V. D. Kuznetsov.

On the Effect of Temperature on the Internal Friction of Solids.

(With 2 pages of diagrams accompanying the article).

§ 1. Introduction.

After it had become clear to me that the coefficient of internal friction of solids is not a constant quantity, but varies within very wide limits depending on the magnitude of the force producing the “flow” of the body, it seemed to me especially interesting to elucidate the nature of internal friction. This elucidation could be carried out by studying internal friction as a function of various factors affecting its magnitude. The most important factor is temperature. The viscosity of solids depends very strongly on temperature. If the internal friction at a given constant temperature were constant and did not depend on the force producing the shear, then the question of studying the effect of temperature on viscosity would be sufficiently clear and simple. It would then suffice to determine the coefficient of internal friction by some method at several temperatures and to express the dependence by a formula. If internal friction is determined by the method of twisting a rod, it would suffice to apply a constant twisting force and to determine, at different temperatures, the stationary rate of torsion. The change of angular velocity with temperature could serve as a characteristic of the effect of temperature on internal friction. But in reality this is not so. Experiments with lead¹ showed that the viscosity of lead is the smaller, the greater the angular velocity of torsion. On this basis the following propositions may be stated:

1. If the twisting force \(P\) remains unchanged in all experiments and if, at temperatures \(\Theta_1, \Theta_2, \Theta_3\), etc., for one and the same specimen, angular velocities \(\omega_1, \omega_2, \omega_3\ldots\) are obtained, on the basis of which, by the formula

\[ \eta = \frac{2L}{\pi R^4}\cdot \frac{Pgd}{\omega}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (1) \]

or by Shvedov’s formula:

\[ \eta = NT + \frac{N\varphi_0}{\omega}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (2) \]

values of the coefficient of internal friction \(\eta_1, \eta_2, \eta_3\ldots\) are obtained, then these values are not comparable with one another.

In formulas (1) and (2) the following notation has been introduced: \(L\)—length of the specimen, \(d\)—arm of the twisting force, \(g\)—acceleration due to gravity, \(R\)—radius of the round specimen, \(N\)—modulus of shear, \(T\)—relaxation time, \(\varphi_0\)—limiting value of the angle of shear at which the phenomenon of relaxation begins.

1. If the structure of the specimen did not change with a change in temperature, i.e. if a specimen heated and then again cooled to the original temperature had, at this latter temperature, before heating and after heating, identical properties, then the numerical values of \(\eta\), obtained at the same angular velocities of torsion but at different temperatures, could be compared with one another in order to find the dependence of \(\eta\) on \(\Theta\).

Metals are aggregates of small crystalline bodies, the so-called crystallites, which have varied forms and are arranged without any definite orientation. The spaces between these crystallites are filled with a more fine-grained substance. Such a structure arises because, when a molten metal solidifies, small crystals begin to form at very many points; these hinder one another in their regular growth, and crystals with irregularly developed faces are obtained. The more rapidly solidification takes place, the smaller the crystals obtained, and the more they are intermingled and entangled with one another. If the metal is annealed, i.e., kept at a sufficiently high temperature, yet still far from the melting point, the small crystals begin to grow. The rise in temperature causes a more vigorous motion of the molecules, which group themselves around the existing crystallites and form large grains. Such crystallization by annealing strongly changes both the structure and the properties of the metal. The longer the elevated temperature acts, the more strongly the properties change.

If, in addition to heating, the metal is subjected to such deformations in which displacement of particles occurs, this promotes a change in structure. Twisting a metallic specimen at elevated temperature more quickly than a mere rise in temperature alone causes the metal to pass from a fine-crystalline to a coarse-crystalline state.

This circumstance must be taken into account when studying the dependence of the internal friction of metals on temperature. If one first determines $\eta$ at a low temperature and then at a higher one, the results may prove incomparable precisely because of the change in structure. If, conversely, one first determines $\eta$ at a high temperature and then at a low one, one may expect greater comparability of the results, although even in this case the structure cannot be identical.

It is impossible to determine $\eta$ with a single specimen at several different temperatures, since the specimen cannot for a long time, without destruction, be subjected to twisting or shearing. Consequently, one has to investigate different specimens, whose structure may differ greatly from one another.

All these circumstances greatly complicate the investigations, and this probably explains why “attempts to measure the internal friction of solid bodies have not yet led to clear and definite results”¹.

§ 2. Some theoretical expectations.

If the coefficient of internal friction $\eta$ of solid bodies were a constant quantity, then, in the torsion of rods, formula (1) would hold, from which it should follow that:

$$ \frac{P}{\omega}=\mathrm{const.}\ . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) $$

If we denote by $T$ the quantity reciprocal to the angular velocity of torsion $\omega$, then instead of formula (1) one may write:

$$ \eta = aPT \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) $$

and for $\eta=\mathrm{const}$ it must be:

$$ PT=\mathrm{const.}\ . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) $$

Experiments with lead showed that formula (5) is entirely inapplicable. Likewise, Shvedov’s formula (2) is not valid either; from it, as a consequence, one obtains:

$$ PT = a + bT \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6) $$

Only within certain limits of $\omega$ or $T$ can this formula be applied.

Exactly the same results are obtained from experiments with tin. At room temperature the following dependence between $PT$ and $T$ is obtained. If $T$ is laid off along the axis of abscissas and $PT$ along the axis of ordinates, then the curve passes through the origin and forms a slight concavity toward the axis of abscissas. Over a certain extent this curve may be taken as a straight line (Fig. 1, curve I).

If the temperature is raised, then it is evident that, in order to obtain the same angular velocity $\omega$ or the same interval $T$, a smaller twisting force $P$ must be applied. Consequently, for one and the same value of $T$ the product $PT$ will be the smaller, the higher the temperature.

At different temperatures there must be obtained curves which 1) will necessarily pass through the origin and 2) will be the less inclined to the axis of abscissas the higher the temperature, and 3) the concavity of the curves near the origin must increase with increasing $\Theta$, since if this were not so, then at some sufficiently high $\Theta$ a straight line coinciding with the axis of abscissas would be obtained, which cannot be. This would be equivalent to saying that at some $\Theta$ the internal friction would be equal to zero (curves II and III).

Now let us suppose that the temperature $\Theta$ is raised so much that the metal has melted and passed into the liquid state. What form should the curve $PT = \varphi(T)$ have for a molten metal? For liquids it is customary to regard the viscosity as constant, at least at velocities below the critical one. If this is in fact so, then the curve must pass through the origin, then bend toward the axis of abscissas, and turn into a straight line parallel to the axis of abscissas (curve IV).

With a further increase in the temperature of the already liquid metal, the viscosity $\eta$, while remaining independent of the velocity of flow, continues to decrease, and the curve $PT = \varphi(T)$, remaining parallel to the $T$ axis, approaches it (curve V), while the range of values of $T$ for which $\eta$ is not constant becomes narrower.

If the considerations expressed are correct, then one may hope, on their basis, 1) to establish a connection between the internal friction of solids and the internal friction of liquids, and 2) to approach an understanding of the nature of internal friction.

One cannot help noticing the resemblance between the curves in Fig. 1 and the curves expressing the dependence of current strength on potential difference for bodies possessing ionic conductivity. It is as though, with increasing temperature, there occurs a gradual transition from Ohm’s law to a saturation current.

It seems to me that, in order to clarify the nature of the internal friction of solid and liquid bodies, it would be interesting to carry out the following experimental work.

1. To study the dependence between the value of the internal friction of a liquid ($PT$) and the velocity of its motion ($T$). For this one may use the method of flow through capillary tubes, causing the liquid to flow under different pressures. It is especially important to follow the portion of the curve at high velocities. In view of the fact that the liquid may heat up during flow, its temperature should be determined before and after the flow.

2. To carry out the experiments indicated in 1° at different temperatures.

3. To follow the gradual change in the form of the curves with increasing temperature for such substances as pass smoothly from the solid into the liquid state (Canada balsam, resin, pitch). One may use the method of spiral torsion between two coaxial cylindrical surfaces.

1. To trace the change in the form of the curves with increasing temperature for such solids as have a definite melting point (metals, fused salts, and crystals in general).

§ 3. Some data on the effect of temperature on the internal friction of lead and tin.

In order to establish the dependence of internal friction on temperature, observations were made on lead and tin at various temperatures within the limits \(12^\circ\)—\(220^\circ\). These experiments are still far from being completed to the extent intended; however, some definite results have already been obtained, which may serve as an illustration of the considerations set forth.

Internal friction was measured by the method of twisting round rods. The apparatus, arrangement, casting of the specimens, and method of observation have already been described many times¹. The dimensions of the specimens were the same as before. The time interval \(T\) sec was also observed, during which 1 cm of the scale passed across the horizontal thread of the telescope; moreover, the initial and final divisions in all experiments were approximately the same.

For heating the specimens a furnace with an electric heater was constructed. This furnace consisted of an iron rectangular box with double walls. On the bottom of the box, between the walls, there was placed a metal plate wrapped in asbestos, with several turns of German-silver wire through which an electric current from the station was passed. A lamp rheostat was included in the circuit for regulating the temperature. The box was cut along a vertical plane passing through the axis of rotation of the specimen. In the side walls cutouts were made with small openings for the axes of the apparatus. After the specimen had been fastened in the apparatus, the two halves of the box were pushed one onto the other. In the upper wall of the box an opening was made for a thermometer; its bulb almost touched the specimen. When the station voltage was constant, the temperature was maintained quite well. If, however, the voltage fluctuated, then in order to maintain the temperature it was necessary constantly to watch the thermometer, switching one of the rheostat lamps on or off. The tube with the scale was placed on a small immovable platform; therefore, despite the fact that sometimes the tube had to be shifted somewhat, it could be assumed that the distance from the mirror to the scale remained unchanged, equal to 165 cm. The inconvenience of the arrangement consisted in the fact that the specimen was not visible; as a consequence, sometimes, if the specimen had a nonuniform structure or a cavity, the observations were wasted, since the incorrect twisting was detected only after the end of the experiment.

Most of the observations were carried out by the student physicists I. I. Popov and F. P. Otradnykh.

In the observations the following data were recorded: 1) the time interval \(\Delta t\) required for \(\Delta n\) cm of the scale to pass; 2) the initial and final division of the scale, the difference of which was \(\Delta n\); 3) the time by the clock at the moment of observation; and 4) the temperature \(\theta^\circ\) inside the furnace.

When the last divisions of the scale appeared in the tube, the ratchet wheel of the apparatus was turned by several teeth, and the beginning of the scale again appeared in the field of view of the tube.

The twisting was begun with a large force, in view of the fact that the experiments with lead had shown the advantage of passing successively from larger to smaller forces. Observations at a given force and a given temperature were ended when, over the course of several minutes, the same values of \(T\) were obtained, or when \(T\) fluctuated about some mean value.

In Table I are given the results obtained for lead. It does not contain the numerical values of \(\eta\), which are easily obtained by multiplying the product \(PT\) by \(\alpha = 2.40 \cdot 10^3\). Some values of \(PT\) at room temperature are taken from the preceding work.

Table I.

Designations:

\(P\) — the applied twisting force in grams.
\(\tau_1\) — the time of action of the force \(P\), in minutes.
\(\tau_2\) — the duration of observation of the stationary state, in minutes.
\(T\) — the time, in seconds, for 1 cm of the scale to pass through the filament of the tube.
\(PT\) — the product characterizing the magnitude of the internal friction.
\(\Theta^\circ\) — temperature in Celsius.
№ — specimen number.

Lead.

In Table II the newly obtained results for tin are given.

Table II.

Tin.

The results for lead are shown in Fig. 2. From a comparison of the tables we first of all obtain the interesting result that the curves \(PT=\varphi(T)\), at one and the same temperature (about \(12^\circ 5'\)), for lead and tin coincide with one another. This means that, at equal angular velocities of twisting, the coefficients of internal friction for lead and tin are the same.

The data presented are not yet quite sufficient to establish the dependence between \(\eta\) and \(\theta^\circ\). However, it is already clear from them that all the curves \(PT=\varphi(T)\) pass through the origin of coordinates.

We obtain an interesting result if we plot (Fig. 3) temperatures along the axis of abscissae, and along the axis of ordinates the product \(PT\), which characterizes the magnitude of the internal friction of lead. The curves at various values of \(T\), i.e. at various angular velocities, show that at comparatively low temperatures the dependence between \(PT\) and \(\theta\) can be expressed linearly

\[ PT = a - b\theta, \tag{7} \]

where \(a\) and \(b\) are certain constants depending on the angular velocity, with both \(a\) and \(b\) decreasing as the angular velocity increases. As the temperature rises and approaches the melting point, the curves become convex relative to the abscissa axis and all intersect it at the point corresponding to the melting temperature (about \(330^\circ\)). The greater the rate of flow, the smaller the effect of temperature.

Further experiments should consist in subjecting one and the same specimen to torsion at different temperatures, but at the same angular velocity, which should be as small as possible, since at low velocities the experimental errors become negligible in comparison with the influence of temperature. The most convenient velocity, with our apparatus, is that for which \(T = 5\)—\(15\) seconds.

Tomsk.

December 1924.

article by V. D. Kuznetsov, “On the Effect of Temperature on the Internal Friction of Solids.”

Vertical axis: Elongation \(P T\).
Horizontal axis: Time interval \(T\), sec.

\(T = 13.0\) sec.
\(T = 10.0\) sec.
\(T = 8.0\) sec.
\(T = 6.0\) sec.
\(T = 3.0\) sec.
\(T = 2.0\) sec.

Vertical axis: Elongation \(P T\).
Horizontal axis: Temperature \(C^\circ\).

Effect of temperature on the internal friction of lead.

Fig. 3.

To the article by V. D. Kuznetsov: “On the Effect of Temperature on the Internal Friction of Solids”

1. V. D. Kuznetsov and L. A. Shvirk. loc. cit. ↩↩↩