Eng.-Mech. G. V. Trapeznikov.

Laws of Residual Deformations

1. Compression.

(The text is accompanied by 4 pages of drawings).

In carrying out work with impact testing in compression, and in processing the results of the experiments, I, when deriving an analytical dependence between the individual elements entering into the phenomena of impact, was compelled to dwell on one of the constants that appeared in the final formula of the relation. The very character of this constant indicated that it is a characteristic of the metal. This prompted me to carry out a small series of experiments on plastic compression, of the same metals, under static loading, in order to determine the value of the indicated constant. The results proved so satisfactory that it became possible to find not only the value of this constant, but, moreover, to give an analytical relation between the forces, on the one hand, and the deformations caused by them, on the other. The latter made it necessary to broaden the work, in order to verify the relations obtained on as many individual metals as possible. This work, carried out by me in the mechanical laboratory of the Tomsk Technological Institute in the summer of 1922, forms the subject of the present article, which is only a very brief report on the work.

The results obtained from the experiments on plastic compression led me to consult the literature on this question, in the hope of obtaining confirmation in experiments that might have been carried out by other investigators. Unfortunately, neither in separate works nor in the journal literature was anything found that could clarify the given question. In works on the strength of materials and in substantial courses it is mentioned, in passing, that there is a dependence between forces and deformations beyond the elastic limit. Most authors say that the analytical dependence in tension and compression, beyond the elastic limit, probably has a very complex character, but that the curve of plastic deformations can be expressed, quite accurately, in the form:

\[ \lambda = A + BP + CP^2 + DP^3 + \ldots, \]

where \(\lambda\) is the absolute deformation, and \(P\) is the force corresponding to it. In other words, by a polynomial parabola, with the coefficients: \(A, B, C, D,\) etc., to be selected separately for each specimen.

Some authors, for calculating the work of residual deformations in the tension of some specimen, propose replacing the curve of residual deformations by a portion of a simple quadratic parabola. This substitution is encountered in Tetmajer, Jonson, Morley, Madamet, Timoshenko, and others.

According to drawing 1, we see that it is proposed to equate the areas: of the diagram obtained on the testing machine, on the one hand, and, on the other: the sum of the areas of the parabolic segment, with height \(CH - HH^{1}\) and base \(AH^{1} = OH +\) the rectangle \(OAH^{1}H\). Thus, the work expended on the residual deformation of the specimen, neglecting the work of elastic deformation, will be proportional to the area:

\[ S = OH \cdot OA + \frac{2}{3}OH(CH - HH^{1}). \]

Calling the load at the elastic limit \(P_0\), the load that breaks the specimen \(P_1\), and the greatest absolute elongation \(\lambda_1\), we shall have the following expression for the work of the residual deformations:

\[ E = P_0 \lambda_1 + \frac{2}{3}\lambda_0(P_1 - P_0) = \frac{\lambda_1}{3}(2P_1 - P_0). \]

As for the work of residual deformations in compression, I have nowhere encountered even a similar substitution. It is true that Unwin and Morley (Strength of Materials) give a diagram of plastic compression for certain metals. The latter is reproduced in drawing 2 and is an exact copy from Unwin’s work. Here there is an indication of the tendency of the curves of plastic compression to approach certain hyperbolas. But, as is evident from a fairly exact copy of such a diagram, this tendency is very, very problematic. In the cited work no information or hints are given regarding the constants of such hyperbolas.

In studying residual deformations in compression, the experimenter must direct his attention to three principal questions:

1) the influence of temperature variation;
2) the influence of friction on the end surfaces of the specimens;
3) the influence of the duration of the experiment (time interval).

1. The influence of temperature variation, under those small fluctuations of it which occurred during the summer month in the mechanical laboratory \((14^\circ\text{--}16^\circ)\), not directly illuminated by the sun at midday, is, of course, negligible. It is known that, for all chemically pure metals, Young’s modulus, the elastic limit, and the temporary resistance decrease with increasing temperature and increase with decreasing temperature (Meyer, Phil. Mag., 1896, v. 41, v. 32, and Dewar, Proc. of the R. J., of 9.13 v. 14). But these changes, within the indicated limits of temperature fluctuation during the experiments, of course lie within the limits of observational errors. As for the heating of the specimen from all-round compression, which occurs in proportional compression, and from the friction of individual particles of metal against one another during the flattening of the specimen, it must be noted that, although here the change in the temperature of the specimen itself will be appreciable, it likewise cannot be taken into account; the comparatively small specimens used for experiments on plastic compression very quickly, almost instantaneously, equalize their temperature, on the one hand because of the large ratio of surface to volume, and on the other because of their extremely close contact with the supporting, very massive, plates and with the solid table of the testing machine.

2. The influence of the supporting surfaces is the most significant of the above-mentioned factors affecting the course of plastic deformations. The frictional forces arising at the ends of the specimen in contact with steel, though very carefully ground, plates considerably distort the course of the plastic deformations of compression. In the presence of surface frictional forces at the ends, the compression deformation decreases the smaller the ratio of the height of the specimen to its diameter. To take this circumstance into account, it was necessary to conduct experiments with specimens having different ratios of heights to diameters. Thus, if the frictional force cannot be determined, it is possible to find that ratio of the specimen height to diameter at which this force may be neglected, or to reveal, by extrapolation, the course of the deformation in its ideal case, i.e. in the absence of frictional forces at the ends of the specimen.

3. The influence of time on the increase of residual deformations is a very serious question, which cannot be passed over without consideration when experimenting beyond the elastic limit. The functional form of the dependence of the growth of residual deformations on time is extraordinarily complex and,

Moreover, it cannot be said that a proper analytical relation has been established here. Professor N. Bouasse, who worked for many years on the study of plastic deformations and wished to relate, in a general function of time, stress and deformation, arrived at inconclusive results. He says: “When one begins to study some group of phenomena, there is a great temptation to find a general theory, a general explanation. Only little by little does one notice the futility of such efforts and decide, for a more or less long time, to divide the phenomenon into subgroups and give them partial explanations.” Therefore he divides all metals into two categories, which, however, are only types admitting intermediate representatives, namely: metals with solid friction (métaux à frottement solide) and viscous metals (métaux visqueux). The hopelessness that sounds in Bouasse’s words is understandable: the phenomenon of residual deformation is too complex to be studied in all its completeness at one time. It is first necessary to arrange experiments in such a way that the influence of time, in the sense of comparing residual deformations for different metals, is, as far as possible, negligible. As is known, deformation lags in time behind the force that produces it, and the law of such lagging or delay is extremely difficult to trace, since this delay, as well as the deformation itself, is a function of the rate of increase of the force with time. A certain analogy for the variation in time of the residual deformation and of its, so to speak, “shift” in time with respect to the force may be furnished by the curve of rise and fall of the current in an inductance as a function of the rise and fall of the potential. The indicated analogy is taken, as is the curve, from Professor B. P. Weinberg’s article “Advances in Solid-State Physics.” In drawing 3, the curve “v” represents the change of potential during the time of the current; “i” represents the change of current strength.

Arrangement of the experiments.

In view of the fact that the problem, in its full scope, was too complex and extensive for the means and materials that were at that time at the disposal of the laboratory, it was necessary to eliminate the influence of time on the growth of residual deformations. For this purpose a separate series of compression experiments was carried out. Nine different metals were tested: mild, non-hardening steel, soft iron, bronze, brass, red copper, aluminum, zinc, tin, and lead. The specimens made from these metals all had the same dimensions and were cylinders of length \(l_0 = 15.00\) mm and diameter \(d_0 = 10.00\) mm. All materials, before specimens were made from them, were heated, being embedded in cement paste, in order, as far as possible, to avoid the presence of residual stresses that had appeared in them during rolling or casting in cold molds. All experiments, both preliminary and principal, were carried out on two machines of the firm Amsler–Laffon. For stronger metals requiring a large load, a 50 tn machine was used, on columns, driven by a rotary oil pump; for weaker metals, a 2-tn pendulum machine. The specimens were loaded at the same and, as far as possible, uniform speed for all specimens, up to the required force, and then the load was removed as follows:

1) specimens of series A were unloaded immediately upon reaching the previously assigned load;

2) specimens of series B were unloaded after the load, having reached the assigned magnitude, had been held constant for \(0'.10\);

3) specimens of series C were unloaded after holding the load for \(0'.25\);

4) specimens of series D were unloaded after holding the load for \(1'.00\).

About \(0'_{25}\) was spent on loading; unloading was carried out almost instantaneously.

Each series included specimens of all the metals, in duplicate, in order to obtain more reliable figures.

The next element that it was necessary to take into account in the experiments was the ratio of the initial length of the specimen to its initial (before the test) diameter:

\[ \alpha_0 = \frac{l_0}{d_0}. \]

If, in compression tests, we were not constrained by the strict necessity of making specimens with a comparatively small \(\alpha_0\) (not more than 3), because of the impossibility, with a more relatively long specimen, of obtaining compression without buckling of the specimen, it would be possible, by conducting tests with ever larger values of \(\alpha_0\), almost completely to eliminate the harmful influence of friction at the ends. Experimentation with specimens for which \(\alpha_0 > 2\) already becomes very difficult if we wish to obtain a considerable compressive deformation, and often the test has to be repeated more than once, sacrificing a new specimen, since the preceding one, although carefully set up in the machine, has buckled. As for specimens for which \(\alpha_0 > 3\), such specimens, without special devices, can almost entirely not be compressed without misalignment.

The influence of friction at the ends of the specimens was taken into account by carrying out fundamental tests, for which, in duplicate, six specimens of each of the metals mentioned above were made.

The diameter of all specimens: \(d_0 = 10.00\) mm.

Length: No. 1 — \(l_0 = 5.00\) mm, \(\alpha_0 = 0.50\)
» No. 2 — \(l_0 = 10.00\) » \(\alpha_0 = 1.00\)
» No. 3 — \(l_0 = 15.00\) » \(\alpha_0 = 1.50\)
» No. 4 — \(l_0 = 20.00\) » \(\alpha_0 = 2.00\)
» No. 5 — \(l_0 = 25.00\) » \(\alpha_0 = 2.50\)
» No. 6 — \(l_0 = 30.00\) » \(\alpha_0 = 3.00\)

The maximum load applied in the compression tests did not exceed 40 tn., which is about \(75\%\) of the full capacity (53 tn) of the above-mentioned Amsler—Laffon press. This has precisely a double raison d’être: first, any testing machine, under loads reaching its limiting load, will give readings less exact and therefore less reliable than under medium loads; and, second, the heights (lengths) of the specimens of the strongest metals, when deformed by a load of 40 tn., are so small that any further reduction of them will lead to less accuracy in their measurement.

The main test with each specimen proceeded as follows. The specimen, resting with its end faces on hardened ground steel plates, was placed under the press; by rotating the oil pump by hand, the load was applied, raising it to the first period of flow of the material, the latter being determined by the settling of the mercury column in the manometric tube; if such was not sharply observed, as was the case in soft plastic metals, then the load was simply raised to a certain small, predetermined limit. After the machine was unloaded, the specimen was removed, its changed length was measured with an accuracy of up to 0.01 mm, and it was again placed under a load, which was increased, at each separate subsequent observation, by an assigned amount; thus only the residual deformation of the specimen was measured. From the numerical data of the observations (the mean of two parallel series), two diagrams were constructed—one in a Cartesian coordinate system, the other in a logarithmic one (logarithmic

nomogram); along the abscissa axes were taken, respectively, the load—\(P\) and \(\lg P\); along the ordinate axis, the length of the specimen \(l\) and \(\lg l\). The latter diagrams, which are now becoming more and more widespread, were used as more convenient for the graphical analysis of curves and for comparing individual experimental curves with one another.

Influence of Time.

The tables presented below have been compiled as follows:

Column 1—the number of the observation.
Column 2—the load on the specimen in tons.
Column 3—has 4 columns, respectively for specimens of series A, B, C, D.

Next to the letters denoting the series of the specimen, the time \(t\) is indicated in minutes, during which the given load was maintained constant in magnitude.

Each column is divided into two subcolumns; in the first is indicated the length of the deformed specimen in mm, and in the second its relative deformation in the form:

\[ 1-i=\frac{l}{l_0} \]

where \(l_0\) is the length of the specimen before deformation, and \(l\) is the length of the deformed specimen.

Influence of Time. Iron. 1st set.

As is seen from the table of tests of iron specimens, the holding time of the deforming load, within the limits of 1 minute, does not produce a noticeable increase in deformation. The same is true also for the other metals tested, excluding lead, tin, and zinc, for which—especially at small loads—the difference in deformations for different periods of holding the load is already noticeable. For lack of space, I cannot present tables and diagrams for these metals. Examination of the logarithmic diagrams for all metals indicates that the straight lines closest in slope are those for which the holding time of the load is equal or close to zero. In other words, if, after reaching the prescribed load, we immediately unload the specimens, then by doing so we place them, in terms of comparing the course of deformations, under identical conditions. Therefore, in all subsequent experiments, both principal and auxiliary, the holding time was taken equal to 0, insofar as this could be carried out technically.

Principal experiments. Red copper. 1st set.

In drawing 4 a diagram of the deformation of copper specimens is given in the decimal system of coordinates, and in drawing 5 in the logarithmic system. In view of lack of space only two curves are presented: for $a_0 = 0.5$ and $a_0 = 3.0$.

From the diagrams it is seen that, together with an increase in the value

\[ \alpha_0 = \frac{l_0}{a_0} \]

the deformation curves become steeper, i.e. they approach the axis of abscissas more rapidly, which is their common asymptote. Such is the case

with curves in the Cartesian coordinate system; as for their logarithmic anamorphoses, then, as has already been said earlier, the latter are straight lines; the inclination of these straight lines to the line of abscissas is the greater, the greater the value of \(a_0\).

The curves of residual deformations for different metals and different values of \(a_0\), for each individual metal, have one and the same character, namely: beyond the elastic limit there begins a curve having a concavity with respect to the line of abscissas, which, after a certain short portion of the curve, turns into a convexity with respect to the same axis; i.e., the curve has a certain point of inflection. In the logarithmic grid the anamorphosed curve is divided even more sharply into two portions. Beyond the elastic limit we have the first, short portion in the form of a curve turned toward the line of abscissas with its concavity, and, instead of the second portion, turned toward the abscissa axis with its convexity, there will be a straight line. The portion of the curve of residual deformations to which this straight line corresponds is overwhelmingly large in comparison with the first, very short, portion immediately following the elastic limit. Therefore we have the right to consider that residual deformations follow a law which is graphically expressed by the second portion of the indicated curve. As for the first portion, then, in a first approximation, it may be attributed to imperfections of the metals taken; this is also indicated in part by the fact that for some metals, as is seen from the diagrams, this portion is reduced almost to zero.

If we consider the equation of the straight line in the logarithmic grid, with coordinates \(\lg P\) and \(\lg l\), then we shall have the following:

\[ \lg l = -n \lg P + \lg A \]

or

\[ \lg P^n l = \lg A \]

or

\[ P^n l = A = \mathrm{const}. \]

Here \(n\) is the absolute value of the tangent of the angle of inclination of the straight line to the line of abscissas, and \(A=\mathrm{const}\) is a quantity characterizing the metal.

Thus, in the most general form, the dependence between the load and the length of the deformed specimen is polytropic in character, with the exponent of the polytrope being a variable quantity depending, above all, on the ratio of the length of the specimen to its diameter, measured before deformation. Leaving aside for the time being the question of the dependence between \(n\) and \(a_0\), let us examine the general properties of the diagram obtained.

First of all, on each of the logarithmic diagrams the circumstance stands out sharply that all straight lines, for different values of \(a_0\), for one and the same metal, intersect at one point, and that this point lies on a straight line parallel to the abscissa axis and at a distance from it equal to \(\lg l_0\); in other words, if we disregard the first, short portion of the deformation curve, the matter is as though the elastic limit were located precisely at this point common to all straight lines. Calling by \(P_c\) the force which corresponds to the coordinate of this point, we shall find the value of the constant of the polytrope, namely:

\[ P^n l = P_c^{\,n} l_0 = A = \mathrm{const}. \]

We shall call \(P_c\) the force at the theoretical compressive elastic limit; accordingly, it will be greater than the force which corresponds to the actual or visible elastic limit of the given metal. If the metal were perfect in the mechanical sense, then residual deformations would begin for a specimen of the same transverse dimension only with a load equal to \(P_c\).

Let us denote by \(k\) the (mean) stress in the metal after deformation, to which the length \(l\) corresponds; by \(k_c\), the elastic limit (theoretical) in compression; by \(i\), the relative deformation, equal to \(\dfrac{l_0-l}{l_0}\); and by \(f_0\) and \(f\), respectively, the cross-sectional areas of the specimen before and after deformation. Then we shall have:

\[ \frac{P^n l}{f^n l_0} = \frac{P_c^n}{f^n} = \frac{P^n}{f^n}(1-i). \]

On the basis of the numerous and thorough experiments carried out by Kicaldy, we are fully justified in assuming, for well-annealed and rolled metals, constancy of volume. The latter may be expressed as follows:

\[ fl=f_0l_0. \]

Substituting into the preceding equation \(l\) as determined from this, we have:

\[ \frac{P^n}{f^n}=(1-i)=k^n(1-i)=\frac{P_c^n}{f^n}=\frac{P_c^n l^n}{f_0^n l_0^n}; \quad \text{or} \]

\[ k^n(1-i)=k_c^n(1-i)^n. \]

Whence

\[ k=k_c(1-i)^{\frac{n-1}{n}}. \]

In determining the work of residual deformations, we neglect the work of elastic deformations, as being small in comparison with the former. Let us compute the work of residual deformations of an ideal metal. It is necessary to note, as is also evident from the diagrams, that this work will be very slightly greater, especially for large deformations, than the work of an imperfect metal.

\[ T_c=\int_{l_0}^{l} P\,dl = P_c l_0^{\frac{1}{n}}\int_{l_0}^{l}\frac{dl}{l^{\frac{1}{n}}} = \int \frac{n}{n-1}P_c l_0^{\frac{1}{n}} l^{\frac{n-1}{n}}\Bigg|_{l_0}^{l} = \]

\[ = \frac{n}{n-1}P_c l_0 \left| \frac{l^{\frac{n-1}{n}}}{l_0^{\frac{n-1}{n}}} \right|_{l_0}^{l} = \frac{n}{n-1}P_c l_0 \frac{ l^{\frac{n-1}{n}}-l_0^{\frac{n-1}{n}} }{ l_0^{\frac{n-1}{n}} } \]

Generally speaking, \(n<1\); therefore, although \(l<l_0\), the work will be positive.

Otherwise, and therefore one may write:

\[ k=\frac{k_c}{(1-i)^{\frac{1-n}{n}}}. \]

\[ T_c= \frac{n}{n-1}P_c l_0 \frac{ l_0^{\frac{1-n}{n}}-l^{\frac{1-n}{n}} }{ l^{\frac{1-n}{n}} } = \frac{n}{1-n}P_c l_0 \frac{ 1-(1-i)^{\frac{1-n}{n}} }{ (1-i)^{\frac{1-n}{n}} } \]

Or, if the work is expressed through the stress:

\[ T_c=\frac{n}{1-n}P_c l_0\frac{k-k_c}{k_c}. \]

Dividing and multiplying the right-hand side by \(f_0\), and having:

\[ v_0=f_0l_0=fl, \]

we obtain:

\[ T_c=\frac{n}{1-n}\,v_0(k-k_c). \]

That is, the work of residual deformations will be directly proportional to the volume of the deforming specimen, which fully agrees with the law of Barba and Kick.

Determination of the Theoretical Elastic Limit for Various Metals

The logarithmic diagram clearly shows the regular increase of the theoretical elastic limit together with the increase in the value of Young’s modulus.

Even a simple determination, from the graph, of the value \(P_c\), and from it the finding of \(k_c\), as

\[ k_c=\frac{P_c}{f_0} \]

gives a linear dependence between Young’s modulus and the theoretical elastic limit for the metals tested. For a more accurate and reliable result, I used the method of least squares to determine the values of \(P_c\), from which the corresponding \(k_c\) were computed. It remained necessary to determine Young’s modulus for the same metals. This latter determination was made much later than the principal experiments. Below is given a table of observations for determining the moduli from tension. The specimens taken for this purpose all had the same dimensions:

\[ d=10.00\ \mathrm{mm};\quad l=110\ \mathrm{mm}. \]

The elastic deformation was measured with Martens’ mirror apparatus, with a lever-arm length (calculated): \(l_0=50.00\ \mathrm{mm}\); the distance, in all cases, from the mirror to the scale was \(L=5000\ \mathrm{mm}\); the diagonal of the prism: \(\delta=4\ \mathrm{mm}\).

The table below represents a general summary for the determination of Young’s moduli for various metals and of the values of the corresponding \(P_c\) and \(k_c\) computed by the method of least squares.

The diagram (Fig. 6) shows that between \(k_c\) and \(E\) there is a direct linear dependence; the greatest deviations from the straight line are given by the points found for bronze and brass, but since the latter are alloys, and not pure metals, even this falling away of the points from the straight line is not a violation of the general regularity of the relation between \(E\) and \(k_c\).

The coefficient of proportionality in the relation:

\[ k_c=\beta_c E \]

is constant for all plastic metals and, on average, if its value for bronze and brass is excluded, will be equal to:

\[ \beta_c=0.00267 \simeq \frac{1}{375}. \]

Thus, just as in elastic deformation, Young’s modulus here too, in residual deformation, will be an equal-rights characteristic of the material. What, then, is the physical meaning of the coefficient \(\beta_c\)? Since it gives the connection between the modulus of elasticity of the first kind and the stress of the metal at the theoretical elastic limit, it is clear that we can find its meaning if we consider the deformation before the theoretical elastic limit, close to the latter. Hooke’s law is written as follows:

\[ k=-E i \]

where \(i\) is the relative deformation.

Here the following relation has been found:

\[ k_c=-E\beta_c. \]

This means that \(\beta_c\) will be that elastic relative deformation in compression of an ideal metal at which residual deformations already begin; in other words, a plastic metal cannot be compressed along its length by more than \(0.267\%\) without residual deformations beginning. All this refers, of course, to a metal that is in its “natural” state, i.e., one not possessing residual stresses.

The polytropic exponent and the relative dimensions of the specimen.

As was already seen from the preceding, the polytropic exponent increases together with an increase in the ratio of the length of the specimen to its diameter. This increase proceeds at different rates for different metals. For most metals, up to the value

\[ \alpha_0=2.0 \]

the increase in \(n\) proceeds very rapidly, and then, dying out, within the limits of the experiment remains a quantity close to

\[ n=\frac{2}{3}. \]

When carrying out my main experiments, I, in my reports, indicated the value of the exponent

\[ =\frac{2}{3} \]

as that toward which this exponent probably tends for infinitely large \(\alpha_0\). Further experiments to determine the relation between \(\alpha_0\) and \(n\) showed that with a still greater increase of \(\alpha_0\), the exponent, although very slowly, nevertheless increases.

In order that it would be possible to compress specimens having \(\alpha_0>3\), I made a simple device, shown in Fig. 7. It consists of a turned cast-iron cylinder, with an iron bottom, on which, in a groove, a hardened plate is placed.

A hardened steel rod enters the longitudinal opening from above. The lower end of the rod and the upper face of the plate were carefully lapped by me to one another, on a machine, with carborundum No. 00000. Thus, now, when

closely fitting into the opening of the cylinder, a rod and surfaces ground to one another, one could expect that a well-prepared specimen, placed on the plate and pressed from above by the rod, would not tilt as easily as occurs when it is set directly on the table of the testing machine. Experience disappointed these hopes; even with a carefully made apparatus it was not possible to compress specimens with \(a_0 > 4.2\) without bending.

In processing the experimental results it turned out that, at the extremely high value \(a_0 = 4.2\), it is already possible to discern a certain grapho-analytical relation between \(n\) and \(a_0\).

All the specimens tested had the initial (before deformation) diameter: \(x_0 = 6.00\) mm. The transition to smaller specimens was necessitated by the fitting of the specimen in the apparatus at its maximum deformation.

The lengths of the specimens and the corresponding \(a_0\) were:

Below, diagrams are given only for iron, red copper, and aluminum.

Aluminum

The diagrams show that there is no linear dependence between \(E\) and \(n\); thus, the curve for aluminum, which has a Young’s modulus smaller than that of copper, lies between the curves for iron and copper. This circumstance can be explained by the dif-

different coefficients of friction for these metals, on the presence of which depends, to a greater or lesser degree, the slope of the logarithmic straight lines of deformation, and hence also the magnitude of the exponent \(n\), for various \(\alpha_0\). The magnitude of the exponent \(n\) is influenced by one further circumstance—namely the size of the load interval: the smaller the interval through which we increase the load, the larger will be the magnitude of the exponent \(n\). This is quite understandable: the smaller the load interval, the less will be the effect of the frictional force at the ends of the specimen, which distorts the basic compressive deformation, and hence, for the same load value, we shall have a larger deformation. In spite of all these circumstances, which hinder, or even make impossible, the obtaining of a reliable and constant dependence:

\[ n = f(\alpha_0) \]

where all the parametric coefficients of the curve would be completely definite, on considering it one sees clearly that the curve is asymptotic.

To verify this I used a graphical method that makes it possible approximately to find its asymptote.

On the diagram, through the origin and each point of the curve rays were drawn up to the straight line:

\[ n = 1.000 \]

From the point of intersection of the ray with this straight line, a segment of a straight line equal to the distance of this point from the center of the rays was laid off downward along the ray. In this way all the points marked by small circles were obtained.

Their aggregate gives, approximately, a straight line passing through the point with coordinates:

\[ n = 1.000;\quad \alpha_0 = 0.0 \]

This will be the parametric straight line of our curve. Here, on the diagram, is shown the final result of the graphical selection of the asymptote, which has the equation:

\[ n = 1.000. \]

In actually finding it, however, it was necessary to take several straight lines parallel to the obtained asymptote; in all those cases the points obtained for the parametric straight line do not fall on a straight line, but give certain, often complex, open and closed curves.

For all metals, despite the different slope of the parametric straight lines, a common asymptote was found:

\[ n = 1.000, \]

i.e., in the limit, when the specimen will have the relative geometrical dimensions:

\[ \alpha_0 = \frac{l_0}{d_0} = \infty . \]

The compression polytrope beyond the elastic limit is transformed into a hyperbola referred to the axes: \(P\) and \(l\), as to asymptotes. Of course, for \(\alpha_0 = \infty\) we may assume that friction at the ends will have no influence on the course of the residual deformation.

The second diagram presented here is constructed from the same experimental points already along the parametric straight lines, whose slope was found by the method of least squares when applying the formulas for this curve (see drawing)

\[ y(ax + by)a^2x \]

here

\[ \begin{aligned} &y = l;\quad a = l_0; \quad \text{in the axes}\\ &x = P;\quad b = \text{parameter of the straight line} \end{aligned} \left\{ \begin{array}{l} \text{in the axes } P, l; \end{array} \right. \]

\[ \begin{aligned} &y = (1-i);\quad a = 1.000; \quad \text{in the axes}\\ &x = P;\quad b = \text{parameter of the straight line} \end{aligned} \left\{ \begin{array}{l} \text{in the axes } P, (1-i). \end{array} \right. \]

The complete agreement of the curves \(n=f(a_0)\), in the sense of their asymptotic approach to a common straight line, nevertheless did not satisfy me. It seemed desirable to carry out such an additional experiment as would confirm the tendency of the polytropy exponent, in the absence of friction at the ends of the specimen, to approach unity.

At first several experiments were made with specimens whose ends were covered, before each application of load, with a lubricating substance. I successively tried lubricants of various viscosities: paraffin, tallow, vaseline, castor oil, oleonaphth, and kerosene. When a lubricant was applied, the polytropy exponent increased in comparison with that for unlubricated specimens, and increased the more, the thicker the lubricant was (for paraffin—max.). This was natural: the thicker and more viscous the lubricant, the less it was squeezed out from under the specimen. Finally, having used lead as a lubricant for the ends of specimens made of hard metals, I obtained an even greater increase of the curve exponent, but still it was not equal to unity. Then I decided to use, as lubricant, the same metal as that being tested. From aluminum, red copper, and iron, specimens were made as indicated in full size in drawing 9.

The test specimen itself was a cylinder with \(d=10.00\) mm, from which conical heads extended, with the diameter of the cone at the base \(D=30.00\) mm. From each metal 3 specimens were made, with the size of the cylindrical part:

\[ l_0 = 10.00\ \text{mm};\quad l_0 = 20.00\ \text{mm.}\quad \text{and}\quad l_0 = 30.00\ \text{mm.} \]

The conical appendages were also made with the calculation that, gradually, beyond the elastic limit, the material of the cylindrical part would pass beyond the elastic limit before the parts of the cones nearest to the cylinder, and thus there would be no radial transverse force, or, if there were, it would be insignificant in magnitude. The cones were made relatively blunt. This is extremely necessary, as my work on the compression of metals showed, so that the stresses in the cone, in that part where it is in contact with the cylinder, should not differ greatly from the stresses in the cylinder itself. The length was measured with a Böhme cathetometer, with an accuracy of up to \(0.01\) mm. The calculated length was taken to be the distance between the cones.

As is seen from the last table, which presents a summary of all the experiments with cylinders having conical appendages, the polytropic exponent becomes equal to unity if the absence of frictional forces at the ends is ensured. Thus, in the absence of frictional forces, the dependence between the force and the residual compressive deformation produced by it is hyperbolic. We write

\[ Pl=P_c l_0 \quad \text{or} \quad P(1-i)=P_c . \]

Whence follows the condition:

\[ k=k_c, \]

i.e., constancy of stress for any magnitude of the residual deformation. The magnitude of the work of deformation will be:

\[ T_c=P_c l_0 \lg \frac{l_0}{l}=v_0 k_c \lg \frac{1}{1-i}. \]

The volumetric specific work will be:

\[ T'_c \gamma=(x)=\frac{T_c}{v_0}=k_c \lg \frac{1}{1-i}. \]

In elastic deformation the volumetric specific work depends on the stress that we have produced in the material:

\[ T=\frac{k^2}{2E}; \]

here, however, with constant stress, the work depends on the relative deformation; together with an increase in the magnitude of \(i\), the expended work also increases.

Such plastic metals as aluminum, lead, tin, red copper, iron make it possible to deform them within the broadest limits. It may even be asserted that, practically, they cannot be destroyed by compression. On the other hand, the expression for the work, by the polytrope or by the hyperbola, in any case gives, for values of \(i\) close to unity, an infinitely large value. Obviously a metal, even a very plastic one, can be deformed only up to a certain limit. Such a limit will evidently be the moment when, at least in some small part of the specimen, there will be found, per unit volume, an amount of work that gives the transition for a solid body into another phase, in other words, that is equivalent to the total heat of fusion of this metal (again calculated per 1 unit of volume). Wishing to determine the deformation of a metal at which work equal to its total heat of fusion is obtained, I turned to the literature on this question, but, unfortunately, found only in Person (Ann. Chim. et Phys.) the formula proposed by him for metals:

\[ \rho=Aq\left(1+\frac{2}{\sqrt{s}}\right). \]

Where \(A=\mathrm{const.}\), \(q\) is the coefficient of elasticity, and \(s\) is the density of the metal, \(\rho\) is the latent heat of fusion.

Making use of the fundamental work of Joseph W. Richards, Metallurgische Berechnung, 1913, I attempted to verify it by substituting the corresponding heats, coefficients of elasticity, and densities, but arrived at unsatisfactory results: the formula was not justified. The same discrepancy of the experimental data when they are substituted into another formula of Person’s, proposed by him for organic compounds, is pointed out by O. D. Khvolson in his course of physics.

In view of circumstances of this kind I decided to process those data which are given by Richards. I took the full heats of fusion of metals presented in his work, referred to 1 kg. The heats of fusion were calculated from tables, taking the full heat of fusion of the metal from absolute zero. Of course, in view of the impossibility of having an exact law of the variation of heat capacities at low temperatures, it was necessary to recalculate the heats according to those dependences which are given for temperatures above zero. Therefore there is no need to claim an exact calculation of the heats. On the other hand, the heat required for heating the metals, given the comparatively small heat capacities of the latter, the relatively high melting temperatures, and the large values of the latent heats of fusion, is small. For Young’s moduli I took mean rounded figures from Holtzon (Course of Physics), Berdov (Course of Machine Parts), and Morley (Strength of Materials).

Below are given a table and a diagram showing the relation between Young’s modulus, the density of the metal, and its full heat of fusion, calculated from absolute zero.

The table has the columns:

\(E\)—Young’s modulus in kg/mm.\(^2\,10^3\);

\(\delta\)—density of the metal in kg/dm.\(^3\);

\(r\)—full specific (weight) heat of fusion in Cal./kg.;

\(q\)—full specific (volumetric) heat of fusion in Cal./dm.\(^3\);

\(B\)—coefficient of proportionality for the straight line obtained, in Cal./kgm.

The diagram has the abscissa axis \(E\) in kg/mm.\(^2\);

and the ordinate axis \(q\) in Cal./dm.\(^3\).

As is seen from the diagram, we obtain a linear dependence between \(q\) and \(E\). Tin gives a large deviation from the general regularity. This straight line will be expressed as follows:

\[ q = B E \]

The coefficient of proportionality \(B\) has been calculated for each metal, and its variation is comparatively not large, if one takes into account that it was obtained not from direct experiment, but calculated from mean values of heats and Young’s moduli.

Since the specific volumetric heat of fusion of the metal will be:

\[ q = r\delta \]

then we can write the dependence in another form:

\[ r=\frac{q}{\delta}=B\frac{E}{\delta}. \]

The mechanical work equivalent to the total heat of melting will be:

\[ T_q=\frac{q}{A} \]

where \(A\) is the thermal equivalent of mechanical energy, or:

\[ T_q=\frac{BE}{A}. \]

The proportionality coefficient \(B\) also has the dimension of the thermal equivalent of mechanical energy.

If we exclude the value of \(B\) for tin, as sharply differing from the values of \(B\) for all the other metals cited, then we obtain for it the mean value:

\[ B=1.55\times 10^{-4}=0.00015\ \mathrm{Cal./kgm}. \]

Taking for \(A\)

\[ A=\frac{1}{427}=0.00234\ \mathrm{Cal./kgr}. \]

We have:

\[ T_q=0.0669E=\frac{E}{15.07}\simeq \frac{E}{15}\frac{\mathrm{kgm}}{\mathrm{m}^3}. \]

Thus we see, insofar as such approximate calculations show us, that Young’s modulus is a measure of the total heat of melting, counting the latter from absolute zero. Young’s modulus itself varies with temperature, and therefore the coefficient \(B\) is also a function of temperature, if we take \(E=\mathrm{const}\) under the ordinary conditions (temperature conditions) of its determination \((15^\circ—20^\circ C)\).

Thus, in order to obtain a liquid metal, it is necessary that it absorb work:

\[ T_q=\frac{E}{15}\frac{\mathrm{kgm}}{\mathrm{m}^3}. \]

Let us try to compare a metal being melted with a metal being plastically deformed.

For this it is necessary to equate the specific volumetric work in its deformation to the work of melting (volumetric specific), after subtracting from the latter the work equivalent to the heat possessed by the metal under the conditions of the experiment. This latter work, let us call it \(T_0\), amounts for all metals to approximately \(25\%\) of the total work of melting, calculated from absolute zero. Thus:

\[ T_c=T_q-T_0\simeq \frac{3}{4}T_q. \]

Let us take the work \(T_c\) for a specimen having:

\[ \alpha_0=2 \]

which requires, during deformation, the minimum specific work,

\[ T_c=kc\lg \frac{1}{1-\frac{1}{2}}=\frac{3}{4}T_q=\frac{3E}{4.15}=\frac{E}{20}. \]

To the article by G. V. Trapeznikov: “Laws of Residual Deformations (Compression)”

Fig. 4.

Red copper.

Axis labels: \(1-i\); \(P_t\).

Curve labels: \(\alpha_0 = 0.5\); \(\alpha_0 = 3.0\).

Fig. 5.

Red copper.

Axis labels: \(\lg(1-i)\); \(\lg P_t\).

Curve labels: \(\alpha_0 = 0.5\); \(\alpha_0 = 3.0\).

To the article by G. V. Trapeznikov: “Laws of Residual Deformations (I—Compression).”

Fig. 6.

Specimen.

Fig. 7.

along \(a—b\)

To the article by G. V. Trapeznikov, “Laws of Residual Deformations (I—Compression).”

Fig. 8.

Axis labels: \(n\), \(\alpha\).

Visible tick labels on the \(n\)-axis: 0; 0.100; 0.200; 0.300; 0.400; 0.500; 0.600; 0.700; 0.800; 0.900; 1.000.

Visible tick labels on the \(\alpha\)-axis: 1.0; 2.0; 3.0; 4.0.

Fig. 9.

Fig. 10.

Axis labels: \(q\), cal/dm\(^3\); \(E\), kg/mm\(^2\).

Visible tick labels on the \(q\)-axis: 0; 1000; 2000; 3000.

Visible tick labels on the \(E\)-axis: 5000; 10000; 15000; 20000.

To the article by G. V. Trapeznikov: “Laws of Residual Deformations (I—Compression).”

Drawing 1.

Compression.

Drawing 2.

Drawing 3.

but \(k_c = \dfrac{E}{375}\), therefore:

\[ \lg \frac{1}{1-i} = \frac{375}{20} = 18.75, \]

\[ \frac{1}{1-i} = 5.625 \cdot 10^{18} \]

or

\[ i = 1 - 1.78 \cdot 10^{-19} \]

i.e., practically, the relative deformation is equal to 1.

Summary.

1. The relation between the load or stress, on the one hand, and the absolute or relative deformation, on the other, in plastic compression is expressed by a polytrope:

\[ P^{n} l = P_c^{n} l_0 = \mathrm{const.} \quad \text{and} \quad k(1-i)^{\frac{1-n}{n}} = k_c. \]

1. The exponent of the polytrope is a function of the properties of the metal and of the relative dimensions of the specimen

\[ \left( \alpha_0 = \frac{l_0}{d_0} \right). \]

1. In the absence of friction on the end faces of the specimen, or for infinitely large \(\alpha_0\), the polytrope turns into a hyperbola.

2. A new concept is introduced: the theoretical elastic limit, common to specimens of the same metal but of different geometrical dimensions.

3. The theoretical elastic limit is linearly related to the Young’s modulus of the metal:

\[ k_c = \beta_c E. \]

For all the metals tested, the coefficient \(\beta_c\) is one and the same:

\[ \beta_c = \frac{1}{375}. \]

1. Young’s modulus is linearly related to the total heat of fusion of the metal,

\[ q = B E \]

where \(B\) is a coefficient common to all metals,

\[ B \simeq 0.000155 \ \mathrm{Cal.}/\mathrm{kgm}. \]

The mechanical work of fusion is related to Young’s modulus:

\[ T_q = \frac{E}{15} \ \mathrm{kgm.}/\mathrm{m}^3. \]