Prof. G. V. Trapeznikov

DEVICE FOR DETERMINING

TRANSVERSE DEFORMATIONS

UNDER TENSION AND COMPRESSION

Tomsk

1928

Prof. G. V. Trapeznikov.

Device for Determining Transverse Deformations under Tension and Compression.

While conducting, in 1925, a series of experimental studies to determine the relation between elastic and plastic deformations, I encountered the need to take readings of transverse deformations under tension and compression with an accuracy of up to 0.00002 and, subsequently, up to 0.00001 mm. The Pinergin–Meyer instrument available in the Mechanical Laboratory of the STI, for determining Poisson’s ratio, could not satisfy me, since it gives, at best, readings to 0.0001 mm. This compelled me to design and build an instrument making it possible to determine deformations with very great accuracy. I now proceed to a description of this instrument, first presenting the description of the design and of work with it, the diagram, and the theoretical justification of the setup.

The diagram shows (Fig. 1) that the instrument consists of a ring C, embracing the transverse section of the test specimen F.

Fig. 1.

Attachment to the specimen is achieved by pressure, on one side by the set screw B and, on the other, by the rod A, pressing—

to the spring \(F\). Two rollers \(E\), with mirrors, are pressed against rod \(A\) by means of flat springs \(G\), as is the case in Bauschinger’s instrument. The ray from the scale, before entering the observation tube, is reflected many times from the mirrors, passing from one to the other and back. In order to determine the best arrangement of the mirrors of the instrument, let us consider the path of the ray in the mirrors.

Fig. 2.

Let (Fig. 2) \(A\) be a certain conventional axis from which we shall measure the angles of the course of the rays. The aperture angle of the mirrors \(B\) and \(C\) is divided by this axis into two angles \(\alpha_1\) and \(\alpha_2\) and is denoted by us as \(\varphi\). The axis of the tube forms with axis \(A\) the angle \(\alpha_o\). We shall measure the angles from the positive \(A\), counterclockwise.

Let the number of reflections of the ray from the mirrors be \(m\), counting from the entrance of the ray into the system of mirrors to its exit from it. Then, for an odd number of reflections, the angle of the ray emerging from the system of mirrors will be:

\[ \alpha=\pi-\left[(m-1)\varphi+\beta+\alpha_1\right]; \]

but

\[ \beta=\alpha_o+\alpha_1 \quad\text{and}\quad \alpha_1+\alpha_2=\varphi \]

therefore:

\[ \alpha=\pi-(m-1)\varphi-\alpha_o-2\alpha_1 \]

for an even number of reflections, we have:

\[ \alpha=\pi-\left[(m-1)\varphi+\beta-\alpha_1\right] \]

or, with the former substitution:

\[ \alpha=\pi-m\varphi-\alpha_o \]

since, for the instrument, it is necessary to have complete insensitivity of it to rotation in the plane of its setting, i.e., to rotation of the system of mirrors as a whole, the rotation of the entire angle of the mirrors \(\varphi\) must not affect the change in \(\alpha\). In which of these two cases does the indicated circumstance occur? With a simple rotation of the angle of the mirrors (\(\varphi=\mathrm{const.}\)), the angle \(\alpha_1\) changes.

For the increments of the angles we have

\[ \Delta \alpha=-2\Delta\alpha_1 \]

for an odd number of reflections, and

\[ \Delta \alpha=0 \]

for an even number.

It follows from this that only in the case of an even number of reflections does rotation of the entire instrument (the specimen or the machine) not affect the reading. Therefore the number of reflections must necessarily be even.

If, during observations—as is almost always done—the scale is attached to the tube, then it is most advantageous to set the axis of the tube parallel to the bisector of the angle \(\varphi\) (Fig. 3). Let us find the angle that the mirrors must have between them, with the incident and emerging rays parallel.

Fig. 3.

For \(m\) reflections we have:

\[ \pi - [(m - 1)\varphi + \beta] = \beta \]

but here

\[ \beta = \frac{1}{2}\varphi; \]

therefore:

\[ \varphi = \frac{\pi}{m} \]

The magnification of the instrument may be obtained as follows:

Let
\(\mu\) — the absolute transverse deformation,
\(d\) — the diameter of the mirror roller,
\(n\) — the number of scale divisions corresponding to the deformation \(\mu\),
\(L\) — the distance from the mirror (the first one) to the scale.

We write:

\[ \frac{n}{2mL} = \frac{\mu}{d} \]

Magnification:

\[ \gamma = \frac{n}{\mu} = \frac{2mL}{d} \]

In my instrument: \(d = 3\ \mathrm{mm}\)

Taking, on average: \(L = 3000\ \mathrm{mm}\).

Then

\[ \gamma = 2000m \]

With a sixfold reflection and a reading capability (normal) of 0.1 mm, we have a deformation reading down to \(\frac{1}{120000}\) mm. As we shall see below, by separating the scale and the tube, we have the possibility, with the same visible size of division, of moving the scale twice as far away; then we shall be able to read \(\frac{1}{240000}\) mm.

The bisector arrangement of the tube has two disadvantages:

1—With a large number of reflections (more than 4), the angle at which the incoming and outgoing rays fall on the mirror is small. Because of imperfect polishing of the mirrors, double reflection from the first and second surfaces of the glass, and loss of light intensity when passing through a great thickness of glass at “grazing” incidence of the ray on the plane of the mirror, the image of the divisions becomes blurred and dark.

2—A disadvantage common to all mirror arrangements with the tube and the scale in one unit. Along the light ray, the scale is located from the tube at twice the distance of the scale from the mirrors. In the calculation, however, only this latter distance is introduced as the arm of the light lever. The tube may be at any distance from the mirrors. This loss of distance manifests itself in the greater difficulty of catching the image of the scale in the mirror when sighting, and in the impossibility of making full use of the resolving power of the tube (most technical laboratory telescopes have a limit of 5–6 m). This wholly unnecessary double path of the ray reduces the optical lever and reduces the scale divisions visible in the tube, thereby making it difficult to determine tenths of divisions by eye.

All the circumstances stated above compel us to separate the scale from the tube; such separation has the additional advantage that sighting the tube is thereby simplified to a minimum. The tube is placed near the instrument, and its axis is arranged almost perpendicular to one of the mirrors (Fig. 4).

Fig. 4.

In this case the light ray makes almost right angles with the planes of the mirrors; the image obtained is bright and sharp, and, at the same distance of the scale from the mirrors, the divisions of the scale are twice (almost) as large.

We now proceed to the description of the construction of the instrument. (Fig. 5).

Fig. 5.

The device consists of a ring A, of trough-shaped cross-section, cast from a specially composed alloy (aluminum, zinc, copper, silver) with a specific gravity of 3.4. This alloy is hard, takes an excellent polish, does not oxidize in moist air, and has a silvery-white color. On two diametrically opposite sides there are bosses. One, B—with a thread (pitch 1 mm)—serves as the nut for the setting steel hollow screw C; at one end of this screw there is a mushroom head D for turning it, and at the other a soldered-on bicycle ball, for the screw to bear against the specimen during installation. The other boss, E, is shaped; it serves as a guiding bushing in which, in a direction radial with respect to the ring, a thin-walled cylindrical cup F, made of very hard steel, moves; the latter is lapped to the bushing. The bushing, on its outer side, has a cover G screwed onto it; through the central opening of the cover passes a steel rod H. At one end it is screwed blindly into the bottom of cup F, with a small tail projecting outward, and at the other it carries a mushroom head J. Inside the cup there is a fairly strong, but long-travel, steel helical spring K. One end of it bears against the bottom of the cup, and the other against the cover of the bushing. The spring is always in a compressed state. The mushroom head J prevents the spring from pushing the cup out of the bushing, into the ring. The sides of the bushing are milled out on two sides to expose the lateral sides of the cup. In front of the windows L formed by the milling, steel flat springs M are riveted. Between the springs and the body of the cup, on either side of the bushing, steel rods—rollers 3 mm in diameter—are inserted; at their ends they have aluminum holders-blades with mirrors.

The device is installed as follows:

The setting screw C, by rotating the mushroom head D, is brought into such a position that the distance between its ball and the tail of the rod of cup F becomes approximately 1.5 mm less than the dimension of the cross-section at which the ring of the device is to be installed. Then, taking hold of the mushroom heads D and J, they are pulled in opposite directions; the spring is compressed, and the ring, placed on the specimen, is set in the selected section; by reducing the force on the mushroom heads, the ball of the screw and the tail of the cup are brought into contact with the diametrically opposite generatrices of the specimen; the ring is now held by friction. Between the side springs and the cup, rods with mirrors are installed, to which, by eye or by means of a template, the required angle of divergence is given. It should be noted that, with the scale carried separately, aiming the telescope at it is unusually simple and takes no more than half a minute.

A device with duplicated mirrors possesses a very interesting property, the theoretical treatment of which, because of the somewhat cumbersome graphical constructions and the complexity of the analysis, cannot be given here, since it constitutes the solution of a purely optical problem. If the mirror rods are not placed exactly perpendicular to the axis of the cup, then the image of the scale, in reflection, is inclined at an angle to its actual (horizontal) position.

With a large number of reflections the image may even turn out to be perpendicular to it. In this case there is no need to take care to correct the skew (provided, of course, that the skew itself is not large) of the rods; it is sufficient to rotate either the scale or the rim of the telescope crosshairs until the crosshairs and the divisions of the scale are mutually parallel. The second, of course,

Device for determining transverse deformations under tension and compression.

easier to make. Despite the skewed image, during the entire operation of the device the scale will not leave the field of view of the tube.

In order to estimate the influence that deformation of the ring itself may have, I shall give a comparison of the deformations of the ring and of the cup spring. In determining the deformation of the ring we neglect the influence of the bosses, with all the more justification since the latter make the ring more rigid.

The separation of diametrically opposite points of the ring under the action of a tensile force will be:

\[ f_k=\frac{(\pi^2-8)R^3}{4\pi EJ}P \]

The deformation for a cylindrical spring will be:

\[ f_p=\frac{8nD^3}{Gd^4}P \]

Here, for the data of the instrument:

Radius of the ring \(R = 43.5\ \mathrm{mm}\)

Young’s modulus of the alloy \(E = 8230\ \frac{\mathrm{kg}}{\mathrm{mm}^2}\)

Moment of inertia \(J = 581\ \mathrm{mm}^4\)

Number of coils \(n = 12\)

Spring diameter \(D = 5\ \mathrm{mm}\)

Shear modulus of steel \(G = 8300\ \frac{\mathrm{kg}}{\mathrm{mm}^2}\)

Wire diameter \(d = 0.75\ \mathrm{mm}\)

we have:

\[ \frac{f_k}{f_p}=\frac{1}{2970} \]

As we see, the ring is 3000 times stiffer than the spring; consequently, we shall have no influence of its deformation on the reading.

The author considers it necessary to point out those advantages which, in his opinion, his instrument and arrangement possess:

1—Solidity and lightness of construction (weight 180 g); absence of delicate and fragile parts in the form of centering axes, knife-edges, levers, which eliminates damage in the form of blunting, breakage, bending, or crushing; absence of play. The most delicate part of the instrument—the cup—is closed on all sides.

2—Absence of delicate parts removable after operation. The rods with mirrors need not be removed.

3—No necessity for alignment.

4—Possibility of working with any degree of magnification by changing the number of reflections.

5—Easy installation. One set screw and the rods, the mirrors, installed at any place on the cup.

6—One tube and one scale, instead of the usual double set of them. Hence, as a consequence, great speed of installation and operation, and therefore also less fatigue of the experimenter.

7—The experimenter’s proximity to the machine and to the instrument during readings, which is of great importance when working alone.