B. P. Weinberg.

On the Study of the Behavior of a Solid Body Beyond the Elastic Limit. 1. Internal Friction of Iceland Spar and Rock Salt.

Since 1903 I have been working on the study of the behavior of a solid body under the action, and after the action, of forces exceeding its elastic limit, or, more briefly, beyond its elastic limit, and continued, together with a number of collaborators, to conduct investigations in this direction until 1915, when my departure for two years to America, and the disruption caused by the German and civil wars, interrupted them. The materials studied were: lead, copper, brass, iron, steel, rock salt, Iceland spar, ice, paraffin, clay, pitch, asphalt, dammar resin, and also various mixtures of the last three bodies, both with considerably less viscous liquids and with neutral solid bodies. The experiments were carried out in the physical laboratories of Novorossiysk, Petrograd, and Göttingen Universities and of the Tomsk Technological Institute, and in the mechanical laboratories of the Institutes of Ways of Communication, the Electrotechnical Institute, the Petrograd Polytechnic Institute, and the Tomsk Technological Institute, as well as in domestic conditions (on Neva ice) and in expeditionary conditions (on the ice of the Gok-Yokh and Gantereis glaciers in Tyrol).

Descriptions of methods of observation and a comparatively small part of the results obtained have been printed—or, in recent years, prepared for printing¹; these relate almost exclusively to typical plastic bodies—ice, pitch, asphalt—for which the elastic limit is so small that it can be judged only from the dependence of the coefficient of internal friction \(\eta\) on the rate of deformation².

As for the material relating to my experiments on solid bodies with a considerable elastic limit, until recently I refrained even from working it up, in view of the probably significant, but not fully clarified, influence of temperature on the course of processes in such bodies under the action of forces exceeding, or having exceeded, the elastic limit³. Although the corresponding experiments were, as far as possible, also carried out in rooms where the daily temperature variations were insignificant, nevertheless the annual course of temperature was still great; and this, given the very long duration of many series of experiments—exceeding three years in one case—reduced the reliability of possible conclusions and compelled me to strive for the construction, at our physical laboratory, of an extensive underground room with a temperature adjustable—at least within small limits—but, if possible, constant, where the influence of temperature on these phenomena would first be clarified by sufficiently prolonged experiments. The impossibility of carrying out this intention in the near future, the considerable revival of interest in questions of the solid body in the physical literature (as well as the recent establishment

at the Research Institute of Applied Physics attached to the Tomsk Technological Institute, whose principal task was the study of the behavior of bodies beyond the elastic limit, prompted me to process these observations as well, introducing into them the corresponding—though not entirely reliable—corrections for the influence of temperature.

For practice, the creation of a theory of the behavior of a metal beyond the elastic limit would be of the greatest importance, if by this one understands the influence of time on the dependence of deformation on force. Ordinary metals are quasi-isotropic bodies, consisting of separate “crystallites” (in Tammann’s terminology), whose crystallographic axes, in a practically isotropic metal, are arranged without any regularity and which are fastened to one another by an isotropic “solder,” or, more precisely, by grains of considerably smaller dimensions—very probably of eutectic structure. Therefore the theory of the behavior of a metal beyond the elastic limit must⁴ have a statistical-geometrical character, and at its basis, for materials in which the solder between the crystallites is stronger than they themselves are, there must lie the study of the behavior beyond the elastic limit of individual crystallites, or of their macromodels—individual crystals or pieces of a crystalline body. In approaching the theory of behavior beyond the elastic limit of those metals in which the solder between the crystallites is less strong than they themselves are, the study of materials composed of a typical plastic body with a certain admixture of neutral solid bodies may serve.⁵

Therefore, from the extensive experimental material not yet published, I choose for this first article experiments on Iceland spar and rock salt—all the more because the successes of X-ray analysis have considerably clarified the structure of crystals (in particular, its changes under their residual deformations) and have brought questions concerning the deformation of crystals into prominence. Spar and salt, in comparison with ice, are both typical solid bodies and typical crystals, since, although the cylinders of river ice that I investigated did have optical axes parallel to their geometrical axes, it was impossible to say whether they were cut from a single crystallite perpendicular to the freezing surface or from several; as for glacier ice, its cylinders undoubtedly consisted of separate crystals with axes directed in various ways and with dimensions of almost the same order as the transverse dimensions of the cylinders.

Since, under the action of forces exceeding the elastic limit, we can, at will, keep only either the force or the deformation constant,⁶ the question of studying the behavior of bodies beyond the elastic limit is reduced to the study of separate regimes, of which I personally have studied only the four simplest.

I. Force = const.
II. Deformation = const. (relaxation).
III. Force = 0 (elastic after-effect).
IV. Rate of deformation = const.

Under regime I, 180 series were observed, with a total duration of 3012 days; under II—61 series (188 days); under III—101 series (1756 days); and under IV—45 series (1499 days), for a total of 387 series with a total duration of

6455 days. Usually observations were conducted on several “century-old” installations simultaneously; moreover, thanks to the kindness of my collaborators, they were not interrupted during my illnesses, departures, etc.; in particular, many observations on spar and salt were made by I. Ya. Tochidlovskii, K. I. Ivanov, E. A. Kirillov, V. I. Zanchevskii, I. A. Smirnov, and also by Prof. N. Katman.

As for the types of deformation, having initially⁷ in mind only the determination of the coefficient of internal friction of solid bodies, I chose, both in developing the methods and in the determinations themselves, such deformations in which, as far as possible, there are exclusively changes of shape and not of volume.⁸ Subsequently we dealt with changes of volume only in studying the resistance of “solid,” or very viscous, bodies to the motion of solid bodies within them.

And now I would state definitely—although on somewhat different grounds—that, for studying the behavior of bodies beyond the elastic limit, it is necessary to resort chiefly to deformations in which only changes of shape occur, or, at least, in discussing the results of the study of other deformations, to single out the magnitude of the tangential stresses in the deformed specimens and of the shears caused by them. On the basis of the totality of our information about solid bodies, one may express the thought that, for changes of volume, there exists, when the volume is decreased, neither a limit of elasticity nor a limit of plasticity, while when the volume is increased these two limits must coincide. Such an assertion, only with respect to the elastic limit, is at variance with the hypothesis I have expressed⁹ concerning the existence of two limits of plasticity (the same I also assumed for elastic limits): one for changes of shape, the other—for increases of volume. I shall note that the confirmation by experimental data of Guest’s “law of compound stresses,” according to which residual deformation or rupture always occurs upon the tangential stress reaching certain values, may be interpreted from this point of view as meaning that, in all combinations of various deformations studied so far, the elastic limit was reached earlier at some point of the specimen, and then the limit of plasticity in shear, than any negative pressure reached the limit of plasticity in all-round extension, which, generally speaking, is difficult to make considerable without considerable shears.

I tested Iceland spar only in shear, and rock salt—in shear and in torsion. The investigated crystal of spar belonged to the physical laboratory of Novorossiisk University; its origin is unknown to me; it was fairly transparent. Such also was the rectangular parallelepiped of rock salt investigated in shear; the rest of the salt was brought by me from the Wieliczka mines, thanks to the kindness of the management, who selected for me an especially transparent piece measuring about \(30 \times 15 \times 15\ \mathrm{cm}^3\). From it, by careful splitting along the cleavage planes, I obtained prisms of rectangular se—

⁷ B. P. Weinberg, Zh. R. F. Kh. O., 36, 47–48, 1903; Proc. Phys. Soc. Lond., 19, 291–292, 472–474, 1905.

⁸ Of the 15 existing methods for determining the coefficient of internal friction of solid bodies (13, if one counts squeezing and flow through tubes as two), analyzed by me in Zh. R. F. Kh. O., 38, 192–203, 1906; in addition, the method of shearing two parallelepipeds has been applied—indicated by me in Zh. R. F. Kh. O. 44, 45, 1912; developed by V. D. Dudzelskii, Zh. R. F. Kh. O., 45, 519–532, 1913—and the method of a body falling into a “liquid”—B. P. Weinberg, Zh. R. F. Kh. O., 42, 167–176, 1910; 45, 252–256, 1912; in 9 there are only changes of shape; of these, 7 were developed or improved by me and my collaborators.

⁹ Zh. R. F. Kh. O., 45, part II, 88–97, 1913.

readings (my experiments in Odessa and Petrograd and V. D. Kuznetsov’s in Tomsk) or plates (my experiments in Göttingen); the latter were polished. Wieliczka salt is distinguished by negligible hygroscopicity, although in the underground room of the Göttingen laboratory, with a very constant temperature, even plates of this salt became covered with so much dew that it was necessary to resort to lubricating them with paraffin oil or Venice turpentine and to placing large portions of calcium chloride under a cover of the instrument.

Shear. The ideal deformation for studying phenomena beyond the elastic limit is shear, as a homogeneous deformation not accompanied by changes of volume; but the realization of pure shear is simple only for bodies with a very small elastic limit. When the method of shearing rectangular parallelepipeds gave quite satisfactory results for var \(^{10}\), I tried to apply it first to lead, and then to crystals, for which the very ascertainment of the gradual increase of deformation with time under the action of a constant force was quite essential.

A specimen was cast from lead and then worked in the form of two thick (10 mm) parallel plates, joined by a comparatively narrow (4 mm) plate perpendicular to them (height 45 mm, length 32 mm). The lower plate was fixed horizontally, while from the upper one a load was pulled, parallel to the length of the narrow vertical rectangular parallelepiped, by means of a cord (or a thin wire rope) thrown over a pulley, arranged so that the tension was parallel to the upper plate. I note that under such a force action on the specimen the rear upper end tends to rise upward, and the front one to sink downward, so that the shear of the layers in the horizontal direction is somewhat complicated by phenomena of shear in the vertical direction.

In view of the probable smallness of the shear angles and the duration of the experiment, it was important to work out a method for detecting and measuring small shear angles. For this purpose, in one of the first series on the shear of a lead parallelepiped, four small mirrors were used, each of which—MMM in Fig. 1—was attached to a brass plate ABCD, and this latter was set in a definite position relative to another, longer plate EFGH by means of three rods \(T_1\), \(T_2\), and \(T_3\) with screw threads and nuts at the ends, passing through holes in the plate EFGH, and three steel springs surrounding these rods and separating the two plates from one another. By means of the nuts it was possible rather easily to direct the ray from the scale even into a separate tube and thus to measure the angle of inclination of the plate EFGH, fastened, for example, by the end GH on a hinge to the lower plate of the lead parallelepiped, and resting with its middle on the upper one or pressed against it by means of a spring or rubber band, or else attached immovably either to the front face of the sheared parallelepiped or to the “stationary” lower plate. In the latter case the mirror was a “control” one, and the shear angle was determined from the difference of the scale displacements in the control mirror and in the “movable” mirror. Owing to the presence of the control mirror, it was possible not to take special care about the immobility of both the entire system and the tube with the scale.

In the series just mentioned, one of the mirrors was the control mirror, and the other three were movable; one of them was attached to the plate ABCD with wax, another with Mendeleev’s cement, and the third—

\(^{10}\) B. P. Weinberg and Iv. Al. Smirnov, Zh. R. F. Kh. O., 44, 3–35, 1912.

was mounted in a metal frame (all the mirrors being simply specially selected pieces cut from thin mirrors). Observations showed the equal reliability of all three methods of fastening, even in very prolonged experiments; as a result I usually resorted, as the simpler method, to fastening with putty.

Experience also showed the great smoothness of the curves representing the dependence of deformation on time, if the readings of the images of the scale in the various mirrors (the tube was placed at a distance of \(5^{1}/_{4}\) m, so that all four mirrors were visible in the field of view at one position of the eyepiece relative to the objective, and the images of the scale in all four—at another position of it) were made after a slight shaking of the string on which the load hung, or a slight tapping on the thick wooden board on which, in a niche of the wall, the specimens subjected to force were installed. This partly compensated the inequality between the tension of the string and the weight of the load, caused both by the variability of the tension of the piece of string between the sheared specimen and the pulley due to changes of temperature and humidity,^11 and by the variability of the force of external friction in the pulley, both in magnitude and in direction. That the force of friction before the body starts from rest may have any magnitude up to that at which motion begins is evident, among other things, from the observations of A. V. Verkhovskii, a preliminary communication on which is to appear in “Communications and Scientific-Technical Work in the Republic”; as for the direction, after the pulley starts from rest and the load, which has begun to fall comparatively freely, is stopped by the string and acts on the pulley with a force exceeding its weight, the force of friction may prove to act in the same direction as the tension of the string. In consequence of all this the motion of the mirror cannot be smooth, but inevitably proceeds by jumps even with the knife-edge pulleys that I usually employed—and the magnitude of these jumps is reduced by tapping before each reading.^12 These circumstances, which became clear only gradually after long observations, play almost no role for comparatively high rates of shearing; they acquire importance for rates of the order of \(10^{-11}\)—\(10^{-10}\) radians per second, with which I had to deal, and later prompted the effort to dispense with the transmission of force by pulleys, without supporting the torsion rings by bearings, and so forth.

For shearing a crystal of spar (and later—rock salt), two identical vertical walls were soldered to a thick brass horizontal plate; in these walls grooves were made for the movement along the saw of the upper plate. Brass strips were soldered to the upper and lower plates, bent so that the lower and upper parts of the crystal just barely entered them. The spar was fixed so that the forces acted on it parallel to one of the natural faces in the plane of the principal section in the direction toward the apex formed by three obtuse angles; on the salt—parallel to one of the cleavage planes.

The spar with which I experimented from 24.7.04 to 28.4.05 was subjected to an ever more considerable stress (and subsequent unloadings), but the insignificance of the changes in the differences of the scale readings in the movable and control mirrors from day to day allows one to regard as established the phenomenological fact of gradual defor—

11) In particular, the opening of windows and the washing of the floor were reflected in the smoothness of the curve; these had to be done, albeit rarely, in the room where this apparatus was installed.

12) For more detail on this, see Zh. R. F. Kh. O., 38, 300, 1906 and 44, 14–15, 1912.

…formation of Iceland spar under the action of a force close to constant; only the last two series under the action of a force of 21 kg, the first time over 34 days, the second time over 91 days.

These series clearly revealed what could also be observed at smaller loads, namely that, upon application of the force, a rather considerable “elastic” deformation is obtained, which gradually increases—at first rather rapidly, and then ever more slowly. The ascertainment and determination of the final, ultimate, and not infinitely small rate of deformation is very difficult, both because of the errors of the observation itself (the question is of tenths of a millimeter of the scale at a distance of \(5 \tfrac{1}{4}\) meters) and because of the accidental displacements of the mirrors, which undoubtedly sometimes occurred in the course of such prolonged experiments. At times these displacements were so considerable that one or another mirror had to be set up anew.

Fig. 2 shows the differences of readings (in mm) of the scale in the movable and fixed mirrors for the second series with a load of 21 kg. In accordance with the experiments described below by V. D. Kuznetsov on rock salt, all deformation rates are reduced to \(20^\circ\), under the assumption that they change by \(0.06\) of their value for each degree of temperature, which remained around \(18^\circ\), with extreme deviations of \(14^\circ.5\) and \(20^\circ.6\). Each point of the diagram represents, in the case of several readings in a day, their mean; moreover, both these means and the individual readings were reduced by simple interpolation to 12 noon. Reliable observations are marked by crosses, unreliable ones by small circles (in particular, the observations of the second half of January, regarding the doubtful nature of which there is a note in the observation notebook, have been completely rejected); cases of probable or undoubted displacement of a mirror in the interval are marked by breaks in the broken line drawn through the separate points of the diagram.

If one takes the most reliable periods from 26.11.04 to 18.12.04 and from 29.1.05 to 21.3.05 and draws straight lines with respect to which the separate points would be distributed as uniformly as possible on both sides, the tangents of the angle of inclination are obtained as \(0.135\) and \(0.059\) mm of scale per day, whence, for the mean angular velocities of shearing \(\varphi' = 1.5\) and \(0.7 \cdot 10^{-10}\) radians per day, values of \(\eta\) are obtained equal to \(0.7\) and \(1.5 \cdot 10^{16}\ \frac{\mathrm{g}}{\mathrm{cm}\cdot\mathrm{sec}}\). The indication \(\varphi'\) is given in view of the dependence of \(\eta\) on \(\varphi'\), observed for bodies with a finite elastic limit, although in the present case the first value of \(\varphi'\) is far from the ultimate final rate, as, probably, is the second.

The parallelepiped of rock salt was subjected to shearing forces and subsequent unloadings from 2.5.05 to 1.7.06,—and I shall confine myself to presenting the diagram (Fig. 3) depicting the longest series (193 days) at a load of 30 kg and extreme temperature limits of \(14^\circ.3\) and \(23^\circ.5\). From these series, if the first three weeks are excluded, there are obtained for the remaining period, divided into three equal parts, values of \(\eta\) of the order \(5.7\), \(5.7\), and \(6.2 \cdot 10^{16}\) (with \(\varphi'\) equal to \(3.3\)—\(3.0 \cdot 10^{-11}\)). A fairly long preceding series at a load of 10 kg (109 days, which was preceded by a series at the same load over 86 days) gave a value of \(\eta\) of \(3.8 \cdot 10^{16}\), with \(\varphi' = 1.6 \cdot 10^{-11}\).

Twisting. Twisting has, in comparison with shear, the advantage of a considerably larger magnitude of the angle of twist in comparison with the angle of shear even of the outer layers of the body being twisted; but, on the other hand, it is a nonuniform deformation, the interpretation of whose results may present greater difficulties, if only because, for the more

of the outer layers the elastic limit may be exceeded, while for the more internal ones the deformation will still remain elastic. To twist thin-walled tubes, for which the differences of the angles of shear of different coaxial layers, for crystals, are almost unattainable, it was therefore necessary to confine oneself to the twisting of prisms and strips. The arrangement of the experiment for twisting them very much resembled the arrangement of experiments on the twisting of ice cylinders⁷. Worth mentioning are only the device of the clamps for the mirrors, which is clear from Fig. 4, and the method of fastening the ends of the rock-salt prisms in recesses made for them in the fixed stand and on the axle connected with the torsion circle. These ends were either filled in with molten alum, or clamped. The latter was preferable, since alum weathered during prolonged standing.

The coefficient of internal friction was computed from the formula

\[ \eta=\frac{12PgRL}{ab(a^2+b^2)\alpha'}=\frac{6PgRL}{a^4\alpha'}, \]

where \(PgR\) (dyne-cm.) is the twisting moment, \(L\) is the length of the specimen, \(a\) and \(b\) are the sides of the cross-section, \(\alpha'\) is the angular velocity of twisting; the second part of the formula refers to prisms of square cross-section (experiments of V. D. Kuznetsov).

The longest series of my observations are the following:

I. A prism (\(L=20\) cm., \(a=2.0\), \(b=1.2\)), placed horizontally, under the action of a twisting moment of 25 kg-cm. (176 days—winter 1905–6 and spring 1906, in Odessa);

II. The same prism in apparatus I for twisting glacier ice, under the action of a moment of 11 kg-cm. (116 days—winter 1905–6, Odessa);

III. The same prism, somewhat shortened, in apparatus II at \(30\text{--}32^\circ\) (25 days at the beginning of the summer of 1907, Petrograd); \(PR=12\) kg-cm.;

IV. The same prism remained in the apparatus under the same load; after 714 days another reading on the scale was made; the temperature was not recorded;

V. A strip (\(L=15\) mm., \(a=5.8\), \(b=1.0\)): \(PR=250\) g-cm.; 12 days (summer 1907, Göttingen).

For lack of space I shall confine myself to giving, for series I, in the table placed below, the readings (in tenths of a millimeter on a scale 525 cm away), counted from the reading with no load after \(t_1+t_2\) days after the beginning of the experiment; all readings have been corrected for displacement of the fixed mirror and for accidental or deliberate displacements of one of the mirrors, and have been reduced to \(20^\circ\), and the values of \(\eta\) for series I—V. These values were obtained from the final velocities for series I—from observations of the last 128 days, for II—from 84, for III—from 7, for IV—from all 714, and for V—from the last 5. For I, II, IV, and V one obtains the values (with \(\varphi'\) of the order of \(5,\ 3\), and \(4\cdot10^{-12}\))

\[ \eta_{20}=2.3\cdot10^{18};\ 1.8\cdot10^{18};\ (1.2\cdot10^{18});\ 4\cdot10^{16}, \]

and for series III (with \(\varphi'=2\cdot10^{-11}\))

\[ \eta_{81}=2.6\cdot10^{17}. \]

Table.

Continuation of the table.

¹) Unloaded.

I note the circumstance that the values of $\eta$ from torsion (if one discards the rather unreliable value of $\eta$ from Göttingen’s short-term experiments, where the frame could slide over salt dissolved in atmospheric moisture) are considerably greater than the values of $\eta$ from shear.

Conclusion. The experiments described fully establish the phenomenological fact of a very prolonged increase in the deformation of a crystal under the action of a practically constant force exceeding the elastic limit, and make it highly probable that the rate of deformation in this case tends asymptotically toward some finite magnitude. The latter cannot be asserted with complete certainty, since a duration even of 5–6 months is far from sufficient.

It is much more difficult to give any mechanical model of the forces of internal friction in a crystal analogous to the model of the transfer of excess quantity of motion by gas particles. The only picture that suggests itself to me personally is a jump—under shear—of whole layers of the crystal from one position of the elements of the crystal lattice into another, analogous one, but displaced relative to the first by an integral number of times the distance between these elements. Under torsion, however, such a jump is impossible, and with residual deformations there must occur a kind of internal fracture. Arguments in favor of such an interpretation are the phenomena of simple and twin gliding, demonstrated by A. F. Ioffe. The preservation of the crystalline structure under other residual deformations, and the internal fractures found both by these experiments and by the observations described below by V. D. Kuznetsov, as well as the fact that $\eta$ from torsion proved, for rock salt, to be greater than $\eta$ from shear. But precisely how the element of time enters into the phenomenon of the jump of a layer, and how the same tangential stress which caused one jump can be exerted by that same layer in its new position, is far from clear to me. Most probably, both in the crystal and in the gas, one should speak not of a force of internal friction, but only of the work required for the shear (or of the power required for shifting) of one layer relative to another, just as in gases one can speak only of the quantity of motion transferred in an analogous shear. Such a finite shear may occur successively in different layers of the crystal being sheared, depending on the random arrangement of the elements of the crystal lattice relative to one another during their motion relative to the lattice vertices, which makes possible such a finite shear. An argument in favor of such an assumption may be the enormous influence of temperature on the internal friction of solids in general and of crystals in particular.

B. P. Weinberg’s experiment.
Study of the behavior of a solid body beyond the elastic limit.
1). Internal friction and [[unclear: continuation cut off at page edge]]

Fig. 1.

Fig. A.

Fig. A.

In the graph:

number of days from the beginning of the experiment

[[unclear: fragment of text at the upper left]] beyond the limits of elasticity,
[[unclear: fragment of text]] of the type and
[[unclear: fragment of text]].

Fig. 3.

Fig. 2.

Sanitary belt.

1. A list of these 37 works, for lack of space, will be placed here in my next article on the internal friction and relaxation of lead. ↩

2. Cf. B. P. Weinberg, Zh. R. F. Kh. O., 38, 189, 1906. ↩

3. For a summary of these data, see B. P. Weinberg, Zap. Nov. Univ., 105, 157–186, 1906; in subsequent times the experimental data increased comparatively little (for ice—B. P. Weinberg, Zh. R. F. Kh. O., 38, 270–273, 319–320, 1906; for pitch and asphalt—B. P. Weinberg, Zh. R. F. Kh. O., 44, 201–229, 1912; for rock salt—V. D. Kuznetsov, the following article; for paraffin—his experiments as well). ↩

4. B. P. Weinberg, “Probable outlines of a future theory of deformation of a solid body (as a preliminary communication),” Proceedings of the Society of Naturalists and Physicians at Tomsk University for 1913, 175–181, 1915. ↩

5. B. P. Weinberg, Proceedings of the 3rd Congress of the Russian Association of Physicists, 24–25, 1923. ↩

6. B. P. Weinberg, Diary of the 2nd Mendeleev Congress, 3, 24–25, 1911; Zh. R. F. O. 45, Part II, 67–115, 1913. ↩

7. B. P. Weinberg, Zh. R. F. Kh. O., 38, 250–281, 289–328, 1906. ↩