MECHANICAL PROPERTIES OF CRYSTALS1

A. F. Ioffe, Leningrad.

Elastic Aftereffect

The electrical theory of crystal lattices constructs a scheme of an elastic body which corresponds, to a certain degree, to the properties of crystals. The true behavior of solids, however, is considerably more complex than this scheme permits one to assume. Some experimental facts apparently contradict the basic conception of a crystal as an equilibrium state of a solid body. To explain these contradictions we must delve into the details of the mechanism of these phenomena.

First of all we shall turn to phenomena occurring below the limit of elasticity, i.e., corresponding to very small deformations. Here, from the very beginning, we encounter difficulty in explaining elastic aftereffect, a phenomenon characteristic, to a greater or lesser extent, of all solid bodies. A force applied to any solid body produces a stress that changes continuously with time. Using sensitive methods of observation, we can detect something like creep, which follows the initial deformation for many months after the force that caused the deformation has been removed. When the force is removed, the main part of the stress disappears with the speed of sound. But a certain residual stress remains and disappears slowly, asymptotically approaching the initial sta—

to its state. After a sufficiently long time the body is completely restored, and no residual properties can be noticed in it. By carrying out the deformation infinitely slowly, we can obtain a reversible process. Conversely, at a finite rate the deformation is irreversible and is accompanied by a loss of energy. In a repeated cyclic process, elastic aftereffect will lead to elastic hysteresis. As a result of elastic aftereffect, oscillations are damped more strongly and the sound becomes duller. Further, Kelvin found that under the prolonged action of oscillations the damping becomes ever stronger; he called this effect elastic fatigue. The original properties are restored either after heating or after a prolonged rest. Thus, for example, an oscillating wire gradually becomes fatigued and returns to its normal state only after some rest.

Similar anthropomorphisms are applied to elastic processes on the basis of the observed analogy between the phenomenon of elastic aftereffect and the properties of human memory. In fact, from the facts described above it follows that the shape and size of a solid are not a single-valued function of the external forces, as our reasoning would require, but depend on its entire preceding history. Every action causes a change that disappears very gradually. In every given state we find traces of the action of forces long since removed, but still continuing to act even now.

It would be quite hopeless to try to explain phenomena like those we have described, since we are considering a generally regular crystalline lattice. It is necessary, however, to note that these phenomena have never been observed in single crystals. The samples usually studied are aggregates of small crystals, distinguished by extreme complexity and inhomogeneity. As Maxwell showed in his time, in materials of this kind phenomena similar to internal friction or elastic aftereffect may occur. In fact, the greatest elastic aftereffect has been found in the most inhomogeneous bodies, such as rubber, wax, silk, etc. This effect

turns out to be considerably smaller in metals and in glass, and especially small in quartz fibers. The question arises whether inhomogeneity is the sole cause of aftereffect. Would a correctly built single crystal not prove to be entirely free of any aftereffect?

For such an investigation I chose quartz, since quartz can easily be obtained in the form of specimens of suitable dimensions. Its elastic limit is very high, and up to the moment of rupture no residual deformation can be observed in it. Its hardness and transparency make it possible to reduce the number of experimental errors. Finally, I was led to the study of quartz precisely by the question proposed to me by Röntgen for my doctoral dissertation. This question was: is deformation or stress the primary cause of piezoelectricity? Observing the appearance of piezoelectricity on the surface of a quartz plate during aftereffect, we must ascribe it to deformation, since neither the load nor the stresses, consequently, change. If, on the contrary, stress were the primary cause of piezoelectricity, charges should not appear under the action of a constant stress during aftereffect. I personally doubted the existence of aftereffect in quartz crystals in general, and began by measuring the magnitude of this presumed phenomenon of aftereffect.

Fig. 1.

In the first method I employed, I used piezoelectricity to measure the aftereffect. A thin quartz plate, cut perpendicular to the piezoelectric axis with the principal axis directed along the width of the plate, was provided with silver electrodes and subjected to a load. The electrodes were separated from the grounded ends by means of narrow insulating strips, as shown in Fig. 1.

One of the electrodes was grounded. The other was first briefly grounded before the load, and then connected to an electrometer. A small charge appearing

in the electrometer, can be readily explained by leakage from the free quartz strips separating the electrodes from the ferrules at the ends. The length of the electrodes was about 80 mm, and the width of the insulating gaps about 0.1 mm each. Assuming that half of the charge produced by the load flows to the electrode, we should expect that about 0.1% of the total charge produced by the load flows to the electrode from the side. Thus the accuracy of the method was about 0.1%, and such were also the deviations of the electrometer. Assuming that, if an elastic aftereffect existed, it would produce charges in exactly the same way as the elastic stress, we must conclude that the observed aftereffect in the plate under investigation does not exceed 0.1% of the deformation.

Fig. 2.

However, the piezoelectric method is connected with a particular crystal and a particular crystallographic orientation of it. Moreover, this method contains a hypothesis which, though not devoid of meaning, is also not free from objections. I therefore turned to a more general method, allowing greater accuracy. I measured the bending of a plate resting on two prisms. In order to exclude the influence of displacement of the edges of the prisms supporting the plate, and deformation of the entire apparatus, I investigated the change in the distance between the middle of the deflected plate and a glass plate resting by means of three screws on the quartz plate. The arrangement is shown in Fig. 2.

Monochromatic sodium light was reflected simultaneously both from the lower surface of the glass plate, coated with a semitransparent layer of silver, and from the upper surface of the quartz, producing interference fringes. During the slow bending of the quartz plate the fringes shift. Each time that the distance between the two plates increases

on half-waves of the applied sodium light, the dark interference fringe shifts to the position of the neighboring fringe. The position of the fringes relative to the optical cross was noted by means of a tube, and from time to time the course of the deformation was measured.

At first the previously described piezo-electric plate was investigated. The total deflection was about 1,000 fringes (500 wavelengths). Contrary to expectation, a slowly creeping deformation was discovered, exceeding the limits of observational error. Before, however, attributing this deflection to elastic after-effect, we must consider possible systematic errors created by secondary phenomena. These phenomena can be predicted on the basis of the second law of thermodynamics.

Thermal expansion requires that every change in dimensions should produce a secondary change of temperature, which in turn diminishes the deformation. While the temperature change disappears, the deformation grows from its initial adiabatic value to its final isothermal value. Thus an effect is produced which simulates the phenomenon of after-effect. The difference between the isothermal \(S\) and adiabatic \(\sigma\) moduli is equal to

\[ S-\sigma = J \frac{a_1 a_2 T}{\rho C_v}, \]

where \(a_1\) and \(a_2\) are the two coefficients of thermal expansion parallel and perpendicular to the axis, \(T\) is the absolute temperature, \(J\) is the mechanical equivalent of heat; \(\rho\) is the density, and \(C_v\) is the heat capacity at constant volume. For quartz we find:

\[ \frac{S-\sigma}{S} = 0.0021. \]

A total deflection of 1,000 fringes will be accompanied by a secondary deformation of 2.1 fringes. However, calculating the rate of this deformation, we find that this phenomenon cannot be considered responsible for the observed elastic after-effect. In fact, the deflection compresses the upper layers and stretches the lower layers of the quartz plate. The first

will be heated, and the second cooled. During deformation there arises a temperature difference of \(2.5 \cdot 10^{-2}\) degrees Celsius between the surfaces, and thereby a temperature gradient is produced in the plate. Calculating the process of equalization of the temperatures through the thermal conductivity of the plate, we find that the remaining gradient, and hence the remaining deformation, fall to \(1\%\) of their value within the first 0.07 seconds. We begin to note the effect only after this time has elapsed. The residual thermal effect is negligibly small and does not exceed the limits of accidental errors.

Apart from the secondary thermal phenomenon, the deformation of a piezoelectric crystal must be accompanied by secondary electrical effects. A calculation based on the second law of thermodynamics determines the electrical deformation as \(0.9\%\) of the initial deformation, i.e. as 9 bands for a total deformation of 1000 bands. The low electrical conductivity of quartz suggests that the secondary electrical deformation will disappear much more slowly than the thermal one. It is in fact known that a charge insulated by quartz drains away gradually over the course of several days.

We can easily imagine the distribution of charges in a bent quartz plate. Let us mentally divide the plate lengthwise into a large number of thin sheets (Fig. 2). The upper sheet will be especially strongly compressed. Downward the compression gradually decreases, reaches zero in some middle layer, and then passes into tension as one approaches the lower surface. Each of the sheets may be regarded as a plate electrified by virtue of the stress, bearing opposite charges on its two surfaces. The charge is proportional to the stress. The boundary between layers will be charged positively by the upper layer and negatively by the lower layer, but the charges will be unequal. A difference charge of one and the same sign will remain on each boundary surface, creating throughout the interior of the crystal plate a volume charge of the same sign. A charge of the opposite sign, but of equal magnitude to the volume charge, arises on the upper and lower surfaces (Fig. 3). In what way can these now be neutralized

charges? The direction perpendicular to the surface, i.e. the direction of the greatest gradient, is at the same time the least conducting direction in quartz (as perpendicular to the principal axis). The direction coinciding with the principal axis has, approximately, a 1,000 times greater electrical conductivity—this is, in the given plate, the direction parallel to the width of the plate. However, the geometrical conditions (path length and cross section) worsen this direction by approximately 1,000 times. It is natural to suppose that, by coating the surface, for example, with a conducting layer of silver, we can facilitate this second path and somewhat accelerate the mutual neutralization of charges after bending. The acceleration of the neutralization process should also accelerate the corresponding deformation. This supposition was confirmed experimentally, though not yet convincingly enough to make it possible to prove with certainty the electrical character of the elastic after-effect in quartz.

Fig. 3.

It was desirable to remove the volume charge by a more direct and more effective method. I therefore tried to increase the internal electrical conductivity of quartz by means of ionizing rays. Just at that time such an experiment was carried out by J. J. Thomson. However, the increase in current observed by him under the action of X-rays was explained by Röntgen as a current passing around the solid dielectric through the air. It was assumed—Röntgen in particular was convinced of this—that no true increase in electrical conductivity occurred here. Initial observations on the effect of radium rays on the rate of after-effect gave, however, a negative result. Nevertheless, I noticed at the same time that, while the approach or removal of the radium had almost no effect on the deformation, the process itself accelerated with each day of experiments with radium rays. It turned out that, under the influence of the rays, the electrical conductivity increases gradually over the course of many

days, approaching a certain maximum value many times greater than the initial electrical conductivity of quartz. By increasing the electrical conductivity to the highest possible value, I succeeded in making the after-effect proceed extremely rapidly and in obtaining a state in which, apparently, the entire after-effect was completed within a measurable interval of time. In this case the whole after-effect proved equal to the value calculated for the secondary electrical deformation; namely, it amounted to 7.2 fringes with an initial bending of 800 fringes. Fig. 4 shows the deformation as a function of time for different states of electrical conductivity. The lower curve corresponds to normal conditions, the uppermost curve to the case of the greatest electrical conductivity, while the straight line corresponds to the calculated purely electrical deformation.

Fig. 4.

The total observed elastic after-effect in a piezoelectric plate never exceeded the secondary electrical effect. Thus the elastic after-effect is altered under the action of ionizing rays, which can hardly directly produce elastic phenomena. In those cases in which I was able to measure the total effect, it proved equal to the electrical one. Consequently, there remains no basis for assuming that there exists an elastic after-effect independent of the expected thermodynamic secondary phenomenon. Nevertheless, the limit for the smallest possible real elastic after-effect still lay insufficiently low. I then investigated quartz plates oriented in various ways with respect to the crystallographic axes, and found that the observed elastic effect is always smaller than the electrical one.

A calculation shows that a plate cut so that its length is parallel to the principal axis should not acquire volume charges under the action of bending. In such a plate secondary electrical deformation is impossible, and therefore the entire measured elastic after-effect must be real. However, in such a plate no after-effect whatever was found. The total deflection was increased to 3000 fringes. Beginning with 1 second and continuing the observation up to 24 hours after deformation, I could not observe a greater displacement than 0.1 fringe. An analysis of all possible accidental errors led to a limit of accuracy of about 0.1 fringe. Hence we may conclude that the real elastic after-effect in quartz, if it existed at all, could not have been greater than \(3\cdot 10^{-5}\) of the deformation. This is in fact a very low limit, if one takes into account the magnitude of the elastic after-effects usually observed, reaching \(3\cdot 10^{-2}\) of the deformation, and sometimes still higher.

This result, obtained in 1904, was confirmed in 1906 by M. Brillouin, who used quartz springs and proved the absence of after-effect. This circumstance has now found wide application in radio engineering. Owing to their negligibly small damping, piezoelectric oscillators made of quartz are distinguished by an exceptionally sharp resonance at high frequencies.

Subsequently this result was generalized. It was shown that single crystals of rutile, tungsten, zinc, copper, and bismuth possess no elastic after-effect of any kind below the elastic limit.

The first contradiction in the theory of crystal lattices that we encountered in the domain of ideal elasticity is now resolved in the sense that all complicating phenomena, such as elastic after-effect, elastic hysteresis, elastic fatigue, etc., are absent in pure single and correctly built crystals. These effects arise from irregularities in inhomogeneous bodies, from the crossing of the limit of elasticity or strength in certain small regions within the body, and from interaction between crystalline grains.

New difficulties confront us as soon as we reach the elastic limit and enter the region of plasticity.

Elastic Limit.

The results of investigations of metals and other fine-crystalline aggregates are usually reduced to the following definition of the elastic properties of a solid body: throughout the entire region in which the body is ideally elastic, it obeys Hooke’s law; in other words, until a permanent change has appeared, the stress is proportional to the load. Deviation from Hooke’s law, according to the definition given above, is observed by the ordinary methods of testing laboratories and indicates the appearance of permanent deformation. It is usually assumed that the limit of proportionality can be identified with the elastic limit. More careful investigations have shown, however, that deviations from Hooke’s law are sometimes observed before the elastic limit is reached, and that the limit of proportionality decreases continuously as the accuracy of the measurements increases. But Hooke’s law is nothing other than the first term of the series representing the force function expanded in \(\Delta r\). The force

\[ f=-\frac{mA}{r^{m+1}} \]

may be represented in the form

\[ f=f_0+f'\Delta r+\frac{1}{2!}f''(\Delta r)^2+\frac{1}{3!}f'''(\Delta r)^3+\ldots \]

Putting \(f_0=0\) and neglecting terms containing \(r\) in powers higher than the first, we obtain Hooke’s law

\[ f=f'_0\Delta r . \]

Thus it becomes evident that the limit of proportionality cannot be detected, since in fact one cannot expect that proportionality exists between force and deformation. The observed limit of propor-

…of proportionality is a function of the accuracy of the measurements and of the magnitude of the experimental errors. The circumstance that this quantity has been measured and used in technical tests shows that, before any deviation from Hooke’s law becomes noticeable in rough measurements, a new phenomenon appears—permanent deformation; in other words: plastic deformation arises under such small loads that the curve representing the relation between $f$ and $r$ has not yet had time to deviate noticeably from a straight line.

In technical tests a third limit is also used, at which an easily observable deformation occurs, a minimum of half a percent. This limit is called the yield point. In Fig. 5 a typical curve is presented, on which both the elastic limit ($E. L.$) and the yield point ($Y. P.$) are visible.

Fig. 5.

Microscopic investigation of the new phenomenon of permanent deformation, occurring when the elastic limit is passed in crystalline aggregates, has shown that yielding consists of a series of small elementary shears either between individual crystalline grains or within single crystals. In the first case we observe, on the polished surface, large steps in the direction of the maximum shearing stress, i.e. at an angle of $45^\circ$ to the axis of the tensile forces. In the second case, slip planes are observed under the microscope within individual crystals, connected with certain crystallographic planes of theirs, such as: the basal plane in hexagonal crystals, the planes of the cube or of the rhombic dodecahedron in crystals of cubic symmetry.

It might have been supposed that the study of single crystals would make it possible to reduce these complex phenomena to simple ones. In contrast to elastic after-effect and elastic fatigue, plastic deformation can, however, also be observed in single crystals, such as rock salt, gypsum, zinc, aluminum, etc. This effect is easily noticed if a crystal of rock salt, heated to 600° C, is subjected to bending or torsion. In its plasticity rock salt then resembles wax. Nevertheless, the bent or twisted crystal continues to remain transparent and seems to retain the wholeness and strength of a normal crystal. Once released, however, it no longer returns to its original form, but remains bent. The existence of a bent crystalline lattice, preserving this state without the participation of external forces, would obviously contradict the conception of the lattice as an arrangement corresponding to a minimum of potential energy.

It should be noted that curvature of the external form by no means signifies that the crystalline lattice is necessarily curved, for, of course, we can cut from a normal crystal a specimen of any desired form without at all changing its internal structure. Indeed, suppose that a series of layers of the crystalline lattice moves during plastic deformation strictly parallel to some crystallographic direction. We may imagine that the crystal continues to exist when the atoms of the slipping layer have come into a new mutual position, analogous to that in the normal lattice, corresponding to a distance equal to an integral multiple of the atomic distances, as shown in Fig. 6.

In this way one can obtain any residual deformation without changing the normal structure of the lattice. Such a mechanism is fully compatible with the theory of lattices. Several simple experiments convince us, however, that the explanation just given does not correspond to the true plastic deformation of crystals. Indeed, the deformation corresponding to Fig. 6 ought to restore the crystal

and return it to the normal state; thus we might expect that the bent specimen possesses the properties of a normal crystal. Further, the slip surfaces must be planes, following the planes of the regular crystal lattice. Instead of planes, however, we find slip surfaces bent parallel to the externally bent form of the crystal. This shows that the rock salt has undergone some internal change. On the other hand, as has already been said, we cannot agree with the simplest picture following from the shear experiments, namely, to imagine that a bent lattice can exist without being held in this form by external forces.

Fig. 6.

Another way of explaining it would be to attribute the curvature to the influence of internal stresses remaining as a result of the nonuniform deformation of bending or twisting. If this explanation were correct, the curvature should diminish if a small piece were cleaved from the bent crystal so that the difference in stresses would not be large; an experimental test of this consequence, however, gave a negative result. We split a bent rock-salt crystal into two curved pieces. One of them we heated to \(700^\circ\) for twenty-four hours. At this temperature even a negligible force produces a residual deformation in the crystal. If there existed in the crystal internal stresses sufficient in magnitude to bend the crystal, these stresses should have disappeared upon heating. After carefully cooling it, we again put both pieces of the crystal together and were able to convince ourselves that both the external form and the internal changes, expressed in the curved

on cleavage surfaces did not disappear upon heating. Although internal stresses do exist in a plastically deformed crystal, they are neither the sole nor even the principal cause of the change in the properties of the crystal.

Consequently, the puzzle of the plasticity of crystals is resolved neither by the mechanism of slip nor by the influence of internal stresses.

We needed to learn what change plastic deformation actually produces in the crystal lattice. As soon as, thanks to Laue’s discovery, a method had been developed for investigating crystal lattices by X-rays, M. V. Kirpicheva and I applied this method to the study of plastic deformations. We did indeed find a substantial change in the Laue pattern from plastically deformed crystals. Instead of separate, sharp spots, we saw a pattern of radial beams issuing from each spot, as is now observed in the study of fibrous structures. The explanation of this phenomenon was difficult because of the complexity of the nonuniform deformation itself.

We therefore began to investigate uniform deformation, such as: one-dimensional compression and tension. Hydrostatic pressure, easily obtained by means of hydrostatic pressure, had to be excluded, since shear is the principal characteristic feature of plastic deformation. Indeed, with uniform deformation the understanding of the observed phenomena proves considerably easier. A second important improvement was the visual observation of the Laue pattern on a fluorescent screen during deformation. In a dark room, after 10–15 minutes, the sensitivity of the eye proves sufficient to observe the brightest spots of the Laue pattern. We used an ordinary X-ray tube at 70 kilovolts and 10 milliamperes. In tension it was not possible to notice visually the load causing plastic deformation. However, by photographing on one and the same plate the patterns of a weakly and a strongly loaded crystal and by passing the X-rays through very thin diaphragms, we were able to...

the plastic deformation can be measured. To study changes in angles it is necessary to use a continuous spectrum; to study changes in dimensions—a monochromatic one. Elastic bending and torsion can be observed and easily measured both on a photographic plate and on a fluorescent screen.

For a qualitative study of the phenomena occurring during plastic deformation, the visual method has the advantage that it shows not the final result, but the entire process as a whole.

Gradually increasing the load on the crystal, we finally reach a certain limit, after which the Laue pattern suddenly changes (Fig. 7). Instead of a system of spots corresponding to reflections from one series of planes of the crystal, several intersecting systems of spots appear in the pattern, corresponding to several crystals. The number of such systems increases as the deformations progress. All these systems, however, have one common point on the pattern—and hence one common plane in the crystal. Thus it can be seen that the single crystal has been broken up into several pieces, despite the fact that the transparency and rigidity of the resulting aggregate give it the appearance of a single crystal. The displacement and rotation of the fragments correspond to slip in a certain definite crystallographic plane (plane 110 in the case of rock salt) and to rotation about an axis perpendicular to the plane that has remained unchanged,

Fig. 7.

corresponding to a bright, unchanged spot. This axis may be either perpendicular to the plane of slip, or it may lie in this plane (for example, in rock salt this is the direction of the axis \([100]\)).

The load at which the first change in the X-ray photograph is observed corresponds to some irreversible destruction of the lattice and may be called the yield point. Further investigation showed that the first irreversible shift does not always occur immediately upon reaching this limit, and that subsequent shifts are separated from one another by intervals of time. The measured limit therefore depends to some extent on the rate of loading. This error progressively decreases when we pass to higher temperatures. But even at room temperature this error in rock salt can be reduced to \(1\%\).

The existence of such a definite limit has been disputed many times. Experiments carried out on metals and glasses led to the opinion that “everything flows more or less slowly.” On this basis the notion of a definite limit was accepted reluctantly. Of course, it is impossible to establish that a given body does not flow, so long as the very term “to flow” is not restricted by some minimum rate. We can, however, assert that at \(500^\circ\) C. a load of \(2\%\) below the measured limit, acting on a specimen for 24 hours, causes not the slightest visible change in the X-ray photograph; whereas a load exceeding this limit by \(2\%\) causes a change after one second. Physically this means that this limit has a definite physical significance, even if the practically measured value exceeds the theoretical limit corresponding to an infinitely long time.

Further, we became convinced that this phenomenon depends on internal stresses, but does not depend on the size of the crystal. We selected crystals of one and the same crystallographic orientation, but with different ratios of cross-section to perimeter. The yield point, measured

at four temperatures between 20° and 600° C, was always proportional to the area of the cross section, but did not depend on the perimeter (Fig. 8). The limit proved also to be one and the same for compression and tension. Rock-salt crystals of different origin, and consequently with different impurities and with a surface treated

Fig. 8.

in different ways, for example: dissolved in water, polished, cleaved, etc., all had one and the same limit, differing by no more than 2%. We may, therefore, conclude that the yield point measured by X-rays has a quite definite physical meaning.

We then studied the dependence of this limit on temperature and found that the yield point rapidly decreases with increasing temperature and reaches zero at the melting point (Fig. 9). Two heteropolar crystals, such as NaCl and NaNO₃, and then the metals Al and Mg, behaved very similarly in this respect. We tried to determine whether the limit

of the yield point at the melting point directly to zero, or else, while retaining a certain definite but small value, it drops abruptly to zero already in the liquid state. Experiment could answer only in the following way: the jump is less than a certain measurable quantity. We established that this jump is less than

\[ \frac{1}{100} \]

of the value of the yield point at room temperature. At room temperature this limit proves to be equal to approximately 3 kg. We therefore loaded the crystal with 30 g and, gradually raising the temperature, observed the change in the pattern until the melting point was reached. The sharpness of the edges proved that we had not yet reached the melting point. In addition to this evidence, the onset of melting is clearly revealed by the appearance of irregularly scattered X-rays and by the general illumination of the fluorescent screen.

Fig. 9.

One may think that the coincidence of the melting point with the point where the fracture limit is equal to zero has a real physical meaning, if one assumes that the mechanism of melting is equivalent to a kind of destruction that causes the existence of a limit at low temperatures. When the smallest shear stress is sufficient to cause destruction, the existence of the lattice becomes impossible, and the heat absorbed by the crystal is expended in bringing the atoms into a new equilibrium position. Possibly this may explain the circumstance that we know of no examples of superheated crystals, whereas all other limits, as a rule, can be exceeded.

In an article published in the summer of 1926 in the Zeitschrift für Physik, Georgiev points out that he found in bismuth a final limit of elasticity at the melting temperature. The form of his curve, which apparently at first descends straight to zero and suddenly, just before the melting point itself, changes its course, suggests that his results were distorted by some process of recrystallization. We are now engaged in a more detailed investigation of the same substance.

The yield point depends on the direction of the tensile forces in the crystal that produce slips in the plane (110). Slip occurs in a definite crystallographic plane, and therefore we may expect that slip is connected not with the shearing stress in this plane, but with the tensile force. In fact, the yield point depends on the orientation of the tension. This is shown by Fig. 9, where curve I corresponds to the direction [100] for the applied tensile force, curve II is measured for the direction [110], and curve III for [111].

It should be noted that all the curves reach zero at one and the same point—the melting point. The numerical value of the ratio of stresses for two directions does not, however, correspond to equal shearing stresses in the planes (110). The yield point must, consequently, depend not only on the magnitude of the shearing stress in the plane (110), but also on the direction of sliding in this plane. This result is nothing unexpected, if one merely recalls the structure of this plane (110) in such a crystal as rock salt. Indeed, in sliding along the direction [110] a row of positive ions slides over negative ones, much like, for example, a train moving along rails. In motion in the direction [100] in the same plane, the positive ions are alternately attracted by negative ions and repelled by positive ions, so that this motion resembles the movement of a cart over bumps and ruts. Although in general this conclusion is satisfactory, we nevertheless, up to...

it has so far not been possible to find the dependence of the yield limit on the orientation in the plane (110), and to study the possibility of shears in other planes besides the plane (110).

Mechanism of Plastic Deformation.

Observation, by means of X-rays, of the boundary of the destruction of a crystal gives the magnitude of the force that causes an irreversible change in the structure of the crystal. The crystal turns into powder, and the small fragments, having become interlocked quite disorderly, slide and rotate relative to one another. Plastic deformation does not impair the transparency of the crystal; consequently, in this case, during destruction, no holes of the size of the wavelength of light or larger are formed in the crystal. Further, the strength of a crystal destroyed in this way is not reduced, but, on the contrary, a greater stress is required to rupture a plastically deformed crystal than to rupture a single crystal. We may therefore conclude that, in plastic deformation, the particles move apart by distances not exceeding the magnitude of interatomic distances. The combination of these conditions with the conditions of slip, which require that slip occur only in one plane, namely, for example, (110), i.e. in the cleavage plane, limits the disorder in the arrangement of the possible fragments. As a result of plastic deformation by compression or tension, we obtain not a compressed powder, but something resembling a fibrous structure, possessing a definite axis or else a definite plane of symmetry. Complete restoration of the specimen after such destruction cannot be achieved by reverse deformation. The process that takes place in plastic deformation is irreversible not only in the sense that the work performed is converted into heat, but also in the sense that, even by performing work in the opposite direction, we cannot restore the initial state.

The undoubted internal destruction that takes place during plastic deformation does not, however, exclude the possibi—

stics of residual phenomena of another type, which we mentioned when considering the bending of rock salt, namely a deformation consisting of a series of slips parallel to one of the crystallographic axes without rotation. X-rays cannot detect such a deformation, since in this case the orientation of the atomic planes reflecting the rays does not change. However, such a deformation can be observed by other methods, such as: precise measurement of the dimensions and shape of the crystal, or else examination of the specimen in polarized light.

M. A. Levitskaya measured the angle between the ends of a bent rod of rock salt by means of a beam of light reflected from three mirrors fastened to the rod. She carefully noted the first appearance of residual deformation and compared it with the load necessary for the appearance of an irreversible change visible in the radiograph. These measurements showed that there exists a preliminary phase of plastic deformations, in which slip occurs without rotation, as is approximately shown in Fig. 6. This plastic deformation, although likewise irreversible, and likewise producing heat, can be eliminated by the action of the opposite sign.

Consequently, we must distinguish two limits:

1. The elastic limit, causing the first residual deformation of slip of the crystalline layers parallel to some one of the crystallographic directions without noticeable rotation.

2. The yield limit, which reduces to irreversible destruction of the crystal lattice. It is reached at greater forces and is detected by X-rays through the rotation of the lattice. It consists in slip along two planes, leading to the rotation of separate fragments about one of the axes. If we imagine, for example, that in rock salt displacement occurs along two planes \((110)\) and \((1\bar{1}0)\) at an angle of \(45^\circ\) to the direction of the force, coinciding with \([100]\), then we obtain rotation about the axis \([100]\), lying in both slip planes simultaneously. In recent years, for many metallic crystals there have been found

shear planes, for the purpose of elucidating the processes of cold working of metals. These investigations show that the indicated properties of rock salt are typical of all crystals.

A plastically deformed crystal is no longer a single crystal, but represents an aggregate of small displaced crystallites fused with one another to such an extent that they constitute, as it were, a single whole. One may ask whether, under plastic deformation, the lattice of each fragment changes separately. Investigation in polarized light shows the presence of stresses in crystals as a result of the mutual actions of the grains upon one another. These stresses, however, can be destroyed by annealing the crystal at high temperature. The structure of the individual fragments was then studied by the Debye method in monochromatic X-rays of silver. This method shows that the atomic distances do not depend on the orientation of the individual crystallites. It was shown that, despite such energetic deformation of the whole crystal, the individual grains remain unchanged.

The apparent contradiction mentioned above between the possibility of plastic deformation and the conceptions of the crystal lattice has thus been clarified. A change of the crystal lattice has never been observed without the presence of external forces balancing the internal stresses created by deformation of the lattice. Thus the deformed lattice must be regarded as an equilibrium state of atoms displaced under the action solely of external forces, as also follows from the properties of the model which we use. Contrary to the opinion of O. Lehmann, who considered it necessary to assume that a crystal subjected to plastic deformation constitutes a new type of crystal (homeotropy of the second kind), it turns out that the fragments of such a crystal differ from the original crystal in nothing except their dimensions.

An extremely powerful and useful method for studying deformation in crystals is the method of observing a crystal in polarized light between two nicols. If

the crystal is transparent and has cubic symmetry, it will not produce double refraction until it is subjected to the action of external forces. The magnitude of the double refraction is proportional to the shearing force or to the difference of two principal forces. The optical method can be applied to the measurement of hydrostatic pressure, i.e., to the determination of the modulus of compression. I. V. Obreimov and L. V. Shubnikov developed a method for measuring small changes in the refractive index caused by a change in density through immersion of the crystal in a liquid having the same refractive index. In this case definite mixtures are used, making it possible rapidly to adjust the refractive index of the liquid to that of the crystal. Applying both these methods, we study volume deformation and shear deformation. In isotropic bodies, or in certain crystals such as rock salt, these two data are sufficient for a complete description of the stresses.

The optical investigation of natural crystals showed that they already possess internal stresses beforehand. When placed between crossed nicols, a natural crystal can never be brought to darkness. Prolonged annealing from 600° to 700° C is necessary in order to destroy all stresses and make the crystal truly homogeneous. The design of the apparatus used for loading was carefully worked out so as to create a uniform tension over the entire cross-section of the crystal. This circumstance was checked by the same optical method. Gradually loading the crystal and observing the pattern in polarized light, we observed the first appearance of residual deformation at loads amounting to only approximately one tenth of those forces which caused destruction of the integrity of the crystal in X-rays. A sharp, bright line appeared on the dark background and remained after removal of the load. Under a constant load slightly exceeding this elastic limit, after some time a second line appeared, parallel to the first, then a third, and so on. All these lines are projections of the planes (110). More often they suddenly appeared

intersect the whole crystal, sometimes starting at one edge and rapidly spreading through the crystal to the other edge.

By compensating the double refraction, it was possible to measure the stresses in the crystal and to show that each such plane divides the crystal into two parts. The upper part is stretched, the lower compressed. Both the compression and the stretching are especially great at the boundary and gradually decrease with distance from the slip plane. If we consider any layer of the crystal between two slip planes, we shall notice that this layer is stretched in its lower layers and compressed in its upper ones. This means that all the layers are subjected to bending forces. Indeed, when the process has already advanced sufficiently far and very thin interlayers have formed, we can easily see the curvature of the slip planes, sometimes reaching even \(30^\circ\). The distribution of the principal stresses, determined on the basis of optical data, is shown in Fig. 10a.

Fig. 10a.

Under a constant load, the intervals of time between the appearances of two successive slips gradually increase, until the appearance of new displacements ceases completely; then, under a load exceeding the initial elastic limit, a new elastic equilibrium proves to have been reached. In order now to continue further the process of the appearance of displacements, it is necessary to increase the load. The greater the number of already existing slips, the higher is that elastic limit upon the crossing of which new slips appear. At every stage of the process the slip planes are distributed quite uniformly. Each stage of the process can be characterized by the number of existing displacements, by the curvature of the slip planes, or by the elastic limit. The general form of the picture described by us is given in Fig. 10b.

The phenomena of plastic deformation of this type consist of a series of displacements parallel to a certain crystallographic-

chanical plane, and are produced by a shearing force in the given plane. In order to have a correct description of this phenomenon, it would have been desirable first to study the elementary phenomenon underlying it—pure homogeneous shear—as a function of the orientation of the shear plane, the magnitude of the stress, and the temperature. It may be expected that even in the cleavage plane \((110)\), shear will depend on the direction of the stress in the given plane. Indeed, a row of ions in the direction \([100]\), lying in the given plane, consists alternately of positive and negative ions. In the normal lattice each ion encounters an ion of the opposite sign in the neighboring layer. If the next layer is displaced relative to the first in this direction, then after a displacement by half the interatomic distance the negative ions will replace the positive ones. At this moment ions of like signs will meet, and the layer which previously was attracted will now be repelled. With a further half-period of displacement the attraction is restored, and opposite ions again meet. Thus shear in the direction \([100]\) will periodically cause attraction and repulsion of the sliding surfaces. Under such conditions the displacement is equivalent to the motion of a cart over the ruts of a bad road.

Fig. 10b.

Considering the direction \([110]\) in the same plane \((110)\), we see that it consists of ions of one sign, alternating with a row of ions of the opposite sign in the next layer. Sliding in this plane will occur under constant attraction between the sliding surfa-

... Only the magnitude of the attraction will change periodically. Such a displacement will resemble the motion of a railroad car along rails.

In Fig. 11 the distribution of ions in the plane (110) is given, and the directions [100] and [110] are indicated. We must imagine the next layer with opposite charges above the lower atomic layer. In order to create a shear in the given direction, we clamped a rock-salt rod between steel holders (Fig. 12). One of them

Fig. 11.  Fig. 12.

(I) was immobile, while the second (II) could move strictly parallel under the action of a load \(P\), applied at a point above the gap between holders I and II.

With such an arrangement the bending of the rod was reduced to a minimum. Bending is proportional to the cube of the length, whereas shear does not depend on the distance between the two pairs acting on the ends. The smaller the distance between the clamped ends, the smaller the bending. However, the forces at the points marked by the arrows \(S\) rapidly exceed the elastic limit; plastic deformation arises, causing an increase in bending. We eliminate this source of error to some extent by reducing the cross-sectional area in the middle by means of a notch corresponding to the area clamped in the holders.

The investigation is not yet finished and has not yet yielded definite quantitative results; however, some qualitative results already obtained shed light on the phenomenon of plasticity. The investigation was begun by Prof. P. Ehrenfest and myself, and was then conducted by M. V. Klassen. We found that the shear of heated rock salt and zinc occurs in small jumps, each of which is accompanied by a noise resembling the ticking of a clock. In a quiet room these ticks are clearly audible and occur at regular intervals of time. Many hundreds of ticks can be noted, and their frequency depends on the applied load. The jumps become noticeable and audible only when the plastic deformation is already considerable. In the initial stage, the shear was observed by the optical method.

Fig. 13.

M. V. Klassen measured the relative displacements of the holders by means of the reflection of light from two mirrors, using a magnification of 10,000. She found that not only are the intervals of time between two ticks remarkably constant, but also the magnitude of the individual jumps under given conditions remains constant within 10%. Thousands of jumps could be observed. The intervals between jumps, however, gradually increase; then the jumps become rare and finally cease altogether. The magnitude of the jump remains constant throughout. Fig. 13 gives a general representation of the change of deformation with time. By increasing the load \(P\), we at the same time increase the frequency of the jumps. The frequency is equal to zero at the elastic limit \(P_0\); therefore we are entitled to regard the frequency as a function of \((P - P_0)\). The magnitude of the jump apparently does not depend on \((P - P_0)\), either when \(P\) is changed or when \(P_0\) increases. For small values of \((P - P_0)\), the interval of time between two

with jumps may reach up to 30 minutes, but the regularity of the phenomenon remains unchanged.

At low temperatures the magnitude of the jumps decreases. We may regard \(S\) as a function of temperature.

As was already indicated above, the elastic limit \(P_0\) increases with the number \(N\) of shifts already produced per unit length \(L\).

In accordance with this assertion we may write

\[ Z=\frac{\Delta N}{\Delta t}=F(P-P_0) \tag{1} \]

\[ P_0=f\left(\frac{N}{L}\right) \tag{2} \]

\[ S=\varphi(T). \tag{3} \]

The rate of plastic deformation may be expressed as

\[ v_p=\frac{\Delta S}{\Delta t}=Z\cdot S=\Phi(P_1 N)\varphi(T). \tag{4} \]

This rate is fundamentally different from the analogous rate \(vf\) in a viscous liquid. The quantity of motion transferred through a unit cross-section during the time \(\Delta t\) in a viscous liquid, where the velocities of displacement of two layers separated from one another by \(\Delta x\) differ by \(\Delta v\), will be

\[ \Delta q=\eta \frac{\Delta v}{\Delta x}\Delta t, \]

where \(\eta\) is the coefficient of internal friction. The frictional force per unit cross-section in this case is equal to

\[ P=\eta \frac{\Delta v}{\Delta x}. \]

If the velocity of one layer is zero and that of the other is \(vt\), then

\[ P=\eta \frac{vt}{\Delta x}, \]

or

\[ vf=\frac{1}{\eta}P\Delta x. \tag{5} \]

Resistance to flow arises from the transfer of momentum and is created by a gradient of velocities in the direction \(X\). Nothing of the kind can be found in plastic deformation, despite the similarity of the mathematical laws. Plastic deformation develops in jumps. In the interval between two jumps, the velocities throughout the crystal are equal to zero. During a jump the velocities do not depend on the acting force. The force determines the number of jumps per second, or the interval of time between two successive jumps. The force, by producing stress in the crystal, acts upon some process that is going on during this time and that prepares the next jump. The explanation of this process is now the most important question in the explanation of plastic deformation. Another task is to explain the constancy of \(S\). Finally, we must explain why the distribution of slip planes \(\dfrac{\Delta N}{\Delta L}\) is so uniform. Neither \(Z\), nor \(S\), nor even \(\dfrac{\Delta N}{\Delta L}\) exhibit a statistical character. Contrary to what might have been expected, they are all determined by strict laws, the deviations from which do not exceed the errors of observation.

Our knowledge of the mechanism of plastic deformation and of the quantitative laws governing it is not sufficient for constructing a definite theory. Nevertheless, we shall list here several possible hypotheses explaining the essence of the mysterious phenomena described here.

The increase of the elastic limit by means of plastic deformation, or, in the language of technologists, by means of cold working, is now quite often attributed to bends of the crystalline layers. It is assumed that slip in a normal crystal with flat surfaces is simpler than in a crystal with curved atomic surfaces, and this explains the strengthening under plastic deformation. Although curvature does in fact exist, it is usually small, and it is difficult to understand how this fact can explain such strong hardening. Slip along a smoothly curving...

of a curved surface should have been almost as easy as along a flat one.

Another hypothesis, which suggests itself, assumes that sliding along some surface destroys the regularity of its atomic arrangement. The surface becomes rough on the atomic scale. This irregularity extends from the plane of sliding into the crystal and gradually decreases with distance. The more irregular the given part of the crystal is, the greater must be the force capable of producing sliding. It is easy to see that a new displacement will occur in the middle of the layer, where the distortion is least, and, moreover, it will occur first in the thickest layer. The elastic limit depends on the distance between the new plane of sliding and the preceding one. This hypothesis explains both the increase of the elastic limit with the number of jumps per unit length and the distribution of the planes of sliding at equal relative distances. Thermal motion will gradually destroy the irregularities and will facilitate the appearance of a new jump after the lapse of a certain interval of time.

A third hypothesis may regard displacement as the discharge of elastic energy accumulated in a unit layer. Displacement leads to the appearance of two new surfaces with a destroyed lattice; such surfaces will possess a higher potential energy than the original crystal. If sliding occurs in less time than is required for sound to pass through the entire crystal, the energy is imparted to a small region of the crystal near the new surface. The former displacements constitute a layer with different velocities of sound, reflecting the elastic waves propagating in the direction toward the plane where the new displacement occurs. Thus it is possible that the energy is taken only from one or several layers. The store of energy \(U\) in a layer of thickness \(D\) and surface \(S\), under stress \(p\), will be

\[ U_1=\frac{1}{2}EDSp^2, \]

and the energy required for the displacement,

\[ U_2 = 2aS. \]

The smallest force \(p_0\) capable of causing displacement, i.e. the elastic limit, is equal to

\[ p_0 = 2\sqrt{\frac{a}{E}}\sqrt{\frac{1}{D}}. \tag{6} \]

Formula [6] may explain the regularity of slip planes and the increase of the elastic limit under cold working. Considering the conditions after displacement as a new state of elastic equilibrium corresponding to the amorphous state, we can find a certain explanation for the constancy of the magnitude of displacements.

For the time being, however, we still do not have more precise data, and until the form of the functions [1], [2], and [3] is clarified by new experiments, further discussion of the mechanism of plastic deformation seems pointless. We may, however, assert that although the model of the crystal lattice in its present form has proved insufficient for explaining plastic deformation, nevertheless in this field no arguments are apparent that speak against the fundamental premises of this model.

Strength.

In greatest contradiction with the predictions of the electrical theory of lattices is the strength of solids. We calculated the maximum magnitude of cohesion in rock salt on the basis of the theory of crystal lattices and found it to be of the order of \(300\ \mathrm{kg/mm^2}\) for all-round tension. Uniaxial tension at room temperature would give about \(200\ \mathrm{kg/mm^2}\). Meanwhile, the observed tensile strength of individual crystals of rock salt does not exceed \(0.5\ \mathrm{kg/mm^2}\). By creating a tensile force uniformly distributed over the cross section, and thus excluding eccentric forces causing bending, we obtain values for the strength that differ

differed from one another by no more than 10%, with an average of \(0.44\ \mathrm{kg}/\mathrm{mm}^2\). Lowering the temperature to that of liquid air does not noticeably affect the strength of rock salt. Heating to \(200^\circ\) C likewise caused no change (curve IV, in Fig. 9). Above this temperature the strength, it seemed, increased.

However, it became clear that at high temperatures the strength was in fact measured not on individual crystals, but on crystalline aggregates. This was explained by the fact that at \(200^\circ\) C the elastic limit of the crystal falls to \(0.44\ \mathrm{kg}/\mathrm{mm}^2\), i.e., is equal to the strength limit. Thus, at temperatures above \(200^\circ\) C, the elastic limit proves to be below the strength limit. By gradually increasing the load, we first reach the elastic limit, whereupon plastic deformation begins, disrupting the crystal lattice. As already mentioned, the process of plastic deformation increases the strength of the crystal many times over.

Fig. 14.

In order to establish a quantitative law for the dependence of strength on the degree of deformation and on distortion of the lattice, we determine this degree of deformation from the change in cross section. We have seen that plastic stretching of a crystal consists in sliding along certain crystallographic planes—for rock salt, along the plane \((110)\). As a result of such shifts (Fig. 14) a cylindrical rod is transformed into a ribbon, whose width \(B\) is equal to the original diameter of the rod, while the thickness \(D\) gradually decreases in the course of stretching. The transverse cross section \(s\) becomes smaller than the original cross section \(S\). The ratio \(\dfrac{S}{s}\) gives a certain measure of the degree of plastic deformation. Plotting the strength as a function of this ratio, we obtained the curve shown in Fig. 15.

For a deformation corresponding to \(\dfrac{S}{s}=35\), the strength reached \(5\ \mathrm{kg}/\mathrm{mm}^2\), i.e., was approximately 12 times greater than

strength of a single crystal. It follows from this that the value of the strength found from tensile experiments does not represent the strength of a single crystal above 200° C, but corresponds to the strength at the given degree of plastic deformation that had occurred by the moment of rupture. But since the development of plastic deformation requires time, it may be expected that, by shortening the interval of time between the moment of passing through the elastic limit and the moment of rupture, we reduce the magnitude of the error.

Fig. 15.

Indeed, by shortening this time, we are able to observe, up to 650° C, a strength much lower than that measured by us earlier. It tended toward a certain lower limit. However, in obtaining this result it was necessary, during loading, to avoid sharp shocks. In such a case we must regard this constant lower limit as the strength limit of a single crystal. As Fig. 9 shows, this strength limit retains, up to 650° C, the same value as at room temperature. Thus, in the temperature range from −180° C to +650° C, the strength of rock salt remains approximately constant.

The intersection of the curves corresponding to the strength limit with the curves corresponding to the elastic limit evidently represents a general property of all crystals. The strength of a single crystal always exceeds the elastic limit when approaching the melting point; here the elastic limit falls to zero. At sufficiently low temperatures the elastic limit increases faster than the strength and exceeds the latter. At higher temperatures the crystal will rupture as a plastic body; at lower temperatures this will be the rupture of a brittle body, i.e., rupture will occur before any plastic deformation appears. Consequently, brittleness and plasticity are not properties of different bodies; rather, both are determined by the temperature and the nature of the deformation of one and the same body. Every body is brittle at low temperatures and plastic at high temperatures. The point of intersection in Fig. 9, for the given type of deformation, determines the temperature at which a brittle body becomes viscous. In doing so we need not assume, as O. Lehmann did in order to explain plasticity, the presence in rock salt of a special homeotropy of the second kind—geometrotropy—which exists at high temperatures and is absent at low ones.

By choosing the temperature appropriately, one can impart precisely the desired mechanical properties to a body. Thus, for example, in order to avoid the occurrence of plastic deformations and deterioration of the crystal when treating individual crystals of lead and zinc, the temperature should be lowered to that of liquid air. In other cases heating may be recommended.

The tensile strength of individual crystals is approximately 500 times less than the calculated maximum for the cohesion forces, and rupture occurs before any substantial deviation from Hooke’s law can be observed. Even plastically deformed crystals have a strength 40 times smaller than would be expected. If the electrical theory is not mistaken in its basic premises, this contradiction should not exist. The ques—

than, it is quite possible that the rupture usually observed under tension has nothing in common with true strength. We can readily understand that gradual partial rupture requires a much smaller force than rupture occurring simultaneously over the entire cross-section. Suppose, for example, that on the lateral surface of a stretched rod there is a small but very sharp crack. The distribution of stresses will be such that at the edge of the crack the stress will be extremely concentrated, as is evident from the diagram in Fig. 16.

If the radius of curvature of the bottom of the crack is sufficiently small, the ratio of the stress at the edge of the crack to the average stress over the whole cross-section may prove to be very large; it may be assumed that this takes place in a crystal. Indeed, calculation shows that the ratio of the stress existing at the crack to the average stress over the cross-section of the specimen may be of the order of several hundreds. Comparing the energy in this state with the energy that will be produced if the crack begins to grow, Griffith and Wolf formulated conditions favoring the growth of a crack; in a cracked crystal this process ultimately leads to complete rupture of the specimen. Griffith succeeded in showing experimentally how much the strength can be lowered by the presence of these cracks.

Fig. 16.

If the low tensile strength of crystals is explained by this, then, obviously, the condition of the surface from which the tear and destruction of the specimen begin must play an essential role; in that case it is necessary to find an additional explanation for the fact that the observed tensile strength is constant within 10%.

The influence of the state of the surface on the strength of rock salt was in fact observed and described by W. Voigt, Sella, and G. Müller.

The first two authors found that the strength of rock-salt crystals depends not only on the orientation of the force, but also on the orientation of the external surfaces bounding the crystal. They found, for example, that the strength of a crystal with (110) surfaces proves to be almost twice as great as the strength of a crystal with (100) surfaces. It must be noted, however, that the sharpness of the crystal edges in the two cases is very different. I tried to avoid this objection by preparing crystal specimens in which the true surface was different for one and the same crystal shape, while the edges were always rounded, as shown in Fig. 17.

Fig. 17.

But in these specimens no effects of the order of magnitude reported by Fort and Sella were found. The observed values of the strength limit in these experiments merely showed scatter over a somewhat wider range than usual.

H. Müller found that the strength, expressed as the tensile force per unit cross section, increases from \(0.5\ \mathrm{kg}/\mathrm{mm}^2\) to \(7\ \mathrm{kg}/\mathrm{mm}^2\) when the cross section is decreased approximately from \(20\ \mathrm{mm}^2\) to \(0.3\ \mathrm{mm}^2\). This would have meant that the strength depends not only on the cross section, but also on the perimeter of the specimen. In contradiction to these assertions, M. A. Levitskaya found that, in the case where crystals prepared in the same way were investigated, when the cross section was varied within the limits from \(36\) to \(0.1\ \mathrm{mm}^2\), there was no effect of the change in perimeter. We suspect that Müller’s results are explained by the treatment of the crystals with water, since, as we shall show below, water has a strong effect on the strength. In Fig. 18, curve I gives Müller’s results, while curve II gives the results of M. A. Levitskaya.

In mica sheets, a reduction of the thickness to \(0.1\) mm does not affect the strength; however, when the thickness is reduced from \(0.01\) to \(0.001\) mm, the strength, as A. Walter has shown, increases by approximately a factor of 4; at the temperature of liquid air this effect is doubled again.

Neither the experiments of F. Fogg and Sella, nor the experiments of G. Muller, prove any substantial influence of the state of the surface on the rupture of rock-salt specimens of ordinary dimensions. If, therefore, cracks are indeed responsible for the rupture of crystals, then they must evidently be created by some regular process, and their depth and sharpness are determined by the structure of the crystal. The increase in strength at thicknesses of the order of \(1\,\mu\) indicates that the cracks have a size from \(0.1\,\mu\) to \(0.01\,\mu\), or from \(10^{-5}\) to \(10^{-6}\) cm.

Fig. 18.

Although the experiments described do not confirm the theory that attributes low strength to the influence of the surface, they do not contradict this hypothesis. Even without knowing the mechanism of crack formation, we may expect that the process of their formation requires time. In other words, a fresh surface must differ from an old surface. By continually renewing the surface by dissolving it in water, we can prevent the appearance of cracks. The more rapidly the old surface is removed by the water, the more sig-

It is natural to suppose that cracks do not have time to form. Some indications of this kind are provided by the property of rock salt of becoming plastic in water; this property means that the limit of strength becomes higher than the limit of elasticity.

M. A. Levitskaya and I therefore tried to rupture crystals of rock salt immersed in hot water, and we found that in this case the strength increased considerably; the rock salt always fractured in the dry section even when the cross-section of the dry part was at least 10 times greater than the section of the part of the crystal immersed in water. By gradually dissolving the crystal, we reduced the section of the wet part until the specimen broke at a neck about \(0.2\) mm thick, while the dry part had a thickness of about \(6\) mm. Calculating the tensile force as the ratio of the applied load to the transverse section of the specimen, we found a strength limit of up to \(30\) and even up to \(160\) kg/mm\(^2\). These values are already sufficiently close to the theoretical strength of \(200\) kg/mm\(^2\).

Before rupture occurred, the elastic limit corresponding at room temperature to \(0.9\) kg/mm\(^2\) was reached, and at the moment when rupture took place the crystal proved to be plastically deformed. The strength measured was thus not the strength of a single crystal, but the strength of a crystalline aggregate. Nevertheless this result is essential for the electrical theory of crystals, for it shows that the internal strength of rock salt is not less than \(160\) kg/mm\(^2\), and that rock-salt crystallites can withstand such stresses. This result very satisfactorily confirms the calculation of cohesive forces given by the electrical theory.

M. Polanyi (M. Polanyi) and W. Ewald objected to our results. They suggested that the effect of water consists not in increasing the strength, but in lowering the elastic limit. Therefore, they pointed out, rock salt becomes plastic and its strength will increase as a consequence of the phenomenon of plastic—

mechanical deformation, as was indicated in Fig. 15. Although such a line of thought may be correct, the actual process does not correspond to these assertions. We measured simultaneously both the elastic limit insofar as it is revealed by X-rays, and the tensile strength of a single crystal under a rapidly increasing load. Contrary to the suppositions of M. Polanyi and W. Ewald, the elastic limit, within 1%, was the same both in water and in dry air. A crystal which did not rupture in water at stresses up to \(5\ \mathrm{kg}/\mathrm{mm}^2\), when subsequently dried, had the normal low strength of \(0.4\ \mathrm{kg}/\mathrm{mm}^2\), corresponding to an undeformed crystal. This direct test of the hypothesis of M. Polanyi and W. Ewald did not confirm it, but, on the contrary, showed that the increase in strength in water is not connected with plastic deformation.

It is interesting to determine how much time is actually required in order to bring a fresh surface with high strength to the normal conditions under which the surface is covered, as I suppose, with small cracks. This can best be observed by testing rock salt in bending. At room temperatures the elastic limit in bending is apparently only slightly lower than the force causing fracture; therefore, under bending forces, pieces of rock salt prove brittle. Under water they become flexible, which indicates an increase in strength. If the specimens are removed from the water, it turns out that some of them remain flexible for many seconds, minutes, and sometimes even days, especially if they are placed in a vacuum. Specimens of different origin behave differently. Some of them become brittle immediately after being removed from the water.

We note that a saturated salt solution or dry oil does not affect the strength. The same proved to be true with respect to hydrogen and carbon dioxide gases, and also when the crystal was kept for many days in an empty space. The strength was always approximately \(0.5\ \mathrm{kg}/\mathrm{mm}^2\).

Although the rupture of rock salt in hot water did in fact show the expected strength, this was not ...

strength of a single crystal. Experimental difficulties did not allow us to measure, on single crystals under water, a force greater than \(10\ \mathrm{kg}/\mathrm{mm}^2\).

But there is another way—to avoid the influence of surface cracks. If the distribution of stresses is such that the surface layers are not stretched, the influence of the surface will be excluded. Neither cracks nor any other surface phenomenon can influence the process of rupture occurring inside, where the stress reaches its maximum value. A test of this kind can be carried out if spheres are subjected to sudden and all-sided heating. The outer layers will expand before the others and will stretch the cold central part, producing all-sided tension. This is precisely the deformation that is considered in deriving the forces of cohesion from the electrical theory of crystalline lattices. It is clear that radial tension can never exist at the surface, since the surface can expand freely and no forces act upon it. The distribution of stresses inside the sphere was calculated by G. A. Grinberg. It consists in a uniform tension \(T\), going from the surface to the center and reaching it after a time \(\tau\), determined by the thermal conductivity. This tension has a maximum

\[ T_{\max}=A(t-t_0), \]

where \(A\) is a coefficient, equal to \(0.08\ \mathrm{kg}/\mathrm{mm}^2\) for rock salt; \(t\) and \(t_0\) are the initial and final temperatures of the sphere.

The time \(\tau\) turns out, for rock salt, to be equal to

\[ \tau=1.9\,r^2\ \mathrm{sec.}, \]

where \(r\) is the radius of the sphere.

In addition to radial tension, a tangential stress must also appear in the sphere, which will lead to compression, different from zero at the surface in the first stages of heating. But compression is not as dangerous in its influence on strength as tension.

Several spheres of rock salt were first cooled in liquid air, and then suddenly immersed in hot water at \(100^\circ\) C or in molten tin at \(600^\circ\) C.

If measures were taken to ensure that the heating of the sphere was uniform, then neither cracking nor rupture was observed. The greatest all-round tension at the center in these experiments must have reached as much as \(60\ \mathrm{kg}/\mathrm{mm}^2\). The central part remained cold, so that the crystal lattice was not destroyed. In the surface layers plastic deformation may have occurred, but during the half-second required to produce at the center a stress of \(60\ \mathrm{kg}/\mathrm{mm}^2\), the deformation could not have developed very far. Indeed, in small spheres no traces of such deformation were found. Therefore one may be certain that the true tension at the center was only slightly less than the values calculated on the assumption of perfect elasticity.

Although the experiment with the sphere does not give a quantitative value for the true limit of strength, it nevertheless proved that this limit exceeds \(60\ \mathrm{kg}/\mathrm{mm}^2\). Thus this experiment showed that the usually observed low strength of \(0.5\ \mathrm{kg}/\mathrm{mm}^2\) has nothing in common with true cohesion. The latter, on the contrary, agrees in its order of magnitude with that calculated from theory.

Questions concerning the strength of solid fine-crystalline bodies, the influence of the size and arrangement of crystalline grains, cold working, etc., apparently depend to a considerable degree on the formation of small cracks both within the grains and at the boundaries between crystalline grains. The strength calculated from the electrical theory therefore gives us the upper limit of the strength of a crystal, which at the same time is in general the highest attainable limit of strength.

The investigation of the elastic properties of crystals in this and the preceding sections has led us to the conclusion that all observed facts, without exception, agree with the theory of crystal lattices. All contradictions concerning the phenomena of elastic after-effect, plasticity, and strength proved to be only apparent contradictions. The general predictions of the theory were confirmed both qualitatively and quantitatively.

It turned out, however, that the theory in its present form is not capable of explaining the mechanism of certain

elastic phenomena, such as: slip in the crystal lattice, the appearance of cracks on the surface, etc. To construct an exact model of the crystal, a detailed study of these phenomena is necessary.

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1. The article is an excerpt from the author’s forthcoming book Physics of Crystals. ↩