Surface Phenomena during Plastic Deformation of Metals*)

A. F. Brown

It is now generally accepted that the appearance of slip bands on the surface of a plastically deformed metal is evidence of the inhomogeneous character of the deformation and of its localization on relatively few atomic planes. Recent microscopic investigations have shown that this conclusion is valid only for the later stages of the process, while the initial, small part of the deformation is almost homogeneous. In the present article, in light of the data of microscopic investigations, theories are considered that explain the existence of the indicated stages of deformation. The validity is also discussed of conclusions concerning the internal processes occurring during deformation that have been drawn on the basis of the study of surface changes. It is shown that the state of the surface affects not only the degree to which internal processes are manifested, but also their course.

The available information leads to the conclusion that, in cases where deformation is inhomogeneous, its course is limited: slip along an active plane tends toward a limiting value, which is reached gradually or by a sudden jump, depending on the nature of the metal and the scheme of the stressed state. The processes that arrest slip along active planes also determine the overall hardening of the metal. However, under the influence of temperature changes and stresses, shear along previously existing slip planes, or near them, may begin again as a result of activation of the process; under favorable conditions it will continue until fracture occurs. Therefore slip bands—sources of hardening—are at the same time the weakest places in the metal.

§ 1. Introduction

1. 1. Inhomogeneous plastic deformation. Ewing and Rosenhain[^1], and also Rosenhain[^2], first showed that the dark bands visible on the surface of a plastically deformed crystal are in fact individual steps arising as a result of the shear of atomic layers relative to one another along clearly expressed slip planes. This process is very similar to the mutual displacement of cards in a deck. The thickness of each layer, determined by measuring the distance between steps, is for most metals of the order of several microns. The distance

) From the supplement to the journal Philosophical Magazine: Advances in Physics 1*, 427 (1952). Translated by V. M. Rosenberg under the editorship of B. Ya. Lyubov.

between neighboring steps can be calculated from the macroscopically observed shear and is on the order of several thousand interatomic distances. Apparently, relative displacement of the atomic planes within a layer does not occur. From this one may conclude that slip in metals is inhomogeneous and is entirely concentrated in several atomic planes situated rather far from one another. However, the fact of deformation inhomogeneity has until recently very rarely been used in analyzing the process of deformation of metals. In most works in this direction, technical methods of investigation were applied. In doing so, macroscopic quantities were measured, and on large specimens the relation between stress and strain was determined. On the other hand, in theoretical works certain successes were achieved in explaining the existence of the yield point and strain hardening during deformation. Researchers in this direction used the concept of dislocations*)—imperfections of the crystal lattice on the atomic scale. For the study of phenomena of an intermediate scale—distortions of the crystal lattice arising as a result of plastic deformation—wide possibilities are opened up by the use of X-ray methods. Until recently such works gave values of averaged deformation in regions of the crystal large in comparison with the width of a slip lamella.

It is surprising that, given the state of affairs described, the inhomogeneity of plastic deformation was accepted to such a degree on faith. If it is assumed that a single crystal, under the action of applied stresses, first flows along some especially weak slip plane, then it is necessary to explain why slip does not continue along this plane until fracture, but leads to the involvement of other planes in the deformation process. Usually the explanation of this fact is reduced to invoking the concept of hardening: displacement does not occur by simple sliding of layers of atoms along smooth planes, so that the remaining crystal lattice remains undistorted. The lattice around the active slip plane is somewhat distorted, and therefore ever-increasing stresses are required for the process to continue along the initial plane. However, deformation may continue along one of the planes that has not yet been distorted. These ideas are insufficient for explaining some observed facts concerning the behavior of slip planes. First, slip becomes visible at displacements over distances of about 1000 interatomic spacings. It seems unlikely that such considerable displacements are possible before the slip plane active at a given moment becomes less favorable for the displacement of atoms than another, as yet undistorted, plane. Second, from Yamaguchi’s work^3 it has long been known that slip bands which formed first can increase in size simultaneously with the formation of new ones. If hardening is caused by distortion of the active shear planes, then one may expect that planes on which slip has not occurred possess a more perfect structure and are

) A survey of recent works on the theory of dislocations is given in two articles by Cottrell, translated into Russian^57,90. A theoretical consideration of the deformation of an atomic chain was carried out in the investigations of T. A. Kontorova and Ya. I. Frenkel^91. Criticism of a number of propositions of the theory of dislocations is contained in a review article^92 and in a paper^93. In the latter, an attempt is made to develop new ideas about the mechanism of plastic deformation associated with the loss of stability of the crystalline lattice. (Ed. note*)

planes of easier slip than those along which it has already occurred *).

1.2. Stress–strain curves. The stress–strain curve is the macroscopic reflection of microscopic processes taking place in the crystal. It is possible to find an analytical expression for this curve and even to write an “equation of state” for a plastically deformed body. However, independently of this, one should study slip processes having scales accessible to investigation with a microscope. It is not necessary to describe slip processes by equations of a purely statistical nature. On the other hand, a study of the mechanism of motion of individual dislocations in a stress field caused by other dislocations does not make it possible satisfactorily to explain the appearance of slip bands, which evidently arise as a result of the simultaneous displacement of several hundreds of dislocations.

An illustration of the influence of purely statistical factors in the study of deformation of metals is provided by the data obtained in investigations of creep (deformation with time under constant stress). As Andrade⁴ showed, in this case there is a definite relation between the magnitude of the deformation and the time. This relation proved valid for a large number of different polycrystalline metals. Since Andrade’s work it has been shown that this same relation, with small changes, can be applied to the description of creep at different stresses and temperatures in many substances: metals and nonmetals, crystalline and amorphous. At present it is clear that the mechanism of plastic flow in these materials must be different, but the analytical expression corresponding to the different mechanisms of the process is the same, because it is based on a statistical consideration and describes an averaged process occurring throughout the whole crystal. It is possible that Andrade’s universal relation will prove incorrect when the micro-picture of creep is studied. Only recently have special investigations been carried out in order to clarify the question of the homogeneity of deformation during creep (Mac-Lin⁵, Trotter⁶). Many attempts have been made to reconcile stress–strain curves for plastic deformation with the equation of state.

§ 2. DATA ON THE DEFORMATION OF CRYSTALS OBTAINED BY MICROSCOPIC METHODS

2.1. Study of slip by means of the electron microscope. Interest in the question of the causes of considerable slip in narrow bands increased after the work of Heidenreich and Shockley⁷, who studied slip bands in aluminum single crystals by means of the electron microscope. These authors found that slip bands visible under a light microscope, when examined in the electron microscope, appear as bundles of fine lines. The latter was interpreted as an indication that the steps found by Ewing and Rosenhain are not steep and produced as a result of slip along one plane, but that they are terraces consisting of smaller steps formed by slip along several parallel planes spaced from one another at distances

) A survey of modern views on the nature and mechanism of plastic deformation of metals is given in the book by Ya. S. Umanskii et al., Physical Metallurgy⁹⁴. (Editor’s note.)

on the order of 100 interatomic spacings. The latter may therefore be regarded as the boundaries of elementary slip lamellae. It turned out that each of the lamellae is displaced relative to its neighbors by a distance of the order of 2000 Å. The experiments of Heidenreich and Shockley were subsequently repeated in many countries (Brown \(^{8}\), Nishimura \(^{9}\), Joukowsky \(^{10}\)). Numerous authors arrived at similar conclusions, at least as regards the external features of the observed picture. There is, however, some divergence in the interpretation of the micrographs. For a reason that will be made clear below, these investigations were not extended to a large number of metals; but for the metals studied the same results were obtained as for aluminum (§ 2.6).

The results obtained by means of the electron microscope must be considered with great caution. First of all, it is well known that metals cannot be investigated directly in an electron microscope. Instead, an exact copy of the surface relief of the specimen—a replica—is prepared. The latter must be sufficiently thin for electrons to be able to pass through it. One of the methods of obtaining such a film is to coat the surface with a lacquer and then evaporate the solvent. The result is a film, usually having a thickness of the order of several hundred angstroms, that accurately reflects the relief of the specimen surface. This film is separated from the specimen and is the object of study in the electron microscope. It is obvious that, when a thin film is separated from the specimen, a spurious structure may arise. In particular, lacquer films shrink in an uncontrolled manner upon drying, and therefore the results obtained by measurements made on electron-microscope photographs of replicas are seldom accurate. The resolving power of a lacquer replica is limited by the dimensions of molecular formations in the substance of the film and is much lower than the resolving power of the microscope itself. Some authors present photographs of the finest structures corresponding to a resolution of up to 400 Å.

However, in the case of aluminum one can obtain a replica with a considerably greater resolving power. Such a film is formed by electrolytic oxidation of the specimen surface and probably has a thickness of about 200 Å. It can readily be separated from the specimen without distortion by chemical action on its surface *). In this process the specimen is destroyed. The resolving power of such a replica is about 50 Å, and the interpretation of micrographs obtained in this way is less doubtful than in cases where other methods are used.

In studying the deformation of metals with the aid of the electron microscope, it was found that the main difficulty lies in establishing a connection with results obtained directly by means of the light microscope. The correctness of the reproduction of the surface structure by the replica no longer raises serious doubts. In the recent work of Haim and Nutting \(^{11}\), by comparing photographs obtained from one and the same place on different specimens with optical and electron microscopes, it was shown that, at least up to the resolutions permitted by the light microscope, the electron microscope gives reliable data. In studying deformation, the quantities with which we are concerned are usually much smaller than the resolution of the light microscope; at the same time it is desirable to be able to compare each electron micrograph with the corresponding optical one. However, the region viewed in the electron microscope is small, of the order of sev—

*) Methods permitting replicas with high resolving power to be obtained are described in works \(^{95*}\), \(^{96*}\), \(^{97}\). (Editor’s note.)

several microns. The number of visible slip bands is also small: it is rarely more than ten. The methods used in the case of aluminum, on which most of the work has been done, require destruction of the specimen. Thus, it is impossible to observe individual stages of the process during deformation of one and the same specimen. Moreover, the light microscope does not make it possible to single out a region of the specimen surface that it is expedient to examine under the electron microscope. There is therefore a tendency to compare data obtained with the aid of the electron microscope with results observed optically on other specimens and with facts reported in the literature. Although this is not a very reliable procedure, apparently it cannot be improved in the present state of electron-microscopy technique without a tremendous amount of special work. It would be necessary to check the results obtained in this field over the last 50 years by means of a combined method of light and electron microscopy.

2.2. Study of slip with the aid of an ordinary microscope. The results obtained with the aid of the electron microscope have led to an intensive search in the literature for data relating to slip bands. A particularly complete review was recently made by Cuhlmann[^12]. The principal conclusion that can be drawn from his review is that there is a considerable discrepancy among the results of observations of the same phenomena by different authors. Most of the results were more descriptive than quantitative. And in those cases where measurements were made, the results proved to be irreproducible. The reason for this situation is now becoming clear: first, slip bands are the manifestation on the surface of the mechanism of processes occurring inside the crystal. Second, the crystal is far from the simple conception of a perfect three-dimensional arrangement of atoms that was assumed by early investigators. The processes leading to appreciable slip over a relatively small number of slip planes are complex and are easily disturbed by small changes in the crystal structure and in the test conditions. Finally, we have noted that slip processes taking place under identical conditions in one and the same crystal may look entirely different on surface regions prepared for observation in different ways. In what follows, in order to formulate propositions that have been confirmed in various experiments, we shall indicate how the data of recent works may reconcile the remaining disagreements. Questions for which there is not yet complete clarity remain for further consideration.

First of all, it is generally recognized that in all metals the number of slip bands increases during deformation. At the same time, the magnitude of the shear on each visible band usually increases. The relative rate of these two processes is different for different metals and varies depending on the temperature and rate of deformation, as well as on the conditions of deformation. In some cases one of the phenomena mentioned may occur too rarely and therefore not be observed. For example, Andrade and Houtkins[^13] and Andrade and Roscoe[^14] showed that, for some metals having a hexagonal lattice, as deformation increases the slip bands increase in depth, but their number almost always remains constant. Some observations that contradict this picture can now be explained by confusion between slip bands that grow in depth and the lamellae forming them, which, as will be shown below (§ 2.4), behave differently.

The only point, apart from that considered above, on which there is general agreement is the question of the influence of a change in the deformation temperature on slip bands: the higher the temperature, the farther apart the bands are and the greater the magnitude of the shear on each of them. In the case of single crystals of pure aluminum subjected to shear deformation, for example by 10% at the temperature of liquid air, the slip bands are thin lines located at a distance of the order of \(2 \mu\) from one another. The same deformation at room temperature leads to the appearance of slip bands which, even when examined in an optical microscope, look considerably wider and are located, apparently, at a distance of \(6 \mu\) from one another. If the deformation takes place at \(200^\circ\text{C}\), very wide bands arise, located at a distance of the order of \(20 \mu\) from one another.

2.3. Interpretation of the fine structure of slip bands. Brown \(^{8}\) attempted to explain the phenomena described on the basis of the fine structure of slip bands discovered by Heidenreich and Shockley \(^{7}\). He found that the lamellae which make up the slip bands do not change noticeably as a function of the degree, temperature, or rate of deformation. The main difference when the listed factors are changed is a change in the grouping of the elements of the fine structure within the bands. At very small amounts of deformation, slip occurs by approximately \(2000\) Å along planes located at a distance of about \(10 \mu\). As the deformation is increased, new slip planes become active, and the slip on each of them has the same magnitude. The higher the deformation temperature, the closer the new slip planes are to the old ones. Figs. I—VI *) show how groups of slip planes look under the electron microscope at various temperatures. When examined in an optical microscope, each group of closely spaced slip planes appears as a separate dark band. At very low temperatures the new slip planes are usually located almost midway between the old ones, which under examination in an optical microscope appear as thin, closely spaced bands (Figs. I—II).

A small difference has been found between the magnitude of the shear on an active slip plane at different temperatures. The average values, given by Brown \(^{15}\), are presented below in the table on the left.

Temperature, °C	Average magnitude of shear, Å
\(-180\)	1600
20	2000
250	2200

It is not clear whether the values found for 20 and \(250^\circ\text{C}\) are substantially different; however, it is evident that the distance between elementary slip lamellae, which can be displaced relative to one another, is somewhat smaller at the temperature of liquid air than at high temperatures. These results are confirmed by a very small number of experiments on the deformation of aluminum which were carried out at the temperature of liquid helium \(^{16}\). Apparently, there is indeed a decrease in the slip distance as the deformation temperature is lowered down to absolute zero. Confirmation of this may be the finding by Yamaguchi and Tōjin \(^{17}\), by averaging the macroscopic deformation over all observed slip bands, of an average slip distance which proved to be approximately 25% smaller at the temperature of liquid air than at

*) Figures designated by Roman numerals are placed in the appendix at the end of the article.

room temperature. Further confirmation of the above proposition follows from Holden’s experiments^18, described in § 2.5.

2.4. Step-like deformation. The method described above for obtaining the average magnitude of shear is based on the assumption that displacement along the active slip plane occurs instantaneously and over some definite distance. In addition, it is assumed that displacement cannot occur again on a slip plane that has already acted. Brown^19 pointed out that, so far as the accuracy of observations with the electron microscope permits, these assumptions are confirmed.

Proof of the correctness of the accepted assumptions is provided by the correct magnitude of the steps of elementary shears observed on a large number of specimens. Steps corresponding to one half or less of an elementary shear are not observed (with the exception of cases considered below, which occur under special conditions on the surface of the specimen). In not a single case was the existence of a step considerably larger than the above-mentioned slip distance discovered. Such a conclusion, if correct, is very important for the development of the theory of the formation of slip bands. Therefore it is useful here to consider additional evidence of its validity.

The first clear proof that shear occurs in the slip plane over a definite distance in very short intervals of time was obtained by A. F. Ioffe^20, who, together with M. V. Klassen-Neklyudova, found that shear in heated crystals of rock salt and zinc “occurs by small jumps, each of which is accompanied by a sound resembling the ticking of a clock.” The magnitude of an individual jump proved, for a given temperature, to be constant, but decreased when the deformation temperature was lowered. Later Maddin, Mathewson, and Hibbard^21, ^22 observed that slip lines on the surface of α-brass acquire a definite depth in immeasurably short intervals of time. Below it will be shown that there are grounds for identifying the slip lines observed on brass by the authors just mentioned with the elementary slip lines visible on aluminum and forming slip bands.

The conclusion that displacement along the active slip plane occurs over a definite distance follows from certain other electron-microscopic investigations. Thus, for example, Brown^23 found that in aluminum single crystals deformed at a temperature of 450° C, at a constant rate of 1% per day, all elementary-shear platelets correspond to a displacement that differs noticeably from the usual value of 2000 Å, despite the fact that deformation takes place over two weeks. Fig. XIII is a photograph of the slip bands formed in these experiments, obtained at low magnification. Each of the bands visible in the figure consists of approximately fifty platelets, some of which are shown in the electron micrograph (Fig. XIV). If slip can take place a second time on slip planes that have already acted, then this should occur under conditions corresponding to the possibility of noticeable destruction.

Finally, in Kahn’s experiments^24, which investigated cross-slip, it was found that when slip passes in a jump-like manner from one active slip plane to another, parallel to the first, along a third plane intersecting them, the platelets intersect one another while remaining separated by a distance that can be resolved with a light microscope. This phenomenon is shown in Fig. X. The macroscopic appearance of cross-slip is shown in

Fig. IX. The theory of transverse slip, described by Mott^25, was developed by Frank, who showed that displacement along the slip plane, coinciding with the direction of the shear stresses, takes place in a narrowly limited region; however, there is no basis for asserting that this does not occur in transverse slip. In the latter case the position is determined to a greater extent by random circumstances. Thus, on the planes of transverse slip, shear lines need not necessarily be observed; instead, a broadly diffuse trace may appear. It was found, however, that apparently in ordinary intersection the plates remain as a single whole (Fig. X). This suggests that each elementary shear occurs as a whole and, probably, over short intervals of time.

2.5. Measurement of the slip distance by means of an interferometer. Tolansky and Holden^26 used Tolansky’s method^27 to study the height of slip steps by means of an interference microscope. By this method, by measuring the displacement of the edges of interference fringes with an accuracy of up to 100 Å, it is possible to determine the height of a step and, consequently, the slip distance. A typical appearance of the surface of an aluminum single crystal, deformed by shear by approximately \(1/2\%\), is shown in Fig. XI. The inclined lines are interference fringes. Each displacement of a fringe is caused by the intersection of the polished surface by slip steps (vertical traces), and the magnitude of the displacement is a measure of the height of the step.

Fig. 15. Growth of slip bands during slow deformation (Holden,^18).

The agreement of the values of the heights of the steps corresponding to different slip bands is striking: the average height is approximately half the distance between the edges of an interference fringe, which is equal to \(1/4\) of the wavelength, or about 1300 Å. Hence, knowing the crystallographic orientation of the specimen, one can calculate the slip distance along the active plane in the slip direction, which turns out to be approximately equal to 2000 Å. In such deformations each slip band is formed by only one elementary slip process. Thus, the value of the slip distance obtained from measurement of the step height proves to agree with the values determined by means of the electron microscope. If

loading is carried out at rates that occur in ordinary tensile tests, it turns out that the steps arise suddenly and, subsequently, as the load increases, grow only slightly. The range of values of the slip distances in deformations corresponding to the possible occurrence of an elementary shear step is \(1400 \div 2200\) Å.

Holden \(^{18}\), carrying out experiments at low rates of deformation, investigated the growth of each slip step. The result of a typical experiment of this kind is shown in Fig. 15. An aluminum single crystal was subjected to the action of a shear stress of \(0.81\ \mathrm{kg/mm^2}\) for a time \(T_0\); in this case two slip bands appeared in the field of view. The shear distance corresponding to each of the bands was measured in the time interval up to \(T_2\), after which the bands ceased to grow appreciably. Then the stress was increased to \(0.92\ \mathrm{kg/mm^2}\), which led to the activation of two more slip bands. Finally, at the moment \(T_3\) the stress was increased to \(1.09\ \mathrm{kg/mm^2}\). After this a fifth slip band appeared additionally. Despite the low rate of deformation, the curves shown in the figure show that all the bands appeared suddenly at a magnitude of elementary shear of the order of \(1000\) Å. In this experiment the average shear deformation, determined from ten slip bands, was \(2000\) Å. All measured values lay within the limits \(1800 \div 2200\) Å.

In parallel experiments carried out at the temperature of liquid air, it was impossible to record continuously the height of the steps, but the mean limiting shear distance for ten bands was \(1750\) Å; all values lay between \(1550\) and \(1900\) Å.

In these experiments the height of a step corresponded to only one act of shear of \(2000\) Å per band and to a distance between bands exceeding \(40\ \mu\). While the resolving power of the interference microscope in a direction parallel to the surface is sufficient to continue the experiment up to a state in which the distance between bands is about \(5\ \mu\), its effectiveness as a means of measuring changes in the surface relief decreases owing to distortion of the latter. Fig. XII shows the edge of the interference fringes on the surface of a specimen deformed by approximately \(1\%\). It can be seen that some steps have a height corresponding to a full interference fringe (\(2700\) Å) and, consequently, to a shear distance of the order of \(4000\) Å, whereas most of the others are half as high.

From Figs. XI and XII it is seen that slip does not occur by the pure displacement of atomic layers over one another in the manner described in the introduction. If this were the case, the interference fringes would have to acquire a stepped appearance, as shown in Fig. 16, б, and not have the saw-tooth form shown in the photographs and in the diagram of Fig. 16, в (in Figs. 16, а, б, and в the interference fringes, for clarity, are shown in a direction perpendicular to the slip bands; in Figs. XI and XII they form an angle of about \(45^\circ\) with them). Instead, the zones between the slip bands rotate through small angles about an axis parallel to the slip planes and perpendicular to the axis of the specimen. The cause of the rotation is the influence of the grips of the testing machine at the ends of the specimen. The curvature of the lattice in the region between slip steps clearly indicates the accumulation of an excess number of dislocations of one sign, captured by each slip plane.

2.6. Experiments with metals (except aluminum)

For the reasons set forth in § 2.1, most electron-microscopic

investigations of slip bands was carried out on aluminum. In aluminum the distance between the elementary slip lines forming slip bands is \(200 \div 800\) Å. Thus, with even the most careful use of the optical microscope, its resolving power proves insufficient for revealing the fine structure of the band. There are a few exceptional cases, such as, for example, transverse slip, when the distance between the slip planes increases. However, even under these conditions resolution is still difficult, since when an optical microscope is used at the highest magnifications the depth of focus may be insufficient to make it possible to register simultaneously more than two steps \(1000\) Å high.

Fig. 16. Study of slip with the aid of an interferometer (Holden, \(^{18}\)). a) Interference lines and the crystal before deformation (the slip planes are shown by hatching); b) interference bands and the crystal after deformation by simple displacement along the slip plane; c) interference bands and the crystal after deformation by tension.

Therefore, in aluminum, slip bands and elementary shear lines are clearly distinguished: the former are located at distances exceeding the resolving power of the optical microscope, whereas the latter, on the contrary, are not. If no increase in the size of the bands and elementary shear lines with increasing deformation were observed, then the difference between bands and elementary slip lines would have to appear to a degree depending on the resolving power of the microscope, the latter being determined in turn by the wavelength of the illumination source.

In some metals such a distinction cannot be established. For example, Straumanis \(^{28}\) found that, with a successive increase in the resolution of the optical microscope, in zinc single crystals more and more shear lines become visible, and the distance between them correspondingly decreases. However, no noticeable grouping of these lines into bands occurs. Recent work carried out with the electron microscope on cadmium has shown something similar: investigation of the surface of a deformed cadmium single crystal by means of a new method of preparing replicas, providing high resolution \(^{16}\), revealed shear lines at distances down to 600 Å (Fig. XVII). These lines correspond to steps with heights varying from several hundred to several thousand angstroms. It is significant that the lines cover the entire crystal, and there are no regions that would clearly correspond to an area free from shears between slip bands in aluminum and other metals with a cubic structure. It is well known that, on deformation of a cadmium single crystal, widely spaced bands appear, which are often visible to the naked eye. When a lens is used, the impression of bands separated from one another is lost and the surface appears to be covered with fine lines. With the aid of the electron microscope an increasing number of lines can be detected, and the crystal assumes the appearance described above. Since in the photogra-

...graphs obtained with the aid of an electron microscope, the minimum distance between the lines considerably exceeds the smallest dimensions that can be distinguished with the available resolving power of the instrument; therefore, taking into account the distortions introduced by the process of making replicas, it seems highly probable that the visible lines form the final structure of slip and that no changes of scale smaller than the magnitude of interatomic distances are possible. It may be thought that, in the final result, the slip lines are component parts of slip bands, and their arrangement is characterized by the appearance of maxima and minima of density. The latter circumstance creates the impression of bands visible to the naked eye. However, the field of view of the electron microscope is too small to confirm this supposition unambiguously. In a few cases, when transverse slip was observed in cadmium (Figs. XVIII and XIX), apparently, definite slip elements passed from one plane to another in the same way as occurs in aluminum.

In the experiments of Maddin, Mathewson, and Hibbard\(^{21,22}\), who studied slip in \(\alpha\)-brass, a fine structure very similar to that observed in aluminum was found. In brass this structure could be revealed by means of the light microscope (Figs. VII and VIII). The authors mentioned found that in brass the fine lines form groups, which they called macrolines (Fig. VIII). The distance between the fine lines increased to 1000 Å, but was maintained considerably less well than in aluminum: in particular, shear lines, apparently of elementary size, were found in places isolated from the macrolines. The magnitude of the shear along each of the elementary lines was of the order of 5000 Å; however, this was not maintained so strictly as in the case of aluminum. There are indications that the fine slip lines in brass, although they are arranged irregularly and can be observed in the light microscope, are in fact elementary shear lines, the boundaries of elementary shear plates, and form part of slip bands, exactly as occurs in aluminum. First, they are not subject to further separation in the electron microscope when replicas giving a resolution of 200–300 Å are used. Second, Maddin, Mathewson, and Hibbard showed that the macrolines, or, in other words, the groups of fine lines, increase in size as deformation proceeds by the addition of more and more new fine lines. With increasing deformation the number of groups increases considerably, but the fine lines themselves do not grow after their appearance. Third, transverse slip in \(\alpha\)-brass occurs by the formation of a fine slip line passing from the point of intersection with one slip plane to another in exactly the same way as occurs in aluminum and, apparently, in cadmium. From Fig. VII, which shows strongly pronounced transverse slip, it is evident that this process takes place between groups of shear lines. Fig. VIII illustrates the picture of internal transverse slip, which consists in the passage of shear between two lines belonging to one group.

In Figs. XX and XXI one can see the distribution of slip bands in a single crystal of tin. The experiments, which will be described below (§ 8.5), were carried out in order to compare slip bands in polycrystalline specimens and single crystals. There are still insufficient data for clarifying the question of the correctness of constructing the fine structure of slip bands. Information relating to other metals is considerably less satisfactory. According to Seitz\(^{29}\), the fine structure of slip bands in copper was discovered by Craig and Chizhevsky. Figs. XXVI and XXVII

represent electron micrographs of slip in a copper single crystal: the bands are not resolved. Measurement of the displacement of one system of bands due to intersection with another, after recalculation to the component of displacement normal to the surface of the specimen, gives a value of 1500 Å for the smaller black band in Fig. XXVI and 3000 Å for the larger one.

2.7. Crystals in which the slip bands apparently have no fine structure. From the foregoing it is clear that it is premature to assert that slip bands in all metals have a fine structure. However, on the basis of the facts established for aluminum and brass, it may be concluded that the essential circumstance is not the existence of a distinguishable fine structure, but rather a more or less definite degree of shear on each of the active slip planes. It may therefore be regretted that one of the best demonstrations of the indicated proposition—the stepwise deformation described in § 2.4—has been observed only in crystals of zinc and rock salt. In neither of these crystals has the existence of a fine structure of slip bands been shown. In zinc the Straumanis lines were found at the limiting resolution of the optical microscope and were arranged quite regularly. However, it is still not clear whether they represent the fundamental elementary structure. Much effort has been spent unsuccessfully on resolving the fine structure of slip bands in rock salt.

In § 7 of this article a suggestion is made as to the reasons why some of the elementary processes lead to the formation of groups in slip bands, whereas others create embryos of new bands. None of these processes requires a distinguishable distance between the regions of grouping of slip lines; they may just as successfully be interpreted as successive elementary shear processes in one and the same plane.

If this view is accepted, then the meaning of the fine structure—the irregularities in the observed distance of elementary shear in cadmium and copper—could be explained as a process of successive elementary shears along slip planes situated so close to one another that they cannot be resolved under the microscope.

2.8. Lateral growth of slip bands. It is often observed that at first slip bands are absent, then they appear and rapidly pass across the surface of the crystal; subsequently, as the deformation proceeds, they broaden laterally. Thus, inside the crystal there is a region in which slip is taking place, surrounded by a medium undergoing no changes. As the load increases, the slip region expands until it either covers the entire crystal or combines with other slip acts on neighboring planes. The described character of the change in the dimensions of the slip region agrees with the assumption that slip begins from a definite source (§ 5.2).

Since lateral growth occurs much more slowly than growth in the direction of the applied force, and is more clearly manifested at the surface, it is considerably more accessible to study. Experiments aimed at observing the lateral growth of slip bands by means of cinematography were first carried out by Yamaguchi^30 and recently repeated by Chen^31. These authors, however, did not take into account the influence of differences in the method of preparing the surfaces of the specimens for the experiment. It was observed that, in specimens subjected to pure electrolytic polishing, slip bands that do not cross the entire crystal or do not merge with other bands are encountered very rarely. Below, in § 3.3, it is indicated that

the slip process can easily cease on reaching the surface if the latter has been distorted by prior mechanical polishing. Thus, it is highly probable that in specimens subjected only to electrolytic polishing, the slip process will be delayed beneath the surface and will be able to propagate in the transverse direction before it appears at the surface in the form of a visible slip band.

In electron-microscopic investigations the ends of slip bands are observed extremely rarely. In the few cases where such investigations can be carried out, the surface step does not decrease gradually, but ends abruptly; the distance from the point where the step has normal height to the point where it ceases to be visible is of the order of a micron.

This is in agreement with the point of view set forth in § 7.4, according to which slip appears at the surface only when a sufficient number of dislocations has accumulated beneath it.

§ 3. DOES THE STATE OF THE METAL SURFACE CHARACTERIZE DEFORMATION IN ITS VOLUME?

3.1. Slip processes inside the metal. Above, data were presented on the nature of slip obtained with the aid of a microscope. However, with a microscope one can observe only the surface of a metal, whereas the theory of slip must take into account processes occurring in its volume. Before discussing slip theories and the question of the extent to which they explain the data of microscopic investigations, it is necessary to determine to what degree the directly observed manifestation of processes on the surface characterizes their course inside the metal.

3.2. Influence of surface treatment on the appearance of slip bands on it. The assumption of nonuniformity of plastic deformation is based almost exclusively on experiments carried out by studying the surface. Doubt as to the correctness of conclusions obtained from the study of surface effects during deformation has arisen recently in connection with the investigation by Brown and Honeycombe[^32], in which it was shown that ordinary slip bands, separated by wide intervals, appear after slight deformation only if the metallic surface had been treated with an abrasive before electropolishing. On parts of the section that were not deformed with an abrasive but were polished only electrolytically, slip bands of the usual type were not found. Instead, blurred strokes were found, parallel to the bands in the zones of the specimen treated with an abrasive. The visible distance between the blurred strokes varied depending on the position of the microscope objective, and they were not revealed at all under the electron microscope. These facts agree with the notion that the blurred strokes are not steps on the specimen surface, but protrusions. As deformation proceeds, the sharp slip traces in the zones subjected to abrasive treatment increase in number and in height. What happens to the blurred strokes has not yet been precisely established, but the general appearance of the surface remains unchanged, except that among the blurred bands there also appear several sharper lines parallel to them. The latter are visible under the electron microscope and are similar to slip bands, but correspond to a very small shear distance, of the order of 200 Å (Fig. XXV). With still greater deformation, the appearance of the treated[^32]: Brown and Honeycombe.

and the unworked zones becomes the same, except that in the unworked zones the slip bands are always located against a background of blurred striations. If the specimen is repolished and all traces of slip are thereby removed, then the bands arising in both zones after subsequent deformation have the usual, clearly expressed form.

3.3. Microslip. Fig. XXIII shows the existence of a boundary between the worked and unworked zones in the case of an aluminum single crystal deformed in tension by 5%. On the left side (the worked region) the slip bands are clearly revealed, whereas on the right side (the unworked region) there are several distinctly visible slip bands and a much larger number of faint striations parallel to them. Figs. XXIV and XXV are electron micrographs obtained from surfaces worked and unworked by abrasion, respectively, on another aluminum crystal. Owing to the use of the replica method, the slip bands appear in the photograph as dark lines running in the figures diagonally from the upper left corner to the lower right. In Fig. XXIV the amount of slip is of the order of 5000 Å per band; in Fig. XXV in some cases it is smaller and is approximately 200 Å. The fine slip bands should be called traces of microslip. The structure resembling a groove running almost vertically across Figs. XXIV and XXV is considered in § 6.3.

Figs. XXVI and XXVII were obtained from regions of a copper single crystal, worked and unworked by abrasion, which had been stretched by 5%. Here slip (a dark band) displaces a scratch (a light band). In Fig. XXVI (the worked region) only two slip bands are visible, the amount of slip along them being of the order of 1500 and 3000 Å. In Fig. XXVII (the unworked region), also with a displacement of 3000 Å, there are many very thin traces parallel to the trace of intersection of the scratch. They may be interpreted as traces of microslip, similar to those observed in aluminum.

Provided that slip inside the crystal occurs along a large number of individual planes, these experiments indicate the role of the undistorted surface as a barrier preventing the exit of atomic displacement processes beyond the boundaries of the crystal. The latter can occur only if the surface is distorted by local surface working or by general working of the whole crystal, capable of producing large slip steps. The cause of the influence of the surface as an obstacle to shear might be considered to be an oxide film, if this effect were not observed on deeply etched, unpolished surfaces of an aluminum crystal (§ 5.9), and also on gold, on which the presence of an oxide film is unlikely. Another explanation might amount to the fact that the destruction of a perfect surface (which permits slip to emerge) requires more energy than in the case of a surface already distorted by prior cold deformation.

Another point of view is that the thin slip bands, always observed on specimens not subjected to prior mechanical distortions, indicate the occurrence of slip at early stages of deformation to a small extent, but along a large number of planes. This quasi-uniform deformation ceases suddenly—as a result of surface working—or gradually—with working of the entire specimen. The mechanism explaining microslip will be considered together with the mechanism of ordinary shear in § 5. For the present it is sufficient to emphasize that microslip gives a small effect and constitutes a noticeable fraction of the total shear only at several pro—

centers of deformation. However, the case described is an illustration of the unreliability of judging, from the state of the surface, the processes taking place inside the crystal.

3.4. Methods of investigating the appearance of slip inside a crystal. It is very difficult to indicate an experiment that would make it possible to decide whether the state inside the crystal is revealed on a polished surface or on a surface not subjected to mechanical treatment. There are various experiments showing that, inside a crystal, in the majority of cases, deformation takes place to a considerable extent along isolated planes. Thus, for example, experiments on transparent crystals have been carried out by Obreimov and Shubnikov \(^{33}\), and also recently by Nye \(^{34}\). In these experiments, crystals of rock salt or silver chloride were examined with the aid of crossed nicols. It turned out that, when the specimen is stretched, the field of view is crossed by light narrow bands corresponding to slip bands on the surface of the crystal. This shows that there are large slip bands passing directly through the crystal. From the data of these experiments it cannot be concluded whether there are thin bands between the large bands. It is unlikely that this method can be made sufficiently sensitive for the detection of small slips that should be expected on a large number of planes situated close to one another, since the distance between the planes of microslip is apparently of the order of \(100 \div 1000\) Å. The latter value is much smaller than the resolving power of the method used in the works mentioned.

There are experiments showing that, under certain conditions, slip planes may appear as a result of etching. Jaquet \(^{35}\), McLean \(^{36}\), and also Burke and Barrett \(^{37}\) observed the appearance of striations as a result of the action of special etchants on the surface of a deformed crystal from which the slip bands had been removed by polishing. These investigators were able to prove that the striations mentioned are traces of slip bands, which thus pass right through the specimen.

Hibbard \(^{38}\) expressed doubt as to the correctness of some conclusions from the results of the experiments described. He found that, at least for small deformations, successful revelation of slip bands by etching was associated with the presence on the surface of steps not entirely removed by polishing. The usual method of electropolishing proved sufficient to remove traces of slip visible under vertical illumination, but a slight curvature of the surface still remains, which can be detected under oblique illumination. This curvature leads to preferential etching at the places where slip bands had previously existed. Even if a method of electropolishing were used that completely erased the steps, it is still clear that this method cannot resolve microslips located close to one another. If, however, by means of the method mentioned one tests a specimen with a surface on which there are polished and unpolished regions, then, for deformations corresponding to the appearance only of indistinct striations on regions of the latter type, the polished regions, in contrast to the unpolished ones, do not reveal traces of deformation during polishing and etching. In order reliably to reveal, by etching, relatively fine slip bands corresponding to such small deformations on polished regions, it is necessary to improve the technique \(^{39}\).

The experiments of Blewitt and Koehler \(^{39}\) on slip in ordering alloys give hope of obtaining a definite answer to this question. The authors indicated consider that, if an ordering alloy is subjected

if the deformation is a shear deformation, then when the layers of metal are displaced relative to one another by an odd number of interatomic spacings, disordering*) and a considerable increase in the electrical resistance of the alloy should occur. Measurement of this effect on macroscopic specimens should show whether the displacement of atoms takes place along a few planes or whether a large number of planes is involved in this process. The authors mentioned obtained a result “not contradicting the conclusions of Heidenreich and Shockley,” who found that in the first stages of deformation it proceeds along isolated planes. If deformation occurred on a large number of planes even to an insignificant degree, then approximately half of these micro-shear planes would have to be displaced by an odd number of interatomic spacings, and a considerable increase in electrical resistance would arise, which could be measured. The experiment described appears decisive. However, in analyzing the mechanism of micro-slip (§ 5.8) it will be shown that there is a possibility that the influence of the process considered may not appear in measurements of electrical resistance.

Some experiments by Paxton et al.^40 indicate another way of investigating this question. It was shown that if a grid is applied, by scratching, to the surface of an iron crystal that has been electropolished, and the scratches are then removed by a second polishing, the grid reappears during deformation in the form of a system of traces of coarse shears located between the blurred strokes. This agrees with the data of Brown and Honeycombe. In ^40, however, it was shown that slip passes continuously from micro-shears to clearly expressed bands: each slip band at the edge of a scratch degenerates into a bundle of fine lines. Analysis of the positions of these lines and of a series of closely spaced parallel scratches can give a clear solution to the question of whether coarse shears are the basic form of atomic displacements in the volume of a crystal. If so, then coarse slip bands crossing several scratches may appear. Such a band simply splits, in passing through the region between scratches, into more or less parallel bundles of fine shear lines. These bundles again merge into a band at the next scratch. If, on the other hand, micro-slip represents the true picture of deformation within the crystal, then there is no reason why the lines, on combining, should form coarse shears in the same way at two successive scratches; in this case it is impossible to trace coarse slip bands crossing the specimen. The picture described by these authors shows the correctness of the latter position, but since it is known that in iron slip bands form with anomalies, more careful investigation is necessary. An impediment to the application of the method considered, as shown by various authors and as will be discussed below, is the fact that the surface is not a passive recorder of the processes occurring inside the metal. It in turn affects the course of these processes. In particular, it will be shown (§ 7.7) that near scratches the conditions for the formation of slip bands change. Therefore it is doubtful whether the results of observations of the distortions of artificially produced scratches can be regarded as an argument for or against the homogeneity of shear. The consideration expressed by Seitz^29 is that the applied marks may be new sites at which slip occurs. It deserves attention only on the condition that defor-

*) Some experimental facts suggesting that, during plastic deformation, short-range order also changes are presented in works 98, 99. (Translator’s note.)

mation accompanying the application of the mark can be eliminated without destroying the surface layer by annealing. Repolishing, of course, leads to destruction of the applied grid.

Summarizing the foregoing, we note that as yet there is no definite proof of the quasihomogeneity of shear in the first few thousandths of a percent of deformation, but there are grounds for such an assumption. If this is so, then the data obtained on specimens with an electropolished surface are correct. However, there is no doubt that at higher degrees of deformation, as has hitherto been believed, slip is nonuniform, and the data obtained by observing both a mechanically treated and an electropolished surface reflect the true state of affairs.

Since the question of the nature of microslip has been resolved, it is not difficult to propose an experiment that would make it possible to study its magnitude and distribution. Thus, for example, Kurnosov, Tronina, and Yakutovich^41 showed by the microinterference method that, at early stages of deformation of a zinc crystal, visible slip bands may account for the appearance of only an insignificant fraction of the macroscopically observed effect. In the experiments of Gölden considered in § 2.5, no indications were obtained of any forms of shear other than slip bands. In this case, however, the surface was necessarily leveled before electropolishing. If subsequently it were subjected to annealing and again to electropolishing, then its shape, revealed with the aid of an interferometer, could indicate the existence of some deformation not appearing as visible slip bands*).

§ 4. THEORIES EXPLAINING THE MAGNITUDE OF THE DISTANCE BETWEEN SLIP BANDS

4.1. The contribution of microscopic investigations to the theory of slip. Very many advances in the understanding of the deformation of metals are the result of microscopic investigations. For example, the concept of dislocations was introduced to explain the low value of the yield point and the existence of hardening in metals; the geometry of shear was developed as a result of the use of methods of X-ray crystallography. It follows from what has been said that it is expedient here to give a review only of those questions of the theory that are consequences of microscopic investigations or can be verified with their aid.

4.2. Regularity in the arrangement of slip bands. Even a not very detailed examination of slip bands on deformed metals leads the observer to two conclusions. First, the distances between bands seem identical, and, second, all bands look very similar. The first proposition in metallophysics is almost a dogma. Several attempts have been made to explain the regularity of the arrangement of bands, among which the works of Orowan^42 and Brett^44 should especially be noted; however, this question has as yet been little studied by statistical methods. Many authors have measured the mean distance between slip bands and, on the basis of their results, drawn conclusions without taking into account how the values of the quantity under investigation are distributed around the mean value obtained. Thus, for example, Yamaguchi^3, Koch and Klepsch^44, Croussard^45, Rosie and Mathieson^46, and also many other investigators related the mean distance between

) An interesting method for investigating plastic deformation in the bulk of opaque crystals is described in work 104. (Editor’s note.)

by slip bands with the value of the yield point. Andrade and Hutin¹³, and also Andrade and Roscoe¹⁴, measured the distance between slip bands in cadmium, lead, and mercury at different deformations. These authors found that the mean distance between slip bands is almost independent of the magnitude of the deformation. Moreover, in lead the quantity studied also proved to be independent of temperature and of the rate of deformation.

The reason for the regular arrangement of slip bands was recently discussed by Orowan⁴⁷. He rejected the assumption according to which the distance between slip bands is predetermined in a pure, undeformed crystal by the existence in it of regularly arranged planes of easiest shear. Since the arrangement of slip bands is regular at any degree of deformation, this latter point of view requires that the planes of easiest shear throughout the whole lattice be situated at the same distance from one another, the next in order of smallness of resistance to deformation—at another distance, smaller than the first, and so on.

On the other hand, Andrade and Roscoe¹⁴, on the basis of their experiments with lead, concluded that the appearance of slip bands in those metals in which they were detected at the beginning of deformation was prepared by accidents of growth; the number of slip bands did not increase as deformation proceeded. In the case of metals behaving in this way, according to Orowan, the existence of potential shear planes is impossible. Another possible explanation of the absence of an increase in the number of slip bands is that near each active plane there is an unhardened region of the metal. This circumstance is considered below (§ 7).

In harder metals it seems logical to assume that the distance between slip bands is determined by the interaction of the bands with one another. A large number of dislocations (§ 4.6) are concentrated in slip bands, producing stresses in the surrounding lattice; this impedes slip in the zone situated on both sides of the band. The difficulty that arises in developing this theory is that the region of action of the stresses due to dislocations lying in a plane is too small to explain even the minimal observed distances between slip planes, except at the beginning of deformation (see, however, Mott⁴⁸).

4.3. Distribution of distances between slip bands. In some recent investigations attempts have been made to represent the distances between slip bands in the form of frequency diagrams (histograms). A typical histogram is presented in Fig. 28. In this example the number of experiments, as in many other cases, is insufficient for the accurate construction of frequency diagrams, but nevertheless one may draw the firm conclusion that the diagram shown probably corresponds to a random distribution for large values of the measured quantity, but deviates from it at small distances between slip bands. The diagram breaks off at a certain minimum value of the distance between bands, depending on the macroscopic deformation; it is the smaller, the greater the deformation. The number of observations of distances between bands smaller than is necessary to bring the observed histogram into full agreement with the curve corresponding to complete randomness is very small, of the order of three for Fig. 28, a. However, the fact that small distances between bands are absent suggests that there is a minimum value of this quantity, depending on the magnitude of the deformation. Substant—

such a value can be explained only by the interaction of neighboring bands: in each slip band there are lattice distortions limiting the possibility of shear in a more or less definite region on both sides of it. The average distance between bands is apparently equal to twice the range of action of these distortions.

Barrett[^49] constructed similar histograms for slip bands observed on specimens of copper and Monel metal. In this case the distance between the bands was determined with the aid of an electron microscope. It turned out that, apparently, in copper all distances from 0.3 μ to 2.5 μ are equally probable. None of them was less than 0.3 μ, although the resolving power of the method made it possible to determine quantities ten times smaller. In Monel metal, however, the histogram had a clearly expressed peak corresponding to 0.3 μ; in addition, values of the distance between bands were found that decreased down to the limit of possible resolution. From this it may be concluded that the value 0.3 μ corresponds to the most probable distance between slip lamellae, while most traces of shear in reality consist of bunches of slip. For aluminum, a similar histogram has a noticeable peak at 200 Å, if slip bands and lines are not differentiated. Apparently, the resolving power of Barrett’s method is insufficient for separating the slip bands in copper.

Fig. 28. Histograms of distances between slip bands in aluminum single crystals.
a) Aluminum single crystal, stretched by 7% at room temperature; the average distance of shear at the surface is 8 μ.
b) Change in the distribution of distances between slip bands with increasing deformation. The distance between the slip bands for four stages of deformation of an aluminum single crystal (each histogram is constructed on the basis of measurement of the same total number of slip bands):

I — tension 0.7%, average distance 25 μ;
II — tension 2.5%, average distance 12 μ;
III — after annealing and further tension to 3.8%, average distance 13 μ;
IV — after secondary annealing and further tension to 7%, average distance 9 μ.

4.4. Theories of hardening.

From what has been set out above it may be concluded that the cause of the separation of slip bands is the stresses arising as a result of the concentration of dislocations on the active planes. These stresses impede the passage of shear through the metal. The macroscopic yield point, therefore, increases as the density of the distribution of slip bands increases, and the metal hardens. It follows from this that the principal cause of hardening, whatever its detailed mechanism may be, is the distortion of the ordered structure of the active slip planes.

4.5. Capture of dislocations by slip planes.

The experiments mentioned in § 2.4 indicate a greater resistance to deformation along the active slip plane in the sense that considerable stresses are required for an additional shear process to pass along them. Even prolonged holding at 450° C

insufficient for the planes along which slip has already occurred to become sufficiently disordered for the shear process to resume along them.

From this one may conclude that, after shear has passed along a slip plane, a high density of dislocations arises on the latter. There is microscopic evidence for the existence of such dislocations captured by shear planes. For example, it is well known that the configuration of slip bands appearing on opposite faces of a flat single crystal of a metal with a cubic lattice is rarely even approximately the same. This means that some slip processes do not pass all the way through and, consequently, dislocations are captured within the volume of the crystal. Cahn has shown\(^ {50}\) that dislocations formed during plastic deformation cannot leave the crystal and form on its surface a network known as polygonal boundaries.

The investigations of Warren and Averbach,\(^ {51}\) carried out using X-ray methods, showed that in a strain-hardened metal there are distortions*) including sharply alternating deformations. Data on the distances between rows of dislocations in a strain-hardened metal are contradictory, but apparently the most reliable estimate belongs to Koehler,\(^ {52}\) according to whom, in a maximally strain-hardened metal, approximately one atom out of every thirty is located on a dislocation line.

4.6. Arrangement and distribution of captured dislocations. The data cited above indicate the existence in a strain-hardened metal of a large number of captured dislocations, but give no indication of their distribution (singly or in groups) or of the places in which they are located. However, some microscopic data permit one to think that captured dislocations are situated on slip planes. Newkirk\(^ {34}\), on the basis of a study of transparent crystals, first showed that the “planes” of slip are curved: this indicates an excess of dislocations at one edge of a region located along the slip planes. The same result for metallic crystals was obtained by Holden (§ 2.5). Further, Castaing and Guinier\(^ {53}\) showed that regions of precipitation of a new phase during aging are formed predominantly on previously activated slip planes, just as they arise on polygonal boundaries (Figs. XXIX and XXX). Since polygonal boundaries are undoubtedly formed by rows of dislocations, the same conclusion may be drawn concerning the nature of slip bands.

The distribution of dislocations over slip planes, as will be shown in § 4.7, apparently is not random; for example, on the average there is one dislocation for every 30 atoms—they gather in groups.

The density of the distribution of dislocations calculated by Koehler is attained throughout the entire volume of the metal only after intensive cold rolling. However, on the basis of the data cited above concerning the hardness of the slip planes along which shear has passed, it may apparently be concluded that the spacing between dislocations of the order indicated above occurs on each plane after slip has passed along it.

*) A series of studies on the investigation by X-ray methods of lattice distortions in metals during plastic deformation was carried out by G. V. Kurdjumov and collaborators.\(^ {100*}, ^{101*}, ^{102*}\) (Ed. note.)

4.7. Interaction of slip bands.

If the distortion caused by the shear process, in passing through a crystal, is limited directly to the region near the slip plane, then it is difficult to understand how these planes, interacting with one another, cause hardening of the metal. The formula usually given for the stress \(\tau\) at a distance \(r\) from a dislocation was obtained by Koehler\(^{52}\) and has the form

\[ \tau=\frac{G}{2\pi}\frac{1}{1-\nu}\frac{a}{r}, \]

where \(G\) is the shear modulus, \(a\) is the interatomic distance, and \(\nu\) is Poisson’s ratio.

If one considers a single dislocation on a slip plane, then, as is easily seen from the formula given above, no appreciable displacements of dislocations can take place at a distance \(d\) from the active plane, determined by the formula

\[ d\sim \frac{1}{2\pi}\frac{G}{\tau_a}\frac{a}{1-\nu}, \]

where \(\tau_a\) is the applied shear stress. The ratio \(\frac{G}{\tau_a}\) is of the order of magnitude 1000; hence \(d\) corresponds to approximately 200 interatomic distances. Thus, the formula given can serve to explain the observed spacing between elements of the fine structure in aluminum, but it is not suitable for this purpose under all conditions. Even the maximum deformations at the temperature of liquid air can lead to a spacing between slip bands in this metal at least equal to 1000 interatomic distances. In particular, such an interpretation of the spacing between bands cannot explain the constancy of this magnitude even at the smallest deformations and its preservation throughout the entire process.

The formula given above, however, shows that the number of slip bands per unit length is proportional to the stresses. This same result was obtained experimentally for aluminum by Yamaguchi\(^{3}\), and also by Koch and Klemm\(^{44}\). Kuhlmann\(^{12}\) arrived at the same conclusion by calculation, using certain experimental data obtained on silver by Gough and Koch\(^{54}\).

4.8. Groups of dislocations.

Apparently, better agreement between the conclusions of the theory and the results of experiments can be achieved if, instead of a row of dislocations arranged closely along the slip plane, one considers a series of groups similar to those proposed by Kahn\(^{50}\) as a model of polygonal boundaries. The stress due to a group of \(n\) dislocations should be approximately \(n\) times greater than that given by Koehler’s formula. Thus, groups containing hundreds of dislocations give satisfactory agreement with the observed spacing between slip bands. In this connection it is interesting to note that, in the above-mentioned experiments of Castaing and Guinier (§ 4.6), the precipitation centers on the slip bands were not arranged continuously along the band, but were separated from one another by approximately 1000 Å. If, indeed, there is one dislocation for every 30 atoms (or 12 per 1000 Å), then the presence of precipitation centers suggests that dislocations may arise on slip bands in the form of small groups. The same spacing of 1000 Å between precipitation centers was also found on polygonal boundaries; this likewise corresponds to the concept of groups of dislocations (Figs. XXIX and XXX).

The preceding discussion indicates the necessity of adopting the assumption that the space between slip bands is filled with something more substantial than elastic stresses. In essence, one cannot avoid the conclusion that dislocations are located not only on the nominal slip planes but also between them. Possible sources of such dislocations are considered below.

§ 5. THEORIES EXPLAINING THE ORIGIN OF SLIP BANDS

5.1. The necessity of clarifying the mechanism of dislocation formation. The most remarkable feature of the slip process is the localization of deformation in slip bands separated by regions in which no shear is detected. From the results of investigations with the light microscope it follows that, in the case of cubic metals, ordinary deformations at room temperature lead to a shear per band of the order of \(1/2\ \mu\). The width of the slip band was found to be less than the resolving power of the microscope. Thus, the distribution of shear within the band, the number of planes participating in the process, and the form of the slip steps produced remained unknown. For example, deformation could proceed uniformly within a band with a shear of the order of 10 interatomic spacings per slip plane. The results of electron-microscopic investigations, in their usual interpretation, show that in aluminum the section of a band has the form of a series of steps. Each step corresponds to a shear of approximately 700 interatomic distances; observations of the sharpness of the step show that this shear is concentrated in fewer than ten neighboring atomic planes. In § 5.9 some recent experiments of Jakutowicz and coworkers \(^{10}\) will be described, giving reason to think that the changes at each step are in fact a zone of concentrated homogeneous deformation extending over approximately 50 neighboring atomic planes. The following exposition is based on the point of view currently adopted.

If it is assumed that a shear step arises as a result of the emergence at the surface of the metal of a dislocation that has passed along the slip plane, then this representation corresponds to the number of dislocations in the lattice from 70 to 700 for each atomic plane. The indicated number in itself is greater than can be accepted for a pure crystal. In addition, one must add the dislocations that have remained on the slip plane after deformation and thus have not emerged at the surface. If it is accepted that there is one dislocation per 30 atoms, then for each slip plane of the crystal a reasonable number of them approaches many thousands.

5.2. Sources of Frank–Read dislocations. From the foregoing it is clear that it is necessary to indicate a mechanism for increasing the number of dislocations, by means of which it would be possible to explain how a small number of dislocations present in an annealed crystal gives rise to thousands of others. Many attempts have been made to indicate such a mechanism; the most promising of them belongs to Frank and Read \(^{55}\). According to these authors, in the slip plane an unlimited shear may take place, occurring as a result of the rotation of a dislocation about an immobile point, which leads to the formation of a dislocation on another plane.

While the results of microscopic investigations of deformation point to the necessity of establishing a mechanism for increasing the number of dislocations, it is difficult to propose a direct experiment, wh—

which would make it possible to test the existing hypotheses. In the case of Frank—Read sources, the idea of the mechanism for increasing the number of dislocations is based on spiral growth of crystals. On the latter question there is a great deal of data, a review of which was recently made by Frank^56.

5.3. Resultant slip

Since Frank—Read sources act inside the metal, they can be observed only through the surface effects conditioned by them, which reduce to the appearance of slip bands. Thus a clear verification of the theory would be proof that slip bands, when detached from their source, cease to grow as deformation increases. The ideas presented can be used to explain the behavior of certain slip bands.

It turns out, especially at high temperatures, that in regions where transverse slip is a common phenomenon, some branches of slip bands are sometimes considerably narrower than they should have been in accordance with the given deformation (Fig. IX). The cause of this phenomenon was discovered by the author jointly with Honeycombe when observing the growth of slip bands with increasing deformation. At present it is still not possible to explain satisfactorily a number of photographs showing nonuniform growth of slip branches. In Fig. 31 the visible picture is shown schematically. As deformation increases, a branch appears on the slip band in Fig. 31, a (Fig. 31, b). The latter in fact represents a new slip band merging with the one that existed at first. As the deformation process proceeds, displacement occurs along the band with a kink ), while the lower branch proves to be in a state of resultant slip (Fig. 31, c). If the surface of the metal is repolished in order to remove all slip bands, and then the crystal is again subjected to deformation, the band with a kink appears again, whereas the branch corresponding to resultant slip, naturally, disappears (Fig. 31, d). From these observations one may conclude that the Frank—Read source was located in the left-hand part of the slip band shown in Fig. 31, a*. This part, directly connected with the source, fed the new branch forming part of the band with a kink. The portion of the original band located on the right in the figure proves to be cut off from the dislocation source and enters a state of resultant slip.

Fig. 31. Diagram of resultant slip.

5.4. Frequency of Frank—Read sources

If the interpretation of the scheme given in Fig. 31 is correct, and if slip bands, in fact, when the power of the Frank—Read source capable of being activated is insufficient, pass into a state of resultant shear, then such sources are encountered inside the crystal relatively rarely. This assertion corresponds to Orowan’s point of view, described in § 5.5, but contradicts the opinion of Mott^25, who came to the conclusion that there should be about \(10^{12}\) sources in one cubic centimeter of metal.

Since the growth of a slip band in aluminum occurs by the formation of lamellae, from these observations another conclusion may be drawn: all lamellae in the band arise from one source or from a group of jointly acting sources. Suppose that this is not so and

* Instead of the term “kink band” — “band with a kink” — in our literature^107, ^108 the term “reset band” is also used. (Editor’s note.)

layers are formed from independent sources that were brought into action because they happened to be close to the existing slip planes. Suppose, further, that the original source ceases to function after it has created enough dislocations for the formation of one elementary shear line. Under these conditions, slip along the branch in the state of indifferent slip is just as probable as along any other.

Another proof that each slip band arises from a single source is provided by experiments carried out under conditions of high temperatures and slow deformations, which were discussed in § 2.4. In Fig. XIV a transverse shear inside a large slip band is shown, which may consist of 50 platelets. Transverse slip is also observed, encompassing whole bands; in such cases all 50, or about that many, layers rotate coherently in a circle in the direction toward the plane of transverse slip and back to the original plane. Since the band is formed by the appearance of individual slip platelets, it may be assumed that the transverse slip appears during the formation of the first platelet, as a result of the combination of two Frank–Read sources located on adjacent planes. If two Frank–Read sources are required to form each platelet with bending, then it is impossible for this to create a macroscopic transverse shear even approximately in one and the same place.

Cottrell^57 showed that if one accepts the proposition that all platelets arise from a single Frank–Read source, then it follows that the full shear in the band occurs mainly along one slip plane, and the different platelets appear as a consequence of obstacles to the slip process. This question will be considered below among the data relating to the source of slip platelets (§ 7).

5.5. Dynamic mechanisms for increasing the number of dislocations. Orowan^47 criticized the theory of Frank–Read sources in its usual presentation on the grounds that this theory does not indicate the reason for the transition of slip from one slip plane to others; this deprives the idea of the emergence of new Frank–Read sources on parallel slip platelets of its basis. According to Orowan, clarification of this question is necessary, since it is impossible to imagine that on every plane capable of being activated at some stage of deformation there is an individual Frank–Read source. Even if one assumes that the appearance of each slip platelet does not require a separate source, Orowan’s objection does not fall away, since, according to what was said above (§ 4.2), the places where slip bands appear are determined by factors other than the presence of “soft” planes.

Other arguments adduced by Orowan in favor of the existence of a dislocation mechanism by means of which slip can spread to parallel planes consist in the existence of the phenomenon of bending and of the “stamp effect.” In bending, a large number of slip planes are apparently involved in the process simultaneously. Slip along each plane may be limited to a small region within the band with bending, and although the shear along an individual plane is so small that its traces cannot be observed microscopically, it cannot be assumed that, before the process begins, there was already a sufficient number of dislocations on all the active planes for it to proceed.

The “stamp effect” was first observed by Gilman^58 on a zinc single crystal cut along its basal planes (Fig. 32). At point \(A\), a hemispherical stamp was pressed into the crystal by impact. It turned out that the imprint of the stamp passed right through the crystal and appeared on its opposite side, retaining its shape but increasing somewhat in size. From this Orowan concluded that in this case bending again occurs; in particular, within the region of shear formed as a result of the action of the stamp, the process must proceed almost simultaneously along many slip planes. Consequently, there must exist a mechanism for increasing the number of dislocations, allowing slip to propagate not only along its own plane but also along parallel planes.

Fig. 32. “Stamp effect” (after Orowan).

A detailed description of the mechanism proposed by Orowan lies outside the scope of the present review. We shall note briefly, however, Orowan’s assertion that if the velocity of motion of a dislocation can approach the critical velocity, approximately equal to half the speed of sound, then the plane of maximum shear stresses in front of it no longer coincides with the slip plane but is inclined to the latter. In the limiting-velocity case, the plane of maximum shear stresses is situated at an angle of \(45^\circ\) to the slip plane. Thus the shear is displaced across the slip plane, forming dislocations on both sides of it. These “lateral steps” can act as Frank–Read sources.

5.6. Microscopic evidence for the existence of a dynamic mechanism of dislocation multiplication.

Leibfried^59 asserts that, owing to thermal braking, a moving dislocation cannot acquire a velocity sufficiently close to the speed of sound required for the operation of the dynamic mechanism of dislocation multiplication. This, however, is disputed by Nabarro^60. It is of interest to consider the experimental evidence for the existence of the mechanism mentioned. Such evidence may consist of the following:

1. Evidence that the shear process can leave the slip plane.
2. Evidence for the existence of dislocations in regions located between slip planes.
3. Evidence for the influence of conditions under which dislocations cannot acquire sufficiently high velocities on the appearance of the slip plane.

Let us consider these points in order.

1. A dislocation that has left the slip plane can move only until its energy is dissipated. It then returns to the direction of maximum shear stresses and continues to move parallel to its original direction or stops. In the first case this should appear under the microscope as internal transverse slip, i.e. the formation of plates of transverse shear within the slip band. The commonly accepted view of macroscopic transverse slip, described by Mott^25, according to which this phenomenon arises as a result of the convergence of slip processes on neighboring planes, each plane having its own Frank–Read source, is inapplicable here, since it is unlikely that any segment of a slip line ob-

had a Frank–Read source capable of being activated. Another case in which a moving dislocation comes to a halt concerns those dislocations which lie outside the band. They may be visible as incomplete short slip lines situated on both sides of the band. In Figs. XXXIII and XXXIV a frequently observed picture is shown, especially in specimens deformed at high temperatures: the slip band is surrounded by a feather-like structure.

1. The presence of dislocations between the bands is evident from the existence of the incomplete slip processes described above. In addition, in § 4 it was shown that, if all dislocations are overtaken by the slip process concentrated in the slip bands, it is difficult to explain the existence of a definite distance between the bands.

The data obtained with the aid of the microscope thus have an indirect character. Seitz\(^{61}\), however, in discussing the work of Warren and Averbach mentioned in § 4.5, pointed out that if the dislocation density in aluminum subjected to cold deformation is as high as in \(\alpha\)-brass, then a large number of dislocations must be located in regions lying between the visible slip elements.

1. A survey of the conditions under which a dislocation cannot acquire high velocities was given by Brown\(^{16}\). His article, written before the appearance of the Frank–Read source theory, summarized the experimental evidence for the ideas then accepted, according to which slip bands appeared as a consequence of avalanche-like shear. It was assumed that a single dislocation, set in motion under the action of the applied stresses, acquires a velocity close to the velocity of sound, and an energy sufficient for creating other dislocations. The latter in turn are accelerated and cause the appearance of new dislocations; this continues until an avalanche-like shear begins along the slip plane. The motion of the avalanche ultimately ceases as a result of the hardening of the slip plane.

At the present time it is believed that slip bands do not necessarily arise as a result of the action of rapidly moving dislocations; they may also be produced by the Frank–Read mechanism by dislocations moving slowly. If, however, we accept Orowan’s conclusion, according to which, for the action of Frank–Read sources, an increase in the number of dislocations by a mechanism requiring the presence of fast dislocations is necessary, then the data given in Brown’s article mentioned above must still be regarded as valid at the present time. If fast dislocations are absent, the mechanism for increasing the number of dislocations does not operate and there is not a sufficient number of Frank–Read sources; then slip bands arise in small numbers or else do not appear at all.

5.7. “Flow without slip.” Any cause that prevents dislocations from attaining high velocities may make impossible the operation of the mechanism for increasing the number of dislocations and thus exclude the appearance of slip bands. In this direction there are the following possibilities:

a) The temperature at which deformation is carried out is too low. In this case the velocity of sound is greater than at high temperatures. However, since the thermal damping is smaller, the amount of slip due to temperature changes is unlikely to fluctuate strongly.

b) Foreign atoms in solution hinder the displacement of dislocations.

c) Centers of precipitation of a new phase hinder the motion of dislocations.

c) The rate of deformation is too low to cause the occurrence of rapidly proceeding shear.

d) The grain size is so small that the accelerating dislocation reaches its boundaries before it attains the critical velocity.

Let us consider the propositions listed, in order.

a) It is known that in aluminum the elementary magnitude of shear decreases with decreasing deformation temperature from 2200 Å at 250°C to approximately 1000 Å at the temperature of liquid helium (§ 2.3). For a given degree of deformation, the density of distribution of slip bands increases as the deformation temperature is lowered.

b) The influence of impurities in solid solution on the distribution of slip bands has not yet been investigated sufficiently carefully. It is known, however, that in specimens of commercially pure aluminum the distance of elementary shear is somewhat shorter than in aluminum of 99.95% purity. In addition, in aluminum specimens containing magnesium, copper, and zinc in amounts up to the solubility limit, slip bands are always present.

c) The influence of precipitation processes on the arrangement of slip bands was studied by Smith and Duigirst 62 in their work on internal oxidation. These authors created conditions under which oxygen diffused into a flat polycrystalline copper specimen containing 0.6% aluminum, in such a way that in the zone situated immediately beneath the surface the aluminum was precipitated in the form of oxide particles about 1000 Å in diameter. When such a metallic, layered specimen was bent, the ordinary slip bands were observed only in the unoxidized region; they ended at its boundary. Slip bands were not detected in the oxidized zone either with the aid of the light microscope or with the aid of the electron microscope (Figs. XXXV and XXXVI).

Similar results were described by Edelanu 63, who found that slip bands present in a solid solution of aluminum are absent, under identical deformation conditions, in a previously aged alloy in which precipitates of a second phase are present (Figs. XXXVII and XXXVIII). In specimens in which the aging process has not proceeded to completion, a small number of slip bands is encountered (Fig. XXXIX).

In the experiments of Smith and Duigirst the specimens were polished mechanically; in Edelanu’s experiments, purely electrolytically. In no case were blurred traces observed in place of slip bands. Thus, deformation in a region free from shear, in the general opinion, may indeed be homogeneous.

Apparently, there are no works in which slip bands have been observed in duralumin subjected to dispersion hardening. It should be noted, however, that the nature of the precipitates in an aluminum alloy with 4% copper is quite different from that in the two cases discussed above. In the alloys mentioned earlier the particles of the second phase were large and had a roughly spherical shape, whereas in duralumin, according to the work of Castaing and Guinier 53, and also of Buynov and Perlman 64, if precipitates exist, then the particles do not have the form of thin and flat plates.

d) and e) Under sufficiently slow creep the acceleration of dislocation motion may be small, and the distance which they must traverse in order to acquire the critical velocity correspondingly large. Therefore, in a fine-grained specimen under creep, slip bands may be absent. This phenomenon was first noted by Hanson and Wheeler 65, and recently studied carefully by Wutt with co-workers 66, 67, 68. The latter

found that, at a given temperature, absent slip bands may appear when the degree of deformation is increased or as a result of an increase in grain size. At a higher temperature, for slip bands to appear a larger grain size is required. Fig. XIII shows that in large single crystals slip bands arise at very high temperatures and very low rates of deformation.

At first glance, these results confirm Leibfried’s theory, according to which the actual velocity of dislocation motion is limited by scattering on thermal waves, and thus is the smaller the higher the temperature. However, experiments carried out on single crystals show that slip bands disappear in the interior parts of the grains of a polycrystalline specimen when the grain size is smaller than the distance between slip bands in a single crystal subjected to the same deformation. It is unknown how slip bands will behave if there is not enough room in the grain for two bands.

Grain size also affects the results of Edelen’s experiments. For example, it turns out that in annealed specimens with grain sizes varying over wide limits, slip bands are not observed in the smaller grains, but are present in the larger ones. Since grain size does not affect the precipitation process, this effect must be due to the different mean free path of dislocations in grains of different size.

The influence of very high deformation rates is considered in § 9.

5.8. Source of microslip. The data obtained from observations of diffuse bands on electropolished specimens have to some extent confirmed the conclusion that, in the initial stages of deformation, slip proceeds on a considerably larger number of planes than had previously been thought. Brown and Honeycombe[^32] believe that the majority of these planes soon after the beginning of the process cease to participate in it, because “the dislocations on the planes are arranged so close to one another that the microslips interact strongly and can subsequently move only on a relatively small number of slip planes”[^99]. They add: “At high rates of deformation this mechanism may also be responsible for hardening in the initial stages of the process, when the coarse bands usually observed are situated too far from one another to interact.” This argument is unconvincing, since it implies the creation of a multitude of dislocations on a large number of closely spaced slip planes. The latter is probably possible only when sources of the Frank–Read type are operating. Apart from the difficulty of creating or activating a large number of sources during deformation of less than 1%, there is no reason for their action to cease at shears considerably smaller than those they usually produce.

A more interesting possibility is the idea that microslip is caused by dislocations already existing in the original lattice. They diffuse, without increasing in number, to the outer surfaces of the specimen, but do not leave its limits and do not form steps. This explains the fact that diffuse slip appears on the surface in the form of bulges. The rapid increase of hardening during deformation is a consequence of the exhaustion of the supply of dislocations in the lattice. Since a decrease in the number of dislocations does not affect electrical conductivity, the slight drop in the latter found in experiments with ordering alloys (§ 3.4) cannot be taken as evidence that only a small number of slip planes

is in the active state. In essence, deformation by means of such a process of exhausting the stock of dislocations may slightly increase the electrical conductivity and diminish the effect of the true slip process.

This theory could be tested if, in a carefully electropolished crystal, the initial part of the plastic deformation up to 1% and the corresponding hardening could be restored by the action of stresses of the opposite sign.

5.9. Combination of concepts of homogeneous and inhomogeneous slip. We have already referred (§§ 2.1, 5.1) to the work of Yakutovich, Yakovleva, Leriman, and Buinov[^10], in which the experiments of Heidenreich and Shockley[^7] were repeated. Since these investigations are still not sufficiently widely known, we shall consider their main points.

Single crystals of aluminum of 99.95% purity were deeply etched with a mixture of sulfuric and hydrochloric acids in order to obtain the usual cubic block relief. The surface was not subjected to any other treatment, for example mechanical or electrolytic polishing. After deformation by tension by 1–2%, using an electron microscope with the oxide-replica technique, it was impossible to detect either slip traces or any other surface changes, such as, for example, distortions of the relief shape. After 4% or greater deformation, slip bands intersected the faces and distorted the edges of the blocks forming the relief. As the deformation increased, the number of slip bands and the degree of shear along each band increased until, at large deformations, the entire field of view proved to be covered with slip traces. At all deformations the slip bands showed a tendency to group together: for example, it turned out that sometimes five bands crossed the face of an etched block over a region with linear dimensions of the order of a micron. In most cases a fine structure could be distinguished in each slip band. As calculations showed, the magnitude of the shear along an elementary slip line was 150–200 Å, and the width of the line was 200–250 Å. From this the authors concluded that each elementary shear line is a region of definite width in which there is concentrated a local deformation equal, in order of magnitude, to one interatomic distance on the slip plane. The appearance of a slip band (in section) should resemble that shown in Fig. 40, a.

The micrographs presented by Yakutovich et al. are similar to those obtained by Heidenreich and Shockley, and also by other investigators, except for the difference caused by deep etching instead of electrolytic polishing. The interpretation of the results by these authors, however, differs from the views of other investigators: first, Yakutovich and co-workers consider that the shear in an elementary slip process amounts to only one tenth of the value adopted in the present article; secondly, they use the data obtained to prove the proposition according to which each elementary slip line is not the consequence of an insignificant shear on one active slip plane, but the result of the summation of a small slip on each of approximately fifty neighboring atomic planes.

In the opinion of the author of the present review, the magnitude of the shear in an elementary slip process has in this case been calculated incorrectly. This magnitude was apparently obtained from the macroscopic deformations and the smallest distances between slip bands, whereas it should be calculated on the basis of the mean distance between bands. This,

however, may be disregarded if it is assumed that the displacement in each elementary slip process occurs more or less uniformly in a zone 50 atoms wide. In this case an elementary displacement of 2000 Å corresponds to a displacement by twelve interatomic distances in each active slip plane. The appearance of a slip band constructed in this way is shown in Fig. 40, c. For comparison, Fig. 40, b shows the structure of the slip band adopted by Heidenreich and Shockley. It is very difficult, on the basis of microscopic data,

Fig. 40. Fine structure of a slip band according to: a) Yakutovich with co-workers\({}^{10}\); b) Heidenreich and Shockley; c) the case in which these two variants are combined; d) scheme of the light contrast when cases b) and c) are observed in electron micrographs.

to make a choice between the two possibilities presented in Figs. 40, b and 40, c, since small steps twelve atoms high are far below the resolving power of any microscope; moreover, the contrast schemes of the image visible under the electron microscope differ insufficiently in the three cases considered (Fig. 40, d). There is only one way to resolve the question of the width of slip lines. This is the method adopted in the work under consideration.

The width of slip lines visible in the electron microscope when oxide replicas are used consists of three parts (Fig. 41): the true width, if it exists; the projection of the free surface of the part of the slip band falling under the electron beam; and the magni-

... approximately equal to the thickness of the film. In some experiments by the author of the present review on fine-crystalline polycrystalline specimens it was found that the slip plane is almost perpendicular to the surface. This means that the plane of slip makes no contribution to the visible width of the line (Fig. 41, b). Under these conditions the measured width proves to be of the order of 200 Å, i.e., somewhat greater than the established thickness of the replica film (100–200 Å). However, the accuracy of the measurement is insufficient to give a definite answer to the question of whether this difference is a consequence of the finite width of the slip zone.

Fig. 41. Contrast pattern when oxide replicas are used: a) the slip plane is situated at an obtuse angle to the surface; b) the slip plane is situated perpendicular to the surface.

If in fact the slip bands have a profile similar to that shown in Fig. 40, c, then the notion of the mechanism by which the number of dislocations increases requires modification. This situation is very similar to that which occurs in the case of the onset of slip, when it is necessary to explain the propagation of slip over a large number of atomic planes, on each of which, however, the shear takes place over several interatomic distances. For this reason the present question deserves further experimental study *).

§ 6. INFLUENCE OF THE STATE OF THE SURFACE ON THE MECHANICAL PROPERTIES OF A METAL

6.1. Griffith Cracks

The theory of surface cracks of Griffith^69 successfully explained the small value of the stress required to fracture brittle materials, in particular glass. It was assumed that small cracks, located mainly on the surface, act as stress concentrators. Owing to this, under relatively small applied loads the magnitude of the stress at the mouth of the crack is considerably closer to the theoretical value of the fracture stress. One of the proofs of the existence of such cracks is provided by observations

) Recently N. N. Buinov carried out work showing that even at large degrees of deformation the latter is more homogeneous than had previously been assumed^104, 105. The question of the homogeneity of plastic deformation of metals is also addressed in study^106. (Editor’s note.)

Griffith, according to which the breaking stresses are considerably higher for freshly drawn glass threads than for the same threads whose surface has been “damaged” by a light touch with some hard body. Reinkober^70 broke a glass thread into two parts, then each half again in half, and so on. Since, as a result of these operations, the probability that cracks weakening the material would enter the specimen gradually decreased, the breaking stresses increased as the fragments of the thread became smaller. In a similar way, Ioffe,^20 by removing surface cracks on rock-salt crystals through continuous dissolution of the surface layers, found that the strength thereby increases hundreds of times. Calculations show that Griffith cracks have a depth of the order of a micron. The width of a crack at the surface is too small for it to be observed with a light microscope; however, Andrade and Martindale,^71 and also Andrade and Tsien,^72 found that metal vapors condense predominantly on the surface of a specimen along certain lines that can be identified with Griffith cracks (Fig. XLII). Until now no reliable electron-microscopic photographs showing cracks have been obtained. The crack-like defects shown in the photographs (Fig. XLIII) cannot be distinguished from scratches.

Attempts to apply a theory taking into account the influence of surface cracks to the explanation of the low magnitude of the yield point of metals were largely rendered obsolete by the success of dislocation theory. For example, Orowan^73 collected a large amount of data showing that the yield point of plastic materials is not determined by the presence of surface defects. All attempts to raise the yield point of metals by methods connected with the removal of surface cracks, similar to the way this is done on glass, either proved unsuccessful, or the results obtained were due to the presence of inclusions. Moreover, the length of a crack which, on the basis of theoretical considerations, is necessary to provide the known yield point of soft metals such as cadmium, often turned out to be greater than the diameter of the specimen.

At the present time, and especially on the basis of the very latest observations—which are usually regarded as the most indisputable objections to the mechanism of the influence of cracks in the flow of metals—it is not impossible that surface cracks or defects may lead to a slight decrease in the yield point. Indeed, if it can be shown that surface defects exist, then they must act as stress concentrators. The influence of the latter circumstance on the yield point may be so small that, in comparison with a specimen having no surface defects, the difference does not exceed the usual scatter of the values of the quantity under consideration when measured on different specimens.

6.2. Thermal etching. The existence of a defective structure in metals was first shown by Graf,^74 who found a network of parallel lines on the surface of metals crystallized in a humid atmosphere. Graf interpreted these lines as traces of the lamellar structure existing in the annealed metal. In metals having a hexagonal lattice, the planes bounding the lamellae that form the indicated structure always coincide with slip planes; in metals with a face-centered cubic lattice such a coincidence occurred only in some cases.

The experiments of Chalmers et al.^75 were a development of Graf’s work and consisted in a careful study of the structure obtained as a result of the thermal etching of pure silver. These authors found—

it has been established that, after heating in an oxygen atmosphere, a complex structure arises on the surface of silver, formed in some cases by parallel grooves and in others by rows of pyramids. The structure obtained disappears upon heating in an atmosphere of inert gases and does not necessarily reappear in its original form under the corresponding conditions. On the basis of the facts presented, the conclusion was drawn that the structure obtained by thermal etching has a lower energy than the “free” surface and is usually formed by \(\{111\}\) planes (i.e., the planes along which slip occurs). No evidence was obtained that the picture described corresponds to any lamellar or other definite structure inside the crystal.

It is obvious that structures similar to those found by Graf, and also by Chalmers and others, must act as surface damage, lowering the yield point by concentrating the applied stresses. This influence on the mechanical properties of the metal may be too small to be noticed.

The connection between the structure obtained by thermal etching and slip bands on cadmium was discovered by Andrade and Randall \(^{76}\). They heated single crystals of pure cadmium under conditions allowing free evaporation. It turned out that elliptical traces, similar to slip bands, then appear on the surface of the crystal. Upon subsequent deformation these traces in fact become traces of slip bands.

6.3. Microetching. The use of the electron microscope and electron diffraction gave additional information about the structure of an etched surface. Brown \(^{15}\) showed that, in electron micrographs of pure electropolished aluminum, a so-called “microetching” structure is visible, which in some cases resembles orange peel (Fig. XLIV), and sometimes, depending on the crystallographic nature of the surface, resembles grooves (Figs. XXIV and XXV). In the latter case the picture is very similar in appearance to the structure obtained by thermal etching by Chalmers and others, although it is much smaller in scale. Of two neighboring grains, one may apparently show a disordered structure resembling orange peel, while the other shows an ordered structure of grooves (Fig. XLV). Structures with intermediate degrees of order were also found. The geometrical relationship between these structures and slip bands is consistent with the viewpoint according to which the grooves are traces of \(\{111\}\) planes. The latter are possible slip planes which, under certain conditions of applied forces, may become active. The conclusion was also drawn that the structure obtained by microetching is the surface of least energy and is formed in the electropolishing bath when the oxide film is temporarily destroyed, and therefore some surface displacement of the atoms of the metal becomes possible.

More debatable are Brown’s \(^{15}\) proposals concerning the dimensions characterizing the structure. It has already been noted that the period of the structure, i.e., the distance between neighboring convexities, is very close to twice the thickness of the plates. Thus, as follows from Fig. 47, it is possible that the traces of thermal etching are the cause of the formation of plates. Regardless of whether this is so or not, it is clear that when the grooves formed during etching are parallel to slip bands, they may mistakenly be taken for slip traces. A case illustrating this possibility is shown in Fig. XLVI. Apparently,

an incorrect interpretation of the structure obtained by microetching led Nishimura and Takamura\(^9\) to the conclusion that visible slip is present in the regions between slip bands.

An electron-microscopic investigation of electropolished cadmium also revealed a fine structure of the microetching, traces of the \(\{0001\}\) planes, and the fact that these planes are parallel to the slip bands. Here confusion between two phenomena is considerably more likely than in aluminum: it is unclear, for example, which of the finest lines in Fig. XVII is a slip trace and which is the result of microetching.

6.4. The Rebinder effect\(^*\). Thus, there is sufficient evidence for the existence of surface defects in metals.

Fig. 47. Relation of the structure obtained by microetching to the fine structure of slip bands: a) before slip; b) after slip.

For the reasons indicated in § 6.1, it is impossible to compare the mechanical properties of specimens having defects with those free of them. However, in creep tests one can observe the influence of a change in the surface state on the rate of steady flow of single-crystal specimens. These experiments were first carried out by Rebinder\(^77\), who established that the creep of cadmium crystals is accelerated by approximately a factor of two if the specimen being tested is immersed in a certain electrolyte. Rebinder’s explanation was that, under the action of the electrolyte, pits are formed on the surface of the specimen, which are traces of crystallographic planes. The pits act as stress concentrators and thus lead to an acceleration of creep. Similar experiments were performed by Ramsley\(^78\), Harper and Cottrell\(^57\), Menter and Hollomon\(^79\), and also Andrade et al.\(^80\). The authors listed unanimously indicate that if, before the tests, the oxide film is carefully removed from the specimen, then the effect described by Re—

\(^*\) For a detailed exposition of works devoted to the influence of surface-active media on the course of plastic deformation, see in \(^{103*}\). (Ed. note.)

binder, is absent. The thickness of the oxide film is no more than a micron. It therefore cannot appreciably strengthen the metal by virtue of its own tensile resistance. Suppose, for example, that the film has been removed from one cadmium specimen and that it has been placed in a creep-testing machine parallel to another, oxidized crystal. Under these conditions the influence of the film on the creep rate should be very small. However, a dense coating on a crystal that is considerably softer than the cadmium crystal can hinder slip at the surface and thus reduce the creep rate.

The results described, however, can also be explained on the basis of Rebinder’s ideas. Suppose that the observed effect depends to some extent on the presence of an oxide film; for example, the oxide is a catalyst in the process by which the electrolyte acts on the surface of the specimen. In this way the formation of defects leading to stress concentration is facilitated, and the creep rate increases.

Consequently, the nature of the influence of the surface state on plastic deformation is uncertain. There is no doubt that it is not enough to raise the question of the extent to which the surface reflects the internal state of the metal. It must be borne in mind that the surface itself can change the internal conditions of deformation.

6.5. Influence of surface scratches. Some experiments of Nishimura and Takamura9, which will be described in § 7.7, show that artificially made shallow scratches affect the distribution of slip bands in the adjoining regions of the metal surface. This is especially important in view of the fact that most specimens for mechanical tests have several marks for measuring the magnitude of surface distortions, and in some microscopic investigations a network of scratches is made in order to detect microdisplacements.

§ 7. NATURE OF SLIP LAMELLAE

7.1. Slip along existing slip bands. In § 5 some theories were considered that explain the inhomogeneity of plastic deformation. Apparently, it is theoretically possible to justify the appearance of a number of dislocations required in order to cause macroscopic slip by the displacement of several thousand atoms on each of a small number of isolated planes. In § 4 it was shown that the magnitude of the distance between these isolated planes can be explained only on the assumption that lattice distortions caused by deformation limit slip in the surrounding region. Nevertheless, it is necessary to explain why, as deformation proceeds further, slip is concentrated in existing bands, usually on planes displaced by several hundred interatomic distances relative to the original slip planes.

7.2. Reality of slip lamellae. The assertion that slip lamellae exist is based entirely on observations of the surface of deformed specimens. Proof that, throughout the entire deformation, slip occurs on existing planes could be obtained by counting the number of slip bands, as was done by Yamaguchi3. Certain data showing that deformation occurs by discrete atomic displacements, each of which is large in comparison with the lattice period and takes place in a very short interval of time, have been obtained from

non-metallographic experiments and were already mentioned in § 2. Confirmation of the assumption that discrete atomic displacements during shear lead to the formation of slip bands separated from one another by several hundred interatomic distances can be obtained only from metallographic observations of the metal surface. Thus, the solution of this question appears very important and must include consideration of the possible ways in which a lamellar structure may arise on the surface without any traces of it within the metal.

In § 6 it was shown that, during electropolishing of specimens in a number of metals, a grooved structure arises. This structure often has elements parallel to the traces of possible slip lines and may easily be confused with them. If, in addition, the metal surface possesses the properties of a shell that prevents its destruction by slip planes, as was the case in the experiments of Greenland[^81], who studied the deformation of mercury crystals, then in each slip band the grooves create the appearance of the presence of slip lamellae.

Fig. 48. Origin of the impression of a fine structure of a slip band owing to irregularities of the surface layers (schematic).
a) View of the surface before slip. The grooved structure may not appear in an electron micrograph, since both sides of each groove are inclined equally with respect to the electron beam.
b) View of the surface after slip. Shear along one of the slip planes has caused an inclination of the surface layers. Inside the profile of the slip band, the sides of each groove are not inclined equally with respect to the electron beam; this creates light contrast.

The effect described is shown in Fig. 48. Similar cases were observed under the condition, indicated above, of surface influence, but they are not typical, since the grooves are not always parallel to the acting system of slip lines. Moreover, in experiments carried out under conditions of high temperatures and low rates of deformation (§ 2.4, Fig. XIV), the distance between lamellae was much greater than the period of the surface relief obtained by microetching. In some cases the grooved structure was mistakenly taken in electron micrographs (§ 6.3) as the result of shear in the region located between slip bands. Figure 47 shows a case demonstrating the possibility of such an erroneous interpretation of the observed picture. At the same time it may be thought that fine shears that occur in the region located between slip bands are usually not detected with the aid of the electron microscope.

7.3. Coarsening of the structure. The thickness of the lamellae obtained in experiments carried out under conditions of high temperatures and low rates of deformation also shows that these lamellae cannot be the result of coarsening (the term introduced by Stranski[^82]—vergröberung) of the observed part of the slip plane. Such coarsen—

... could occur in the manner shown in Fig. 49: small steps are formed by the surface displacement of atoms, which continues until a surface is formed with an energy lower than the energy of the slip surface. During deformation under room-temperature conditions such displacement is impossible without the aid of an electrolyte, although most electrolytic micrographs of the fine structure in slip bands have been obtained from replicas prepared using electrolytic etching; the lamellar structure is also visible on replicas prepared by other methods. Fig. XLIV shows the fine structure of a slip band. This photograph was obtained from an aluminum polycrystal by the lacquer-replica method (Brown^15). It also illustrates the proposition according to which the structure arising as a result of microetching has the appearance of an “orange peel” and is not associated with the presence of slip bands.

Fig. 49. Formation of an impression of a fine structure as a result of roughening.

7.4. Formation of slip lamellae as a consequence of subsurface phenomena. Cottrell pointed out^57 that the complete shear leading to the formation of a slip band occurs near the source, at least on one slip surface, and that probably the greatest part of the shear is concentrated inside the crystal, with the exception of the layer lying immediately beneath the surface. When slip reaches an obstacle, which may be a free surface or, probably, a kink band, the band breaks up into a number of parallel slip planes, situated close to one another and appearing at the surface in the form of a group of lamellae. The transfer of slip from one plane to another occurs as a result of slip on another shear system, as shown in Fig. 50. In the usual evaluation of maximum shear stresses this latter system may remain inactive; however, at some point it is activated because the propagation of the stress field caused by dislocations is arrested at the obstacle in almost the same way as slip within a kink band on planes inclined

Fig. 50. Formation of fine structure (schematic):
a) formation of the first step;
b) the step is carried outside the plane in which the source is located by the formation of a new step, due to slip on another slip system. Slip in the first system continues;
c) continued slip from the first source creates a second step.

to one another and appearing at the surface in the form of a group of lamellae. The transfer of slip from one plane to another occurs as a result of slip on another shear system, as shown in Fig. 50. In the usual evaluation of maximum shear stresses this latter system may remain inactive; however, at some point it is activated because the propagation of the stress influence caused by dislocations is delayed at the obstacle almost in the same way as slip within a kink band on inclined planes...

with respect to the ordinary slip system. The latter does not exclude the possibility of a transition of slip from one plane to another by means of transverse slip, shown schematically in Fig. 51. In this case also it is necessary that there be an obstacle in the path of advance of the slip front. Such a mechanism can explain the influence of temperature: at higher temperatures there is a larger displacement along each slip band, since the work hardening is then smaller, and at the same time transverse slip is a more frequent phenomenon both between bands (Fig. IX) and within them (Fig. XIV).

Fig. 51. Fine structure arising as a result of transverse slip. a) A source producing dislocations that form a shear step. b) Dislocations from the same source deviate to another plane by a certain obstacle lying on the second step.

A positive contribution to the proof of the validity of Cottrell’s proposition is provided by observations that, in the case where the slip process encounters an obstacle, it is in fact smeared out. Of course, with the aid of a microscope it is impossible to observe the changes that occur in a slip band when it reaches the surface; however, it was often found that, in the vicinity of bands with bending of the plate, the band positions are at considerably greater distances than usual (Figs. LII and LIII). Bands with bending are not seen with high contrast in the electron microscope, but can be revealed as the sites of the above-mentioned retardations of the slip process (see Fig. 54). The notion of Cottrell’s mechanism can be used to eliminate two difficulties that arise in the theoretical analysis of planar slip. First, the analysis requires a specific distribution of Frank–Read sources, if each plate has its own source; in Cottrell’s mechanism all the plates arise from one and the same source. Second, there is the difficulty in explaining how the slip process can be directed across the slip plane, situated close to the first slip plane, i.e., through the already hardened region.

Fig. 54. Behavior of slip bands at points of intersection with bands having bending.

As for the first proposition, some reasons why all layers must be generated by one source have already been given (§ 5.4). The second difficulty, however, includes the assumption that the shear is not stopped by the hardening of the acting slip plane; for if it were retarded, then the stress that could again activate slip on the same plane would, of course, be no less than the stress necessary for activating a new slip plane at a distance of hundreds of interatomic spacings from this place.

Thus, Cottrell’s mechanism assumes not discontinuous slip, as was accepted in § 2, but a continuous displacement of atoms along each of the active planes. If the stresses have reached a value sufficient to bring the Frank–Read source into action, the latter creates dislocations that move continuously in the direction toward the surface as long as the applied stress acts. These dislocations are held up by certain obstacles and are not realized until enough of them have accumulated (in the case of aluminum, apparently about 700) to raise the local stress to a value corresponding to the activation of the mechanism that transfers slip to another plane. In this case the dislocations appear on the surface in the form of steps 2000 Å high. Thus, for a slip band visible on the surface, there is also, on the average, a slip distance of the order of half the distance between the slip lamellae hidden beneath the surface. This hidden slip makes the macroscopic deformation larger than that obtained in measurements by determining the heights of the slip steps. At larger deformations, when the amount of slip attributable to a band is of the order of a micron, the hidden slip constitutes a very small part of the total effect and therefore cannot be detected. At small deformations, however, a larger shear must occur than that following from the visible slip bands. Thus, Yamaguchi could not detect any slip lines until the deformation reached a value of the order of 1%, when a certain number of lines suddenly appeared (about 50 per mm). According to the results of the experiments of Kurnosov, Tronina, and Yakutovich, to which reference has already been made in § 3.4, it is possible that part of the deformation associated with the absence of visible slip lines could be explained in the manner described.

One may suppose that the slip processes were held up beneath the surface while awaiting the accumulation of a sufficient number of dislocations necessary to ensure their emergence at the surface. It is possible that the number of held-up dislocations required to bring Cottrell’s mechanism into action is nearly constant for aluminum, but is not preserved in the case of other metals. This explains the apparent regularity of the slip spacings in aluminum.

Thus, dislocations retained beneath the surface must retain mobility and, consequently, recovery of the fraction of the deformation that they represent is possible when the sign of the stress is changed. If the process of “non-slip” is not detected, then it is possible that there are internal obstacles, similar to bands with a kink, which do not allow the dislocations to return to their initial position (§ 8.3).

Cottrell’s theory is based on the assumption that the slip process can pass from plane to plane. If the occurrence of transverse or some other type of shear, leading to the appearance of non-rectilinear slip bands, is unlikely, then the effect of the surface reduces to stopping or slowing the slip on the band under consideration; the process may also transfer locally to another band. Thus, straight slip bands should consist of a smaller number of lamellae, while there will be more of the straight bands themselves at a given deformation. Precisely this situation has been established for technically pure materials tested at low temperatures. Some of the most interesting examples of band formation concern hexagonal metals, in which a transition of slip from one plane to another is rarely encountered, since in this case there is only one-

slip system. Obvious cross slip in cadmium is shown in Figs. XVIII and XIX. The figures presented, as well as other photographs obtained from the same specimen, exhaust the recorded cases of cross slip in a hexagonal metal. The results of experiments carried out on tin are described in § 8.5, where it is shown that when tin behaves as a hexagonal metal (single-crystal specimens), bands are formed, but there is no cross slip; when tin behaves as a cubic metal (polycrystalline specimens), cross slip often occurs, but bands are not formed.

So far theory has not clarified the nature of the factors that hinder slip. It seems unlikely that any obstacle to shear could be equally effective at 500°C and at −180°C. However, the distance between lamellae, which has been interpreted as a measure of the effectiveness of the obstacle, increases only slightly with temperature. Likewise, in metals with impurities, where one might expect the accumulation of a larger number of dislocations before they emerge at the surface, the distance between slip lamellae turns out to be smaller.

If the view according to which slip in a band occurs only along a single atomic plane is correct, then in cases where this plane coincides with the free surface, the width of the band should be unresolvably small. In reality, the width of such bands, as can be seen at points where the surface has been damaged by some markings, is always several hundred angstroms. In Fig. LV two slip bands of measurable width are visible, with shear directions almost coinciding with the free surface; they are distorted by a number of bands almost perpendicular to the surface. However, in no case, when observing shear parallel to the surface, was it possible to distinguish, at damaged places, small steps corresponding to slip lamellae.

7.5. Mechanism of the recovery occurring in slip lamellae

Another explanation of the nature of lamellar slip was proposed by Brown[^23]. According to Brown, the external appearance of the surface is regarded as the true indicator of the processes occurring inside the metal: the slip lamellae are preserved throughout the entire volume (although they do not necessarily remain parallel), and each lamella arises as the result of an avalanche-like shear occurring in a very short interval of time. A shear avalanche arises when the applied stresses reach a value sufficient to cause an increase in the number of dislocations, and its motion is retarded as a result of work hardening. The hardening that stops the shear avalanche also causes macroscopic hardening of the crystal, since the distance between slip bands is determined by their interaction. Thus it is assumed that the disordering of the metal structure, which is the result of the passage of a shear avalanche, leads to stresses in the surrounding medium, making it hardened: larger stresses are required in order to produce new slip. The resistance to slip is greatest in regions immediately adjacent to the slip plane. Therefore the most probable site for the occurrence of a new slip plane is situated midway between the existing bands. To explain the observed slip on planes that are very close to, but do not coincide with, the existing active slip planes, it is assumed that the region surrounding the slip bands becomes a region of easy slip as a result of recovery. The basic assumption is that

in the majority of highly stressed sites of the lattice, a low temperature is required for recovery to take place.

The assumptions concerning the interaction of slip bands located at distances of several microns from one another, and concerning the occurrence of the recovery process in the region situated between the bands, require the presence of dislocations outside the slip planes. In the case where all dislocations were concentrated on the slip planes, then, as was indicated in § 4.7, the region over which hardening extends would have linear dimensions much smaller than the distance between the slip bands. Moreover, if the indicated assumption were valid, the softening sufficient to ensure slip over the surface of the crystal would require the passage of the recovery process on a plane that had arisen earlier. Since removal of the load is associated with the release of stresses throughout the entire volume, recovery must encompass the whole region between the slip planes. Thus, if dislocations exist only on the planes that are activated immediately, then after recovery on planes close to the initial slip bands there will be no preferential conditions for slip. Proof of the existence of dislocations in the regions between the bands, as well as an assumption concerning their nucleation, were already given in § 5.6.

The hypothesis that takes recovery into account explains the influence of the deformation temperature on the slip bands by the dependence on temperature of the critical stresses required to cause spontaneous annealing. At high temperatures, thermal activation promotes the onset of spontaneous annealing. Thus, under conditions of high temperatures and very small stresses, slip occurs at sites located close to existing bands. At the temperature of liquid air, and still more at the temperature of liquid helium, the influence of the stresses leading to spontaneous annealing is not intensified by the action of temperature, and therefore lamellar slip is not observed until deformations of the order of 40–80% are reached. (Compare Fig. II, corresponding to a deformation of 20%, with Fig. LV, showing the results of deformation exceeding 80%.)

The annealing mechanism has been invoked to explain the features of lamellar slip, which represents a shift within existing slip bands and not slip along new planes. Of course, it is not necessary that there be an explicit difference between newly formed and older slip bands, just as it is not necessary that they differ at all. All that is required for this mechanism to manifest itself is a shear process in which, once it has begun on a slip plane, it must stop after a greater or lesser deformation and resume when conditions become more favorable. Thus, in order to confirm the existence of the mechanism described, the main point is not to detect lamellae in all metals, but to prove the fundamentally discontinuous character of deformation discovered in the experiments of Ioffe, and also of Holden (§ 2.4). Apparently, the assumption of the creation of dislocations between slip bands in the process of shear is needed in order to explain the magnitude of the distance between the bands. If the arguments set forth in §§ 5.5 and 5.6 are taken into account, then dislocations arise in these regions rather as a result of the action of the mechanism of dynamic (i.e., avalanche) slip than as a result of a calmer, continuous shear.

7.6. Slip bands as places with low resistance to shear. In the mechanisms proposed by Cottrell and Brown, the slip bands are surfaces along…

by which slip is readily effected. According to Cottrell, the cause of this phenomenon is that the source of dislocations begins and must continue to operate as long as the externally applied stresses are present. According to Brown, the unstrengthening of slip bands occurs as a result of reverse slip. If the sign of the applied stresses is changed, then, according to Cottrell’s mechanism, the source generates dislocations of the opposite sign and, consequently, the original slip band is unlikely to be the site of slip along the very same crystallographic slip planes, but in the opposite direction. According to Brown, in parts of the crystal strengthened as a result of reverse slip, slip is possible in any direction.

It is now known that in specimens deformed by alternating stresses, some slip bands have a cross section not in the form of straight lines, as in the case of stresses of one sign, but in the form of inclined steps. The latter means that not all the little steps are identical; this is shown in Fig. LVI, obtained from an aluminum polycrystal deformed by bending and then straightened again. The steps “leading upward” (dark) are distinguished from the steps “leading downward” (white) by shading. In this case the evaporated metal is deposited on the “rising portions” of the band relief, while the “falling portion” is, as it were, in shadow. The external appearance of these inclined slip steps is shown schematically in Fig. 57. On examination at high magnification it turns out that in such a specimen there is an accumulation of slip bands consisting only of lamellae forming the “rising” part of the relief, and of others formed only by steps “leading downward.” In addition, many cases are observed—too many for it to be accidental—where the relief consists of these and other types of slip bands.

Slip planes      Slip planes

Fig. 57. Bundle of slip lines arising as a result of the action of stresses of the opposite sign (schematic).

7.7. Slip bands arising as a result of fatigue stress. The hypothesis according to which the active slip planes are planes having low resistance to shear and capable of developing up to fracture is not new (Orowan^73). This proposition was demonstrated especially convincingly recently by Duce^83, who studied slip lines on specimens of nickel, copper, magnesium, aluminum, and other metals subjected to fatigue stresses in torsion. A series of Duce’s photographs, obtained from one and the same place in a nickel polycrystal at different stages of this test, is given in Figs. LVIII–LXVI. Fig. LVIII shows the electropolished and undeformed specimen at the beginning of the tests. In Fig. LIX (5000 cycles) thin slip bands have appeared; their direction is evidently not connected with the maximum shear stresses. In Fig. LX (10,000 cycles) the bands situated more favorably with respect to the applied stresses have begun to broaden; along them slip takes place to a greater extent. As the number of cycles increases, new bands are scarcely formed, but slip along each of the already existing ones steadily increases until just before frac-

destruction (Fig. LXV, 270,000 cycles) the slip bands become very wide and similar to cracks. In Fig. LXVI, obtained from another region of the specimen, part of the main source of fracture is shown, and it is seen that, while the fracture is mainly intercrystalline in character, intracrystalline ruptures occur in some places on the slip bands that had arisen earlier. (It should be noted in connection with Fig. LXVI that the slip bands in the region of the source of fracture arose under conditions of very nonuniform stresses and therefore need not necessarily be oriented in accordance with the visible direction of the maximum shearing stresses.)

Figures LXVII and LXVIII show electron microphotographs of slip bands on an aluminum specimen that was deformed in a similar manner. Since obtaining these photographs involves disturbing the surface of the specimens, it was not possible to show a series of photographs obtained from one region of a specimen after different numbers of cycles; however, the appearance of additional shear on each band in the form of new slip lamellae is clearly visible, which occurs as a result of the action of one or another of the mechanisms under discussion.

Figure LXVIII shows how a considerable shear along a slip band in one grain distorts the boundary between grains, which may serve as a cause of intercrystalline fracture. In magnesium specimens, in which only one slip system is possible in each grain, this effect is especially clearly expressed and already in the early stages of testing (Fig. LXIX) leads to intercrystalline cracking.

7.8. The influence of local slip. Since slip simultaneously causes hardening of the metal and, apparently, leads to the formation of paths of easy slip, it is interesting to note the effect of slip occurring in sharply limited regions. Traces of such slip are visible at the edges of thin scratches (Fig. LXX), and also near such surface depressions as impressions made in hardness measurements.

Nishimura and Takamura \(^{9}\) made scratches approximately \(10 \mu\) deep on undeformed single crystals \(1.5\ \mathrm{mm}\) thick. In those cases where the scratch was located perpendicular to the direction of the expected slip traces, subsequent stretching of the crystal led to only a very small increase in the density of distribution of slip bands in the region surrounding it (Fig. LXXI).

If the direction of the scratches was parallel to the expected direction of the slip bands, then there were clearly expressed regions on both sides of the scratch that were free of slip bands (Fig. LXXII). These regions are also visible on the opposite (unscratched) side of the crystal.

The probable cause of this phenomenon can be understood on the basis of the picture shown in Fig. LXX, which was obtained by Whitehead \(^{84}\) and depicts fine scratches made on aluminum by a steel sliding indenter carrying a load of only \(50\ \mathrm{mg}\). On each side of the scratch, fine slip lines appeared in places, and the active slip bands were situated exactly at an angle of \(45^\circ\) to the direction of motion of the indenter. The indicated slip bands testify to hardening of the region through which they pass, while at the same time each such band is a site of further slip, i.e., it includes a plane with a small resistance to shear, situated in the middle of the hardened region. Thus, if the slip that leads to elongation of the specimen can proceed along planes

of the kind described, and this is undoubtedly so, then this process leads to a considerable change in the distribution of slip bands. However, if the slip planes that have become operative under the influence of tensile stresses are not planes of local shear, then slip is concentrated in the region around the scratch. This occurs because the indicated region has already been hardened and only planes with a small resistance to shear in it are incorrectly oriented. The consequence of such a situation is a somewhat greater strength of the crystal with respect to slip along potential slip systems than along the operative ones. The opposite result was obtained by Rom and Kochendörfer^85, who deformed single crystals of aluminum by pure shear along one system of slip planes, then divided the specimen into parts so that the resistance to shear along another system of planes could be measured. The latter experiment evidently gives a more direct result, although there nevertheless arises the question whether the very process of dividing the crystal into parts affects it.

§ 8. SLIP AND HARDENING PROCESSES

8.1. Single crystals. One of the best-known diagrams in the physics of metals was obtained by Schmidt and Boas^86. It shows that the “stress—strain” curves of metal single crystals fall into two clearly expressed groups (Fig. 73). The figure shows that metals having a cubic structure harden to a significantly greater degree than metals with a hexagonal lattice. The results for different metals can be compared more clearly if the “stress—strain” curves are represented not at one absolute temperature, but at a temperature in each case referred to the melting point. This procedure brings the curves for both groups of metals closer together, but their clear separation remains.

Fig. 73. “Stress—strain” curves for various metals.

In § 2 data were presented according to which there is a substantial difference between the slip processes in the aforementioned groups of metals. In cubic metals slip occurs along a large number of slip bands located close to one another. In each of the bands, shear occurs only over a distance of several thousand interatomic spacings. In hexagonal metals most of the slip bands appear at early stages of deformation, and along them the shear increases continuously from the beginning to the end of deformation. In each case the total shear in a band is distributed among the platelets formed by the slip planes. In hexagonal metals many platelets are involved in the process, whereas in cubic metals only a few are. Without offering any explanation for such a division of the total slip in a band, it seems meaningful (§ 5) to suppose that the shear proceeds entirely from a single source. Therefore, in hexagonal metals several

of sources in a position to generate an unlimited number of dislocations, most of which may leave the slip-plane surface to such an extent that they are capable of causing fracture along the given slip plane. The small hardening observed permits one to assume that some dislocations remain within the plane. In cubic metals, however, dislocations accumulate on the active slip planes and near them until a distribution density of the order of one dislocation per thirty atoms is reached. Thus, in cubic metals the number of trapped dislocations per millimeter of slip plane is about \(10^6\).

8.2. Polycrystals of metals with a cubic lattice. An estimate of the distribution density of dislocations is of interest because it makes it possible, in each particular case, to compare the form of the “stress—strain” curves for single crystals and polycrystals of one and the same metal with the behavior of slip bands. Fig. 74, borrowed from Schmid and Boas, shows that in the case of cubic metals (aluminum) the degree of hardening does not change appreciably in passing from large single crystals to fine-grained polycrystalline specimens (the results of experiments on the deformation of single crystals by pure shear are not taken into account here and will be considered below). This fact may be regarded as an indication that the distribution density of trapped dislocations is independent of grain size. Thus, for a given deformation, apparently, in many cases a large number of dislocations is retained on the slip planes of a single crystal having a large surface. The same is true on the very small surfaces of the slip planes of a polycrystal. The amount of slip per band, i.e., the number of dislocations emerging at the surface over the whole range of grain sizes, changes by no more than a factor of five. Here the region of complex slip occurring near grain boundaries is not considered. Thus, in metals with a cubic lattice the number of dislocations emerging at the surface is almost independent of the total stock of dislocations. This can be explained by the fact that the surface acts as a valve, opening at a certain pressure of dislocations, analogously to the concept proposed by Cottrell to explain the nucleation of slip bands (§ 7.4). It remains unclear, however, how this valve closes when about 99% of the dislocations are still within the crystal. It is more probable that the dislocations which emerge at the surface are drawn not from the whole slip plane, but only from the part of it close to the surface. The length of this part should be, in order of magnitude,

\[ l \sim s \cdot n, \]

where \(s\) is the distance between dislocations on the slip plane and \(n\) is the number of dislocations appearing at the surface. For the values of \(s\) and \(n\) adopted above, \(l\) is approximately equal to \(3 \times 10^4\) interatomic distances, or about \(10\,\mu\).

Fig. 74. “Stress—strain” curves for aluminum single crystals of various orientations. For comparison an analogous curve is given for a polycrystal.

Since the slip must be exactly the same in all parts of the slip plane, and it is impossible to suppose that a crystal or crystallites (grains) consist of a core in which slip occurs to an unlimited extent, separated from the surface layer by an impenetrable wall, the above consideration leads to the conclusion that the entire crystal breaks up into regions separated by walls impervious to dislocations, situated at a distance \(l\) from one another.

8.3. The nature of walls impervious to dislocations. The assumption according to which the grains of a metal are divided into parts by walls impervious to dislocations, situated perpendicular to the slip plane, was first made by Taylor\(^{87}\) in order to explain hardening during deformation. In his model such walls already exist in the annealed metal and do not change during deformation. In each part of the crystal there are dislocations which move freely under the action of the applied stresses within the given region, but cannot pass into another. Hardening during deformation is explained by the fact that the number of dislocations in each isolated part of the crystal increases during deformation as a result of their generation at the dividing surfaces, while the motion of each dislocation under the influence of the applied stresses gradually becomes more difficult because of the increasing stress field due to other dislocations.

Taylor’s theory includes many assumptions that are not entirely valid in the light of modern experimental data, but it leads to an analytical expression for the “stress—strain” curve that agrees quite well with experimental results. Recently Orowan\(^{47}\) collected the principal objections to Taylor’s theory. One of the most serious objections lies in the assumption that obstacles to slip already exist in the undeformed crystal, and are not, contrary to the presently accepted point of view, the result of damage inflicted on the lattice during deformation. There is also no experimental proof of the existence, at any stage of plastic deformation, of the walls described by Taylor. They cannot, for example, be identified with the boundaries of mosaic blocks, since slip has been observed to pass, without visible disturbance of the course of the process, through any boundaries of this type.

At present it is natural to regard the impenetrable walls described above as planes with kinks. The properties of the latter were described by Honeycombe\(^{88}\). These planes are bent regions of the lattice, formed during deformation on surfaces perpendicular to the slip planes. They are evidently impenetrable to shear, since Honeycombe showed that slip bands which form after the appearance of bands with kinks pass through them only with great difficulty. On the other hand, it turned out that after repeated polishing of a specimen that removes the bands with kinks, the old slip bands reappear and can pass in any direction. The thickness of bands with kinks changes from approximately \(3\,\mu\) after 20% elongation of an aluminum crystal (purity 99.5%) to approximately \(10\,\mu\) for the same elongation of an aluminum crystal (purity 99.95%).

In general, according to Honeycombe, the purer the specimen, the greater and less regular are the distances between kink bands. This is in accordance with the assumption that the visible slip bands arise from regions close to the surface and having dimensions

\[ l \sim s \cdot n, \]

since the shear on the slip plane, which is proportional to \(n\), is smaller in less pure materials.

Honeycombe also investigated the effect of temperature and rate of deformation on bands with kinks. It was found that the distance between them was two to three times greater in specimens rapidly deformed at a temperature of 450°, than in the case of deformation at the same rate but at room temperature. It is well known that the distance between slip bands, i.e., the shear attributable to a slip band, also increases with increasing temperature, and although the effect of deformation temperature on the distance between dislocations is not known, these results are in qualitative agreement with the assumptions set out above.

As for the effect of the rate of deformation on the appearance of a band with a kink, the data relating to this are more contradictory. Whereas there is a small difference between the degree of development of bands with kinks in aluminum specimens of 99.5% purity, one of which was deformed rapidly at room temperature and the other at a rate of \(1/2\%\) per day, specimens of high-purity aluminum deformed at these same rates at 300° C differ substantially. On specimens deformed slowly at 300° C, clearly visible slip bands developed, situated at a large distance from one another. It is now well known that slip bands formed in the process of deformation at a high rate are located closer to one another, i.e., have a smaller amount of shear attributable to each band at room temperature, than in the case where they are formed at a low rate and elevated temperatures, but this difference is small (§ 9). It is probably insufficient to create a noticeable difference in the distance between bands with kinks. On the other hand, under conditions of very slow deformation and high temperature the slip bands, as was shown in § 2.4, are located far apart and arise as a result of enormous displacements.

8.4. Crystals without bands with kinks. Honeycombe showed that the appearance of X-ray asterism can be attributed to bands with kinks. In cubic metals, irrespective of the thoroughness of preparation and of tensile deformation of single crystals, X-ray asterism and bands with kinks appeared in the initial stages of deformation. On the other hand, in hexagonal metals asterism in single crystals was detected by sensitive X-ray methods only in those cases where the crystals had first been deliberately bent. It turned out that the absence of X-ray asterism corresponds to the absence of bands with kinks. If only bands with kinks hinder the emergence of dislocations to the surface of the crystal, then one may expect a considerably larger amount of shear attributable to each band in hexagonal metals than in cubic ones. At the same time, hexagonal metals at a given deformation should harden considerably less, since a significantly smaller fraction of the dislocations formed during deformation will be trapped inside the crystal.

8.5. Polycrystals of hexagonal metals. Fig. 75 shows that a polycrystal of a hexagonal metal (magnesium) hardens during deformation to a significantly greater extent than a single crystal having any orientation. The true degree of hardening of the polycrystal is comparable with the degree of hardening of cubic metals. Slip bands on hexagonal polycrystals are thin and closely spaced, similar to what occurs in cubic-

metals. The reason for this phenomenon lies in the impossibility of unlimited slip along any slip band, since this is hindered by neighboring grains. The process takes place in the following way. Slip begins on the slip plane when a dislocation source is activated by the applied stresses. The dislocations created move toward the grain boundaries, and some of them are able to pass through the boundary if the neighboring grain can deform slightly. Since in hexagonal metals there is only one slip plane, such deformation of neighboring grains is limited, and all dislocations that arise later on this slip plane must remain within the grain, which leads to hardening. An increase in stress causes activation of a larger number of sources and, thus, the number of slip bands increases. This process for tin is illustrated in Figs. XX–XXII. Tin does not have a hexagonal structure, but its crystals behave like hexagonal ones. For example, in the absence of twinning a tin single crystal deforms by one system of slip planes, and the “deformation—stress” curve for it belongs to the lower group (Fig. 73). In Fig. XX it is seen that the slip bands on the single crystal are located far from one another and break up into bundles, exactly as in cadmium single crystals. Fig. XXI is an enlarged image of such complex bands. In polycrystalline tin, considerable slip along slip bands located far from one another is impossible, but tin, unlike cadmium, has other available equivalent slip systems, and slip has the form shown in Fig. XXII. Thin, closely spaced, branching traces of slip correspond to a high degree of hardening.

Fig. 75. “Stress—strain” curves for magnesium single crystals having different orientations. For comparison, a similar curve is given for a polycrystalline specimen.

Whereas on X-ray photographs taken from deformed polycrystals of hexagonal metals after plastic tension, asterism is clearly visible, it is unknown whether they also have slip bands with bending. However, Honecombe’s experiments make it possible to think that the existence of such bands is possible. If this is so, then the connection between bands with bending and the degree of slip emerging on the surface is obvious.

In the experiments of Roma and Kochendörfer ^85 it was shown that on X-ray photographs of aluminum single crystals deformed by pure shear, asterism is not observed. Thus, according to Honecombe, in this case there are also no bands with bending. Since Roma and Kochendörfer did not polish their specimens, it is impossible to say definitely what influence deformation of this type has on slip bands. However, the photographs of unpolished specimens published by them show that slip is limited to a very small number of bands. Taking into account the mechanism described above, this means that a large part of the dislocations formed during deformation was able to leave the crystal, and, consequently, the hardening must be less—

APPENDIX

Illustrations to the article by A. F. Brown

I. Slip in Aluminum at Various Temperatures

Fig. I — −270° C. (×20,000)

Fig. II — 180° C. (×10,000)

Fig. III — 20° C. (×25,000)

Fig. IV — 250° C. (×15,000)

Fig. V — 350° C. (×20,000)

Fig. VI — 500° C. (×24,000)

All specimens were rapidly deformed in shear by approximately 200%.
Figs. I–VI are electron micrographs obtained by the oxide-replica method.

II. TRANSVERSE SLIP

Fig. VII. Pronounced transverse slip in $\alpha$-brass after deformation of 0.09% ($\times 2000$).

Fig. VIII. Internal transverse slip in $\alpha$-brass. Shear deformation 0.1% ($\times 1000$).

Fig. IX. Aluminum crystal stretched by 9% at 400°C ($\times 100$).

Fig. X. Aluminum single crystal stretched by 3% at room temperature ($\times 2350$).

III. CONSTANCY OF THE ELEMENTARY MAGNITUDE OF SLIP IN ALUMINUM

Fig. XI and XII. Aluminum single crystals, stretched by 0.5% (Fig. XI) and 4% (Fig. XII), showing displacement of the edges of interference fringes as a result of slip.

Fig. XIII. Wavy slip bands on an aluminum single crystal, deformed by 12% at 450°C at a constant deformation rate of 1% per day (×6).

Fig. XIV. Part of the fine structure of one of the slip bands shown in Fig. XIII. Electron-microscopic photograph from a shadowed oxide replica (×25,000).

IV. Slip in Cadmium

Fig. XVII. Fine structure of the slip band of a cadmium single crystal \((\times 25{,}000)\).

Fig. XVIII. Direct slip in a cadmium single crystal \((\times 15{,}000)\).

Fig. XIX. Direct slip in a cadmium single crystal \((\times 15{,}000)\). (Figs. XVII—XIX are electron micrographs of replicas obtained by the polymerization method.)

VI. THE INFLUENCE OF THE METHOD OF SURFACE PREPARATION ON THE APPEARANCE OF SLIP BANDS

Fig. XXIII. Transitional region on a single crystal of aluminum ($\times 250$).

Figs. XXIV and XXV. Electron micrographs from zones, mechanically ground and unground, respectively, on an aluminum single crystal. Elongation 5%. Replicas obtained as for Figs. XVII–XIX ($\times 10\,000$).

Figs. XXVI and XXVII. Electron micrographs from zones, mechanically ground and unground, respectively. Copper single crystal. Elongation 5%. Replicas obtained as for Figs. XVII–XIX ($\times 16\,000$).

V. SLIP IN TIN

Fig. XX. Slip bands located far from one another in a tin single crystal \((\times 10\,000)\).

Fig. XXI. Fine structure of a slip band in a tin single crystal \((\times 25\,000)\).

Fig. XXII. Slip and transverse slip in a tin polycrystal \((\times 10\,000)\).

(Figs. XX–XXII were obtained in the same way as Figs. XVII–XIX.)

VII. POSITIONS OF DISLOCATION GROUPS REVEALED AS A RESULT OF THE PRECIPITATION PROCESS

Fig. XXIX. Specimens of aluminum with 4% copper, aged for three hours at 190°C, show selective precipitation along the slip plane (×15,000).

Fig. XXX. Specimens of the same alloy, aged for 12 hours at 150°C, show selective precipitation along subgrain boundaries (×15,000).

VIII. SLIP BANDS

Fig. XXXIII. Fine slip lines located on both sides of an ordinary slip band \((\times 25\,000)\).

Fig. XXXIV. The same \((\times 50\,000)\).

Fig. LII. Slip bands at the point of intersection with a twin having a kink \((\times 25\,000)\).

Fig. LIII. The same \((\times 25\,000)\).

Fig. LV. Slip bands formed at a temperature of \(-180^\circ\mathrm{C}\), distorted by another shear process lying in a direction almost coincident with the external surface of the specimen. Deformation about 80% \((\times 15\,000)\).

IX. FLOW WITHOUT SLIP

Fig. XXXV. Slip bands terminating at the boundary of the internal oxidation zone. Copper–aluminum alloy. Optical micrograph \((\times 1500)\).

Fig. XXXVI. Same as XXXV. Electron micrograph. The replica was obtained by the decoration method \((\times 5000)\).

Fig. XXXVII. Al + 7% Mg \((\text{all Mg in solid solution})\). Slip bands visible after 5% elongation \((\times 1000)\).

Fig. XXXVIII. The same alloy, aged for 48 h at \(200^\circ\text{C}\). No visible slip bands after 5% elongation \((\times 1000)\).

Fig. XXXIX. The same alloy, aged for 24 h at \(200^\circ\text{C}\). After 5% elongation, several weakly expressed slip bands are visible.

Fig. XLII. Griffith cracks on a quartz tube etched with sodium vapor (×39).

Fig. XLIII. Crack in viscous glass. Electron micrograph obtained by pressing aluminum foil into glass, followed by oxidation of the aluminum and use of an oxide replica. The structure, having the appearance of orange peel, arises on account of the aluminum (×35,000).

Fig. XLV. Change in the character of the microetching structure at the boundary of an aluminum grain. Oxide replica, heavily shadowed with a gold–palladium alloy (×25,000).

Fig. XLIV. Surface of aluminum with a slip band. The replica was obtained by a polymerization process and shows both the fine structure and the microetching pattern (×25,000).

Fig. XLVI. Microetching structure of an aluminum surface parallel to (100) and similar in appearance to the fine structure of a slip band. Oxide replica shadowed with a gold–palladium alloy (×25,000).

Fig. LVI. Under the action of stresses of the opposite sign, all slip steps in the band slipped uniformly (see Fig. LVII). Shaded oxide replica (×20,000).

Fig. LXX. Fine slip at the edges of a weak scratch. Aluminum. Oxide replica (×40,000).

Fig. LXXI. A scratch perpendicular to the presumed direction of the slip bands has little effect on it (×100).

Fig. LXXII. If a scratch is applied parallel to the presumed direction of the slip bands, then an area free from shears appears, located on both sides of the scratch (×100).

XII. Slip bands arising under the action of alternating stresses

Fig. LVIII. Nickel, initial state (×400).

Fig. LIX. Nickel after 5000 cycles (×400).

Fig. LX. Nickel after 10,000 cycles (×400).

Fig. LXI. Nickel after 25,000 cycles (×400).

Fig. LXII. Nickel after 50,000 cycles (×400).

Fig. LXIII. Nickel after 100,000 cycles (×400).

XIII. SLIP BANDS ARISING UNDER THE ACTION OF
ALTERNATING STRESSES

Fig. LXIV. Nickel after 200,000 cycles (×400).

Fig. LXV. Nickel after 270,000 cycles (×400).

Fig. LXVI. Nickel, more than 270,000 cycles. Part of the main fracture zone (×400).

Fig. LXVII. Aluminum, 5000 cycles. Electron micrograph (×8000).

Fig. LXVIII. Same as Fig. LXVII (×8000).

Fig. LXIX. Magnesium, 70,000 cycles (after fracture) (×350).

Fig. LXX, LXXI, and LXXII on Plate XI.

XIV. SLIP BANDS IN ALUMINUM DEFORMED AT A VERY HIGH RATE

Fig. LXXVI. Fine structure of a slip band \((\times 2000)\).

Fig. LXXVII. Region in which the traces are situated close to one another \((\times 10\,000)\).

Figs. LXXVI and LXXVII were obtained from an aluminum single crystal deformed by 4% at a rate of 100% per sec. Both photographs were taken in an electron microscope using an oxide replica.

more than in specimens deformed in ordinary tensile machines. In essence, the authors mentioned found that the indicated situation does in fact occur.

§ 9. THE INFLUENCE OF THE RATE OF DEFORMATION ON THE PROCESS OF SLIP

If the temperature is kept constant, then very slow deformation leads to the appearance of slip bands that are situated at large distances from one another, and the bands themselves are wider than those formed at rates used in ordinary mechanical tests. Thus, a decrease in the rate of deformation leads to a considerable shear within each slip band. A comparison of Figs. XIII and XIV with Figs. V and VI shows that, at least in the case of aluminum, this additional shear leads to an increase in the number of slip lamellae within the band.

However, at rates of deformation much greater than those usually used in tensile tests, the situation becomes more complicated.

Parker and Smith \(^{39}\), and also Crossar \(^{45}\), for example, reported that at very high rates of deformation occurring in impact tests, slip bands, although situated closer to one another, appear broader and include a larger number of active slip planes. There are several experiments confirming this situation. In Figs. LXXVI and LXXVII are shown the results of experiments on aluminum single crystals stretched at room temperature by \(4\%\) at a rate of deformation of about \(100\%\) per second. At a normal rate of deformation such conditions should have led to a spacing between slip bands approximately equal to \(10\,\mu\), each band consisting of one or two ordinary lamellae \(2000\,\text{Å}\) thick. From Fig. LXXVII it is evident that the spacing between slip bands is from \(0.5\) to \(1.5\,\mu\). Fig. LXXVI (obtained from another crystal) shows the fine structure of one of the bands. There are four lamellae, and the slip distance for each is much less than \(1000\,\text{Å}\). However, the fine structure of the overwhelming majority of slip traces formed in this way is not resolved, and they appear simply as a broad band.

Thus, at very high rates of deformation, slip processes are transferred more readily to neighboring slip planes, and a smaller shear is observed on each of the active planes. This is very similar to what occurs in bending and in the “die effect” (§ 5.5): in these cases deformation also occurs extremely rapidly.

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