B. P. Weinberg.

Toward the Study of the Behavior of a Solid Body Beyond the Elastic Limit

II. Internal Friction of Lead

Lead, as the “softest” of the common metals, was the first typical solid body to show me quite convincingly that, under the action of a constant force exceeding the elastic limit, changes in its shape continue over considerable intervals of time and considerably exceed the elastic deformation; and, somewhat less convincingly, that in this process the rate of deformation tends asymptotically to a certain limiting value different from zero.

For this reason lead was the object of the most varied and prolonged experiments; the present work is devoted to a review of these experiments, together with a statement of the results relating to the stationary regime under the action of a constant force. In it I confine myself to giving only the experimental data.

The lead specimens studied by me were pieces of bell tubes (for air bells) or water pipes, or were obtained by casting from the material of water pipes, and had the following shapes and dimensions (the latter data, like all distances, are in centimeters):

I. Rectangular parallelepiped: \(L = 3.18,\ a = 0.394,\ h = 3.52\), described in detail in the preceding article of this same series¹.

II. Tube (“short horizontal”): \(L = 9.0,\ R = 1.588,\ r = 1.278\).

III. Tube (“thin vertical”): \(L = 269,\ R = .2535,\ r = .1794\).

IV. Tube (“thick horizontal”): \(L = 65.7,\ R = 1.179,\ r = 0.935\).

V. Tube (“thick vertical”): \(L = 275,\ R\) and \(r\) the same as in IV.

VI. Tube: \(L = 51.5,\ R = 1.406,\ r = 1.010\).

VII. Tube: \(L = 46.0,\ R = 3.063,\ r = 2.555\).

VIII. Tube: \(L = 30.7,\ R\) and \(r\) the same as in VII.

The history of the force actions on these specimens may be summarized in the following table, in which the constant force is denoted by \(F\), the constant moment by \(M\), the constant angle of twist (in radians) by \(\alpha\), and the constant angular velocity of twisting (in radians per second) by \(\alpha'\); days, months, and years are separated below by points.

Table I.

III. a) 24—26.1.04, M = 181 g cm; b) 26—28.1 M = 0; c) 28.1—1.2, M = 14; d) 4—7.2, M = 35; e) 7—11.2, M = 0; f) 11—13.2, M = 56; g) 13—16.2, M = 0; h) 16.2—3.3, M = 35; i) 3—13.3, M = 56; k) 13.3—27.5, M = 78; l) 27.5—21.6, M = 25; m) 21.6.04—29.3.05, M = 110; n) 29.3.05—10.9.06, M = 0.

IV. a) 4—5.2.04, M = 2.5 kg cm; b) 5—11.2, M = 6.3; c) 11—12.2, M = 0; d) 12—21.2, M = 8.9; e) 21.2—22.5, M = 12.7; f) 22.5—21.6, M = 0; g) 21.6.04—9.4.05, M = 51; h) 9.4—9.9, M = 0.

V. a) 24.3—6.5.04, α = 4.2·10⁻⁴; b) 6—18.5, M = 0; c) 18.5—11.6, α = 8.9·10⁻²; d) 21—23.6, M = 0; e) 23.6—5.7, α = 1.2·10⁻⁴; f) 21—22.7, M = 0 kg cm; g) 21.7, short-term twisting at constant speed; h) 22—23.7, M = 0; i)—r) 22.7—3.8, 8 short series of twisting; s) 6—24.8, M = 0; t) 25.8, short series of twisting; u)—ts) 2—3.12, 4 short series of twisting; ch) 3.12, α′ = 3.0·10⁻⁶; sh) 5.12, α′ = 2.1·10⁻⁶; shch) 14.12, α′ = 2.1·10⁻⁵; y) 4.1.05 α′ = 1.5·10⁻⁶; ʹ) 11—15.4, M = 0; ʺ) 15.4—21.9, M = 0; yu) 2.10—8.11, α = 2.3·10⁻⁴; ya) 18.11.05—31.5.06, M = 4; e) 31.5—10.9, M = 0.

VI. a) 14.1.05, α′ = 3.3·10⁻⁴; b) 14.1, α = 1.2; c) 14.1, α′ = 1.3·10⁻²; d) 14.1, α = 6.9; e) 14—17.1, α = 6.9; f) 17.1, α′ = 1.1·10⁻².

VII. a) 17—18.1.05, α′ = 5.0·10⁻⁵; b) 18—21.1, α = 3.6.

VIII. a) 27.1.05, α′ = 1.8·10⁻⁴; b) 27.1, α = 0.41; c) 27—28.1, α′ = 4.4·10⁻⁵; d) 28—29.1, α = 3.2.

The experiments with specimen II were of an orienting character (I note that, with the short-term application between series II-b and II-c of a moment = 25 kg cm, the twisting proved so rapid that the load had to be removed); their arrangement was analogous to that of the experiments with specimen IV, after making sure that the movable and immovable mirrors, attached at the ends of the screwed-in short tube, could be simultaneously seen in a tube located at a distance of 230 cm.

For twisting the thin vertical tube (specimen III), it was fastened by its upper end to the ceiling (by casting it into a lead piece in the form of a truncated quadrangular pyramid—compare the leaded ends of the tubes in Fig. 1), and to its lower end was attached a light wooden torsion circle (R = 5.32); two threads from it were passed over two light aluminum pulleys on points and were furnished with cups (wooden in series III-a and thin aluminum in the remaining series) for placing loads on them. The angles of twist were observed by means of a mirror fastened (on three rods—see Fig. 4 of work 44—as in all the other cases) to the torsion circle near the place where the twisted tube was attached, with two tubes with scales 100 cm long, set at a distance of 181 cm from the twisted tube so that the scales fitted closely against each other. At first, in the left tube (the twisting took place counterclockwise) only its scale was visible; then the beginning of the scale of the right tube appeared in its field of view, and at the same time the scale of the left tube appeared in the field of view of the right tube. After this, readings were made on both scales in both tubes until the scale of the right tube left the field of view of the left, and in the field of view of the right tube its own scale appeared. From that time the readings were made in this one tube, now becoming the left one, since the one that had until then been the left was moved to the right of the one that had previously been the right. In the longest series (III-m), such rearrangements had to be made 14 times, since the total angle of rotation of the lower end during this series alone amounted to 7½ radians (and during the time from the beginning of the experiments with this tube, more than 9 radians).

The thick horizontal tube (specimen IV) was inserted with one end fitting into the corresponding opening of the immovable wall, and with the other—into

the mandrel at one end of a steel axis of the torsion circle, which rested on agate bearings. To measure the angles of torsion there were three mirrors A, B, C (Fig. 1; dimensions not observed): the middle one was fastened to the middle part of the specimen by means of three copper screws with blunted ends, which clamped the middle cross-section of the tube, while the two outer ones were attached to the ends of thin-walled glass tubes, the other ends of which—by means of the same three screws each—were fastened to sections of the specimen close to the fixed post and to a section close to the torsion circle. In order not to clamp too tightly these screws, which had to be reset rather often—especially for mirror B—the glass tubes were slightly reinforced near the mirrors on top with steel strips (Fig. 1); similar strips, not shown in the drawing, placed between the middle mirror and the ends of the glass tubes, protected the specimen from bending under the influence of gravity.

The distance of the tube with the scale from the mirror was 523 cm. The action of the tube on the torsion circle was transmitted by a cord thrown over a pulley.

The installation of the thick vertical tube (specimen V) was made so as to avoid frictional forces in the pulleys or bearings. For this purpose its upper end was fixed immovably at the ceiling, and the lower end was fastened—exactly as in the vertical post—to the torsion circle, which was used, when necessary, to act on the specimen by a pair of forces from two loads acting through cords thrown over pulleys; the lower part of the torsion circle could be fastened by means of two mutually perpendicular and horizontally arranged pins to the upper end of a square steel rod, which was also positioned vertically under the center of the torsion circle and whose lower end could be turned by hand or by a motor. The angle of rotation of the lower end of the steel rod was given by a mirror placed on the upper end of the vertical glass tube, which coaxially surrounded the rod and was fastened to its lower end, while the angle of torsion of the lead tube, equal at the same time to the angle of rotation of the upper end of the steel rod, was given by a mirror placed at the bottom of the torsion circle. The difference of the readings of the one and the other mirror thus gave the angle of torsion of the lead rod. The tube with the scale was at a distance of 112 cm from the mirrors.

All these installations were in the basement floor of the Physics Institute of Novorossiysk University.

The remaining tubes (specimens VI—VIII) were loaded on Amsler machines in the mechanical laboratory of the Petrograd Electrotechnical Institute.

The processing of the observations was carried out by the same method as for observations on rock salt and Iceland spar: all intentional or accidental changes in the position of one or another mirror, or of the tube with the scale, and all differences of readings over a certain interval of time (except for differences corresponding to the initial “elastic” deformations), when the mean temperature was \(t^\circ\), were reduced to the temperature \(20^\circ\) by the formula derived (4,184) on the basis of the treatment of series III, namely:

\[ \Delta n_{20}^{\circ}=\Delta n_t\cdot 1.03^{20-t}. \tag{1} \]

In the case of regimes of constant or zero force and constant rate of deformation, the readings \(n\) served as a measure of deformation; in the case of observations on relaxation, as a measure of the gradually decreasing force, but the reduction to \(20^\circ\) was in this case made by formula (1), since it may be assumed that a decrease in the coefficient of internal friction of lead by 0.03 of its value when the temperature is raised by \(1^\circ\) must correspond to an increase by the same fraction in the rate of decrease of force at constant deformation.

From the 20 values of \(\eta\) thus obtained (obtained by summing successive \(\Delta \eta_{20}\)), the values of \(\eta\) at equidistant moments were then calculated, which were subsequently represented in the form of tables and graphs and subjected to further processing.

As regards the methods of actually carrying out the regimes \(F=\mathrm{const.}\), \(F=0\), \(M=\mathrm{const.}\), \(M=0\), \(\alpha=\mathrm{const.}\), \(\alpha'=\mathrm{const.}\), enumerated in Table I, only the regime \(M=0\) was carried out exactly in the case of twisting vertical tubes (specimens III and V); the remaining regimes were carried out only approximately—with oscillations of \(F\), \(M\), and \(\alpha\) about certain mean values and with a certain constant mean angular velocity of twisting.

The constancy of the mean value of \(\alpha'\) in the experiments at the Electrotechnical Institute, where the pulley of Amsler’s machine was driven by an electric motor into very slow rotation by means of several belt transmissions, could, owing to the sufficient steadiness of the voltage, be considered quite satisfactory. The transmission of rotation from the pulley to the frame in which the twisted end of the specimen was clamped (the other end was clamped in a frame on the axis of the pendulum-like part of the machine, which produced and measured the twisting moment) was, of course, effected by small jerks—for the same reasons as those set forth in detail in work 44.

For the same reasons, only approximately were attained both the regimes \(M=\mathrm{const.}\) when twisting vertical tubes (specimens III and V), where the action of forces was transmitted through pulleys, and the regimes \(M=\mathrm{const.}\) and \(M=0\) when twisting horizontal tubes (specimens II and IV), and the regimes \(F=\mathrm{const.}\) and \(F=0\) in the shear of specimen I, where, besides friction in the pulleys, friction in the bearings also played a considerable role. The latter circumstance, for the majority of the corresponding forces or moments, when weights suspended on a thread or wire acted, made the force or moment not equal to zero, but either increasing up to some positive value until the moment of its kind of “tearing away” from its place, or decreasing from some initial negative value to zero after such a tearing away. In the experiments on twisting a thin vertical tube, the hygroscopicity of those rather long, “severe” threads on which the cups were suspended also had an effect—and the variability of the moment acting on the tube itself is clearly revealed by the sharp difference in the degree of smoothness of the graph of the change of deformation with time under the action of a “constant” moment and under the action of a moment equal to zero, when even the threads were removed from the pulleys.

Turning now to the regime \(\alpha=0\), i.e., to observations on relaxation, I shall point out at once that, if the stress existing at a given moment is measured not by the optical—the only, perhaps, faultless in this respect—method, but mechanically—by the magnitude of the forces balancing these stresses—then, if only because in order to judge the presence of such equilibrium it is necessary to change the magnitude of the deformation until the forces balancing the relaxed stresses are ascertained by trial, constancy of deformation cannot be realized with complete rigor. Such deviations from constancy of deformation only during the very measurements of the magnitude of the balancing force—measurements which took a very short time compared with the interval of time (in long series, for the most part a day) between two observations—were carried out only in series V-d. In this series, a screw was attached to the twisting circle which, when the angle of twist of the tube increased to a certain value, came into contact with a contact fixed motionlessly at the side; this was judged by the sound of an electric bell connected into the same circuit. Owing to this, a certain part of that constant over the interval of time between two moments—

observations the moment of the twisting force of the lower steel rod was transmitted to this immovable contact,—and at every moment only that part of this moment acted on the lead tube which corresponded to the stresses in it that had undergone a certain registration. But even in this series the deformation did not remain constant, but gradually increased, since the stand itself with the contact was gradually displaced under the action of the constant force.

In series V-a and V-b, however, we achieved an approximately constant deformation. For this purpose, knowing from the preceding days how much the angle of twist of the lead tube increased per day when the twisting moment was constant (the decrease from the equal reduction of the angle of twist of the steel rod could be neglected), after determining that angle of twist at which the deformation of the lead tube at the moment of observation was equal to the previously assigned constant value, we twisted the steel rod so that the deformation proved to be less than this constant value by half the change expected over the following twenty-four hours, and left the apparatus in this state until the following day. By the time of the next observation the deformation exceeded the norm by almost the same amount,—and we again reduced it by half the expected change; and so on.

When twisting the tubes on Amsler’s machine I was satisfied with stopping the motor, since the angle of rotation of the “immovable” end, fastened to the measuring pendulum, was considerably smaller than the angle of rotation of the end twisted by means of the motor, and, consequently, the decrease of the first angle owing to relaxation of the lead tube had very little effect on the total angle of twist of the tube. In processing the observations, a correction was introduced for the increase of deformation during the interval of time between two readings of the balancing twisting moment.

Under the action of an approximately constant force or an approximately constant moment, one should expect an asymptotic approach of the rate of deformation to some constant limiting value; but the experiments described do not make it possible, even in the longest series—I-3 and I-k, III-k and III-m, IV-d and IV-zh—to establish with complete certainty the attainment of the limiting rate of deformation and to indicate its exact value. A duration of 2–3 months (I-3, I-k, III-k, IV-d) must clearly be considered insufficient for lead. With still longer twisting (8–9 months), and with comparatively large twisting moments, the angle of twist becomes so large that at first there begin to appear, locally, in small sections of the tube, larger angles of twist per unit length, and then “twists.” The latter are local flattenings, which in certain parts of the tube transform the cross-section from circular into elliptical and thereby increase the moment of inertia of the section. As a result of this, on the one hand, the twist spreads in both directions from the place where it first appeared, while, on the other hand, the rate of deformation becomes lower than the limiting value that might have been expected from the general course of its changes. This is clearly visible in Fig. 2, in which the results of series III-m are represented by the graph \(n=f(t)\) of the change in the angle of twist and by the graph \(\frac{dn}{dt}=f'(t)\) of the change in the rate of deformation; moreover, in view of the influence on the magnitude of the acting force of friction in the pulleys and of changes in the tension of the threads, and the considerable fluctuations thereby caused in the mean daily values of the angular velocity of twisting, the averages for each five days were taken for constructing the second graph. This graph shows that, after an approximate constancy of the rate over 2–3 penultimate months, there occurred a noticeable decrease in the rate in the last month of this series. Let us recall that all rates have been reduced to one and the same temperature.

The same local shifts and flattenings (and then also destruction at the place of greatest twist) were also obtained under very—comparatively—short-duration twistings, but ones occurring with large angular velocities, on Amstler’s machine, specimens VI, VII, and VIII, as well as tubes of other metals.

This is seen in Fig. 3, which is a photograph of the most characteristic twisted tubes. If, onto a plane, one unfolds the surface of a tube that has not yet undergone even local shifts, then the straight lines that have turned into helical lines forming the cylindrical surface would, under such an unfolding, give straight lines inclined to their original direction at the angle of shear of the outer layers (Fig. 4a). If, however, one unfolds the surface of a tube that has undergone a local shift, then the generatrices will turn into broken lines (Fig. 4b) made up of two parts at the same angle of inclination, connected by an intermediate piece much more strongly inclined to the original direction.

If one takes the intervals of time when the rate of deformation could be regarded as established, then we obtain the values indicated in Table II, in which by $\eta$ is denoted the coefficient of internal friction, by $\varphi'$ the angular velocity of shear of the middle layers of the tube, and by $T$ the mean value of the tangential stress experienced by the transverse layers of the specimen.

Fig. 3.

In processing the observations with specimen I, the means of the differences of the scale readings in each of the three movable mirrors (44, 42–43) and in the “fixed” one were taken, while in processing the observations with specimen III the angles of twist per unit length of the tube were calculated; for this purpose the sums of the differences corresponding to the middle and end mirrors (B and V in Fig. 1), of the values $n$ and of the values $n$ for mirror A, were divided by the sum of the distances of the points of attachment to the lead tube of the middle mirror and of the glass tube fastened to mirror B from the point of attachment of the glass tube of mirror A. The magnitude of the frictional force on the pulleys and in the bearings was taken to be 7% of the load in the experiments with specimen III and 10% in the experiments with specimens I, II, and IV.

Prof. B. P. Weinberg. “Intra-atomic Friction of Light.”

Fig. 4.

a)

b)

Fig. 5.

\(n = f(t)\)

\(e = f(t)\)

days

Table II.

An examination of this table reveals sufficient agreement between the results obtained by the sagging method and those obtained by the method of twisting tubes, and at the same time clearly indicates a decrease of \(\eta\) with increasing deformation rate. If this decrease were expressed by the formula

\[ \eta_{\varphi'}=\eta_{\infty}+\frac{f_e}{\varphi'}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (2) \]

(where \(\eta_{\infty}\) is the coefficient of internal friction for an infinitely large angular rate of displacement, and \(f_e\) is the limit of elasticity in shear), which follows from the relaxation law of Maxwell\(^1\), to which, however, the relaxation of lead does not conform, if judged from our experiments and from those of Trouton and Rankine\(^2\), then one could find the shear elastic limit of lead. Comparing the values of \(\eta\) at high rates of deformation (\(\varphi'\) of the order \(10^{-9}\)) for series Iз, Iк, and Iж with the values of \(\eta\) from series IIIк, IIIм, and IVд (\(\varphi'\) of the order \(10^{-10}\)), we obtain

\[ f_e\sim 3\cdot 10^{6}\,\frac{\mathrm{dyn}}{\mathrm{cm}^{2}}\sim 3\,\frac{\mathrm{kg}}{\mathrm{cm}^{2}}.\ldots\ldots\ldots\ldots\ldots (3) \]

Hence follows the indication that, for determining the coefficient of internal friction \(\eta_{\infty}\), series in which \(T\) has values less than (3) should be of little use, however many there may be.

But, on the other hand, series with too large values of \(T\) are scarcely suitable either, since (at any rate, with the rapid twisting of tubes) along with them enter the phenomena discussed on p. 31, and consequently one has to be satisfied with initial stages, showing the action of quasi-impact. As evidence I shall cite the results of treatment of very short-time—from fractions of an hour to several hours, as far as can be judged by the value of \(\alpha\)—series of twisting of lead tubes on Amsler’s machine.

Table III.

\(^1\) Compare 5, 188—189; 19 and 39.

\(^2\) Trouton and A. O. Rankine, Phil. Mag. 8, 538—566; 1904.

The values of \(\eta\) given in this table are wholly incomparable with one another and clearly show the unsuitability of such an arrangement of experiments for determining the coefficient of internal friction of lead (the same applies to other metals as well).

Thus here too medio tutissimus ibis—and the best results could have been obtained at comparatively large angular shear velocities—of the order of \(10^{-10}\)—although with them it is impossible to obtain so long a duration of the experiment as to be certain of the stationarity of the process.

A considerable part of the very laborious computations forming the basis of paper 44 and, in part, of the present one, was carried out by calculators (Yu. V. Grdina and V. A. Pletnev) placed at my disposal from February to July of this year by the Administration of the Tomsk Technological Institute, to which I offer my sincere gratitude for this assistance in my scientific work; while the greater part of the remaining computations (even more extensive) for the present work and for the treatment of observations on other regimes of lead were carried out by the permanent (Yu. V. Grdina and G. N. Sokolov) and temporary calculators of the Institute of Applied Physics, without whose help the treatment of these observations would have been extremely delayed.

1. 1. 23.

III. The steady regime of a plastic body under the action of a constant force from the relaxation point of view.

Maxwell put forward the supposition that, when the deformation \(\varphi\) of a plastic material is held constant (for definiteness we shall speak of the deformation of shear, so that \(\varphi\) will be the angle of shear), the force \(f_t\) exerted by it after an interval of time \(t\) elapsed since the beginning of the deformation is proportional to the initial value of the force \(f_0\) and decreases geometrically as the time \(t\) increases arithmetically. Hence it follows that

\[ f_t = f_0 e^{-\frac{t}{T}} \qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots (1), \]

if by \(T\) we call the “relaxation time”—the time during which the force decreases by a factor \(e\).

From this supposition concerning the regime of constant deformation Maxwell derived the law of the steady regime of constant rate of deformation, proceeding from the idea that, at a constant rate of deformation \(\varphi'\) (for shear, \(\varphi'\) will be the angular velocity of shear), there must ultimately be established an equality between the increment of force caused by the increase of deformation during the time \(\Delta t\), equal to

\[ \Delta f_t = N \varphi' \Delta t \qquad \cdots\cdots\cdots\cdots\cdots\cdots (2), \]

where \(N\) is the modulus of shear, and the decrease of force due to relaxation, equal to

\[ -\Delta f_t = -\frac{f_t}{T} e^{-\frac{t}{T}} \Delta t = -\frac{f_0}{T}\Delta t \qquad \cdots\cdots\cdots (3). \]

Equating (2) and (3) for \(t=\infty\), Maxwell obtained the magnitude of the force \(f_\infty\) in the steady regime of constant rate of deformation, namely

\[ f_\infty = NT\varphi' = \eta\varphi' \qquad \cdots\cdots\cdots\cdots\cdots (4), \]

where \(\eta\) is the coefficient of internal friction.

Shvedov, replacing Maxwell’s law (1) by the assumption that only the excess of the force over the elastic limit \(\lambda\) decreases from relaxation, obtained, instead of formulas (1) and (4),

\[ f_t-\lambda=(f_0-\lambda)e^{-\frac{t}{T}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5), \]

\[ f_\infty-\lambda=NT\varphi' \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6). \]

Applying the same reasoning to the establishing regime of constant rate of deformation, I have obtained formula (5, 224), expressing the law of increase of force in this regime:

\[ f_t=f_0e^{-\frac{t}{T}}+(NT\varphi' + \lambda)\left(1-e^{-\frac{t}{T}}\right)\ldots \ldots \ldots (7). \]

Formulas (4) and (6) are applied tacito consensu also to the case of an established regime of constant force, although the method of reasoning adduced is not applicable to the initial stages: the decrease of force is expression (3) under Maxwell’s hypothesis, and under Shvedov’s hypothesis,

\[ -\Delta f_t=-\frac{f_0-\lambda}{T}e^{-\frac{t}{T}}\Delta t = -\frac{f_t-\lambda}{T}\Delta t \ldots \ldots \ldots (8), \]

with constancy of the force \(f_t\) will be constant, and, consequently, the rate of increase of deformation too must at once become constant, which does not correspond to observations showing that after the application of the force the rate \(\varphi'\) gradually decreases, asymptotically approaching the value corresponding to (4) or (6).

In order to pass to the establishing regime of constant force with relaxation, let us divide the finite interval of time \(t\) into a very large number \(n\) of small equal intervals of time

\[ \Delta_1 t,\ \Delta_2 t,\ \Delta_3 t,\ \ldots\ \Delta_n t \ldots \ldots \ldots (9) \]

and let us assume that the individual increments of force

\[ \Delta_1 f,\ \Delta_2 f,\ \Delta_3 f,\ \ldots\ \Delta_n f \ldots \ldots \ldots (10) \]

causing the increases of deformation

\[ \Delta_1 \varphi,\ \Delta_2 \varphi,\ \Delta_3 \varphi,\ \ldots\ \Delta_n \varphi \ldots \ldots \ldots (11) \]

over the intervals of time (9), decrease from relaxation each independently.

The increase of force over the interval \(\Delta_n t\) will be equal to

\[ \Delta_n f=N\Delta_n\varphi=N\varphi'_n(t)\Delta_n t \ldots \ldots \ldots (12), \]

where by \(\varphi'_n(t)\) we denote the rate of deformation over the interval \(\Delta_n t\). To obtain the decrease from relaxation over this interval of time, we shall compile the following table (based on Maxwell’s law; an analogous table could be compiled also for Shvedov’s law) for the available forces arising at equal intervals of time (9), and we shall take their magnitude to be identical and equal to \(\frac{1}{n}t\), and denote it by \(\Delta\).

Table (13).

Since the trace remains constant, the sums of all the members of each row must be identical; consequently, if we denote the applied force (per unit surface) by \(F\), then

\[ \Delta_1 f\cdot e^{-\frac{t-\Delta}{T}} +\Delta_2 f\cdot e^{-\frac{t-2\Delta}{T}} +\Delta_3 f\cdot e^{-\frac{t-3\Delta}{T}} +\cdots+\Delta_{n-1}f\cdot e^{-\frac{\Delta}{T}} +\Delta_n f =F. \tag{14} \]

Hence, using formula (12), we obtain

\[ N\left[ \varphi(\Delta)\cdot e^{-\frac{t-\Delta}{T}}\Delta +\varphi(2\Delta)\cdot e^{-\frac{t-2\Delta}{T}}\Delta +\cdots +\varphi((n-1)\Delta)\cdot e^{-\frac{t-(n-1)\Delta}{T}}\Delta +\varphi(n\Delta)\cdot e^{-\frac{t-n\Delta}{T}}\Delta \right] \]

\[ = N\int_0^t \varphi'(z)\cdot e^{-\frac{t-z}{T}}\,dz =F. \tag{15} \]

Equation (15) can give the form of the function \(\varphi'(z)\), whence one may also find the dependence of the deformation \(X\) on time by integration

\[ X=\int_0^t \varphi'(z)\,dt . \tag{16} \]

March 1919.

P. S. (22. 12. 23). For small \(t\) it is necessary to take into account the forces of inertia, which in this period must predominate over the elastic forces that have not yet had time to develop; thus, to determine the form of the function \(\varphi(z)\), the lower limit of the integral (15) must be replaced by \(t_0\), where \(t_0\) is the interval of time during which, on the contrary, the forces of inertia will be negligible in comparison with the elastic forces. A more curious—purely paradoxical—conclusion to which equation (15) leads will form the subject of one of the following papers of the present series.

1. No. 44 of the list of works, which is placed above on pp. 1–3 and to which further references will be made. I take the opportunity to correct misprints that have crept in and distort the meaning in that work: p. 40, line 20 from the top, “Podkhodim” instead of “Podkhodom”; p. 42, line 15 from the bottom, “otdel'nuyu” instead of “otdalennuyu”; p. 45, line 4 from the top, after “sloev” the word “nichtozhny” has been omitted; p. 46, line 19 from the bottom, “Ioffe. Sokhranenie” instead of “Ioffe sokhranenie.” In addition, on p. 42, line 20 from the top, “45” should be replaced by “35.” ↩