DEFORMATION, FRACTURE, AND STRENGTHENING OF CRYSTALS¹

M. Polanyi, Berlin-Dahlem.

A solid body, when subjected to mechanical action, changes its shape. If, after the force ceases to act, it again assumes its original form, then this body has been changed elastically; if, on the contrary, some change of form is retained, then overloading has taken place. Such overloading may either tear the body apart or merely deform it; only in those cases where the body is very brittle does it break without a preliminary change in its shape.

Deformation and fracture have hitherto been investigated on very varied material. Thus, Griffith successfully studied fracture in amorphous bodies—glass and quartz; the extensive technical experience in the processing and strength of materials relates to crystalline substances, such as stone and metals, which also served as a model for the mathematical theory of plasticity (Prandtl, Hencky, Nadai). By contrast, the physical analysis of the phenomena of deformation and strengthening was carried out primarily on individual crystals, namely crystals of rock salt and, in particular, large metallic crystals.

My report today will deal mainly with investigations on metallic crystals; in doing so, of course, use will also be made, on individual occasions, of data available to engineering practice.

¹ A report read at a conference of the Faraday Society devoted to the nature of cohesive forces. Published in Trans. Faraday Soc., Vol. 24, Part 2, Feb. 1928, and in Naturwiss. of 27/IV 1928. Translated by S. T. Konobeevsky.

1. Single-crystal wires and rods.

Since Laue discovered the diffraction of X-rays in the spatial lattice of a crystal, the very concept of a crystal in physics has undergone a significant change. Now, when speaking of a crystal, we no longer think of a regular external form, of crystalline faces and edges, which were formerly considered the chief distinguishing features of a crystal. The more essential content of the concept of a crystal has become its internal structure, that regular arrangement of its molecules which is called the spatial lattice and which has become visible to us thanks to the phenomenon of diffraction.

If a sphere or a column is turned from a cube of rock salt, then its crystalline faces no longer exist; however, since the sphere or column has the same lattice as the rock-salt cube from which it was made, we also regard these bodies as crystals of rock salt, calling them “single-crystal bodies.” Thanks to the spatial lattice underlying their structure, single-crystal bodies contain, in hidden form, crystalline faces—the planes of densest arrangement of atoms—which appear as cleavage and slip planes when the single-crystal body is fractured or deformed. The same applies to crystalline edges, which exist in the lattice as directions of densest arrangement of atoms and are revealed upon fracture of the body in the form of boundaries between cleavage planes, and, in the case of deformation, in the form of directions of displacement of the slip planes.

There exist not only such single-crystal bodies which have lost their original crystalline shape merely as a result of external working, but also those which, at the very moment of their formation, have a shape that bears no relation whatsoever to their crystalline nature.

Such are single-crystal metals. These wires and rods consist of metal which, in its structure, in each case represents one whole crystal, but the external

the shape of them reveals none of this. But here too the hidden crystalline faces and edges immediately appear as soon as the body is subjected to overloading. Single-crystal zinc wires, carefully studied by me together

Fig. 1.                                                                     Fig. 2.

Fig. 1, 2. Hidden crystalline faces appearing upon rupture of a single-crystal wire.
In Fig. 1 the basal plane (0001) of a zinc crystal is seen; it is distinguished by its luster and by the fine triple striation from the prism face \((10\overline{1}0)\) in Fig. 2, which is matte and has only a coarse striation.

with Mark (H. Mark) and Schmid (E. Schmid), rupture at the temperature of liquid air along planes which most often prove to be the basal planes \((0001)\) of the hexagonal lattice of zinc. As a simple rupture, the prism plane of the first order \((10\overline{1}0)\) is also encountered. Sometimes, upon rupture, both these planes are formed, and then, where they intersect, an edge appears corresponding to the crystalline edge \([10\overline{1}0]\) (the diagonal axis of the 2nd order), which had previously existed in hidden form.

Fig. 3 shows a hidden crystalline edge revealed upon rupture in the form of the line of intersection of two rupture planes.

It should be noted here that the rupture plane in different wires of one and the same metal is inclined differently to the axis of the wire, i.e., the atomic planes (and consequently the entire crystal lattice) have in each case a special orientation with respect to the axis. Since, because of this, an effect is exerted on the spatial lattice in different directions, this explains why in different wires the role of the planes

rupture are played by different faces. In general, the plane that is closer to the plane of the cross section is formed preferentially.

The deformation of single-crystal zinc wires, which arises especially easily when they are stretched at ordinary or elevated temperature, also leads to the appearance of hidden crystalline faces, manifested here as

Fig. 4. Model of the stretching of a zinc crystal: a—crystalline zinc wire, cut along an oblique section. The long arrow is the major axis of the elliptical slip layer; in its direction the greatest shearing stress is produced during stretching. The short arrow is the edge \([10\overline{1}0]\) of the crystal, serving as the direction of slip. b and c are front and side views of the model in the stretched state. Attention should be paid to the shape of the band and to the slip lines. The vertices of the ellipses recede from the middle line of the band. Also visible is the widening of the band in comparison with the original thickness of the wire. The short arrow in this figure is the direction of slip of the layers; it is parallel to the axis \([10\overline{1}0]\).

slip planes, traces of which can be observed on the side surface of the stretched crystal in the form of separate slip lines (Fig. 5). In this case one of the axes of the spatial lattice of the crystal crystallographically coincides with the direction of slip in the stretched wire. We shall now consider the phenomenon of slip in somewhat more detail.

M. POLANYI

2. Slip in single crystals.

The results of experiments on stretching single-crystal zinc wires can be represented visually with the aid of the wooden model shown in Fig. 4. A body, initially of cylindrical form, as a result of stretching is transformed into a ribbon with an elliptical cross-section. This occurs as a consequence of slip along parallel

Fig. 5, a and b show a stretched single-crystal zinc wire: a perpendicular to the broad side of the ribbon, b parallel to it. Below, toward the arrow, the crystal remains unstretched, and its originally approximately cylindrical form is unchanged. Starting from this point upward, the width of the ribbon somewhat exceeds the thickness of the original wire. In Fig. 5, c another stretched zinc wire is shown, with clearly visible slip lines; the latter plainly deviate from the middle line of the ribbon.

planes, the traces of which, in the form of slip lines having the shape of ellipses, can be seen on the lateral surface of the stretched ribbon (Fig. 5). The upper one of the planes of the model representing the slip planes bears a regular hexagon; this is intended to show that the slip plane is, for the most part, the basal face—the same face which predominantly also appears as the fracture plane.

A long arrow, drawn on the upper plane in the direction of the major axis of the ellipse, indicates the direction in which the tensile force creates the greatest shear stress. However, on examining the model more closely, one can see that slip occurs not in this direction, but parallel to the short arrow.

This is revealed, first, by the fact that the vertices of the ellipses formed by the slip lines do not lie on the center line of the stretched strip, and, second, by the fact that the width of the strip turns out to be greater than the thickness of the original wire.

The reason why the direction of slip deviates from the direction of the acting force lies in the internal structure of the crystal. In the model this is reflected in the fact that the direction of slip, indicated by the small arrow, runs parallel to the side of the hexagon; it must therefore be recognized as coinciding with one of the hidden crystallographic edges, namely with the already mentioned lattice axis along which the basal plane intersects a line of the first kind and which is denoted crystallographically by the symbol \([10\bar{1}0]\).

Thus, slip is characterized not only by the fact that it occurs in a plane that is always crystallographically predetermined, even if this plane deviates considerably from the plane of greatest stress (which in all cases must lie at an angle of \(45^\circ\) to the axis), but also by the fact that in the given plane slip occurs only in a direction that is likewise crystallographically determined, and that may deviate from the direction of the greatest driving force in this plane. Of course, the direction of the acting force must have the effect that, of the three crystallographically equivalent straight lines \([10\bar{1}0]\) (corresponding to the three sides of our hexagon, Fig. 5), the one chosen as the direction of slip will be that which forms the smallest angle with the direction of the force. This regularity also becomes clear from consideration of the model.

The mechanism described here, by means of which stretching occurs, leads to a change in the orientation of the pro-

of the spatial lattice with respect to the axis of a single-crystal wire. Consideration of the model convinces us that in this case the slip plane, in the present case the basal plane, as elongation proceeds forms an increasingly acute angle with the axis of the wire. If the stretching could be continued indefinitely in the manner described, then the basal plane would, in the end, coincide with the axis. And indeed, in the stretching of zinc crystals such a position of the lattice is reached that the basal plane forms an angle of only 5–6° with the longitudinal axis. At the same time, owing to the fact that the direction of slip in the basal plane is crystallographically determined, one crystallographic axis also tends to assume an increasingly parallel position with respect to the axis of the wire. This is precisely the direction \([10\bar{1}0]\), along which slip takes place.

Thus, during deformation there occurs a special, regular rotation of the lattice, and as a result of stretching we have a crystal in which the axis \([10\bar{1}0]\) lies almost parallel to the direction of the tensile force. Therefore, for a group of zinc crystals of different initial orientation, one can obtain an identical arrangement of the lattice (with respect to the axis) by subjecting the crystals to stretching. A case in which this occurs, i.e. in which many zinc crystallites are simultaneously stretched, is the drawing of an ordinary microcrystalline zinc wire. In this process, indeed, the lattice in all the crystals rotates in the same way and, in this particular case, so that the direction \([10\bar{1}0]\) becomes established approximately parallel to the longitudinal axis.

Such an ordering of the crystalline structure, which leads to a structure called—by analogy with the structure of natural wood fibers—a fibrous texture, almost always appears in the drawing of metallic wires; moreover, of course, depending on the different type of lattices, different arrangements also arise (3). In cadmium, whose lattice is similar to that of zinc, the direction \([10\bar{1}0]\), as in zinc, becomes established in the direction of the axis. In metals with a face-centered cubic lattice,

(for example, W, Mo, Fe)—the small diagonal \([110]\) of the cube. In metals with a face-centered lattice (for example, Cu, Ag, Al) two crystalline groups arise: in one, parallel to the axis of the wire, becomes the large diagonal \([111]\) of the cube; in the other—the edge of the cube. Tetragonal tin is an exception, since in it a fibrous texture does not arise at all under tension.

The agreement between the character of the fibrous texture and the result of rotation of the lattice of stretched single-crystal wires, which we have already indicated in the case of zinc, was also established for cadmium (E. Schmid), and likewise for one of the metals with a centered cubic lattice, namely tungsten (Gaucher, 4). However, aluminum, which has a face-centered cubic lattice, represents a deviation from this rule, as yet unexplained. The orientation established during stretching of the crystal proves, only in one exceptional case (namely during hot drawing), to be the same as that observed for the fibrous texture (19). For the most part, however, the final orientation in single-crystal aluminum is such that the direction \([211]\) is established along the axis, which, incidentally, is also comprehensible from the point of view of the mechanism of stretching discovered by Taylor and Elam [Taylor and Elam (5)]. Why in this case the crystallites of a microcrystalline wire arrange themselves differently than a separate large crystal remains for the present not entirely clear.

Incidentally, concerning the above-mentioned work of Taylor and Elam, it should be noted that our ideas about the stretching of single-crystal wires, owing to the choice of zinc as an example, prove to be simplified to a certain degree. The circumstance that in zinc a plane endowed with exceptional capacity for glide is one that is not repeated multiple times in the lattice (there is only one basal plane in the hexagonal crystal) leads to the fact that glide here is carried out almost exclusively along this single plane. If, on the contrary, one turns to cubic crystals, then there emerges

to the fore is the competition among many unambiguous planes and directions of slip (for example, in the case of aluminum there are 4 octahedral groups and 6 small diagonals of the cube). The final orientation toward which the lattice of a stretched crystalline wire tends is no longer determined by the fact that the plane and direction of slip are established by the direction of stretching. On the contrary, the conditions for this final position, owing to the simultaneously occurring slip along various unambiguous planes and directions, are such that no considerable rotation takes place here. The final position is a certain stationary state in which the rotations caused by slips along the individual differently oriented slip planes mutually balance one another.

Fig. 6. Stretched tin crystal in the form of a ribbon, covered with slip lines that are not closed and have fringed edges.

Simultaneous slip along several planes is also accompanied, externally, by a phenomenon different from that observed in the case of zinc. Instead of the closed sharp slip lines appearing in a single metal, here the competition of different planes leads to the appearance of mutually intersecting serrated slip traces, at times unrecognizable.

In Fig. 6 one can see slip lines of tin, having an uneven edge as if trimmed with fringe (b), arising

TABLE I.

Slip directions, slip planes, and cleavage planes in various metals¹).

¹) The numbers given in the columns “slip plane” and “cleavage plane” for zinc, bismuth, and tellurium (under the letters \(S\) and \(Z\)) represent the values of the shearing or tearing force, determined by Schmid and his collaborators (see 5).

which consequently slip simultaneously along different planes in one common direction (see Table I). Such simultaneous participation of several planes in slip occurring in one direction may have served as the basis for Taylor and Elam finding for iron a definite direction of slip, but not being able to discover which plane is the slip plane (7).

3. Slip surfaces, slip directions, rupture surfaces, and lattice structure.

If, in a given crystal, only certain planes and lines are always observed as paths for slip, then the reason for this, one must suppose, lies in the fact that these paths possess so exceptional a capacity for slip that other planes and lines, possessing this capacity to a considerably lesser degree, may be disregarded. Only with such an orientation of the lattice, when the preferred planes and lines are in a particularly unfavorable position with respect to the axis, can slip paths of secondary importance be brought into action. In cubic crystals, owing to the presence of equivalently situated planes and lines, such an unfavorable orientation of the lattice cannot occur, but in hexagonal and tetragonal crystals it may be encountered. Therefore, in crystalline bodies of the regular system, the principal plane and the principal direction of slip always participate in slip; in other crystals, planes and directions of the second rank are also brought into action.

This can be seen from the table presented (p. 747), which also contains the following regularities:

1. In similar lattices (zinc–cadmium) and especially in identical lattices (for example, in all face-centered or body-centered lattices), the same planes and directions of slip occur.

1. The most prominent paths along which slip occurs are, for the most part, planes and straight lines of the lattice with the densest arrangement of atoms.

We shall return to this last point later, in the chapter on strength considered in connection with lattice theory. With regard to the first, however, let us note that, according to the latest observations, the similar behavior of identical lattices cannot likewise be extended to the various (equivalent) slip planes competing with one another. Thus Elam (9) found that slip in brass crystals (a face-centered cubic lattice) deviates somewhat from the behavior of crystals of pure copper. Likewise, according to Schlosser and Wassermann [Wassermann (35)], the fibrous textures of the metals Ag, Cu, Au, which have the same lattice, although they coincide qualitatively, nevertheless prove to be different in the distribution of crystallites between the two orientation groups (see above). Hence one may conclude that even in metals of identical structure the relation of their slip capabilities (and the corresponding strengthening) for different planes and straight lines is not one and the same (see Ch. VII).

Fracture of a crystal for the most part occurs as a result of gradual fraying without the formation of a fracture surface. Cases in which actual separation is observed in crystals (with the formation of a fracture plane) are given in the table. The fracture planes, like the slip planes, are the densest in the lattice (or next after them) and coincide with the slip planes in the case of zinc, bismuth, and tellurium; this, however, is not the case for iron (tungsten).

4. Strength and the Theory of the Space Lattice.

One might now think that the close connection of the phenomena of slip and fracture of crystals with the structure of the lattice—a connection expressed in that minimum strength with respect to shear and tension which distinguishes planes with the closest arrangement of atoms—is

with the indication that the strength of crystals can theoretically be considered and calculated from the point of view of the structure of the lattice. Unfortunately, however, this expectation is not justified.

This is already evident from the fact that, in rupture and slip, the forces binding the crystal are, generally speaking, not balanced by the imposed external forces. If the latter were the case, then at the moment when the crystal yields plastically to the force, its atoms would have to be in a state of indifferent equilibrium. The extensibility in the direction of the critical stress would have to become infinite or, in other words, the modulus of elasticity (respectively, of shear) would have to vanish. Since this does not occur—on the contrary, up to the point at which overloading sets in, almost no deviation from Hooke’s law is observed—one must regard the yielding of the metal to the mechanical force as “premature.” It occurs at a tension significantly lower than that theoretically required for the destruction of the ideal lattice of the crystal.

This indirect conclusion can, for rupture, be confirmed by calculating the theoretical value of the strength and comparing it with genuinely reliable quantities (13, 14). The theoretical value lies around several hundred kg/mm², as against 0.2–0.5 kg/mm² found for rock salt and zinc. The experimental value thus amounts to less than 1% of the theoretical one.

Attempts have been made to explain the contradiction by assuming certain factors that facilitate the onset of “premature fracture of the crystal.” At first there was a tendency to consider small injuries to the crystal (cracks) as the cause (especially after Griffith’s successful experiments with amorphous bodies); now, however, there is a greater inclination to ascribe this phenomenon to thermal motion. An interesting discussion on this question between A. Smekal [A. Smekal, (16)] and R. Becker [R. Becker (17)], proceeding from two different points of view, has not yet led to definitive results. Incidentally, a completed structural theory of rupture and slip must also encompass the case of the formation of twin—

..., which has remained here, in the course of the exposition, unexamined. The point is that this phenomenon, too, occurs in the region of Hooke’s law and consequently also proves to be “premature.”

5. Strength under different orientations of crystals.

The inability to solve the central problem of strength compels us for the time being to confine ourselves to the study of actually observed regularities, in the hope that this will help in the further solution of the fundamental problem. A fruitful way to do this is to measure the resistance to slip and to fracture of metallic crystals for different orientations of the lattice with respect to the axis of a single-crystal wire. In a number of extensive investigations Schmid (11, 12, 15) established on crystals of zinc, bismuth, and tellurium a simple law: each slip plane begins to slip when a definite shearing stress is reached, independently of the magnitude of the force acting at the same time normally to this plane; conversely, fracture is caused by a definite magnitude of the tension perpendicular to the fracture surface, while the tension acting in the plane of fracture has no influence whatever. Schmid was able to show that earlier measurements of strength, made by Voigt [Voigt (18)] on differently oriented rods of rock salt, also obey this regularity.

The same law was also found by Sachs [G. Sachs (19)] for the slip of aluminium.

6. Hardening with respect to shear and fracture.

A number of other regularities, though rather of a qualitative character, can be found by investigating in single crystals the phenomenon of hardening under cold working (Kaltreckung), which is of such great importance in technology. A complicating circumstance here is that drawing changes the orientation of the crystals, so that the strength of a crystal may increase even without any real change in its properties, solely by virtue of the fact that the necessary for attaining

of the required stress for the given slip plane or fracture load increases as a result of the rotation of the lattice. However, the author, together with Schmid (20, 21, 22), succeeded in excluding the influence of this circumstance and in proving that the resistance to shear of slip planes under tension in crystals of tin, and also of zinc, increases severalfold (hardening with respect to shear).

On the other hand, investigations carried out by me together with Masing (G. Masing) also showed (23) that the resistance to fracture for rolled zinc placed in liquid air reaches a value 200 times greater than that which, under the same conditions, is observed for single-crystal zinc (hardening with respect to fracture). The very considerable increase in the tensile strength of rock salt upon fracture of a crystal under water, discovered by Joffe (24), likewise reduces to a hardening effect caused by the still quite enigmatic increase in the plasticity of wetted rock salt.

Taking the preceding paragraph into consideration, one may now draw the following conclusion from the facts presented: tension produces changes in the crystal, owing to which fracture and slip occur not so “prematurely”; in other words, the strength of the crystal approaches more closely its theoretical value.

7. Bending of a Crystal.

In what way, then, one may ask, can a change in the properties of a crystal occur as a consequence of its tension? It is clear that this phenomenon requires some broadening of our ideas about the structure of the lattice, since the lattice of a deformed crystal must be something different from the lattice of an undeformed one.

I believe that a correct conception of this question can be reached by considering the bending of a crystal. Suppose that we bend a single-crystal rod at its middle. In what way is this change of shape accomplished? What will be the state of the lattice in the deformed region?

The deformation mechanism, according to Mügge [Mügge (26)], may be reduced to so-called bending glide (Biegegleitung)—a phenomenon observed, for example, by this author in the bending of gypsum plates. In this mode of glide, the layers of the crystal sliding over one another are not displaced parallel to one another, but are twisted along the surface of a cylinder whose axis is perpendicular to the direction of glide (Fig. 7).

As regards the state of the crystal at the place of bending, it must first of all be considered impossible that the lattice here should have remained uncurved. Although the peculiarities of this modified state of the lattice are unknown to us, the following is in any case beyond doubt:

Fig. 7. Bending of glide layers in tin. The place where the wire passes into the ribbon can be seen; the ribbon itself has been torn off.

1. In layers bent as a result of glide there are elastic stresses. Namely—on the convex side there is tension, on the concave side—compression.

2. Between two neighboring layers the lattice is interrupted, since here a stretched layer borders on a compressed one.

The described state of the lattice is represented schematically in Fig. 8, which depicts two bent glide layers. It is easy to see that the separation surfaces arising as a result of glide differ fundamentally from ordinary intergranular boundaries, since here crystallographically identical surfaces are in contact. Because of this peculiarity of theirs, we shall call such boundaries “internal separation surfaces.” It should be noted that the elastic curvatures shown in the figure are greatly exaggerated; in reality they do not exceed 0.5%.

Let us now ask ourselves whether the picture given here of a bent crystal can make more comprehensible the cause of its

hardening? It seems to me that this question can be answered in the affirmative.

A bent crystal must harden because its spatial lattice along the internal surfaces of separation proves to be disturbed and, consequently, the paths of slip (the directions and surfaces of slip) which ought to have existed at the place of these surfaces of separation are blocked. It also follows that after slip has occurred in one system of parallel planes, any slip in a new direction crossing the first must be impeded, which must further increase the resistance to change of shape.

Fig. 8. Schematic representation of two adjacent slip layers, bent elastically. The lattice is represented by a network, and it can be seen that the convex side of the layer is stretched, whereas the concave side is compressed. Along the surface of contact \(TT\) the straight lines and planes of the lattice are distorted. They constitute the “internal surface of separation” of the crystal.

Of course, one might object that this picture of the phenomenon of hardening is inadequate, since in reality hardening occurs not only with such slips, when the surfaces that are formed cease to be parallel to the original ones, but already at the very beginning, when the slip planes are only being brought into action.

To such an objection one may, on the other hand, oppose the fact of the undoubted existence of preferential hardening of hidden slip planes. This phenomenon, first established in zinc, where it is expressed most sharply, is observed, moreover, to a slight degree in aluminum, and also, according to recent investigations, appears

is clearly absent in brass ¹) (28). But if, thereby, the fact of the preferential strengthening of slip paths that appear only later may be considered established—a fact following directly from ideas about the lattice of a bent crystal—then it must also be added that one cannot speak of the strengthening of each of these paths separately. This means that we must accept that slip does not occur only in a single system of parallel paths, but that, from the very beginning, other small shears also arise, intersecting at an angle the paths of the principal slip; although they are of little significance for the roughly measured, total change in the shape of the body, they nevertheless lead to an increase of resistance in the paths of the principal slip. Indeed, it is extremely unlikely that slip could ever be limited to one system of parallel planes. For aluminum crystals, Caks and Karnop (19) were able to show, by means of precise measurements of the deformed specimen, that several planes act together from the very beginning of stretching. Even in zinc, where the preference for one of the planes (the base) as the slip plane seems most clearly defined, one can find, upon fracture of the crystal, striations on its fracture surface which are traces of numerous mutually intersecting slips (2).

It could also be objected that, in the case of bending of a crystal, we are dealing with a certain special case of plastic deformation, which cannot be equated with those cases of stretching in which only parallel shears of the slipping layers occur. However, these objections are unfounded, since even with such stretching, which apparently proceeds quite uniformly,

¹) The observations on the basis of which Schmid (29) came to the conclusion that the hidden slip planes in zinc are less hardened than the planes already brought into action should, according to new observations by Mathewson (30) and by Schmid himself, be interpreted in another sense.

Laue diagrams show the presence of a certain curvature in the planes of slip (31).

However, the interpretation given here applies only to one of the kinds of hardening, namely to that in which the resistance to shear increases. Tensile hardening remains unexplained. One can only point to its kinship with another, likewise mysterious phenomenon: namely, that fine-grained metal has a considerably greater tensile strength than coarse-crystalline metal, or, to an even greater degree, than a single-crystal wire (23). Since, when a crystal is deformed, owing to the formation of internal surfaces it is divided into parts, it is understandable why it then begins to behave like a fine-grained metal.

8. Restoration of the Former Elastic Properties of the Crystal1

Deformed crystals contain stresses and have internal surfaces of separation. In this state, therefore, they are richer in energy than the original crystal, and the increase in energy is twofold—on the one hand, as a result of the presence of stresses; on the other hand, owing to the energy of the new internal surfaces of separation. Such a state is a constrained state and must tend to pass into the normal state; hence it should be expected that, when occasion permits, it will return to it of itself. In polycrystalline metals, a similar transformation of strengthened material into an unstrengthened state is well known, occurring with the passage of time and especially under the action of elevated temperature (annealing); it should be assumed that this phenomenon is connected with the proposition stated above concerning the capacity of a deformed crystal for reverse transformation.

The first step necessary for this connection has already been taken, namely, proof has been obtained of the existence of softening (a phenomenon opposite to hardening) in single-crystal bodies. At the same time, two different ways have been found in which the softening of stretched zinc crystals can occur: 1) gradual softening, the more complete the higher the temperature and the longer the annealing time, and 2) a second kind of softening, in which the initial strength of the crystal returns, but without intermediate stages (at least noticeable ones). In the first case, no change in the structure of the single crystal is observed. This kind of softening is called recovery (Erholung, Vergütung). In the second case, after annealing the wire proves to consist of two or more crystals—here, consequently, a disintegration of grains, recrystallization (Rekrystallisation), takes place. Since both recovery and recrystallization are also observed in polycrystalline bodies (although the first phenomenon, owing to the complexity of polycrystalline bodies, is only assumed, and could not be precisely established), the desired connection between single- and polycrystalline bodies is established in a qualitative respect.

If we now accept, as was done in the preceding paragraph, that hardening is caused by the appearance of internal surfaces of separation, then softening, on the contrary, must be based on the elimination of these surfaces of separation, as a result of which the subdivision of the crystal into parts either disappears or, at least, decreases, so that the regions of undamaged lattice become larger. Such restoration of the lattice leads not to the appearance of new crystalline grains, but solely to the repair of those already existing. Moreover, the greater these repairs, the stronger the softening of the crystal.

The force causing this restoration of the lattice should be regarded as the tendency of the atoms lying along the boundary of separation to return to their normal position. Since the movements caused by these forces can take place only within atomic distances, they are not capable of bringing about a noticeable (to the eye) straightening of the bent layers.

slip. Therefore, owing to annealing, the state of stress of the crystal is only slightly weakened and continues to serve as the driving force for recrystallization, which occurs with a further rise in temperature. Recrystallization, however, should be understood as a further equalization of stresses, which takes place through grain growth.

It seems to me that the following, fairly important, observations are in good agreement with this hypothetical picture.

1. During annealing, as X-ray investigation shows, the curvature of the slip surfaces is preserved [Sachs (19)]. On the contrary, recrystallization (since it is not accompanied by the appearance of secondary stresses) leads to the formation of sound grain.

2. When a single crystal is stretched, the effect of stretching on strength and on the capacity for recrystallization proves to be different depending on whether the stretching is carried out in the form of simple elongation, as we illustrated above on the model, or by rolling, or by forging. In elongation, a much lower strength and a lower recrystallization capacity are obtained than in rolling and forging (thus, a tin crystal stretched fivefold recrystallizes only near the melting point (210°), whereas one rolled out to the same degree does so already at 60°). The difference is caused by the fact that, under simple stretching, the sliding layers are bent only slightly and in the final result remain almost flat. In rolling or forging, on the contrary, a considerable squeezing of the layers occurs, together with the corresponding accumulation of stresses and abundant formation of internal surfaces of separation.

3. If a partially stretched crystal is heated, then recrystallization begins to spread from the place where the elongated ribbon of the crystal borders on the unstretched part of it (Fig. 9). Thus the greatest recrystallization capacity turns out to be not in the region of the greatest stretching, but in a certain intermediate region. The peculiarity of this transition region is that here the orientation of the lattice, as it is given in the original wire, passes into the orientation of the crystalline ribbon, otherwise

generally speaking, there is a bending of the crystal here. However, the layers that are strongly bent at this point, in the further course of stretching, when they in turn pass into the crystalline ribbon, again become flat, and no curvature can any longer be noticed in them by eye (21).

This means that, in the process of stretching, during which the individual layers of the crystal bend one after another and then straighten again, the recrystallization capacity reaches a maximum at the places where bending occurs, and again falls, despite the increasing stretching, when the slip layers straighten out once more. This agrees with our conception that recrystallization is caused by elastic stresses accumulating at the bends of the (coarser) slip layers.

Fig. 9. Partially stretched (on the left the unstretched part, on the right the stretched part) tin crystal subjected to recrystallization. The formation of new grains begins at the boundary between the stretched and unstretched parts of the crystal. An arrow points to this place. Magnified 10 times.

The mechanical straightening of bends in the slip layers should not in any way affect the internal surfaces of separation, if, as we understand it, the latter can disappear only as a result of thermal motion. Therefore strengthening must occur upon further stretching, even if the layers are thereby straightened. This conclusion is also confirmed (from the analysis of stretching curves), namely: despite a decreasing recrystallization capacity, the strengthening increases with further stretching.

The phenomenon described here of an inflection in the course of the recrystallization capacity, in contrast to the continuous growth of strengthening when the slip layers are straightened, may serve as an explanation of the following simple experiment of Czochralski [J. Czochralsky (32)]: a crystalline rod of

aluminum is twisted. In the crystal there then arises a recrystallizing capacity, and at the same time it is strengthened. Upon reverse untwisting, the recrystallizing capacity diminishes, while the strengthening, on the contrary, continues to increase, as was shown by Sachs (37).

1. Finally, one should also point to another remarkable observation made by Schmid (33), namely that upon annealing there disappears not only the increased resistance to change of shape (up to now we have spoken only of annealing of this kind, which changes the resistance to shear), but there also occurs a restoration of the former tensile strength (Reisserholung); that is, without the onset of recrystallization there disappears the increase in tensile strength which resulted from the preliminary deformation. This phenomenon (still quite mysterious in its basis) is in agreement with our assumption that strengthening in tension is determined by the existence of internal surfaces of separation.

Mono- and polycrystal.

Thus single crystals, quite similarly to polycrystalline bodies, undergo strengthening, and moreover both with respect to slip and with respect to rupture. The state of strengthening, also in the case of a single crystal, proves to be a constrained state, passing upon heating into the normal state, either by means of annealing (Erholung), or by recrystallization. So matters stand qualitatively. But quantitatively there are large differences. Let us compare, for example, the tensile curves of a single- and a polycrystalline body, shown in Fig. 10. It is clearly seen how much more rapidly the strengthening of a single crystal occurs under tension. The same difference is also observed in the recrystallizing capacity caused by preliminary deformation. With the same degree of deformation, a single crystal recrystallizes with considerably greater heating than does a polycrystalline body. If our conceptions of strengthening and recrystallization are correct, then the difference must lie

consists in the fact that in the grains of a polycrystalline body subjected to tension, internal separating surfaces and stresses existing at the points where the slip layers bend arise in greater number than under equal conditions in a single crystal. That this must indeed be so is shown by a single glance at the crystalline

Fig. 10. Tensile diagram. A—annealed monocrystalline metal. B—single crystal.

grain of any polycrystalline body, compressed on all sides by its neighbors: here, of course, excluded is that regular change of form (formation of a band, etc.) in which alone the formation of an elongated crystal is possible, having exclusively only flat slip surfaces. Moreover, precisely this mutual connection

Fig. 11. A column consisting of several large crystals after tension. Formation of a knot at the boundary of contact of two crystals (after Ventzel), \(2/3\) natural size.

at the common boundaries between grains necessarily causes forced changes of shape in the grains, leading to good intermixing of the polycrystalline body.

One may expect that this constraint, acting chiefly in the region of intergranular boundaries, manifests itself most strongly in their immediate neighborhood. Indeed, the boundaries between grains are the places of greatest hardening.

This is evident, for example, from the fact that when a wire consisting of two crystals is stretched, a node always forms at the place where they touch (Fig. 11). Likewise, during recrystallization the boundaries between grains prove to be the place from which new formation predominantly begins, as was shown long ago by Chappel [Chappel (36)] and also confirmed by experiments with a bicrystalline wire (34). It is remarkable that, as Sachs (19) pointed out, the difference in the behavior of a single crystal and a polycrystal in the case of metals with a cubic lattice is insignificant. This is probably because here several slip planes and directions, intersecting one another, always participate simultaneously, and therefore more rapidly bring the crystal to hardening.

State of the Question

The geometrical mechanism of the change in the shape of crystals, as it is set forth here, adjoins earlier mineralogical investigations and may apparently be considered, in the main, clarified. What remains unclear, on the contrary, is the question of the “premature” occurrence of rupture, slip, and twinning shear. These three kinds of change in shape are closely connected with one another and can find their explanation only together.

As for hardening, then, proceeding from the state of the crystal lattice under bending, one can create a clear picture of it, which will contain only the necessary conclusions. It therefore seems to me permissible and desirable to develop these conceptions still further—to the limits within which they prove useful.

The indicated picture is capable of giving an explanation for some essential aspects of the phenomena of hardening, tempering, and recrystallization.

Of course, one may mention much that either cannot be explained at all with the aid of the given picture or is explained by it only with difficulty. Such, for example, are the change of the electrical and magnetic constants under deformation, the increase in the colorability of rock salt by sodium vapors [Pri-

DEFORMATION, FRACTURE, AND STRENGTHENING OF CRYSTALS

Pribram (Pribram) and Smekal), the influence of water on the elastic properties of rock salt, etc. It may be that one phenomenon or another points to additions or changes which our theory still requires.

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1. The content of this chapter is based on work carried out by the author jointly with Schmid. See (34). ↩