r_0} \right)\), where \(r_\infty\) and \(r_0\) are the radii of the outer and inner boundaries of the medium. By calculations of this kind, Keller\(^{19}\) obtained an expression for the “self-energy” \(W_s\) per unit length of a dislocation (excluding the energy of the most highly stressed region inside \(r_0\)) in the form

\[ W_s=\frac{G\lambda^2}{4\pi(1-\nu)}\lg\frac{r_\infty}{r_0}. \tag{7} \]

We see that a unit dislocation in an unbounded crystal may have infinite strain energy. On the other hand, in a real crystal it is reasonable to assume that the radius of action of a unit dislocation is limited owing to the presence of other disturbances, and this indeed occurs for a pair of dislocations of opposite sign.

INTERACTION OF DISLOCATIONS

The strain energy of a pair of dislocations in a medium is not equal to the simple sum of their self-energies, even if one assumes that the stress at any point is the sum of the superposed stresses produced by the action of each individual dislocation. If \(\sigma_1\) and \(\sigma_2\) are stresses, then the strain energy has the form

\[ (\sigma_1+\sigma_2)^2=\sigma_1^2+\sigma_2^2+2\sigma_1\sigma_2. \]

Integrating the first two terms gives the value of the two self-energies, but the product \(2\sigma_1\sigma_2\) gives a new energy, depending on the positions of the dislocations relative to one another. This term represents the interaction energy of the dislocations and is due to the forces acting between them. The energy and the forces can be found\(^{19}\) by integrating the strain over the whole field (excluding a cylindrical strip of radius \(r_0\) around each dislocation), but this integration is difficult. The interaction energy can be obtained by a simpler method: from consideration of the work performed against the forces associated with the existing disturbance in forming the second disturbance\(^{22}\).

Consider in Fig. 8 a positive linear dislocation near the origin of the coordinates of the medium and suppose that at the point \(x_0 y_0\) a parallel positive or negative dislocation is introduced. For convenience of calculation this second dislocation may be regarded as formed by a cut parallel to the \(YZ\) plane, extending from \(y=y_0\) to \(y=+\infty\), into which an additional half-plane of atoms is inserted (positi-

tive dislocation) or a half-plane is removed (negative dislocation). In this case the boundaries of the cut region are displaced in the direction of the \(X\)-axis, and then, owing to the presence of the stress field of the first dislocation, the \(X\)-component of the force does work. Work is also done against the normal elastic resistance of the medium, but this constitutes the self-energy of the second dislocation, which is not of interest to us here. Obviously, the interaction energy appears as a result of the work done by the forces of the first dislocation, and we can proceed to calculate it. The work per unit thickness in the \(x\)-direction, performed by the forces acting on the area element \(dy\), in the formation of a negative dislocation, is approximately equal to \(-\sigma_{xx}\lambda\,dy\), and the corresponding change in the energy of the system will be \(\sigma_{xx}\lambda\,dy\). Consequently, the interaction energy \(V\) from equations (6) is

Fig. 8. Work performed in introducing a second dislocation into the field of an already existing dislocation.

\[ \int_{y_0}^{y_\infty} \lambda \sigma_{xx}\,dy = -\frac{G\lambda^2}{2\pi(1-\nu)} \int_{y_0}^{y_\infty} \frac{3x_0^2y+y^3}{(x_0^2+y^2)^2}\,dy = \]

\[ = -\frac{G\lambda^2}{2\pi(1-\nu)} \left[ \lg (x_0^2+y^2)^{\frac12} - \frac{x_0^2}{x_0^2+y^2} \right]_{y_0}^{y_\infty}. \]

If we denote

\[ R^2=x_0^2+y_0^2,\qquad \cos\alpha=x_0/R \]

and put

\[ y_\infty\to\infty \quad\text{and}\quad (x_0^2+y_\infty^2)\to r_\infty^2, \]

then

\[ V= -\frac{G\lambda^2}{2\pi(1-\nu)} \left(\lg r_\infty-\lg R+\cos^2\alpha\right), \tag{8} \]

then we obtain the radial force \(F_R\) and the tangential force \(F_\alpha\) in the form

\[ \left\{ \begin{aligned} F_R&=-\frac{\partial V}{\partial R} =-\frac{G\lambda^2}{2\pi(1-\nu)}\,\frac{1}{R},\\[4pt] F_\alpha&=-\frac{\partial V}{R\,\partial\alpha} =-\frac{G\lambda^2}{2\pi(1-\nu)}\,\frac{\sin 2\alpha}{R}. \end{aligned} \right. \tag{9} \]

Consequently, dislocations of opposite sign attract one another with a force varying inversely proportionally to the distance between them. For dislocations of the same sign, in the above expressions the signs are changed to the opposite ones and the dislocations repel one another. If the self-energies (equation (7)) are added to the interaction energy, then we obtain the total deformation energy of a pair of dislocations near the center of a large cylindrical crystal of radius \(r_\infty\). For dislocations of opposite sign we obtain:

\[ 2W_s+V=\frac{G\lambda^2}{2\pi(1-\nu)} \left(\lg\frac{R}{r_0}-\cos^2\alpha\right). \tag{10} \]

The term \(\lg r_\infty\) is no longer present in the expression for the energy, so that a pair of dislocations of opposite sign has a finite value of the deformation energy. This result leads to the conclusion that at very large distances from the pair of dislocations the superposed stresses \(\sigma_1\) and \(\sigma_2\) mutually cancel and the material remains stressed. The deformation energy is thus localized in the region adjacent to the dislocations.

Fig. 9. Dislocations of opposite signs in one and the same slip plane (after Orowan).

A special case of dislocations of opposite sign in one and the same slip plane (Fig. 9) is important for considering the merging and mutual annihilation of dislocations of opposite sign, as well as the possibility of the formation of dislocations in pairs in the slip plane of a crystal under local shear under the action of shear stress. For such dislocations there is no tangential force, and the force acting between them in the slip plane is \(F_R\), determined by equation (9).

THEORY OF DISLOCATIONS IN A CRYSTAL LATTICE

Table I

Force between dislocations of opposite sign in one and the same slip plane

Distance between dislocations, cm	\(10^{-7}\)	\(10^{-5}\)	\(10^{-3}\)
Force \(F\) between dislocations per unit length, dyn/cm	600	6	0.06
Equivalent shear stress \(\sigma = F/\lambda,\; g\,mm^{-2}\)	\(2.5\cdot 10^{5}\)	\(2.5\cdot 10^{3}\)	25

The data of Table I were obtained for \(G=4\cdot 10^{11}\ \text{dyn}/\text{cm}^{-2}\), \(\nu=0.34\), and \(\lambda=2.5\cdot 10^{-8}\ \text{cm}\). These values correspond to copper. The equivalent shear stress given here is the stress which, when applied to the crystal, produces a force acting on the dislocation equal to the force produced by another dislocation. This equivalent stress is calculated by formula (1). We see that, in the absence of other stresses in the crystal, dislocations approach one another under the action of attractive forces and mutually annihilate, provided that they are at sufficiently small distances for these forces to exceed the force holding them in equilibrium positions between the lattice layers, as will be discussed in the next paragraph. These attractive forces may be balanced by an external shear stress acting in such a direction that the dislocations move apart; this stress is equal and opposite to the equivalent shear stress given in Table I. It is important to note that the equilibrium is unstable. If, for example, an applied stress of the order of \(2.5\cdot 10^{3}\ g\,mm^{-2}\) is applied, the dislocations approach one another provided that they are at a distance of not more than \(10^{-5}\ \text{cm}\), and move apart at larger distances.

In order to create in a crystal, under the action of an external shear stress \(\sigma\), pairs of dislocations, it is necessary to impart to them an activation energy sufficient to remove them at least to the critical distance corresponding to the unstable equilibrium; otherwise the external stress is incapable of keeping them at a distance. Since the critical distance depends on the value of the applied stress, obviously the activation energy also depends on this same quantity. The activation energy cannot be represented by equation (10) alone, since moving the dislocations apart to a distance \(R\) causes displacements of the halves of the crystal and, in doing so, work is performed by the external forces. This work must be subtracted from equation (10). If we consider a crystal of unit thickness in the direction \(Z\) and of width \(L\) along the slip plane, the formation of a pair of disloca-

and their removal to a distance \(R\) cause a displacement \(R\lambda/L\), and the work done is equal to \(\sigma L \cdot R\lambda/L=\sigma R\lambda\). Since, at equilibrium, \(\sigma=-F_R/\lambda\), the work done by the external forces is equal to

\[ \frac{G\lambda^{2}}{2\pi(1-\nu)} \]

and, consequently, subtracting it from equation (10), we obtain the activation energy per unit thickness:

\[ W=\frac{G\lambda^{2}}{2\pi(1-\nu)}\left(\lg\frac{R}{r_0}-2\right). \tag{11} \]

Using the previous numerical values and taking \(r_0=6\cdot10^{-8}\ \text{cm}\), we obtain the results given in Table II.

Table II

Activation energy for the formation of a pair of jogs
under local slip

Applied stress \(\sigma\) (in \(g\ \text{mm}^{-2}\))	25 000	2500	250	25
Critical distance \(R\) (cm)	\(10^{-6}\)	\(10^{-5}\)	\(10^{-4}\)	\(10^{-3}\)
Activation energy, electron-volts per atomic plane	0.77	2.9	5.1	7.3

In calculating these energies we are still excluding from consideration the energy of the highly strained regions within the core of the jog. Although a rigorous calculation of this energy is impossible, it can be estimated approximately in several ways, and in each case a value of the order of one electron-volt per jog per atomic plane is obtained. The deformation at a distance \(r\) from the jog is of the order of \(\lambda/2\pi r(1-\nu)\). Koehler \(^{19}\) excluded a region \(r_0=6\cdot10^{-8}\ \text{cm}\) around the jog for a measure \(\lambda=2.5\cdot10^{-8}\ \text{cm}\); taking the deformation at half this distance as the mean value for the whole excluded region, we obtain an average deformation of about 0.2. Assuming that Hooke’s law is applicable in this region, we obtain an average stress of about \(0.2G\) and a deformation energy:

\[ \frac{1}{2}\times \text{stress}\times \text{deformation}\times \text{volume} =0.02\,G\pi r_0^{2}d, \]

where \(d\) is the interatomic distance along the jog. For \(G=4\cdot10^{11}\), \(r_0=6\cdot10^{-8}\), \(d=2.5\cdot10^{-8}\), the core energy of one jog per atomic plane is about 1.4 electron-volts. A somewhat different estimate was given by Bragg \(^{23}\) and Huntington \(^{24}\). Bragg, using the fact that the deformation energy within the excluded region cannot exceed the latent heat of melting, obtained an energy of about half an electron-volt,

whereas Huntington, calculating the energy due to the short-range electrostatic forces between atoms in the core of a dislocation in a rock-salt crystal, obtained a value of about one electron-volt.

We thus see that the activation energies for the formation of a pair of dislocations given in Table II must be increased by almost two electron-volts. For annealed single crystals of pure metals the shear stress in flow is from 10 to \(50\, g\,\mathrm{mm}^{-2}\); work hardening can raise this value a thousandfold. Thus, in practically all cases the activation energy amounts to several electron-volts for each atomic plane perpendicular to the dislocation. The formation of a pair of dislocations, even very short ones (100 atoms in length), therefore requires several hundred electron-volts. Such large activation energies cannot be supplied by thermal fluctuations, and therefore we must exclude the possibility of the formation of a pair of dislocations by thermal fluctuations in a stressed crystal.

A more detailed theory of dislocation pairs was given by Nabarro\(^{18}\). Koehler\(^{19}\) also considered the possibility of the formation of dislocations on free surfaces. In this case the activation energy also proved to be large (about half of the values quoted above), and therefore its acquisition by thermal fluctuations is no more probable than in the example cited above.

SHEAR STRESS REQUIRED FOR THE MOTION OF A DISLOCATION

In the theory set forth above, the energy of an isolated dislocation (a dislocation or other distortions remote from free surfaces) does not depend on its position; this means that the dislocation proves capable of moving under the action of the smallest external shear stresses. This agrees with the qualitative consideration of dislocation mobility, where it was shown that, to a first approximation, the interatomic forces assisting or opposing the external force mutually balance. However, more accurately the atomic structure determines the dependence of the dislocation energy on its position, i.e. the energy depends on whether or not the plane of symmetry (\(AA'\) in Fig. 6), marking the center of the dislocation, passes through an atomic plane. During the motion of a dislocation this energy must change periodically, so that at least one position of stable equilibrium occurs for each interatomic interval along the line of motion of the dislocation. Consequently, there is a force holding the dislocation in the equilibrium position, and in order for a shift to occur, the external force must пре-

introduce this restraining force. It is clear that it must be very small, since in the first approximation it is equal to zero; because a higher approximation is needed in order to reveal the existence of this force, even a rough calculation of its magnitude requires a delicate analysis. The first attempt to solve this problem was undertaken by Peierls^31; later Nabarro^18 continued his calculations.

In this work it was shown that the most important property of a dislocation is its width, i.e.—for a linear dislocation—the extent of the most highly stressed region along the direction of slip. The width is determined by the ratio of the forces acting on the slip planes to the forces acting in each half of the crystal. Thus, in Fig. 2 the atoms in the plane \(P\) are under the action of two forces:

1. The compressed atoms in the upper half-space of the crystal tend to spread the pressure uniformly along the plane \(P\) and to widen the dislocation.

2. The atoms in the lower half-space of the crystal, and especially the atoms in the adjacent layer \(Q\), tend to draw together the atoms in \(P\) and to narrow the dislocation.

The equilibrium width corresponds to the equilibrium of the forces. According to Peierls and Nabarro, the forces drawing the atoms together are very large and, consequently, the dislocation is extremely narrow; the region in which the displacements reach one quarter of the interplanar spacing is itself not much larger than the interplanar spacing; however, as the authors acknowledge, in their calculations the width may have been underestimated. The drawing-together force is very sensitive to the width of the dislocation and decreases rapidly as the dislocation broadens. This can be seen from consideration of the limiting case in which the distortions are uniformly distributed along the entire direction of slip, i.e. in the case where a row of \(N\) atoms with uniform spacing \(L/(N-1)\) lies opposite a row of \(N+1\) atoms with uniform interatomic spacing \(L/N\). Then, if \(N\) is very large, the interaction energy of the rows will change only slightly when one row moves relative to the other; in the limit \(N=\infty\) the interaction energy becomes constant, and the drawing-together force becomes zero. According to Nabarro’s calculations, the stress required for motion of a dislocation by overcoming the drawing-together force is equal to

\[ \sigma_t \frac{4\pi}{1-\nu}\, e^{-2\pi/(1-\nu)}, \tag{12} \]

where \(\sigma_t\) is the theoretical shear stress for a perfect lattice \((\simeq G/10)\), and \(\nu\) and \(G\) were defined earlier. This gives a critical shear stress of about \(1000\ \mathrm{g\,mm^{-2}}\), which is ten to fifty times greater than the observed values for annealed pure single crystals. It was pointed out, however, that as a result

permitted in calculating the approximations, the width of the dislocation is underestimated by two or three times, and then the critical shear stress is overestimated by 1000 or more times. The difficulties of such an estimate are so great that one can only conclude that the tensile stress is probably less than the observed elastic limits for metallic crystals.

The existence of a tensile force slightly changes the conclusions drawn earlier concerning the mutual attraction of dislocations of different sign on the same slip plane. Obviously, if the dislocations are separated by a distance greater than that at which the tensile force is equal to the interaction force, the dislocations will come together. Nabarro estimated this distance to be of the order of 10,000 atomic spacings; however, it is assumed that this result, depending on the value of the tensile force, is no more accurate than the estimate of the latter. It is also assumed that, since a distance of 10,000 is approximately equal to the width of mosaic blocks in a crystal, their existence may be the cause of the appearance of this characteristic length.

GROUPS OF DISLOCATIONS

In a certain sense a dislocation is a very “economical” lattice defect; dislocations make it possible, in the simplest way and with the least expenditure of energy, to eliminate a mismatch between two halves of a crystal. It is therefore not unreasonable to suppose that most macroscopic irregularities in crystals, such as mosaic structure and the fragmentation of crystals in the cold-worked state, are formed by groups of dislocations. The simplest groups of dislocations to consider are those forming transition surfaces between mosaic blocks or crystallites inclined to one another at small angles; this question was examined in detail by Burgers \(^{11,12}\) and was also discussed by Taylor \(^{25}\), Bragg \(^{26}\), and Leonard-Jones \(^{27}\). Figure 10 shows one of the simplest cases of a transition surface, in which two crystallites are connected by a row of parallel linear dislocations of equal—

Fig. 10. Transition surface between crystallites, formed by a row of parallel linear dislocations.

of the same sign, correctly arranged along the plane \(OYZ\) at distances \(h\). The crystallites are rotated relative to one another about an axis parallel to the \(Z\) axis through an angle equal to \(\operatorname{arctg}\lambda/h\). Burgers showed that in this case the crystallites, at distances from the transition surface that are large in comparison with the magnitude \(h\), are free of stresses. Rotation about the \(Y\) axis is obtained when the dislocations are arranged parallel to this axis, and from consideration of two intersecting rows of linear dislocations in the transition surfaces, parallel respectively to the \(Y\) and \(Z\) axes, one obtains a rotation of the crystallites about any arbitrary axis lying in the \(YZ\) plane. Such rotations, however, are not general, since they do not contain a component about the \(X\) axis. This rotation cannot be produced by any arrangement of linear dislocations in the transition surface; it can be obtained by introducing screw dislocations. Figure 11 shows a simple transition surface formed by screw dislocations between two crystals that are displaced in opposite directions; rotation can be obtained by introducing a second row of screw dislocations intersecting the first row.

Fig. 11. Transition surface formed by screw dislocations.

Thus, it proves geometrically possible to join two crystals, arbitrarily oriented relative to one another, by means of a complex transition surface formed from a number of linear and screw dislocations. The stability of such surfaces has not yet been studied sufficiently fully. Burgers considered only the simplest cases. From Fig. 10 one can see qualitatively that dislocations of the same sign repel one another, and conclude that the transition surface will spontaneously break down when the dislocations are mutually stretched along the slip planes of the joined crystallites. However, this conclusion is incorrect. In deriving the stress formulas (6) and the subsequent formulas for the interaction, it was assumed that in an infinite perfect crystal there is only one or several

to dislocations, whereas the structure in Fig. 10 is a part of two bounded or unbounded crystals joined to one another at an angle. The assumptions made in the previous theory restrict the consideration of the behavior of a dislocation introduced into a lattice in which the atomic “planes” of slip are planes in the strict geometrical sense. In bounded crystals or polycrystals one must consider dislocations in “bent” atomic slip planes.

This is evident from the fact that the stresses determined by equation (6) become equal to zero only at infinity, which is incompatible with the boundary conditions for a bounded crystal free from the action of external forces. This conclusion is confirmed by a qualitative consideration of the influence of a positive linear dislocation in a crystal of finite thickness \(h\)

Fig. 12. Bending of a bounded crystal by a dislocation.

in the \(Y\) direction, as shown in Fig. 12, a. A dislocation located at the origin divides the crystal into two halves relative to the slip plane \(y = 0\); the upper half tends to expand in the direction of the shear, but is restrained by the lower half, which tends to contract it. This is analogous to the situation in a bimetallic strip, and in the absence of external forces the crystal bends, as shown in Fig. 12, b. An equilibrium position will be reached when the tendency of the material to bend near the dislocation (between \(A\) and \(A'\)) is balanced by the tendency of the more remote regions (from \(A\) to \(B\) and from \(A'\) to \(B'\)) to remain undeformed. The angle of bending decreases with increasing \(h\) and in the limit as \(h \to \infty\) becomes zero, which leads to the case discussed earlier. It can be seen qualitatively that \(\sigma_{xx}\) is a tensile stress in the region from \(A\) to \(B\) and a compressive stress from \(A'\) to \(B'\); consequently, if a positive dislocation is introduced in \(AB\), inserting an additional part of an atomic plane into the plane \(YZ\), or in \(A'B'\), removing part of the plane, the stress will do work and the energy of interaction of the dislocations will be negative. Thus, in a bounded crystal free from external actions, disloca-

of the same sign tend to group on transition surfaces.

Experimental confirmation of these propositions was obtained by Cahn[^28]. For some plastically bent crystals, after their annealing, a splitting of the X-ray asterism into separate spots is characteristic. This is explained by the curvature of the slip planes during bending, whereas annealing allows the crystal to rid itself of this bending by the formation of crystallites (polygonization). Figure 13 shows a picture explaining this process by means of the theory of pile-ups. In the first scheme (a) the bending of the lattice is caused by arbitrarily distributed pile-ups of one sign, indicated by crosses. During annealing these pile-ups group into layers, causing the crystallites to be inclined to one another at small angles, as is shown

Fig. 13. a — bending of a crystal by arbitrarily distributed pile-ups of the same sign; b — formation of crystallites when pile-ups collect into layers.

in the second scheme (Fig. 13, b). If this is correct, then one may expect to obtain microscopic confirmation of the existence of transition surfaces, which must be situated approximately perpendicular to the plane and direction of slip.

Such confirmation was obtained by Cahn and is shown in photograph III (see inset, p. 184). It presents microphotographs of a zinc crystal which, after bending and annealing, was cut, etched, and studied microscopically. With this treatment, separation lines are revealed whose orientation with respect to the slip planes and the axis of bending coincides with the positions expected for transition surfaces.

Proof of the existence of transition surfaces composed of pile-ups was also obtained by Bragg and Nye[^13], who showed that in two-dimensional soap “crystals” boundaries often appear corresponding to such surfaces, as is seen in photograph II (see inset).

SEGREGATION OF DISSOLVED ATOMS AROUND DISLOCATIONS

Owing to the presence of a stress field, dislocations interact with other defects that are sources of internal stresses in a crystal, such as precipitated particles, crystallite boundaries, and other dislocations. The most highly developed questions in the theory of dislocations are connected precisely with this interaction, and thanks to their solution the processes of aging and tempering have been understood. However, one question—namely, the segregation of dissolved atoms around dislocations—is the most fully clarified, and it is useful to consider it first.

As shown in Fig. 2, atoms located above a positive edge dislocation are compressed, whereas below it they are stretched. The deformation energy associated with this distortion can be reduced if the lattice spacing in the upper region is decreased and in the lower region increased; then only deformations will remain, but there will be no stresses. It has recently been shown²² that such a phenomenon occurs during the segregation of dissolved atoms around a dislocation. In substitutional solutions, large solute atoms collect in the expanded region and small atoms in the compressed region*). If an atom in an interstitial solution causes a local expansion of the lattice, it will move into the expanded region, and conversely. However, if the interstitial atoms create nonsymmetric local distortions, then the segregation process may cause a reduction of shear stresses, in addition to a reduction of hydrostatic stresses, owing to the change in volume. In most cases, atoms in substitutional solutions cause symmetric distortions and, consequently, can reduce only hydrostatic stresses. For this case it was shown²² that an atom of the solute of radius \(r_a(1+\varepsilon)\) in a solvent with atomic radius \(r_a\), situated under stress, has an interaction energy in the stress field

\[ V=-\frac{4}{3}\pi \varepsilon r_a^3(\sigma_{xx}+\sigma_{yy}+\sigma_{zz}), \tag{13} \]

where \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{zz}\) are the normal components of the field stress. Using equations (6) and introducing the polar coordinates of Fig. 2,

) This phenomenon, sometimes called uphill diffusion and leading to a redistribution of alloy components toward an increase in the concentration gradient, was discovered by S. T. Konobeevskii and Ya. P. Selisskii in 1932 in an X-ray study of the effect of annealing on cold-worked magnesium alloys with aluminum. An explanation of this phenomenon and a generalization of the theory of diffusion to the case of variable concentration fields and internal stresses were given by S. T. Konobeevskii. (Editor’s note.)*

we obtain the interaction energy with positive linear locking

\[ V=\frac{4}{3}\frac{1+\nu}{1-\nu}\,G\varepsilon r_{\alpha}^{3\lambda}\frac{\sin\theta}{r}, \tag{14} \]

where \(G\), \(\lambda\), and \(\nu\) are quantities defined earlier. Note that \(V\) is positive on the upper side \((0<\theta<\pi)\) of the locking for a large atom of the solute \((\varepsilon>0)\) and negative on the other side, which agrees with the qualitative picture of repulsion of the large atom from the compressed region and attraction into the expanded region. Since \(V\) is a function of \(r\) and \(\theta\), both a radial \(\left(-\dfrac{\partial V}{\partial r}\right)\) and a tangential \(\left(-\dfrac{\partial V}{r\,d\theta}\right)\) force act between the atom and the locking. The magnitude \(V\) is easily investigated. For a solute readily entering into the solution, the value \(\varepsilon=0.05\) is acceptable. Taking the usual values for the constants and \(\theta=\dfrac{\pi}{2}\), we obtain \(V\) equal to \(kT\) at a distance of about \(6\cdot10^{-8}\,\text{cm}\) from the center of the locking. Hence it is evident that considerable local segregation of dissolved atoms may occur and, consequently, a change in the properties of the locking.

If time is allowed for the displacement of atoms, then around an unchanged locking there should form an equilibrium distribution, or atmosphere, of dissolved atoms with a density determined by \(V\). For a rarefied atmosphere this density is

\[ c=c_0 e^{-\frac{V}{kT}}, \tag{15} \]

where \(c_0\) is the mean concentration. However, this formula is not applicable to dense atmospheres; the atmosphere may become saturated when it contains enough dissolved atoms for complete relaxation of the hydrostatic stresses, as a result of which further displacement of dissolved atoms in the direction of the locking ceases. It is important to note that an extremely small amount of solute is required for the formation of the atmosphere. For a metal with the highest attainable density of lockings \((10^{13}\,\text{cm}^{-2}\), see below), only about \(0.1\%\) solute is required in order that there be one dissolved atom for each atomic plane of each locking. In annealed metals the density of lockings must be considerably lower, and therefore in such cases there can be no question of a shortage of dissolved atoms for the lockings, even in so-called pure metals \((99.999\%)\).

This theory can be used to explain the well-known effect of plastic deformation on the rapid and localized decomposition of supersaturated solid solutions in the planes-

in shear networks. The forces attracting dissolved atoms to a locking cause a systematic displacement of atoms toward the common center, and this process must occur considerably faster than the random joining of chaotically moving atoms; therefore decomposition should occur first of all in lockings.

Other expected effects are connected with the tendency to produce plastic deformation through the motion of lockings surrounded by atmospheres. Under the action of external forces a locking begins to move and leaves its atmosphere behind; the energy of the atmosphere therefore increases, and this means that there is a force binding the locking to the atmosphere. If the applied force is insufficient to overcome the binding force, the locking cannot slip away. However, it can move slowly, simultaneously with the process of displacement of the dissolved atoms, whose rate is determined by the rate of diffusion. It can be shown that this slow displacement process is sufficiently fast to cause slow creep; it has been suggested that this process occurs in microcreep^23 and in the blue brittleness of iron^39. Microcreep, studied by Chalmers^30 on tin crystals, is a slow creep occurring at very small stresses. It is characterized by the fact that the initial rate of deformation is proportional to the stress and is equal to zero if the stress is zero, and also by the fact that the rate of deformation decreases as deformation increases, becoming almost equal to zero at a total deformation of about \(10^{-5}\). This is in agreement with the theory set forth. It can be shown^31 that, for small stresses, a linear dependence of the rate of deformation on the stress should be observed. Consequently, if creep is caused by lockings initially present in the specimen, and these lockings move too slowly to create other lockings, then they will gradually be used up and the rate of deformation should decrease with deformation, as is indeed observed.

In order to produce rapid deformation, the lockings must be freed from their atmospheres by applying a force exceeding the force binding them to the atmosphere. An important feature here is that lockings, once freed, need not necessarily be under the action of large external forces, and will then be appreciably accelerated; consequently, rapid deformation under the action of smaller forces will become possible. This resembles the phenomenon of a sharp yield point, which is a characteristic feature of iron and mild steel, and to a somewhat lesser degree of some other materials^32. Some investigations^33, ^34, ^35 have shown that the yield point of iron is associated with the presence of small amounts of carbon in the metal; this permits

to assume that in the present case the atmosphere consists of carbon. This is in agreement with theory, since carbon atoms strongly distort the lattice of \(\alpha\)-iron, the consequence of which is the appearance of a large interaction energy and of the yield point. A recent estimate\(^{36}\) of the force necessary to free a pinning from the carbon atmosphere in \(\alpha\)-iron gives the correct order of magnitude for the yield point. This theory also explains the aging of deformed iron. A freshly deformed specimen of iron does not exhibit a yield point, but after a certain interval of time (depending on the temperature) the yield point returns. This is explained by the migration of carbon atoms to the freed pinnings and the formation of new atmospheres. Nabarro\(^{29}\) showed that the activation energy in the aging of deformed iron is the same as in the diffusion of carbon, and that the time required for this process is of the same order as should be expected in the formation of an atmosphere.

RAPID PINNINGS AND THE FORMATION OF SLIP BANDS

As was indicated above, a free pinning is accelerated under the action of sufficiently large forces to a considerable velocity. The excess energy obtained from the work done by the forces is accumulated in the form of kinetic energy of motion. There is not the slightest doubt that such pinnings produce rapid shear, in which individual slip bands are formed over very short intervals of time. It cannot, however, be assumed that a rapid pinning will accelerate without limit, since two phenomena prevent it from reaching the velocity of sound in the material. First of all, when the atoms in the core of the pinning slide past one another, they move in the periodic force field existing between the planes \(P\) and \(Q\) (see Fig. 2); this must produce oscillatory motion in the direction \(Y\), additional to the translational motion along the \(X\)-axis. The collisions of atoms in the slip plane caused by this motion will lead to the dissipation of energy in the form of elastic waves propagating from the pinning into the rest of the crystal; the amount of energy dissipated at each step increases with increasing velocity of the pinning. It may therefore reach such a value that all the excess energy obtained from the external force is dissipated, as a result of which the pinning will no longer accelerate. This process was briefly considered in the works of Orowan\(^{37}\) and Frenkel and Kontorova\(^{38}\).

The second phenomenon, discovered independently by Frenkel and Kontorova\(^{38}\) and by Frank, is a kind of analogue of the relativistic behavior of fast particles. The complexity of the dynamics of rapid pinnings leads to the necessity of applying a simplified

models of a dislocation for considering this phenomenon. Let us suppose, therefore, that in Fig. 6 the upper row of mobile atoms slides along the lower row of fixed and uniformly distributed atoms, and let \(u_k\) be the displacement in the direction of shear of the \(k\)-th atom of the upper row. The potential energy of this atom is determined by its distance from neighboring atoms in the upper row and by its position in the periodic field created by the lower row. Assuming that the elastic forces between atoms in the upper row obey Hooke’s law, and also that the field of the atoms of the lower row is sinusoidal, the potential energy of the system may be written in the form

\[ U=\sum_k A\left(1-\cos \frac{2\pi u_k}{d}\right)+\frac{1}{2}\alpha\sum (u_{k+1}-u_k)^2; \tag{16} \]

where \(d\) is the interplanar distance along the row, \(\alpha\) is the coefficient of the elastic force between atoms of the upper row, and \(A\) is the amplitude of the periodic field created by the lower row. Then the equation of motion of the \(k\)-th atom is

\[ m\frac{d^2u}{dt^2}=-\frac{\partial U}{\partial u_k}, \tag{17} \]

where \(m\) is the mass of the atom. This equation is solved under certain simplifying conditions, as a result of which we obtain:

\[ u_k=\frac{2d}{\pi}\operatorname{arctg}\left\{C_k\exp\left(\pm\frac{2\pi}{d}\sqrt{\left(\frac{-A}{m'}\right)}\,t\right)\right\}, \tag{18} \]

where \(C_k\) is a constant, \(m'=m-\alpha d^3/v^2\), and \(v\) is the dislocation velocity. When \(m'>0\), the solution corresponds to oscillatory motion, in which the atoms oscillate about fixed positions. However, when \(m'<0\), i.e. \(v<d\sqrt{\alpha/m}\), the displacement changes monotonically from \(u_k=0\) to \(u_k=d\) (or conversely) as the time changes from \(-\infty\) to \(+\infty\); this corresponds to the propagation of slip along the row. In the limiting case, when \(m'=0\), \(v\) assumes the value \(v_0=d\sqrt{\alpha/m}\), equal to the speed of sound or to the velocity of propagation of longitudinal waves along the row. The motion of the dislocation is determined by the condition \(v<d\sqrt{\alpha/m}\), from which it follows that \(v<v_0\), i.e. the velocity of propagation of the shear is always less than the speed of sound. By further development of the theory it can be shown that the total energy of the system is

\[ W=\frac{4d^2\alpha}{\pi}\sqrt{\frac{A}{m\left(v_0^2-v^2\right)}}. \tag{19} \]

This energy becomes infinite when the velocity approaches the speed of sound.

Frank \(^{40}\) recently showed that fast dislocations are capable of using their high kinetic energies to create new dislocations, and considered a possible mechanism of this process. Let us consider the emergence of a dislocation to the free surface of a crystal. When a dislocation emerges from the crystal, the atoms sliding one after another at the end of the slip plane tend to jump off and create a dislocation of the opposite sign. When the new dislocation is formed, it begins to move under the action of the external force along the slip plane into the crystal, thus continuing the sliding process begun by the first dislocation. The course of the process depends on whether there is sufficient energy in the required ordered form for the formation of a new dislocation. One of the sources of energy is the stress field of the first dislocation, but this energy alone is still insufficient, since some part of it is scattered in the form of elastic waves when the first dislocation reaches the surface, and is also expended in increasing the surface energy of the crystal when a step is formed on the surface at the end of the slip plane.

Consequently, the jump process probably cannot take place for a slow dislocation; however, a fast dislocation approaching the surface possesses a large and ordered kinetic energy which, together with the deformation energy, can create an excess of energy sufficient for the formation of a new dislocation. This second dislocation, in turn, becomes a fast dislocation and causes the formation of a third dislocation on the opposite surface of the crystal. Thus, rapid shear in the slip plane apparently continuously sustains itself, and this may explain the localization of rapid and intense shear in the narrow slip bands observed in metallic crystals. From the same considerations Frank concluded that if two fast linear or screw dislocations of opposite sign collide in the slip plane, they must pass through one another or be reflected from one another; these two processes, of course, are indistinguishable. This increases the likelihood of a mechanism by which dislocations can multiply within the crystal, as shown in Fig. 14. In the diagram of Fig. 14a it is assumed that a fast dislocation approaches an obstacle in the slip plane coinciding with the plane of the drawing. If the dislocation is stopped or slowed down in the immediate vicinity of the obstacle (diagram b), a loop is formed. If the remaining part of the dislocation moves very rapidly, then a jump occurs when the loop closes, and a region of the slip plane is formed with double the amount of shear (diagram c). This region then expands, and the process of shear multiplication continues.

The question of the cessation of rapid shear in the slip plane is considered in Frank’s theory from the point of view of the formation, in this plane, of a high density of dislocations owing to the multiplication process. Arguments are given in favor of the fact that the final structure formed in the slip plane consists of a crossed network made up of two intersecting rows of parallel screw dislocations. When the mesh size becomes very small, the interaction of dislocations is the cause of “capture,” preventing further shear in the slip plane. This question is discussed in the following section.

Fig. 14. Multiplication of a fast dislocation upon meeting an obstacle. The numbers 0, 1, 2 in this diagram indicate the magnitude of the shear in different regions of the slip plane.

Finally, it may be pointed out that the crossed network of screw dislocations forms a transition surface between crystals rotated relative to one another about an axis perpendicular to the transition surface. Heidenreich and Shockley\(^ {41}\) consider that they experimentally observed such a rotation of crystal halves separated by a slip band.

STRUCTURAL HARDENING DURING DEFORMATION

As Mott and Nabarro\(^ {14}\) emphasize, it is necessary to consider two different mechanisms of hardening during deformation: structural hardening and hardening as a result of exhaustion. Structural hardening is a well-known type

hardening during cold working; hardening as a result of exhaustion will be considered later.

The increase in resistance to shear in a crystal with increasing prior plastic deformation means that either it becomes more difficult to form dislocations in the crystal, or it becomes more difficult to move them. It is usually assumed that hardening is connected with the difficulty of motion of dislocations; experimental facts confirm this^14, since the internal stresses that arise, as is known, in deformed crystals must promote the formation of dislocations and impede their motion. The starting point for most modern theories of structural hardening is Taylor’s assertion^2 that the yield-point stress is determined by internal stresses that impede the motion of dislocations; these stresses increase as the deformation grows. Limited shear occurring in a given slip plane means that the slip process leaves behind a residual change in the structure of the plane, which is the source of internal stresses impeding further shear. On the other hand, a single dislocation passing through a crystal leaves no such residual change; according to Taylor, we thus arrive at the conclusion that, with the exception of initial dislocations, which are sources of avalanche formation of dislocations, the latter do not pass completely through the crystal, but become stuck inside the crystal. It is precisely the sticking of a dislocation that is the source of stresses impeding the motion of other dislocations.

The simplest arrangement leading to the sticking of dislocations is shown in Fig. 15. Here two dislocations of opposite signs in parallel slip planes try, under the action of an external force, to pass by one another; the cross denotes a positive dislocation moving to the right, and the circle a negative dislocation. From equation (9), the component of the force in the slip plane, referred to unit distance between dislocations, is equal to:

\[ F_x=-\frac{\partial V}{\partial x} = \frac{G\lambda^2}{2\pi(1-\nu)}\frac{h^2x-x^3}{(x^2+h^2)^2}, \tag{20} \]

where \(x\) and \(h\) are the quantities indicated in Fig. 15. The dependence of \(F_x\) on \(x/h\) is given in Fig. 16. We note that for small \(x\) the dislocations repel one another along the slip planes and that the point \(x=0\) is a point of unstable equilibrium for the force component \(F_x\); when the dislocations are in this position, the strong attraction between dislocations of opposite sign is determined by the component \(F_y\), and not by \(F_x\). For large values of \(x\)

\(F_x\) changes sign and the dislocations attract one another along the slip planes. It must be remembered that the interaction energy of dislocations shown in Fig. 16 has been calculated only approximately; Koehler\(^{19}\), using various approximations, obtained a curve qualitatively, but not quantitatively, similar.

Fig. 15. Formation of a pair of dislocations.

Fig. 16. Force acting in the direction of sliding between the dislocations of Fig. 15.

In order to move the dislocations apart, it is evidently necessary to exceed the maximum value of \(F_x\), equal to \(\dfrac{G\lambda^2}{8\pi(1-\nu)h}\), which is equivalent to a shear stress

\[ \sigma=\frac{G\lambda}{8\pi(1-\nu)h}. \tag{21} \]

Using the previous values of the constants, we obtain the numerical values given in Table III.

Table III

Shear stresses required to move dislocations apart in adjacent slip planes

Distances between planes \(h\), cm	\(10^{-7}\)	\(10^{-6}\)	\(10^{-5}\)	\(10^{-4}\)
Shear stress, g/mm	60 000	6000	600	60

Consequently, we arrive at the conclusion that the density of dislocations in the cold-worked state should be about \(10^{13}\ \mathrm{cm}^{-2}\). On the same basis the density in a fully annealed crystal can scarcely exceed \(10^8\ \mathrm{cm}^{-2}\); otherwise the yield point would be greater than the observed one.

The justification of a pinning density of \(10^{12}\ \mathrm{cm}^{-2}\) for cold working is very important; it is therefore essential to know how this density agrees with other properties. Thus, some investigators\(^{19,42,43}\) compared the energy of a row of pinning points with the energy absorbed during cold working. Taylor and Quinney\(^{44}\) showed that in cold-worked copper an energy of the order of \(43\cdot 10^7\) ergs per \(\mathrm{cm}^3\) is accumulated. If the pinning energy is taken as 5 electron-volts per atomic plane, then in one cubic centimeter of copper, containing \(10^{12}\) lines of pinning points, the energy of pinning will be \(32\cdot 10^7\) ergs, which gives the correct order of magnitude. Brown\(^{30}\), explaining the magnetic hardness of cold-worked ferromagnetic metals by means of a row of stuck pinning points, showed that the experimental data agree with a density of \(10^{12}\ \mathrm{cm}^{-2}\).

Fig. 17. Pressure of pinning points at an obstacle, causing the formation of local stresses and rotation of the lattice.

Returning to the discussion of Fig. 15, we note that partial slip may occur in planes passing through the nodal region, but the pinning points are unable to break through until large local stresses have been created; thus, a distribution of the type shown in Fig. 17 may be formed. In connection with this scheme it is necessary to note three points. First, all pinning points which approach the obstacle from one and the same side have the same sign. Second, as a result of the pressure of the pinning points along the plane, the density of pinning points must increase as the disturbance is approached. The pressure \(P\) acting on the obstacle, produced by a row of \(n\) dislocations of the same sign, the \(n\)-th member of which is fixed by the obstacle, may be calculated in the following way: let the forces acting on the \(i\)-th pinning point from the side of the \(k\)-th pinning point, the external stress \(\sigma\), and the obstacle be \(F_{ik}\), \(\lambda\sigma\), and \(P_i\), respectively. Then \(P=-\Sigma P_i\); if \(i>k\), then \(F_{ik}\) acts in the direction \(\lambda\sigma\), i.e. opposite to \(P_i\), and conversely. The equilibrium condition for the \(i\)-th pinning point under the action of these forces is as follows:

\[ \lambda\sigma+\sum_{k=1}^{k=n}F_{ik}+P_i=0. \]

If we sum this expression over all dislocations and note that

\[ \sum_{i=1}^{i=n}\sum_{k=1}^{k=n} F_{ik}=0, \qquad \text{since } F_{ik}=-F_{ki}, \]

then we obtain:

\[ P=n\lambda z . \tag{22} \]

The pressure produced by the dislocations gives rise to a large local stress near the obstacle, exceeding the external stress by a factor of \(n\). Thirdly, we shall point out that dislocations tend to rotate the lattice, as shown in Fig. 17. Burgers\(^{12}\) suggested that this may explain the formation of crystalline fragments and the rotation of the lattice, which according to X-ray studies is characteristic of the state of cold working.

From the arguments given above it follows that, in the process of deformation, the density of dislocations gradually increases, giving rise to large interaction forces and resistance to flow. In order to give a definite model of this phenomenon, suitable for qualitative interpretation, Taylor\(^{2}\) considered an ideal hardened structure containing a regular row of dislocations, one form of which is shown in Fig. 18. The properties of such a row of dislocations were subsequently analyzed by Koehler\(^{19}\). Taylor proposed that the ratio \(a/b\) of the distances between rows remains constant during deformation, and was able to explain the formation of parabolic stress–strain curves observed in metals with a cubic crystal structure. From equation (21) it follows that the stress required for the displacement of the dislocation planes has the form

Planes of slip

Fig. 18. Parallel rows of dislocations (after Taylor).

\[ \sigma=\frac{K}{a}, \]

where

\[ K=\frac{Gb k}{8\pi(1-\nu)} \]

and \(k\) is a constant, the exact value of which depends on the type of dislocation row; since the interaction between

nearest neighbors is the principal one, then \(k\) must be of order unity. In calculating the deformation one must remember that \(\gamma=\rho l\lambda\), where \(\rho=\dfrac{1}{ab}\) is the density of pinning points and \(l\) is the mean distance traversed by a pinning point. If \(L\) is the greatest distance traversed by a pinning point before being stopped, then \(l=\dfrac{L}{2}\), and the shear strain

\[ \gamma=\frac{\lambda L}{2ab}. \]

Taking \(a=b\) and combining the expressions for \(\sigma\) and \(\gamma\), we obtain:

\[ \sigma^3=\frac{2K^2}{\lambda L}\,\gamma; \tag{23} \]

this also gives a parabolic stress–strain curve.

The presence of the length \(L\) in this expression seems somewhat strange. One may suppose that \(L\) is the width of the crystal, but this cannot be correct, since it would mean that the stress–strain curve depends on the size of the crystal, which contradicts experimental facts. The order of magnitude of \(L\) can be determined from the relation \(\gamma=\dfrac{\rho l\lambda}{2}\), if we take \(\gamma=1\), \(\rho=10^{12}\); then \(L\) is about \(10^{-4}\) cm. The agreement of the values of \(L\) obtained for some metals with the above magnitude led Taylor \(^{25}\) to the conclusion that the cause of the braking of pinning points is the boundaries between mosaic blocks, whose size, as is known, is about \(10^{-4}\) cm. The nonuniform distribution of pinning points in the material must strongly affect the value of \(L\). Seitz and Read \(^{42}\) pointed out that the plastic deformation of a crystal reduces to slip bands, and if equation (23) is applied to the material inside the bands, then \(L\), in order to correspond to the large local values of the stress, must be increased considerably. Orowan \(^{45}\) cast doubt on Taylor’s conclusion concerning \(L\); he pointed out that the observed similarity of the plastic properties of rock-salt crystals possessing different degrees of perfection, as shown by X-ray investigations, does not support an explanation in terms of mosaic-block boundaries. Orowan suggested that obstacles may be junctions of pinning points of the type shown in Figs. 15 and 17.

Another point casting doubt on Taylor’s theory is that hexagonal metals obey a linear hardening law, instead of the parabolic law of equation (23). To obtain a linear dependence between \(\sigma\) and \(\gamma\), the distance between pinning points in the direction of slip must be constant, which is difficult to understand. As Andrade and Roscoe \(^{46}\) have shown, it is possible that all metals obey a linear hardening law. For hexagonal metals it is characteristic that the number of bands

slip is constant during plastic flow, whereas in aluminum slip bands are formed continuously, the spacing between them being approximately inversely proportional to the stress. If one assumes that within the slip bands the basic linear law of hardening holds, it is easy to show that this will lead to a linear stress–strain curve for hexagonal metals and to a parabolic curve for aluminum.

A somewhat different picture of hardening during deformation was introduced by Burgers^47 and developed by Kochendörfer^48 and Laurent^49. It is assumed that locks are formed by the combined action of stress and thermal motion at defect sites that cause local increases of stress. The most probable sites are taken to be crystal boundaries. The locks created at these sites propagate through the crystal until they encounter obstacles, where they accumulate and, according to Burgers, cause rotation of the crystallites. It is also assumed that, in the aggregate, these locks form a stress field that neutralizes the local stress around defects and makes further formation of locks difficult; the explanation of work hardening is based on this. Kochendörfer developed this idea, assuming that the crystallite boundaries, besides being a source of locks, are also an obstacle for locks, and that only thanks to thermal motion does a lock held at the boundary pass through it and become annihilated. In this way it becomes possible to introduce dynamics into the theory and to discuss the influence of temperature and rate of deformation on the stress–strain curve. Despite the fact that the flow equations derived on the basis of these considerations can be brought into agreement with some experimental facts, the fairly large number of assumptions introduced here makes this theory not entirely convincing.

Fig. 19. Shear in a crystallite (after Bragg).

A third approach to the problem of hardening during deformation was made by Bragg^23, 43, 50, who investigated the question of whether the energy of deformation in a crystal increases or decreases during slip.

Let us consider a square crystal \(ABCD\) with sides \(t\) (Fig. 19), sheared by an amount \(\gamma\) by a stress \(\sigma\); the crystal is fixed in this position \(AB'C'D\) by external forces at \(A\), \(B'\), \(C'\), and \(D\). Further, if slip occurs over an infinitely

a small distance \(\lambda\) along \(PQ\), the configuration of the crystal takes the form outlined by the dotted line. Bragg was interested in the question whether the strain energy increases or decreases after this elementary slip, as compared with the energy before the slip; if the energy is greater, then the slip is unstable and cannot occur. It is assumed that slip occurs if the strain energy is lowered; at the same time it is implied that the large activation energy required for the initial slip will in some way be overcome. From Fig. 19 it is clear that if the shear \(\gamma\) is small, i.e. \(BB' \ll \lambda\), the crystal will be considerably more deformed after slip than before it, and the slip will not be stable. When \(BB' \gg \dfrac{\lambda}{2}\), the strain energy in the crystal will decrease. The condition of stability is therefore the condition \(BB' = \dfrac{\lambda}{2}\), and since \(BB' = \gamma t\) and \(\sigma = G\gamma\), the corresponding critical shear stress is

\[ \sigma = \frac{G\lambda}{2t}. \tag{24} \]

If one tries to compare this condition for a decrease in the stress energy with the condition for slip, then equation (24) will determine the yield-point value. An important quantity is the crystal size \(t\). According to X-ray data \(^{51}\), in difficult-to-work metals with a high melting point \(t\) is about \(10^{-5}\) cm. Taking the previous values for \(G\) and \(\lambda\), we obtain \(\sigma \simeq 5\ \mathrm{kg/mm^2}\), which is somewhat small for a difficult-to-work metal, although it has the correct order of magnitude.

As Bragg showed, in this simple treatment it is neglected that a crystallite located in a matrix formed by other crystallites must adapt itself to its neighbors. This means that in reality the crystallite is not held along \(DA\) and \(B'C'\) by rigid bonds, since it is in an elastic and, consequently, deformable medium. Hence localized slip must create two defects \(P\) and \(Q\) at the ends of the slip plane—these defects are, of course, pinning points. Reconsidering the question from this point of view, we arrive at the conclusion that only the part situated near the boundary restricts the slip region to an interval of distance \(t\) in the direction of slip, and that, with the exception of the defects near \(P\) and \(Q\), the conditions along the boundary of the crystallite are not connected with this question, since, in general, the elastic properties of the medium on each side are the same. In two dimensions the question therefore consists in the forma-

formed in the crystallite; the pair of dislocations composing it move in opposite directions along the slip plane until they are arrested near the boundaries of the crystal. The question of the stability of a pair of dislocations in one and the same slip plane was considered earlier; it was then shown that slip becomes stable when a critical distance separating the dislocations is reached; this distance is determined by the equilibrium of the external force and the force of attraction between the dislocations. From Table I we see that for \(t=10^{-5}\ \mathrm{cm}\) the minimum shear stress for stable slip is \(\sigma=2.5\ \mathrm{kg}\ \mathrm{mm}^{-2}\), which does not differ greatly from the value given by the simple theory*). Here various conditions for the stability of slip due to Bragg\(^{23}\) were used; if it is assumed that slip is unstable so long as the energy at the end of the slip process is less than at the beginning, then the minimum shear stress increases by a factor of about ten.

A difficulty for the theory is the question of how such a large activation energy for the occurrence of slip can be acquired if thermal fluctuations are not sufficiently large. It is necessary to assume that slip begins at weak spots, where there is no need to apply the large energy required to create a dislocation in an undistorted lattice. Bragg suggested that these weak spots may be dislocations held by the boundaries of the crystallites. This question, however, belongs to the general theory of dislocations and will be discussed below.

Bragg also established a further connection between crystallites and dislocations in worked metals\(^{43}\). Assuming that all dislocations formed in the process of cold working are concentrated at the transition surfaces between crystallites and that the crystallite size is \(10^{-5}\ \mathrm{cm}\), it can be shown that for every \(2.5\) atomic spacings along the transition surfaces there is one dislocation. In the two-dimensional model this gives a difference in the orientation of neighboring crystallites of \(30^\circ\), which is the maximum possible for close-packed crystalline structures having 60-degree symmetry. If, consequently, there is a crystallite whose size does not exceed the limiting size of \(10^{-5}\ \mathrm{cm}\), as Wood\(^{51,52}\) supposed, then the upper limits of the dislocation density and of the energy of cold working can be explained, since the transition surfaces are saturated to the limit with dislocations. However, Taylor’s argument, explaining the resist—

*) It should be noted that there is no accidental coincidence here. Both equation (24) and the equation by which Table I was compiled determine the yield as the greatest multiple of \(t\).

strengthening of strain-hardened metals as the result of a closer approach of the pinning points and, leading, independently of the energy consideration, to a value of the density \(10^{12}\ \text{cm}^{-2}\), shows that the pinning points are packed at least quasi-regularly and are not all concentrated on the transition surfaces.

AGING AND ANNEALING OF WORKED METALS

As was shown in the preceding section, a cold-worked metal is in a thermodynamically unstable state, since it contains a large number of pinning points. Thus, a consistent theory must explain the spontaneous changes occurring during annealing after cold working from the point of view of the processes associated with these pinning points. Two such processes are possible—each of them leads to a decrease in the free energy—(a) the accumulation of dissolved atoms around pinning points (this question has already been discussed) and (b) the displacement and dissolution of pinning points. In practice, three types of changes may be observed during annealing of a worked metal: strain aging, recovery (restoration), and recrystallization, although in many cases one or two of the first two may either be absent or overlap with recrystallization.

Strain aging is the name given to an increase in hardness, sometimes caused by annealing at temperatures below the range in which recovery and recrystallization predominate. In iron this process is accompanied by restoration of the yield point and was explained earlier by the segregation of carbon atoms around pinning points. A very similar effect of thermal strengthening, discovered by Orowan\(^{37}\) in zinc crystals, and also observed in cadmium\(^{53}\), can be explained in the same way. Some other changes occurring during low-temperature annealing are different manifestations of one and the same general process. Freshly worked metal exhibits certain “anelastic” properties\(^{54}\), of which the most important are high internal friction\(^{55}\) and low plasticity under stresses lying below the normal yield point\(^{56}\). Zener\(^{54}\) attributed this to the motion of pinning points in the slip planes of the worked material. This explanation is reasonable, since a local redistribution of pinning points can occur under the action of very small stresses, even if large stresses are required to overcome the forces of interaction between neighboring pinning points and to create appreciable plastic flow. Low-temperature annealing greatly reduces internal friction and causes a return to a truly elastic state. An explanation of this phenomenon, from the point of view of the retention of pinning points by atmospheres of dissolved atoms, agrees with the observations

Rhid and Tindall’s\(^ {57}\) high sensitivity to impurities of inelastic effects in zinc crystals.

At annealing temperatures somewhat higher than those required for strain aging, the worked metal is softened in the processes of recovery and recrystallization. Recovery occurs at lower temperatures than recrystallization, causes partial softening, and is not accompanied by visible changes in the microstructure of the material. Recrystallization can cause complete softening and is associated with the nucleation of new, undistorted crystalline grains, which grow at the expense of the surrounding deformed material and by consuming one another. Consequently, recrystallization is completed when the entire material consists of new grains, although the sequence of changes taking place during annealing of a worked metal may be incomplete, since grain growth may continue. From the point of view of pinning theory, it is natural to explain these processes of thermal softening by the motion and dissolution of pinning points. As we have already seen, some justification for such a view is provided in Cahn’s work on polygonization\(^ {58}\). The great sensitivity of thermal softening to the presence of small traces of impurities\(^ {58,59}\) agrees with the concept of pinning, since a small amount of dissolved substance can, through the formation of an atmosphere, substantially reduce the mobility of pinning points.

Thermal softening has been discussed from the standpoint of pinning by many authors\(^ {6,19,42}\) and was considered in greater detail by Burgers\(^ {60}\). Burgers proceeded from a picture of the cold-worked state similar to that proposed by Bragg, i.e., from an aggregate of crystallites, elastically deformed by internal stresses, joined by transition surfaces consisting of pinning points. It is necessary, however, to assume that some pinning points can also exist within the crystallites. In addition to the simple types of transition surfaces considered earlier, Burgers also considers more complex surfaces several atoms wide, in which there exists an arrangement of pinning points as in Fig. 18.

In this case the transition layers contain pinning points of opposite sign, and the difference in orientation between neighboring crystallites inevitably leads to a predominance of pinning points of one sign.

Two softening processes are considered. In the first, associated with recovery, the decrease in the number of pinning points along the boundaries takes place without any displacements of the boundaries as a whole or without any rotation of neighboring crystallites relative to one another. The latter condition leads to the removal of pinning points from the boundary in pairs of opposite sign; Burgers

proposed two possible ways of accomplishing this. If dislocations of opposite sign are located in one and the same slip plane, they can move together and annihilate one another. If they are not located in one plane, then under favorable circumstances they are capable of shifting along the corresponding slip planes into the adjacent crystals under the action of local internal stresses, which may arise as a result of the dissolution and redistribution of nearby dislocations. This mechanism is difficult to understand when the dislocations are in close contact with one another, since then the interaction forces are very large. It is therefore to be expected that dislocations in one plane annihilate one another almost instantaneously, without requiring thermal vibrations for this purpose; dislocations located on neighboring planes form a node in which they are so firmly bound that they cannot be separated by internal stresses. Burgers pointed out, however, that other nearby dislocations can appreciably influence the processes indicated. A possible mechanism for the dissolution of a node follows from the fact that the geometrical insertion or removal of part of the atomic plane connecting two dislocations leads to the correct crystalline structure. If the dislocations are arranged with their compressed sides inward, then part of the plane must be removed, and conversely. Since in a tightly bound node the dislocations are separated by only a few atoms, the required result can perhaps be achieved in a sufficiently short time owing to the transfer of atoms by their displacement into, or out of, the region between the dislocations. Koehler \(^{19}\) considered a process that may bring about recovery as a result of the dissolution of isolated dislocations. He showed that an inevitable consequence of the theory of elasticity is that a dislocation near a free surface must experience an image force*), equivalent to the force arising from the existence of a dislocation of opposite sign at an equal distance on the opposite side of the surface. Under the action of this image force the dislocation may move to the free surface and be annihilated.

Whatever the mechanism of the process of dissolution of pairs of dislocations may be, it is incapable of completely eliminating the effect of cold working, since dissolution must cease when each transition surface consists entirely of dislocations of one sign and not a single dislocation is contained inside the crystallites. Recovery is regarded as the completion of this stage, and further change must proceed as a result of

*) Analogously, in electrostatics the image force characterizes the interaction of an electric charge and a conductor in the form of a half-space. (Ed. note.)

of another process. A characteristic feature of recrystallization is the growth of a crystal; Burgers, taking this into account, invokes the mechanism of the process proposed by Bregman^26, in which the transition surface moves as a whole as a result of the simultaneous motion of all the dislocations composing it along the corresponding slip planes. As a result of this process one crystallite increases at the expense of another, which leads to crystal growth. This can occur when the resulting internal stress, acting in the direction of slip, reaches a certain value. To obtain a systematic displacement of the transition surface, it is necessary that the state of stress in the adjoining crystallites be different; it is rather difficult to see how this condition is maintained during continuous crystal growth. In Fig. 20, according to Burgers, are shown the conditions under which a crystallite can act as a nucleus of recrystallization.

Here the middle crystallite is located at the point of inflection of the bent region of the lattice, and therefore the internal stresses in it are small. Since it is less deformed than its neighbors, it may be expected that it will grow at their expense.

Fig. 20. Crystallites connected in a deformed crystal. The middle crystallite is not stressed and can serve as a nucleus of recrystallization (after Burgers).

Senn, May, and Burgers^61 describe one phenomenon that promotes crystal growth and confirms these ideas.

Analysis of the shape of crystals grown in recrystallized aluminum plates showed that some crystals do not begin to grow until other crystals, already growing, reach their nuclei and stimulate their activity. This phenomenon apparently explains the observation of Anderson and Mehl^63 concerning the increase in the rate of formation of nuclei in recrystallized aluminum with increasing time.

It is very interesting that the stimulating and stimulated crystals are always oriented so that they have a common slip plane (111) and a slip direction [110] in this plane. Burgers supposes that when the growing crystal reaches a crystallite with a parallel arrangement of slip elements, the dislocations retained in the crystallite “discharge” into the interior of the crystal, thereby freeing the crystallite from internal stresses and inducing it to grow, as shown in Fig. 20.

A. H. COTTRELL

FLEXIBLE LOCKING AND DISPERSION HARDENING

It is well known that a freshly quenched alloy in the form of a supersaturated solid solution is soft and that it hardens if it is made to decompose. If aging continues, the size of the precipitated particles becomes larger and disordering occurs (overaging). Thus, the yield strength of an alloy is a function of the dispersion of the particles in the material, as is shown in Fig. 21. In this diagram \(\Lambda_0\) is the mean distance between neighboring particles in the freshly quenched alloy; in this case the particles are individual dissolved atoms, and \(\Lambda_0\) for alloys of typical composition is equal to approximately three interatomic distances. The maximum hardness occurs at the critical size of the dispersion \(\Lambda_{\text{crit}}\), which, as has been shown experimentally\(^{64}\), is approximately from 25 to 50 interatomic distances. In overaging the degree of dispersion is such that the individual particles are detected microscopically, so that \(\Lambda\) in the softening region has a size of the order of 1000 distances and more.

Fig. 21. Change in yield strength during dispersion hardening of an alloy as a function of the distance \(\Lambda\) between particles.

The theory of dispersion hardening on the basis of locking was developed by Mott and Nabarro\(^{65,66,67,14}\). In their first works\(^{65}\) two ideas were put forward which form the basis for the subsequent development of the theory:

1. The inclusion of particles in the solvent matrix is the cause of the appearance of internal stresses. In a solid solution the distortion is determined by the parameter \(\varepsilon\), which is measured by the difference between the atomic radii of the solvent and of the solute, \(r_a\) and \(r_a(1-\varepsilon)\), respectively. On precipitation, the dissolved atoms group together, forming larger particles; however, the degree of distortion remains unchanged, since a spherical region containing \(N\) atomic sites changes its radius from \(R\) to \(R(1+\varepsilon)\), when solvent atoms are replaced in their positions by solute atoms.

2. The cause of hardening is the resistance offered by these internal stresses to the passage of locks. The motion of the locks must encounter such regions of the lattice where the local stress counteracts the external stress. For the motion to continue, the internal stresses must be overcome; it is assumed that the criterion for this is the attainment by the external stress of the mean value of the internal stresses.

Mott and Nabarro showed that this condition for flow determines the order of magnitude of the yield stress in the form

\[ \Gamma = E \varepsilon f, \tag{25} \]

where \(E\) is the elastic tensile modulus for the matrix and \(f\) is the ratio of the total volume of the particles to the volume of the matrix. An important feature of this result is that it predicts here an independence of the yield stress from the distance between particles \(\Lambda\) and a dependence only on the magnitude \(f\)—the total amount of material producing internal stresses. This is due to the invariance of the degree of distortion during decomposition. Calculations based on the theory of elasticity show that the intensity of the internal stresses depends on \(\varepsilon\), but not on \(\Lambda\), so that decomposition increases the wavelength of the stresses, but not their amplitude.

Thus, one may expect that dissolved atoms contribute to the strengthening effect for dispersion states of all kinds in an alloy. However, from other considerations it is assumed that internal stresses produce almost no strengthening effect. If pinning is regarded as a rigid linear discontinuity of length \(L\), where \(L \gg \Lambda\), then along its length there are arbitrarily located \(L/\Lambda\) local fields of internal stresses, some of which assist the motion of the pinning, while others impede it. In the first approximation these alternating forces acting on the pinning cancel out, so that the resultant effect is zero and no strengthening occurs.

The key to resolving the question and overcoming the difficulties is the important idea, subsequently expressed, \(^{63,67}\) that a pinning is not a rigid discontinuity, but is flexible from the macroscopic point of view, i.e., in comparison with dimensions of atomic order; a long pinning may be regarded as a smoothly bent discontinuity lying in the slip plane, its curvature being caused by stress fields. Let us consider an element of length \(l\) of such a pinning in a region where the local stress is \(\sigma_i\). The force acting on it is \(\sigma_i \Lambda l\); this means that, owing to the stress, the element has a potential energy \(V(x)=\int \sigma_i \Lambda dx\), where \(x\) determines the position of the element in the slip direction relative to the local stress field. Consequently, one can specify the potential energy of the pinning at all points of the slip plane; the resulting diagram is a contour map of chaotically distributed peaks and valleys of the potential in the slip plane, corresponding to the local stress fields, as shown in Fig. 22. In this diagram the circles enclose regions in which the pinning has high energy. In the stable configuration these regions bend and za-

the dislocation lies along the valleys, as indicated by the dashed line. If the dislocation were ideally flexible, it would always wind completely around the peaks. In reality this is not so, and, if the peaks and valleys alternate very frequently, the dislocation cannot follow the sharply curved lines and must assume the form shown in Fig. 22, б. This is explained by the fact that, as the number of loops increases, the dislocations lengthen, so that their energy, associated with their stress field, increases. Mott and Nabarro\(^{14}\) showed that the energy of a dislocation is approximately equal to \(Gd^3\) per atomic plane, where \(G\) is the shear modulus and \(d\) is the interatomic distance; this energy is of the order of several electron-volts. As a result, one may imagine that a dislocation has a tension

\[ T = Gd^2, \tag{26} \]

acting along its length and tending to shorten it. Here there is an analogy with a one-dimensional soap film. It may be shown that the form of a flexible dislocation in a field of internal stresses is determined by the expression

Fig. 22. Flexible dislocations in a field of internal stresses. The slip plane corresponds to the plane of the drawing. Dislocations are indicated by dashed lines; circles indicate regions with high potential energy.

\[ \frac{T}{\rho} = \sigma_i d, \tag{27} \]

where \(\rho\) is the local value of the radius of curvature of the dislocation line in the region with internal stress \(\sigma_i\). Eliminating \(T\), we obtain:

\[ \rho = \frac{Gd}{\sigma_i}, \tag{28} \]

so that the radius of curvature, in interatomic distances, is equal to the ratio of the shear modulus to the magnitude of the internal stress.

Let us now compare \(\rho\) with \(\Lambda\), the distance between the centers of the stresses. It is clear that if \(\Lambda > \rho\), the dislocation can follow the outlines of the stress field, and the configuration is similar to that shown in Fig. 22, а. In this case the dislocation forms loops with wavelength and amplitude of order \(\Lambda\), and, since at such a radius of curvature it is quite flexible, each loop acts more or less independently of the others. Consequently, in order to cause slip, each loop must be drawn across the potential

to the apex without the aid of other parts of the pinning. In order to do this, the external stresses must exceed \(\sigma_i\) and the alloy will be hard (the case of overaging, when \(\Lambda \gg \rho\), is discussed below). On the other hand, if \(\Lambda < \rho\), the pinning cannot follow all changes in the stress field and must have the form shown in Fig. 22, \(b\). This is an approximation to the case of rigid pinning, for which the internal stresses are balanced and do not cause hardening. It is clear that this example is analogous to the case of a freshly quenched hard alloy, in which \(\Lambda\) is of the order of several interatomic distances. In order that the curvature of the pinning reach this magnitude, the internal stresses must be of the same order as the shear modulus, whereas in reality the stresses around a dissolved atom cannot appreciably exceed \(\frac{G}{100}\). Thus, we have obtained a basis for understanding the cause of the hardening of an alloy when it changes from the state shown in Fig. 22, \(b\), to the state of Fig. 22, \(a\). This hardening must be completely finished when the particles grow to the size \(\Lambda = \rho\), which is the critical size of dispersity \(\Lambda_{\text{crit}}\). To determine this quantity it is necessary to know \(\sigma_i\). It is known from theory that \(\sigma_i\) must be almost the same as the yield limit \(\Gamma\) for a fully hardened alloy. Experimentally, \(\Gamma\) in this case is about \(\frac{G}{100}\), whence we find that \(\Lambda_{\text{crit}}\) must be of the order of 100 interatomic distances. This is in excellent agreement with Guinier’s experimental data\({}^{64}\).

According to the theory, the yield limit increases proportionally to \(\Lambda^{1/2}\) when the alloy passes from the freshly quenched state to the state of complete hardening. Let us consider a pinning of length \(L \gg \Lambda\). It is divided by the internal stress field into \(\frac{L}{\Lambda}\) elements, in each of which the stress is \(\sigma_i\), while the sign of \(\sigma_i\) changes arbitrarily from one element to another. Consequently, the pinning is subjected to the action of \(\frac{L}{\Lambda}\) disordered positive and negative forces, the magnitude of each of which is \(\sigma_i \lambda \Lambda\). From statistical theory it is well known that the probable mean value of \(n\) disordered positive and negative identical actions is equal to \(n^{1/2}\), multiplied by the magnitude of one action. Thus, the resultant force over a length \(L\) is on the average equal to

\[ \sigma_i \lambda \Lambda \left(\frac{L}{\Lambda}\right)^{1/2}, \]

which corresponds to an average shear stress

\[ \sigma_i \left(\frac{\Lambda}{L}\right)^{1/2}. \]

The resultant internal stress (and, consequently, the yield limit) thus increases as \(\Lambda^{1/2}\).

The mechanism of softening under overaging was considered by Orowan^68, who showed that it can be explained as a necessary consequence of the concept of flexible pinning. Consider in Fig. 23 a dislocation line held by broad regions with high potential energy. Under the action of an external stress the dislocation bends into the regions between obstacles (line \(A\)), and this bending increases as the applied stress increases (\(B\)). Eventually the dislocation breaks away from the obstacles (\(C\)), leaving them surrounded by small dislocation loops. The yield stress is then that which is necessary for pushing a dislocation line through a row of obstacles spaced a distance \(\Lambda\); to achieve this the dislocation must be transformed into loops with a radius of curvature of approximately \(\dfrac{\Lambda}{2}\). The shear stress required to bend the dislocation to such a curvature is given by equation (28) in the form

Fig. 23. Passage of dislocations through widely spaced obstacles (after Orowan).

\[ \Gamma = \frac{2Gd}{\Lambda}, \tag{29} \]

so that in the region of overaging the yield point should decrease inversely proportional to \(\Lambda\).

An additional effect that should be taken into account is due to the pressure of dislocations held up by obstacles. We have already seen that a dislocation which is held in equilibrium with an obstacle, under the action of the pressure of \(n-1\) other dislocations lying behind it, is subject to balancing forces equivalent to a local stress that is \(n\) times the magnitude of the external stress. It may therefore be expected that, when a sufficient number of dislocations is formed in the slip plane, any obstacle can be bypassed at an arbitrarily small external force. This is true only if the slip plane extends without limit behind the obstacle, in which dislocations can be concentrated. In reality, new obstacles arise at distances \(\Lambda\) in the plane, and the pressure exerted on the dislocation near the obstacles depends on how many dislocations can be concentrated within such a distance. With increasing \(\Lambda\), more dislocations accumulate; this means that a greater pressure can act on the dislocation, pushing it into the region between obstacles. Thus, an increase in \(\Lambda\) not only facilitates

facilitates the passage of dislocations through obstacles, but also makes it possible to attain a higher pressure, for a given external stress, for pushing them through.

Mott and Nabarro have recently developed their theory, taking into account thermal fluctuations that promote the formation of dislocation loops around the peaks of the potential energy in the slip plane[^14]. In a completely hardened alloy, a loop of length \(\Lambda\), in the absence of an external stress, overcomes a potential barrier
\(\int \sigma_i(x)\lambda\,dx\), where \(\sigma_i(x)\) is the average stress along the loop when the latter is displaced by a distance \(x\) from the position of the trough. The distance which the loop must travel before reaching the next trough is of order \(\Lambda\); assuming that the internal stress varies sinusoidally in this region, we may take

\[ \sigma_i(x)=\sigma_i\sin\left(\frac{2\pi x}{\Lambda}\right). \]

Integration with respect to \(x\) gives the potential energy of the peak

\[ \sigma_i\lambda\Lambda^2\pi . \tag{30} \]

If we take \(\sigma_i=\dfrac{G}{100}\), \(\Lambda=100\), \(\lambda=d\), and \(Gd^2=5\) eV, then the energy barrier is from 50 to 200 eV. Thus no thermal fluctuations are capable of producing such an activation energy. Quite often, at room temperature, they can provide an energy of only about one electron-volt or less. It is clear that an external stress \(\sigma\), nearly equal to \(\sigma_i\), must be applied in order for the barrier to be reduced to this value, and this would lead to the yield point’s being almost independent of the temperature of the experiment. Although no special experimental check of this assumption has yet been made on alloys hardened by dispersion hardening, the fact that a strong temperature dependence is observed in many annealed metals and single crystals is not discouraging. Mott and Nabarro came to the conclusion that certain additional factors may be operative, for example the temperature dependence of \(\sigma_i\).

Assuming that flow occurs when the external stress \(\sigma\) is nearly equal to \(\sigma_i\), we find the activation energy for the motion of a dislocation loop from one trough to another:

\[ U(\sigma_i)=0.15\,\sigma_i d\Lambda^2\left(1-\frac{\sigma}{\sigma_i}\right)^{3/2}, \tag{31} \]

and, at least for dispersion hardening, this may replace the well-known Becker–Orowan formula for the activation energy:

\[ U(\sigma_i)=\frac{v(\sigma_i-\sigma)^3}{2G_j}, \tag{32} \]

where \(v\) is the volume in which the thermal fluctuation causing slip occurs. Experimentally, the difference between these formulas should be established on the basis of their different dependence on \(\sigma_i\); the available data, however, are not sufficiently accurate for this purpose.

UNSTEADY CREEP

In addition to microcreep, two other principal types of creep are known: unsteady creep and quasiviscous creep. Unsteady creep occurs in a metal loaded beyond its yield point. After the initial instantaneous elongation, the loaded specimen continues to stretch at constant stress; the plastic strain \(\gamma\) increases according to the law:

\[ \gamma = \beta t^{1/3}, \tag{33} \]

where \(t\) is time and \(\beta\) is the creep coefficient. The initial rapid unsteady creep overshadows the other creep processes, but with time it slows down and, in most cases, a second process is observed, occurring simultaneously and obeying the relation

\[ \gamma = kt, \tag{34} \]

where \(k\) is the coefficient of flow. This is quasiviscous creep. These laws were first established by Andrade\(^{69}\) on polycrystalline metals and subsequently confirmed on crystals of zinc\(^{70}\) and aluminum\(^{71}\). A number of experimental facts show that these two flows are entirely different. Thus, quasiviscous creep depends strongly on temperature and disappears at low temperatures, whereas unsteady creep can be very rapid at extremely low temperatures.

Quasiviscous creep is still not fully understood\(^{42,71}\), and it is probable that in different cases the mechanism of this process is different. In polycrystalline metals it is usually caused by motions at the boundaries or near the grains\(^{29}\), while in single crystals it is probably connected with motions in slip planes where the preceding flow has caused a certain disorder in the arrangement of atoms\(^{73}\). Zener and Rhines\(^{43}\), and also Kauzmann\(^{73}\), considered certain possible processes on the basis of interlocks that may lead to quasiviscous creep.

Recently, the interest of researchers has focused on unsteady creep, and a theory has begun to be developed based on two important features of this type of flow. There is rapid crystallographic flow associated with instantaneous plastic deformation occurring under loading and, consequently, almost certainly caused by rapid interlocks. The fact that the initial rate of flow is very large even at the lowest experimental temperatures means that the energy

activation required for the motion of dislocations, which vanish at the beginning of unsteady creep.

The probability that a dislocation loop will jump over a region of unfavorable stress per unit time can be written in the form

\[ \alpha=\nu e^{-U(\sigma_i)/kT}, \tag{35} \]

where \(U(\sigma_i)\) is the activation energy for a jump—given either by equation (31) or (32), and \(\nu\) is the frequency of oscillation of the dislocation in the potential well. Mott and Nabarro\(^{14}\) showed that \(\nu\) is equal to about \(10^8\ \mathrm{sec}^{-1}\). Orowan interpreted unsteady creep from this point of view and substantiated his proposal by the fact that below the region of softening temperatures the character of the plasticity of a metal is determined by the stress–strain curve, which is an expression of structural hardening during deformation, i.e., of the successive increase of the internal stress \(\sigma_i\) under the action of deformation. Thus, in Fig. 24 the application of the stress \(\sigma\) causes an instantaneous deformation \(OA\), and at point \(P\) the external and internal stresses \(\sigma\) and \(\sigma_i\) are equal. At \(0^\circ\mathrm{K}\) no further flow should occur, since beyond this point \(\sigma_i>\sigma\); but at higher temperatures, fluctuations of thermal stresses may overcome the difference \(\Delta\sigma\) between \(\sigma_i\) and \(\sigma\), thus allowing flow to continue at constant stress.

Fig. 24. Explanation of unsteady creep from the stress–strain curve (after Orowan).

At first, since \(\Delta\sigma\) is vanishingly small, very small fluctuations are sufficient for rapid flow to arise even at very low temperatures. This constant deformation leads to an increase in \(\Delta\sigma\), and subsequent fluctuations must overcome an ever-widening gap between \(\sigma_i\) and \(\sigma\). Favorable fluctuations are therefore encountered more rarely, and creep slows down. If we write \(\Delta\sigma=h\gamma\), where the creep strain \(\gamma\) is measured from point \(A\) (Fig. 24), and if we assume that each fluctuation produces the same increment of strain, then the creep rate, as can be shown\(^{71,14}\), will be:

\[ \frac{d\gamma}{dt}=\mathrm{const}\, e^{-vh^2\gamma^2/2GkT}. \tag{36} \]

The expression (36) was obtained using formula (32). Using formula (31), we obtain an analogous expression with $\gamma_i^2$ instead of $\gamma^2$.

Equation (36) gives a flow that starts from the point $P$ and only with a finite velocity; this is in contradiction with the fact that there is a continuous transition from instantaneous spreading to unsteady flow. To overcome this difficulty, Orowan suggested that when $\Delta s$ is small, the motion of one catch brings others into motion, causing a total shear increment proportional to $1/\gamma^3$. Under this assumption he obtained the formula for creep

\[ \frac{d\gamma}{dt}=\frac{\mathrm{const}}{\gamma^2}e^{-\nu h^2\gamma^2/2GkT}. \tag{37} \]

In the initial period of creep, approximately $d\gamma/dt=\mathrm{const}/\gamma^2$; integration of this leads to the formula $\gamma=\beta t^{1/3}$.

In Orowan’s theory it is assumed that at any point of the deformation—stress curve only one value $\sigma_i$ is effective and that structural hardening occurs during unsteady creep, i.e. $\sigma_i$ increases as $\gamma$ increases. Mott and Nabarro^14 and Smith^74 developed a somewhat different picture of the process, in which there is no assumption of the uniqueness of the value of $\sigma_i$. If there is some region of internal stresses, then at some moment certain catches will be in positions where $\sigma_i$ is small and can move easily, while others will encounter large stress barriers and their motion will require larger thermal fluctuations. Weakly bound catches will, obviously, move first and, if they continue their motion until they are stopped in places where $\sigma_i$ is large, creep will gradually decrease as a result of the exhaustion of weakly bound catches.

Consequently, this mechanism leads to a decrease in the rate of creep without requiring an increase of $\sigma_i$, i.e. without structural hardening during deformation, and in accordance with this it has been called exhaustion hardening. If it is assumed that during unsteady creep no structural hardening occurs, then it can be shown that hardening due to exhaustion leads to a deformation—time relation of the type

\[ \gamma=\mathrm{const}\{\lg(\nu t)\}^{x}, \tag{38} \]

where $\nu$ is about $10^8\ \mathrm{sec}^{-1}$ and $x$ varies in different theories from $2/3$ according to Mott and Nabarro to unity according to Smith. The creep curve takes a form similar to the curve $t^{1/3}$, and from it there follows a large initial flow rate without introducing the assumption that,

that the motion of a single dislocation initiates a slip process causing an increment proportional to \(1/\gamma^3\).

It is characteristic of the exhaustion theory that the strain occurring in the first few seconds must be large compared with the subsequent strain. Thus, for \(t = 1\) sec. the factor \(\{\lg(\nu t)\}^{3/2}\) is of the order of 7, whereas for \(t = 10^6\) sec., i.e. about ten days, it is only about 10. Since all the flow occurring in the first second may be regarded as instantaneous, the instantaneous deformation must always exceed the non-steady deformation for any reasonable duration of the experiment. In reality the opposite is usually observed, and consequently the concept of exhaustion ordering cannot be fully accepted for non-steady creep. It is clear that structural hardening must be taken into account as an additional factor, and thus it is possible that a combination of Orowan’s theory with the theory of exhaustion hardening will lead to a more complete picture.

ORIGIN OF DISLOCATIONS

The justification for discussing the question of the origin of dislocations only at the end is that this question is still the most obscure part of the theory of dislocations. The large energy of their formation leads to the conviction that dislocations cannot exist as stable structural details of a crystal in equilibrium and can scarcely be formed by thermal fluctuations, even in a highly stressed but otherwise perfect crystal. The question of the origin of dislocations is therefore reduced to explaining how this large energy can be obtained. The fact that we still have no definite answer is not an objection to the conception of dislocations and the conviction of their existence. The argument that dislocations must arise as a consequence of the crystallographic nature of plastic flow is convincing. If we refuse to accept it, assuming that the slip planes move rigidly relative to one another, we again encounter the same question, only in that case the energy must be still greater.

The explanation reduces to the possibilities of the formation of dislocations either at the ends of cracks, where large stress concentrations exist, or at crystal boundaries \(^{48, 42, 66, 23}\), if these boundaries are regarded as transition surfaces made up of dislocations. The difficulty of the crack hypothesis lies in the extremely great depth of cracks, of the order of \(1\) mm, required to obtain sufficiently intense stress concentrations needed for the formation of dislocations at the observed yield limits of soft single crystals \(^{76}\). Consequently, one should expect that

plastic flow at the base of the crack will distort it and weaken its action in promoting further flow. It has been supposed that the existence of slip bands supports the crack hypothesis more than other hypotheses, since in this case one may expect the formation of a large shear near the crack, whereas in the crystal-boundary hypothesis one may expect that each slip plane creates only a few obstacles and, consequently, gives homogeneous flow without visible traces of slip. However, Frank’s work on the formation of slip bands by rapid obstacles shows that slip bands can arise without the participation of cracks. Recently the crystal-boundary hypothesis has tended to be accepted.

Recently Frank \(^{39}\)*) , analyzing the conditions determining the growth of a crystal around a nucleus from vapor or melt, showed that a nucleus is not capable of growing if it does not contain obstacles. If this is true, then the question of the occurrence of obstacles has, in essence, already been solved, since obstacles, being thermodynamically unstable, must naturally occur as structural details of real crystals. Whether these obstacles are concentrated in transition surfaces or not is the second unresolved question.

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