Engineer A. M. Rosenberg

Experimental Study

of the Process of Formation

of Metal Chips

Tomsk
1929

Eng. A. M. Rosenberg

Experimental Study of the Process of Metal Chip Formation.

INTRODUCTION.

In 1848 the Frenchman Coquillhat¹ conducted experiments on a drilling machine, with the aim of determining the expenditure of work required to convert a unit volume of various materials into chips. This was the first attempt recorded in the literature to investigate phenomena connected with the machining of metals by cutting. Since then, over the course of 80 years, a large number of varied experiments and investigations have been carried out in this field.

As a result of all these investigations we now have a science called the “theory of metal cutting.” However, on closer acquaintance with this science, one can become convinced of the almost complete absence in it of “theoreticality.” It is nothing other than a collection of experimentally obtained and rigorously verified facts, linked by a chain of empirical dependences that more or less fully cover the question under consideration, i.e., continuous empiricism, without any sufficiently solid theoretical basis.

There have been attempts to provide a theoretical foundation for certain phenomena occurring during the cutting of metals, but none of them has been successful; the reason for this must be sought in the fact that, up to now, we still do not have sufficiently exhaustive information about the very basis of the entire process, namely about the phenomena occurring in the metal during the formation of chips from it.

Experiments conducted in this direction have not yielded consistent results in the interpretation of individual questions, nor have they provided a general coherent picture of the process of chip formation. Some theoretical investigations of the question that have taken place, being based on insufficiently complete experiments by predecessors, were saturated with the authors’ hypothetical premises concerning the course of the process, a priori assumptions, and generalizing coefficients; therefore they could give only relatively valuable results, and are of interest from the standpoint of the history of research on this question.

Since, in chip formation, all phenomena occur far beyond the limit of elasticity, which to this day is the limit of our knowledge in the science of strength of materials, investigation of the process is possible for the time being only by experiment; and only when sufficient experimental material has been accumulated will it be possible to proceed to the theoretical treatment of the question and to the establishment of a theoretical basis.

From this point of view, every more or less carefully arranged experiment is valuable if it extends the limits of our knowledge, or even merely confirms propositions found earlier.

The authors of the earliest experiments considered the phenomena of chip formation so simple that, in their opinion, a certain number of experiments was sufficient to clarify both the general picture of the phenomenon and its details. A series of subsequent works, revealing ever new and uninvestigated aspects of the question, showed how complex it is in its very essence.

In the present work we by no means intend to cover the process of chip formation completely, and we shall be quite satisfied if our work contributes at least to some clarification of the phenomenon under investigation.

Before proceeding to the description of our experiments, we shall give a brief historical account of the works known to us that touch upon the process of formation of metal chips.

Historical review.

Experiments of Prof. Time in 1870.²) The mining engineer, later Professor I. Time, carried out his experiments at the Lugansk works, using a planing machine for this purpose. The materials for the experiments were iron, steel, bronze, cast iron, lead, and zinc. Cutters were used with cutting angles: $\alpha = 45^\circ, 55^\circ, 65^\circ, 75^\circ$ and $90^\circ$. In machining iron, chips were removed with a thickness from $\dfrac{0.75''}{16} = 1.19$ mm to $1/4'' = 6.35$ mm; in machining steel, from $1/8'' = 3.18$ mm to $\dfrac{1.75''}{8} = 5.56$ mm.

Machining of iron and steel was carried out with lubrication of the cutter; the other materials were machined without lubrication.

Time, on the basis of his observations, gave a theory of chip formation. In his opinion, the resistance of metals to cutting may be regarded as the sum of elementary resistances to shearing. In the case of a ductile metal, the process of separation and formation of the chip is represented as follows:

Drawing No. 1.

Under the influence of the acting force, the cutter, in its motion, is pressed into the piece of metal being machined, compressing it with its front face (Drawing 1). Compression continues until shearing of the first chip element occurs in a definite direction, “ab.” This occurs at the moment when the component, in the direction “ab,” of the compressive force on the plane of contact between the cutter and the chip becomes equal to the resistance of the metal to shearing in the section “ab.” After the shearing of the first element begins, compression of the second element arises and subsequent shearing occurs in the same direction “ab” (Drawing 2). In the compression of each subsequent element, owing to the bulging of the metal, rotation of the preceding element of the chip takes place, its separation from the plane of the cutter occurs, and the phenomenon of curling of the chip is produced.

Drawing No. 2.

According to the author’s observations, the direction of shear “ab” is always quite definite and depends on the magnitude of the cutting angle $\alpha$, changing only very slightly, so that the angle $\beta_1$ (Fig. 2) is an approximately constant quantity. Observing the direction of shear on the side surface of a piece of metal, Time came to the conclusion that compression of the metal ahead of the cutter occurs only up to the line “ab,” within the limits of the angle $\beta$; beyond the line “ab,” within the limits of the angle $\beta_1$, the metal is completely undeformed. From the same observation he established the following values of the angles $\beta$ and $\beta_1$ as functions of the cutting angle “$\alpha$” for various materials being machined:

Table No. 1.

Angle	Iron	Iron	Iron	Iron	Iron	Lead	Lead	Lead	Lead	Lead	Zinc	Zinc	Zinc	Zinc	Zinc
$\alpha$	45°	55°	65°	75°	90°	45°	55°	65°	75°	90°	45°	55°	65°	75°	90°
$\beta$	100°	90°	85°	75°	65°	100°	90°	80°	70°	62°	100°	85°	85°	75°	60°
$\alpha+\beta$	145°	145°	150°	150°	155°	145°	145°	145°	145°	152°	145°	140°	150°	150°	150°
$\beta_1$	35°	35°	30°	30°	25°	35°	35°	35°	35°	28°	35°	40°	30°	30°	30°

The author notes the phenomenon of chip shrinkage (the removed chip proves to be shorter than the path traveled by the cutter), and considers its principal cause to be the relative displacement of the elements during the process of its formation.

The length of a chip consisting of “$n$” elements may be expressed as

\[ l_1 = n \cdot e_1, \]

where $e_1$ is the length of one element;

The length of the path traveled by the cutter:

\[ l_0 = n e_0 \]

where $e_0$ is the length of an element in the direction of motion of the cutter (see Fig. 3).

The shrinkage coefficient will be:

\[ K_0=\frac{l_1}{l_0}=\frac{e_1}{e_0}=\frac{\operatorname{Sin}\beta_1}{\operatorname{Sin}\beta} =\frac{\operatorname{Sin}\beta_1}{\operatorname{Sin}(180^\circ-\alpha-\beta_1)} \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1) \]

According to Time’s data, the shrinkage coefficient varies for iron and a cutting angle $\alpha=45^\circ$ from 0.59 to 0.65; for $\alpha=75^\circ$ from 0.45 to 0.48. The fluctuations in the values of the shrinkage coefficient $K_0$ for the same values of “$\alpha$,” with one and the same material, Time explains by very slight fluctuations in the values of $\beta$ and $\beta_1$ when the thickness of the layer being removed “$\delta$” changes; and at larger “$\alpha$,” also by the separation of the chip elements from one another.

Prof. Afanas’ev³) in 1888 carried out a theoretical investigation of the cutting process. The main aim was to determine the cutting force analytically. In passing, the author enters into an analysis of the process of chip formation. The course of this process is conceived by him as follows:

During the motion of the wedge—the cutter—the length of the chip lying on its inclined face gradually reaches the greatest possible value, while the acting forces attain their maximum value and can be found from the condition of chip failure occurring at that moment. Assuming that the pressure at the end of the chip opposite the cutting edge reaches the magnitude of the compressive strength of the material, failure of the chip occurs in one of three possible ways: either it fractures, or it shears, or else a crack is produced in the material, coinciding with the direction of motion of the cutter. The last case may occur when wood is cut along the fibers with small cutting angles.

As a result of a whole series of calculations the author, finding the position of the resultant force acting on the chip, assumes that the shearing or fracture of the chip must occur in a plane parallel to this resultant, and thus determines the angles of inclination of the shearing planes to the direction of motion of the cutter; according to his data, their value depends only on the magnitude of the cutting angle and on the coefficient of friction of the chip on the cutter. If the latter is taken as equal to 0.46, then the values of the angles of inclination of the planes are obtained as very close to those obtained in the experiments of Prof. Time. Their magnitude is constant for a given material and an unchanged cutting angle.

Basing himself on the conclusions of M. Tresca ⁴), the author believes that the principal cause of chip contraction is compression of the material in front of the cutter. The expression for the chip-contraction coefficient is given by Afanas’ev in the form:

\[ K_0=\frac{l_1}{l_0}=\frac{\cos(\alpha+\Theta)}{\cos\Theta}\ldots\ldots\ldots\ldots\ldots\ldots\ldots (2) \]

where \(\pm\Theta=\pm(\beta-90^\circ)=\pm(90-\alpha-\beta_1)\), according to the notation of Drawing No. 3. Substituting these values for \(\Theta\) in the expression for the contraction coefficient of Prof. Afanas’ev, we obtain formula (1), given by Prof. Time, which we have already cited earlier.

The author comes to the conclusion that the motion of the chip along the cutter occurs in the direction toward the cutting edge of the cutter, and the frictional force arising thereby between the chip and the cutter is directed in the direction opposite to the motion of the chip, i.e., away from the cutting edge of the cutter.

Drawing No. 3.

Prof. Zvorykin ⁵) in 1893 carried out an experimental and theoretical investigation, as a result of which he gave an analytical expression for the cutting force and the work required for separating the chip.

The author assumes, in the formation of a chip, either the presence of bending deformation or of shearing deformation; but since the phenomenon of bending far beyond the elastic limit is extremely complex and little studied, he considers it possible to examine only the formation of the chip by means of shearing deformation.

Taking as the basis of his theory of chip formation, in general similar to the theory of Prof. Time, the assumption that the shearing plane, inclined at an angle \(\beta_1\) to the line of motion of the cutter, will

always so situated that a minimum force is required for chip formation, Zvorykin analytically finds the position of the shear planes and the expression for the angle \(\beta_1\) in the following form:

\[ \beta_1 = 90 - \frac{\varphi + \varphi_1 + \alpha}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

Here: \(\alpha\) is the cutting angle, \(\operatorname{tg}\varphi = f\), \(\operatorname{tg}\varphi_1 = f_1\), where \(f\) is the coefficient of friction of the chip on the tool, and \(f_1\) is a certain coefficient depending on the elastic and plastic properties of the given material, on the magnitude of the internal friction of the particles within it, and indicating what part of the force compressing the chip is transmitted to the shear plane normally to it, thereby producing an additional magnitude of internal friction on the shear plane and increasing the resistance of the chip to shearing.

The author assumes for iron and steel \(f = 0.44\) and \(f_1 = 0.34\); from formula (3) the following values of the angles are then obtained for various cutting angles (see Table No. 2).

Table No. 2.

Angles	Material	Iron and steel	Iron and steel	Iron and steel	Iron and steel	Iron and steel
\(\alpha\)		\(45^\circ\)	\(55^\circ\)	\(65^\circ\)	\(75^\circ\)	\(90^\circ\)
\(\beta_1\)		\(46.5^\circ\)	\(41.5^\circ\)	\(36.5^\circ\)	\(31.5^\circ\)	\(24^\circ\)
\(\alpha + \beta\)		\(133.5^\circ\)	\(138.5^\circ\)	\(143.5^\circ\)	\(148.5^\circ\)	\(156^\circ\)

Thus the magnitude of the shear angle depends only on the cutting angle and on the properties of the material being machined.

Pogorzelskii\(^{6}\), in 1907, by means of purely theoretical reasoning, assuming that the position of the shear plane is determined by the presence of a maximum shearing stress under the action of a constant force from the tool, found values of the shear angles which are quite close to Timme’s data and depend on the cutting angle, the coefficient of friction between the chip and the tool, and the coefficient of internal friction in the material.

Sokolov\(^{7}\), in 1914, proceeding from the process of chip formation, sought the most advantageous profile of a roughing tool from the standpoint of eliminating the possibility of chatter. The author accepts as indisputable Prof. Timme’s theory of chip formation. Having measured the shrinkage of the chip, Sokolov found that shrinkage decreases as the thickness of the chip increases; moreover, the number of chip shear elements increases as the chip thickness decreases and as the cutting angle decreases. Sokolov disputes the applicability of Prof. Timme’s shrinkage formula for a shear chip, since, in his opinion, the form of the chip element actually obtained is not similar to that which would satisfy the expression for calculating shrinkage.

Usachev⁸) in 1914 undertook experiments with the aim of determining the temperature of the working part of the cutting-tool blade. The study of the process of chip formation was undertaken incidentally, for a general elucidation and coordination of the phenomena. Experiments on studying the structure of the chip were carried out on a planing machine whose table was driven by a lead screw. The chip was removed and left on the workpiece by reversing the direction of motion of the table; the author assumes that the speed at the final moment of chip formation is equal to the cutting speed at the middle of the table stroke.

By breaking the chips obtained, Usachev observed that the fracture does not occur along the line separating the elements, nor parallel to it, but in a direction making an angle of approximately 30° with the former. From this he concludes that, in the individual elements, shears occur in this direction.

Measuring the temperature at the tip of the tool and at some distance from it, Usachev obtained results similar to the conclusions of Taylor and Nicolson: the temperature of the tip usually proved to be lower than the temperature of a point located somewhat away from the tip. Usachev attributes this phenomenon not to the location of the center of pressure of the chip on the tool at a certain distance from the end of the tool, as the above-mentioned authors state, but to the fact that the very tip of the tool is protected, as he thinks, from overheating by the built-up edge formed on it. Having arrived at this conclusion, the author studies the conditions for the formation of the built-up edge on the tool.

In Usachev’s opinion, the built-up edge consists of material extremely compressed before the working surface of the tool at the place where its blade separates the chip. “The nature of the structure,” writes the author, “shows that this compressed layer belongs neither to the chip nor to the workpiece and moves together with the blade during cutting.” The author explains the formation of irregularities (serrations) on the chip and on the machined surface by the presence of the built-up edge.

Usachev observed that plastic deformations ahead of the tool extend beyond the limits of the angle \(\beta\) and also spread into the machined surface; this was observed by him on the ground side surface of the specimen.

Usachev, the first among Russians, used the metallographic method to study the structure of chips.

Zadneprovsky⁹) in 1918 repeated Prof. Time’s experiments with lead. Observations were made on the side surface of the specimen. It was noted that the plastic deformation ahead of the tool goes beyond the limits of the angles \(\beta\) given by Time and also extends under the tool. Measurements were made of the shrinkage coefficients for lead at a cutting angle \(\alpha = 45^\circ\) and various chip thicknesses.

Klopstock¹⁰) in 1923 investigated the applicability and comparative advantage of a new profile of cutting tools. Part of the work was directed toward studying the process of chip formation, but in view of the fact that the main objective was different, too little attention was devoted to this part.

Chip formation was carried out on a planing machine, and an attempt was made to record successive stages of the process by means of a motion-picture film. But the experiment showed that the number of images obtained per unit time was too small to divide the process into separate stages. Therefore, in order to fix the desired stage of the process, it was necessary to set the machine in motion by hand, photo-

photographing, at the required moments and with slight magnification, the lateral surface of the chip and of the specimen, with the deformations then taking place.

The photographs fully confirmed the basic proposition of Prof. Time that the cutting process consists of the successive sliding of chip layers.

In addition to observing the phenomenon on the lateral surface of the specimen, microsections were prepared from chip cross-sections and photographed. As a result, it was established in all that the process is divided into three stages: 1—the scoring of the material by the tool, 2—the propagation of a crack running ahead of the tool point, 3—the sliding off of the chip element that has been formed.

Herbert E. G ^11) 1923. The aim of the experiments was to determine the capacity for strain hardening of the material being machined during cutting, and to establish a special method for testing the material for the above-mentioned capacity for strain hardening. Seven different materials were taken for the experiments, including iron, steel, cast iron, and copper. Hardness tests were carried out by means of a special pendulum apparatus invented by the author, unknown up to that time. The methods of testing were worked out by the author himself. On a lathe, cutting was performed at a depth of cut \(t = 6.4\ \text{mm}\), a feed \(s = 0.94\ \text{mm/rev}\), and a cutting speed of \(19.8\ \text{m/min}\). The machine tool was suddenly stopped so that the chip remained unseparated at the place of cutting from the material being machined. From the chips, together with the workpiece, a section was prepared on which the hardness was determined in different places; its value was then compared with the initial hardness of the material. The investigation mentioned is of interest to us only in that part which is directly related to our work. On examining the sections obtained, the author noted the presence of a built-up edge (edge appendage) at the tool point and made certain assumptions about its formation and its role in metalworking. The author attributes its formation both to the separation of metal from the chip and to the separation of such metal from the main mass of the specimen. Being strongly compressed and subjected to the action of the high temperature developing during cutting at the tool point, the wedge-shaped built-up edge acquires an extremely high hardness, far exceeding the hardness of the base material. The chip, separated from the base material, as a result of plastic changes in form likewise acquires a high strain hardening, though one lower than that of the built-up edge; moreover, the wave of increasing hardness propagates ahead of the chip in front along the path of the tool in such a way that the degree of strain hardening increases as the tool nose is approached.

W. Rosenhain and Styrney 1925 ^12). The experiments were undertaken with the aim of elucidating the structure of the chip and the process of its separation. The experiments were carried out on a lathe. In a blank of steel or of cast brass, grooves were drilled in advance in such a way that the beads remaining between the grooves could be cut off during the experiment and serve as material for chip formation. The chip was removed at a speed of 1.27 meters per minute with a theoretical tool having a straight-line profile. The dimensions of the tools were \(1/2'' \times 1/4''\). The top angle of the tool was varied from \(30^\circ\) to \(0^\circ\), in accordance with which the cutting angle varied from \(60^\circ\) to \(90^\circ\). The clearance angle varied from \(0^\circ\) to \(20^\circ\). The depth of cut (chip thickness) varied from \(0.002''\) to \(0.05''\), i.e., from \(0.05\) to \(1.27\ \text{mm}\).

After the chip had formed and the machine had been stopped, the chip was cut off together with the adjoining part of the material, sectioned lengthwise, ground, and prepared for examination by etching.

The authors divide all chips obtained, according to their appearance and the character of their formation, into three types:

1. the tear chip type,

2. the shear chip type,

3. the continuous chip type.

The first type is formed, in the authors’ opinion, in the case of a large cutting angle and a large depth of cut. The second type is formed in the case of a large cutting angle and a small depth of cut. In this type, the predominant action is no longer tearing, but sliding (shearing) along a certain plane. In the chip, near the surface of the tool, a zone of large deformations is formed; its upper boundary, separating it from the rest of the chip, is always perpendicular to the direction of shearing. This type of chip is the most common. The continuous-chip type is characterized by the fact that the principal zone of deep deformations is more clearly separated than in the other types. The plane separating the zone of deep deformations from the rest of the chip forms a constant angle of \(60^\circ\) with the direction of motion of the tool, and \(90^\circ\) with a direction which, judging from the photographs, is the boundary between the deformed chip and the material being machined. With a decrease in the cutting angle, the zone of deep deformations diminishes, and, when the cutting angle reaches \(60^\circ\), it disappears completely. Thus the authors express the idea that the chip always tries to create for itself a cutting angle equal to \(60^\circ\), since the zone of deep deformations, in the authors’ opinion, is identical with the built-up edge and constitutes a single whole with the chip being removed. The authors also explain the phenomenon of chip curling by the presence of the zone of deep deformations. They consider that the chip slides along the built-up edge and, on meeting the face of the tool, is bent by it. A lesser role in the curling of the chip is played by the displacements of its successive parts, which, however, when the cutting angle is less than \(60^\circ\), i.e. when the zone of deep deformations is already absent, causes the chip to curl.

In recent times the optical method of studying stresses has come into use and has gained some currency. The essence of this method is based on the fact that transparent isotropic bodies, when elastic or plastic deformations are present in them, become birefringent and, in polarized light, give a vivid colored picture of the distribution of stresses in them under the influence of the acting forces\(^{13}\). In this way it is possible not only to recognize the sign of the stress acting at some point of the body, but also to determine its magnitude. It has been proved that, in the case of a plane problem, the picture of stress distribution obtained in glass, xylonite, or another isotropic body can also be extended to the materials used in machine-building practice, in particular to metals; and thus it became possible, by preparing from glass or xylonite an accurate model of a part or of a plane mechanism and placing it under conditions similar to those in which the part operates, to obtain a picture of the stress distribution, determine the weakest places of the part, and, often,

to find the proper approach to the solution of one or another practical problem proposed, for which there is no solution in theory.

Some tests were also carried out in the field of metal cutting.

Professor C. G. Coker[^14] in 1922 conducted experiments with lathe tools, which he later also carried out for planing tools and milling cutters,[^15] for which he designed a special apparatus with a polarization setup. Cellulose served as the material for the experiments. In addition, some results were checked directly on metal (brass and steel), for which a shaper or a lathe was used.

It was found that, ahead of the tool, compressive deformation spreads in front of it; beneath the tool and behind the tool there is tensile deformation, and the wave of elastic deformations, both in the one and in the other directions, propagates quite far from the cutting zone.

In the case of milling, when several cutters are in action simultaneously, the compressive stresses in front of the cutter, gradually decreasing, change sign and become tensile stresses as one approaches the place of action of the preceding cutter. The magnitudes of these stresses were calculated for several cases, and their variation was represented graphically. With cutters having small cutting angles, a catching up and drawing upward of the machined surface was observed; with large cutting angles this phenomenon disappeared.

Using a glass tool and machining cellulose materials with it, the distribution of elastic stresses in the tool as well was found.

Professor Zaitsev[^16], among numerous other experiments, also tested the action of tools. Observations showed that the smaller the cutting angle, the more easily chip formation proceeds; moreover, a tool with a small angle cuts off the chip, while one with a large angle rolls it off. At the beginning of cutting, the load on the tool and the deformation in the chip are greater than later, when the process has become established. At the beginning of cutting, the greatest pressure is concentrated at the very edge of the tool; later, however, the center of greatest pressure shifts somewhat upward from the edge. In experiments on cutting xylonite, the limiting angle of the region of residual deformations proved to lie between \(135^\circ\) and \(155^\circ\) with the direction of motion of the tool.

Having completed the historical review, we turn to the description of the experiments we carried out.

Purpose and method of the investigation undertaken.

The original aim of our work was:

1) Verification of Prof. Time’s data concerning the shear angles \(\beta_1\), their constancy at a constant cutting angle “\(\alpha\),” and the nature of their changes with variation of “\(\alpha\).” This question was of interest because in all earlier investigations the angles \(\beta_1\) were measured on the lateral surface of the specimen and could be somewhat different, owing to distortions, from those within the body of the material. We intended to measure them in the middle section of the specimen.

2) Determination of the depth of propagation of plastic deformations ahead of the tool and beneath it, in the already machined surface, at different cutting angles and thicknesses of the layer being removed.

Along the way, the question of chip shrinkage was touched upon, and the dependence of the shrinkage coefficient on various causes was clarified, as were certain other phenomena connected with the process of chip formation, which will become clear from the exposition that follows.

If we turn our attention, in the investigations set forth above, to the methods used by the experimenters, we shall be able to trace the gradual change in experimental technique; moreover, this change will be characterized by an increase in the accuracy of the experiment, on the one hand, and in confidence in the reliability of the results obtained, on the other. Time²), Zvorykin⁵), Zadneprovsky⁹), and some others observed the course of the chip-formation process on the lateral, previously polished surface of the specimen, the cutting being carried out at a speed close to zero, in order to make it possible to fix the individual stages of the phenomenon. In 1905 Prof. Kurein¹⁰) was the first to apply, for investigation of the structure of the chip, the method of metallographic examination under the microscope, with preliminary revelation of the structure by etching. Usachev⁸) applied both the first and the second of the described methods; subsequent experiments by Sokolov⁷), Herbert¹¹), and Rosenhain¹²) used exclusively the method of metallographic examination. Klopstok¹⁰), along with metallography, attempted to apply the method of cinematographic filming of the dynamics of the cutting process on the surface of the specimen. Finally, Prof. Coker¹⁴) and Prof. Zaitsev¹⁶) applied the most delicate method of optical investigation; however, in so doing they were constrained in their choice of the material to be machined.

The method of metallographic investigation undoubtedly has a number of advantages over the older method of observing the course of the phenomenon on the surface of the specimen. In view of the fact that, when removing a chip of some finite width different from zero, we inevitably encounter phenomena of change in form occurring not only in the direction of motion of the cutter, but also in the direction perpendicular to it—namely, in the direction of the width of the chip—it is natural to assume that the picture of the course of deformations inside the specimen will be somewhat different, although similar to that which we have on the lateral surface. The presence of transverse widening of the chip, to which Time had already drawn attention, distorts the phenomenon observed on the surface of the specimen, thereby confirming the validity of our supposition and indicating, moreover, the impossibility of using the optical method of investigation, which is applicable only in the case of a “plane problem” (absence of transverse deformations). Therefore, in our work we adopted the metallographic method of investigation; and, in order to reveal the structure, we used etching by Fry’s method, which made it possible, in addition to the basic structure, to obtain an exact picture of the plastic deformations in any section of the specimen.

Fry’s method, which appeared in the literature in 1921¹⁸), was in subsequent years subjected to rigorous verification by a whole series of works, in which the best conditions for its application and use were found. Its essence is as follows:

A plastically deformed specimen is annealed for 30 min. at a temperature of 200° C.,¹³) after which it is cut at the place of interest, ground down to emery paper No. 00000 and, finally, etched with a reagent of the following composition (in the case of a macrosection):

Water — 100 cm³
concentrated hydrochloric acid — 120 cm³
cupric chloride (\(\mathrm{CuCl_2}\)) — 90 g.

Etching continues from 0.5 to several minutes, after which the residues of the reagent are washed off the section with hydrochloric acid¹ (to avoid deposition of copper on it); the section is washed with water and dried. Often, for greater clarity of the picture, the described etching is alternated with careful polishing with aluminum oxide on a cloth disk, with the etching repeated after polishing, and so on several times.

A half-hour anneal at \(t = 200^\circ\mathrm{C}\) serves to reveal, during the subsequent etching, the deformed zone; without it, etching with the indicated reagent proves ineffective, except in the case when deformation occurred at a temperature of \(150^\circ\)–\(300^\circ\). Owing to the necessity of annealing, everything that occurs after it is not revealed on the section, so that cutting the specimen after annealing, or even planing it on a machine, filing and grinding it, are not reflected in the picture revealed by etching, although they do produce certain plastic changes of form in the body of the specimen. Thus it becomes possible to exclude the influence of all deformations, unavoidable in preparing the section, on the final picture.

The method described is sufficiently sensitive and accurate; a deformation of 0.5% (calculated over a length of 10 mm) is already revealed on the section¹. Unfortunately, not all metals give a deformation pattern on etching. The supposition has been put forward, and confirmed by corresponding experiments, that all ferrous metals with a body-centered lattice, possessing great toughness and a sharp drop of the curve in the tensile diagram after the elastic limit, give positive results on etching².

The material taken for our experiments was soft, low-carbon strip iron (see Fig. No. 1), which, when etched by the method described, gives a clear picture. From it specimens were prepared with dimensions \(180 \times 50 \times 6\) mm, and in part with dimensions \(210 \times 50 \times 6\) mm. The experiments were carried out mainly on planing machines. For this purpose, partly a large planing machine with a separate motor was used, and partly a small planing machine driven from the transmission. Part of the experiments was also carried out on lathes.

The preliminary preparation of the specimens consisted of the following. In the middle, along each specimen, at equal distances, three holes \(7/16''\) in diameter were drilled; these served for fastening the specimens on the table of the planing machine. Then all the specimens, by means of bolts passed through the drilled holes, were clamped between two pieces of angle iron in such a way that the edges of the specimens rose above the vertical flanges of the angle iron by approximately 10 mm. The pieces of angle iron, with the specimens clamped between them, were pressed by their horizontal flanges to the table of the planing machine. In this way a firm fastening was achieved, which made it possible to remove chips of considerable size. The specimens thus mounted were first planed in order to obtain a surface parallel to the plane of the table; after this they were removed from the machine and subjected to annealing, the purpose of which was to normalize the material and eliminate stresses and deformations, both those previously present in the iron and those that had arisen during the manufacture of the specimens. For this purpose all the specimens were placed in a seg-

of an iron tube were filled with scale and tightly sealed with refractory clay. All this protected the iron from oxidation during the subsequent annealing. The annealing was carried out in a crucible furnace. The tube with the embedded specimens was heated for \(2^{1}/_{2}\)—3 hours to a temperature of \(950^\circ\)—\(1000^\circ\), after which it cooled slowly together with the furnace to room temperature over 12—15 hours. As a result of such annealing, all the specimens acquired a uniform, fairly homogeneous structure (see Fig. No. 1) and gave a full guarantee of the absence of any preliminary stresses. Only a very thin layer of scale formed on the surface of the specimens.

After annealing, the sides of the specimens intended for chip removal were filed with a personal file against a straightedge, and, for the removal of very thin chips, were polished with emery paper; this completed the preliminary preparation.

Drawing No. 4.

In the experiments the cutters used were made of high-speed steel with a theoretical profile, a straight cutting edge and a flat front face, with cutting angles \(\delta\) (see Drawing No. 4) of \(45^\circ\), \(55^\circ\), \(65^\circ\), \(75^\circ\), \(85^\circ\). The clearance angle for all cutters was \(\gamma = 8^\circ\)—\(10^\circ\). The cutting angles were checked at each setting of the cutter by means of a protractor, with an accuracy of up to \(1^\circ\).

For the experiment, each specimen was clamped between strips of angle iron and fastened on the table of the planing machine in such a way that one of the previously filed edges of the specimen, 180 mm long and 6 mm wide, was parallel to the plane of the table and was intended for chip removal. The parallelism of the specimen’s setting with the direction of motion of the table (or, what is the same, of the cutter) was carefully checked. The cutter was then set in such a way that its blade (edge) was strictly horizontal and exactly opposite the set specimen. The length of the cutter blade was always greater than the thickness of the specimen, and the cutter could therefore remove a chip with a width equal to the thickness of the specimen (6.20 mm); moreover, in view of the fact that the blade was set horizontally, the chip thickness was the same over its entire width. The depth of cut for this case was equal to the thickness of the layer being removed. In this way we carried out a more simplified process of chip separation, as compared with the usual one, just as was done by Prof. Zvorykin, Sokolov, and others. After setting the cutter, the cutting angle was checked with a protractor and the desired thickness of the layer to be removed (depth of cut) was set. The actual thickness of the removed layer obtained as a result of the experiment usually always differed from the set value, since, despite the power of the machine construction, the massiveness of the cutters, and the strength of the fastening of both them and the workpiece, it is impossible to avoid certain elastic deformations throughout the system. The actual chip thickness was measured subsequently, as will be indicated in due course.

After the indicated preliminary operations, the machine was started and the cutter removed the chip. The length of the stroke was usually 80—120 mm, after which either the table was suddenly stopped by switching off the motor, or, by the appropriate setting of the cams that shifted the belt, it received a reversal of the direction of motion. In both cases the removed chip remained connected with the main

of the specimen. After this the specimen was removed from the table of the machine and the next one was installed. Thin chips were removed on a small machine driven by a transmission, thick ones on a large machine. Both gave a cutting speed of about 8 meters per minute. From each side \((180 \times 6\ \text{mm})\) of the specimen only one chip was removed, since it was considered that a surface from which a chip had already been removed had undergone certain plastic deformations and therefore, in its mechanical properties, differed somewhat from the base material and was not suitable for the next experiment.

The specimen with the chip removed was subjected to measurement. The length of the path of the cutter \(l_o\) and the length of the chip obtained \(l_1\) were measured (with an accuracy up to \(0.5\ \text{mm}\)); this subsequently served for calculating the coefficient of longitudinal shrinkage \(\left(K_o = \frac{l_1}{l_o}\right)\). The length of the chip was measured with a flexible thread, always along the side adjacent to the cutter (i.e., along the outer side). After measurement, the chip, together with the adjoining part of the material, was cut off with a hacksaw and subjected to a half-hour anneal at a temperature of \(200^\circ\text{C}\); for this purpose the small chips were placed in a boiling naphthalene bath \((t = 218^\circ\text{C})\), while the large ones were annealed in an electric resistance furnace with an iron winding, of the “Hereus” type. After annealing, the specimen with the chip was cut lengthwise, ground, and etched, according to the procedure indicated. In the longitudinal cutting of specimens with thin chips there was reason to fear fracture of the chip. To prevent this, the specimen together with the chip was embedded in Wood’s metal (melting temperature \(160^\circ\)—\(180^\circ\)), so that the chip was surrounded on all sides by the alloy. Longitudinal cutting was carried out together with the alloy; then filing and grinding were performed on coarse grades of abrasive paper, and only after this was the section freed from the alloy surrounding it, for which purpose the whole piece was lowered into a molten bath of the same alloy. There were concerns that heating the specimen while freeing it from the alloy would lead to the revealing of deformations produced in the specimen by its cutting; however, experience showed that, either because of the short duration of the heating or because of the insignificance of the deformations, the concerns were not justified. It was also necessary to ensure that the temperature of the alloy did not exceed \(200\)—\(220^\circ\text{C}\), which could have adversely affected the clarity of the picture. After this the actual thickness of the layer being removed (the depth of cut) was determined; for this purpose the section was placed in a projection microscope on the ground glass of the microscope’s photo-camera at a magnification of 10–20 times, making it possible to carry out measurement with an accuracy up to \(0.025\ \text{mm}\). There, too, all the other measurements of interest to us were made, and in most cases the section was photographed.

The largest number of experiments was carried out for cutting angles \(a = 55^\circ\) and \(65^\circ\).

Deformations in the Chip.

1. Continuous chip.

In the order of exposition we shall first of all touch on the question of chip shrinkage. The circumstance that the chip removed is shorter than the path traversed by the cutter was noticed long ago, and by a number of experiments (the first were carried out by Prof. Time) it was established that co-

the coefficient of longitudinal contraction \(K_o=\dfrac{l_1}{l_o}\), generally speaking, changes with changes in the quality of the material, the cutting angle, and the thickness of the chip, and is always less than unity; nevertheless, despite the apparent simplicity of the phenomenon and the ease of studying it without touching on the internal mechanism of the contraction phenomenon, there is still no single opinion as to the dependence of the coefficient of contraction on the above-mentioned factors.

In Time’s opinion, the coefficient of longitudinal contraction decreases (i.e., the contraction increases) with an increase in the cutting angle; with the cutting angle held constant, it changes extremely little, increasing somewhat with an increase in the chip cross-section. Thus, for iron and for \(\alpha=45^\circ\), it fluctuates within the limits \(K_o=0.59—0.65\).

Sokolov’s experiments\(^{7}\), agreeing with this, state, in addition, that the greater the viscosity of the material being machined, the smaller the coefficient of longitudinal contraction.

Zadneprovsky\(^{9}\) establishes an increase in the contraction coefficient with an increase in the chip cross-section (for lead).

In the widely circulated book by Prof. Savvin\(^{22}\), however, we find: “The contraction of the chip is the greater (i.e., the coefficient of contraction is the smaller), the more viscous the material, the larger the cutting angle, and the thicker the layer being removed.” The last proposition, as we see, is directly opposite to those cited above.

In order to clarify the dependence of the coefficient of longitudinal contraction on all the causes mentioned above, we measured almost all the chips removed that were intended for the preparation of microsections and for etching. In addition, additional experiments were carried out, the purpose of which was to establish the dependence of the coefficient of longitudinal contraction on the depth of cut (the thickness of the layer removed, “\(\delta\)”) at all cutting angles. In these experiments chips of various thicknesses were removed, beginning with \(0.03\) mm. In order to eliminate the influence of the variable chip thickness at the beginning and at the end of cutting—since in the first case the tool enters deeper into the material, while in the second it usually tears off extra pieces of material—cuts were made with a hacksaw at a distance of 10 mm from the edges of the specimen, which limited the chip to a uniform thickness. Naturally, in calculating the contraction coefficient, the length of the tool travel \((l_o)\) was taken as the length between the hacksaw cuts. In view of the fact that, with very thin chips, direct measurement of the chip thickness could give significant errors, the thickness was determined by weighing the chip obtained. The weight was determined on analytical balances with an accuracy of up to 0.005 gram. Having determined the weight of each chip, and also having determined in advance the specific gravity of iron, it was possible to find the volume of material contained in the chip; knowing also the length of the tool pass (the length between the hacksaw cuts) and the width of the specimen, it was not difficult by calculation to obtain the exact value of the thickness of the layer removed (the depth of cut). The specific gravity of the iron was obtained by us as the average of several weighings, equal to 7.85; moreover, within two decimal places there was no difference between the specific gravity of the base material and the specific gravity of the chip. For some cases, in addition to the indicated measurements, the width of the chip obtained was measured. The results of the measurements carried out for the cutting angle \(\alpha=45^\circ\) are given in Table No. 4, for the angle \(\alpha=55^\circ\)—in Table No. 5, and for \(\alpha=65^\circ\)—in Table No. 6. At angles of 75° and 85° we,

as a rule, a whole chip was not obtained; it was usually divided into separate elements, which hindered measurement of the chip length.

Table No. 4.

Chip shrinkage at $\alpha = 45^\circ$.

No. in order	Specimen No.	Thickness of removed layer in mm, “$\delta$”	Length of cutter pass, mm, $l_0$	Length of removed chip, mm, $l_1$	Shrinkage coefficient $K_0=\frac{l_1}{l_0}$	Chip width before chip shrinkage, mm, $b_0$	Chip width after chip shrinkage, mm, $b_1$	NOTE
1	46	0,07	80,0	40	0,50	6,2	6,2	Depth of cut (thickness of the removed layer) was determined by the weighing method.
2	40	0,08	”	41,5	0,52	”	6,2	Depth of cut (thickness of the removed layer) was determined by the weighing method.
3	47	0,13	”	41,5	0,52	”	6,22	Depth of cut (thickness of the removed layer) was determined by the weighing method.
4	42	0,16	”	43,5	0,54	”	6,22	Depth of cut (thickness of the removed layer) was determined by the weighing method.
5	41	0,18	”	43,0	0,54	”	6,24	Depth of cut (thickness of the removed layer) was determined by the weighing method.
6	45	0,20	”	46,0	0,57	”	6,24	Depth of cut (thickness of the removed layer) was determined by the weighing method.
7	43	0,29	”	45,0	0,56	”	6,3	Depth of cut (thickness of the removed layer) was determined by the weighing method.
8	44	0,52	”	47,0	0,59	”	6,4	Depth of cut (thickness of the removed layer) was determined by the weighing method.
9	77	0,5	”	41,0	0,51	”	—
10	74	0,35	87,0	46,0	0,53	”	—
11	73	0,8	46,0	27,0	0,58	”	—
12	103	1,0	136,0	83,0	0,61	”	—
13	35	1,2	98,0	49,5	0,51	”	—
14	101	1,6	140,0	91,0	0,65	”	—
15	106	1,75	99,0	67,0	0,68	”	—
16	104	1,85	122,0	89,0	0,73	”	—
17	105	2,2	126,0	92,0	0,73	”	—
18	107	4,0	66,0	46,0	0,70	”	—

Table No. 5.

Chip shrinkage at \(\alpha = 55^\circ\).

Trial No.	Specimen No.	Thickness of removed layer, mm, \(\delta\)	Length of cutter travel, mm, \(l_0\)	Length of chip removed, \(l_1\)	Shrinkage coefficient \(K_0=\frac{l_1}{l_0}\)	Width before chip removal \(b_0\)	Width of chip removed \(b_1\)	NOTE
1	13	0.03	194	67	0.35	6.2	6.22	Thickness of the removed layer, \(\delta\), determined by weighing.
2	20	0.04	”	73.5	0.38	”	6.3	Thickness of the removed layer, \(\delta\), determined by weighing.
3	14	0.07	”	78.5	0.40	”	6.34	Thickness of the removed layer, \(\delta\), determined by weighing.
4	30	0.08	”	79.5	0.41	”	6.35	Thickness of the removed layer, \(\delta\), determined by weighing.
5	146	0.13	123	55.0	0.45	”	—
6	29	0.16	88	40	0.45	”	6.45
7	15	0.2	194	93	0.48	”	6.4	Thickness of the removed layer, \(\delta\), determined by weighing.
8	28	0.31	88	40	0.45	”	6.5	Thickness of the removed layer, \(\delta\), determined by weighing.
9	23	0.44	194	93	0.48	”	—	Thickness of the removed layer, \(\delta\), determined by weighing.
10	12	0.45	”	90	0.47	”	6.7	Thickness of the removed layer, \(\delta\), determined by weighing.
11	22	0.70	88	47	0.54	”	6.9	Thickness of the removed layer, \(\delta\), determined by weighing.
12	148	0.25	37.5	16.5	0.44	”	6.25
13	147	0.30	46.0	21.5	0.47	”	6.4
14	151	0.60	52	24	0.46	”	6.47
15	149	0.78	46	22	0.48	”	6.62
16	26	1.1	124	66.5	0.54	”	--
17	25	1.35	142	83.5	0.59	”	—
18	21	1.45	156	87.0	0.56	”	.—
19	24	1.8	143	82.5	0.58	”	—
20	108	2.0	71	40	0.56	”	—
21	20	2.0	100	53	0.53	”	—
22	1	2.1	109.5	58.5	0.54	”	—
23	19	2.3	120	72	0.6	”	—
24	166	2.5	101	67.5	0.67	”	—
25	2	2.6	78	47.5	0.61	”	—
26	109	2.65	139	80	0.58	”	--	Second layer.
27	3	3.2	102.5	67.5	0.66	”	—
28	23	4.1	65	44	0.68	”	—
29	79	0.4	92	43	0.47	”	—

Table No. 6.

Shrinkage at \(\alpha = 65^\circ\).

No. in order	Sample No.	“\(\delta\)”	\(l_0\)	\(l_1\)	\(K_0\)	\(b_0\)	\(b_1\)	NOTE
1	48	0,1	99	35	0,35	6,2	6,23
2	52	0,1	”	33	0,33	”	6,3
3	53	0,1	”	26	0,26	”	6,21	Divides into elements.
4	51	0,11	”	30	0,3	”	6,3
5	50	0,2	”	32	0,32	”	6,75
6	49	0,3	”	26	0,26	”	6,5—7,5	Divides into elements.
7	154	0,2	71,5	27,0	0,38	”	6,40
8	156	0,4	70,5	21,5	0,30	”	7,25	Divides into elements.
9	75	0,5	45	15	0,33	”	—
10	157	0,5	68	15,5	0,23	”	6,9	Divides into elements.
11	155	0,55	37,5	12,5	0,32	”	7,35	” ” ”
12	158	0,55	73,5	21,5	0,29	”	7,2	” ” ”
13	38	0,6	156	42	0,27	”	—	” ” ”
14	110	1,6	123	49	0,40	”	—
15	13	1,85	111	51	0,46	”	—
16	111	2,1	112	57	0,51	”	--
17	18	2,15	113,5	59	0,52	”	—
18	114	2,7	90	58	0,64	”	--
19	115	3,7	93	73	0,78	”	—
20	17	4,3	66	41	0,62	”	—
21	90	0,95	60	24	0,40	”	—

Note: For Nos. 1, 2, 3, 4, 5, 6, “\(\delta\)” was determined by the weighing method.

Using the data of Table No. 4, the graph in Drawing No. 5 was constructed, where the shrinkage coefficient \(K_0 = \frac{l_1}{l_0}\) is plotted on the ordinate axis, and the thickness of the removed layer (depth of cut) “\(\delta\)” in millimeters is plotted on the abscissa axis. In exactly the same way, from Table No. 5 the graph in Drawing No. 9 was constructed for \(\alpha = 55^\circ\), and from the data of Table No. 6—the graph in Drawing No. 13, for \(\alpha = 65^\circ\).

From examination of the tables and the corresponding graphs, one can first of all discern a quite definite tendency toward an increase in the shrinkage coefficient with an increase in the thickness of the layer being removed.

Thickness of the layer removed “\(\delta\)” in mm.

Drawing No. 5.

Thickness of the layer removed “\(\delta\)” in mm.

Drawing No. 6.

Angle \(\beta_2\)

Thickness of the layer removed “\(\delta\)” in mm.

Drawing No. 7.

Angle \(\beta_1\)

Thickness of the layer removed “\(\delta\)” in mm.

Drawing No. 8.

The increase in the shrinkage coefficient is quite clearly expressed by the graphs, and for small values of “\(\delta\)” it proceeds more rapidly than for large ones. At a cutting angle \(\alpha = 65^\circ\), we observe a certain fluctuation in the values of the coefficient

shrinkage for small values of “$\delta$.” This fluctuation is not difficult to explain by the fact that, at $\alpha = 65^\circ$, some of the thin chips obtained by us consisted of elements linked with one another (joints); such chips

Thickness of the layer removed, “$\delta$,” in mm.
Fig. No. 9.

Thickness of the layer removed, “$\delta$,” in mm.
Fig. No. 10.

Thickness of the layer removed, “$\delta$,” in mm.
Fig. No. 11.

Thickness of the layer removed, “$\delta$,” in mm.
Fig. No. 12.

had very great shrinkage, a very small value of $K_0$. Another part of the thin chips, however, had a completely continuous structure, and the shrinkage coefficients for them had a higher value.

One may note the error of Prof. Savvin\(^{22}\), who assumes that shrinkage increases with increasing thickness of the layer being removed. We see a relationship of a completely opposite character. At the same time, the shrinkage coefficient varies within much wider limits than

\[ K_0=\frac{l_1}{l_0} \]

Thickness of the removed layer, “\(\delta\)” in mm.

Fig. No. 13.

\[ K_1=\frac{\delta}{\delta_1} \]

Thickness of the removed layer, “\(\delta\)” in mm.

Fig. No. 14.

Angle \(\beta_2\)

Thickness of the removed layer, “\(\delta\)” in mm.

Fig. No. 15.

Angle \(\beta_1\)

Thickness of the removed layer, “\(\delta\)” in mm.

Fig. No. 16.

Prof. Time assumed this for iron, and this is especially noticeable at \(\alpha = 65^\circ\), where \(K^\circ\) ranges from 0.23 to 0.78. At \(\alpha = 55^\circ\) we have a variation from 0.35 to 0.68, and at \(\alpha = 45^\circ\) from 0.5 to 0.73.

From a comparison of the graphs with one another it is evident that, with an increase in the cutting angle, the shrinkage coefficient falls, and does so quite appreciably, especially at small values of “\(\delta\).”

We shall later have to return once more to the phenomenon of chip shrinkage, and there we shall cite some additional experiments in this regard and attempt to draw certain conclusions; for the present we turn to the study of the structure of the chip.

For this purpose, by the method already described, polished sections were prepared from all the chips, except those that served for calculating shrinkage by the weighing method. When etching the polished sections by Fry’s method, attention is first of all drawn to the circumstance that the etching process is sharply divided into two stages. During the first period, lasting up to \(1\)—\(1\frac{1}{2}\) min., the etching reagent reveals the macrostructure of the specimen, and sharply separates the region of the strongly deformed chip from the rest of the metal of the specimen, which is comparatively less deformed than the chip. If the etching process is stopped at this point, we can obtain a clear representation of the structure of the chip, judge the boundary of the transition of the metal into the chip, and judge the deformations taking place in the latter. Etching interrupted in this way gives no indications of the deformations that have occurred in the main mass of the metal. Here there is observed only a certain uniform darkening of the polished section, with the revelation of the macrostructure, inclusions, blowholes, and so forth. If etching is continued further, then after some time, sometimes quite a long time, the deformed region in the main mass of the metal is revealed. But in this case the metal of the chip is so strongly over-etched that it appears as a very dark, uniformly colored field, with an absence of any details that would make it possible to draw one conclusion or another.

For these reasons, for all polished sections we had to stop the etching process in the first stage, make the necessary observations, measurements, and photographs, and then, continuing the etching and thus carrying out the second stage of the process, obtain a picture of the deformation of the main metal.

Let us consider the results of the first stage of etching. For chips removed by cutters with cutting angles \(\alpha = 45^\circ\) and \(\alpha = 55^\circ\), no qualitative difference was obtained in the pictures produced; with a further increase in the cutting angle, however, certain changes arose. Therefore we shall begin with the angle \(\alpha = 55^\circ\), considering along with it also the case of cutting at \(\alpha = 45^\circ\), and then proceed to the larger angles.

When examining the photographs (see Figs. Nos. 2, 3, 4, and 5) relating to the cutting angle \(\alpha = 55^\circ\) and arranged in order of increasing thickness of the removed layer (depth of cut), a sharp boundary between the chip and the base metal is striking. For thin chips this boundary is almost exactly rectilinear; in thick chips it passes into a broken line consisting of two straight lines, one of which goes from the tip of the cutter in the direction of its motion, while the other intersects the first at a certain angle and bounds it. In thin chips the boundary is seen more clearly and is more easily revealed by etching than in thick ones, where it can be revealed only after several alternating polishings and etchings, and the possibility of over-etching is not excluded.

and finally lose the sharpness of the image. A schematic representation of these boundaries is given by us as the straight lines ABC in drawing (17), where other notations needed by us later have also been placed.

Then it is very clearly seen that the whole chip is, as it were, divided into thin, frequent layers parallel to one another, extending through almost the entire thickness of the chip and inclined at a certain angle to the rectilinear boundary between the chip and the parent metal. It is not difficult to notice that the boundary of the chip is the geometric locus of the end points of these layers, and is actually formed by them. At high magnification (100–150 times) one can distinguish that the layers consist of extremely strongly deformed grains of metal, elongated in the direction of the layers; therefore there is no doubt that the direction of the layers gives us the direction of sliding of the metal grains in the process of chip formation.

Drawing No. 17.

Thus, in the present case, the process of chip formation consists of successive shears of the metal in the direction of the angle \(\beta_2\) (drawing 17). From consideration of Figs. Nos. 3 and 4 it may be concluded that the shear lines inside the chip arise at the place where the machined surface passes into the chip (point A, drawing No. 17) and propagate in the direction opposite to the motion of the cutter and at the angle \(\beta_2\) to it. The boundary of their propagation is the side of the angle \(\beta_1\). It is then evident that, at the surface of the cutter, the shear lines lose their rectilinear character, bending in the direction opposite to the motion of the chip along the cutter and, at the limit, sometimes assuming a direction parallel to the front face of the cutter; in thin chips the bending of the layers is not always noticeable, while in the case of great thickness it is always present. This phenomenon is especially clearly visible in Fig. No. 6. From this same photograph, as well as from others, it is also seen that the surface of the chip facing the cutter is not even and smooth, but is covered with considerable serrations, with their sharp ends directed toward the motion of the chip along the cutter. In the case of a small layer thickness, an almost complete absence of serrations is sometimes observed, but for the most part, even if only in an incipient state, they can nevertheless be detected (Fig. No. 2).

On the surface from which the chip has already been removed (we shall hereafter call it the under-cutter surface), we likewise have the presence of serrations, with their points directed in the direction opposite to the motion of the cutter (see Figs. Nos. 3, 4, and 6); they are similar to those found on the chip, but have a more regular, uniform form and are located at almost equal distances from one another.

The photographs (see Figs. Nos. 7, 8, 9, 10, 11) give an idea of the process at a cutting angle \(\alpha = 65^\circ\) and various values of “\(\delta\).” With small “\(\delta\)” we obtained either chips consisting of separate elements (Figs. 7 and 8), or of elements joined together but showing a distinctly articulated structure; cases of completely continuous chips were also not infrequent. With an increase in the thickness of the layer being removed, the chip assumes a definitely continuous structure (Figs. Nos. 9, 10, and 11). In all these cases we again clearly distinguish both the boundary separating the chip from the main body of the specimen and the directions of shears in the chip itself;

as also at smaller cutting angles, here the shears in the chip, on approaching the plane of the tool, lose their rectilinear character and bend in the direction opposite to the motion of the chip along the tool (Fig. No. 9); on the side of the chip adjacent to the tool and on the surface under the tool we also have burrs, which here are especially pronounced and, at large thicknesses of the layer being removed, sometimes attain a very considerable size.

Thus, in essence there is no difference from what we had earlier, except that thin chips sometimes divide into separate elements, and drawing No. 17 is equally suitable for a schematic representation of the process. From examination of the photographs in Figs. Nos. 7 and 8 it may be concluded that the division of the chip into elements occurs for the most part by no means in the direction of the shear lines (angle $\beta_2$); rather, one may suppose that the boundary of an element is formed in the direction of the side of angle $\beta_1$, and that separation of the element takes place along the line bounding the chip. From Fig. No. 7 it is evident that the shear lines in the elements that have already separated make a very noticeable angle with the lines of division of the elements. We shall dwell in more detail later on the chip consisting of separate elements.

A scheme of the process of chip formation was given by Prof. Time and served as the basis for subsequent work in this field. Zvorykin, Afanasiev, Pogorzhelsky, and others, basing themselves on his conclusions, tried in one way or another to coordinate the results of their work with them and to connect them with them. Usachev, observing cases of chip fracture, introduced certain changes into Prof. Time’s theory; however, it must be said that this was not reflected in later works, despite its substantiality. Many theoretical investigations and attempts at analytical treatment of the question of chip formation are based entirely on the unconditional acceptance of Time’s theory as reliable; and even in the most recent and sufficiently extensive experimental and theoretical works we find references to it.^10), ^16), ^23), ^24).

Time assumed that in the case of cutting a ductile material all chips, irrespective of the cutting angle and the material, owe their formation to a process of successive sliding of the metal, in a direction quite definite and strictly constant for each cutting angle. The sliding in this case propagates rectilinearly from the tip of the tool to the point of transition of the machined surface into the chip (point A in drawing No. 17), and each successive line of sliding constitutes the boundary of the propagation of deformations in the specimen.

From examination of our photographs we reached different conclusions. Considering for the present only the case of a continuous chip, it may be said that Time’s conception is far from corresponding to reality. We have no cleavage in the direction from the tip of the tool to the point of transition of the machined surface into the chip. In this direction there is only the boundary of the propagation of deformations in the chip, but even it has a rectilinear character only in thin chips, turning, at more considerable values of $\delta$, into a broken line. The shear lines forming the chip are always directed at a certain angle to the boundary mentioned, never coinciding with it, and they propagate not forward from the tool, but, on the contrary, backward toward the tool, arising at the point of transition of the machined surface into the chip; moreover, sliding occurs simultaneously in many planes.

When examining the sections obtained, we were interested in the magnitude of \(\beta_1\) and \(\beta_2\) (Fig. No. 17), the angles of inclination of the boundary of the deformations in the chip, and the directions of the shear lines relative to the line of motion of the cutter. The studies available on these questions, identifying both angles according to Time’s conclusions, arrive at the conclusion that the shear angle \(\beta_1=\beta_2\) is strictly constant for each cutting angle and, when the cutting angle changes, varies within very narrow limits, decreasing as the latter increases. An exception is the opinion of Hankins \(^{24,25}\), who simplifies the question still further by asserting that, in general, for all materials and all cutting angles, \(\beta_1=\beta_2=26^\circ=\mathrm{const}\). Finally, Rosenhain and Styrney \(^{12}\) suppose that \(\beta_1=30^\circ\) in the case of machining steel and brass.

We measured the angles of interest to us in the following way. On the image of the section, enlarged 10–20 times on the ground glass of the photographic camera of the projection microscope, a sheet of very transparent tracing paper was placed, on which an angle of \(90^\circ\), divided into individual degree divisions, had been drawn. In this way the magnitude of the angles \(\beta_1\) and \(\beta_2\) was estimated; with a carefully etched section this could be done with an accuracy of up to \(2^\circ\). The results obtained are collected in tables: for the cutting angle \(\alpha=45^\circ\), Table No. 7; for \(\alpha=55^\circ\), Table No. 8; and for \(\alpha=65^\circ\), Table No. 9. In these same tables are included certain quantities needed below, obtained by measurement: \(s\)—the horizontal distance, in mm, from the tip of the cutter to the point where the machined surface passes into the chip (see Fig. No. 17); \(\delta_1\)—the thickness of the removed chip; \(h\)—the height of the serrations on the underside surface; \(s_1\)—the average distance between serrations. In addition to these quantities, in these same tables we have included the computed value of the coefficient

\[ k_1=\frac{\delta}{\delta_1}, \]

which is thus the reciprocal of the chip-thickening coefficient and will always be less than unity, since the removed chip, as a result of longitudinal shrinkage, undergoes transverse thickening and proves to be thicker than the layer removed: \(\delta_1>\delta\); \(\frac{\delta}{\delta_1}<1\).

Here, too, for comparison, we have included the coefficients of longitudinal shrinkage \(k^\circ\) obtained earlier.

Table No. 7. \(\alpha=45^\circ\).

No. in order	Specimen No.	\(\delta\), mm	\(\delta_1\), mm	\(S\), mm	\(\dfrac{h}{S_1}\)	\(\beta_1\)	\(\beta_2\)	\(k_1=\dfrac{\delta}{\delta_1}\)	\(k_0=\dfrac{l_1}{l_0}\)	Note
1	76	0,18	0,27	0,35	—	\(32^\circ\)	\(48^\circ\)	0,67	—
2	74	0,35	0,6	1,2	—	\(32^\circ\)—\(34^\circ\)	\(50^\circ\)	0,64	0,65
3	77	0,5	1,1	2	\(\dfrac{0,1}{0,5}\)	\(26^\circ\)	\(45^\circ\)	0,45	0,51
4	73	0,8	1,6	3	\(\dfrac{0,1}{0,5}\)	\(26^\circ\)—\(28^\circ\)	\(44^\circ\)	0,5	0,58
5	36	1,0	—	2,8	—	\(28^\circ\)	\(46^\circ\)	—	0,48

No. in order	Sample No.	$\delta$ m/m.	$\delta_1$ m/m.	$S$ m/m.	$\dfrac{h}{S_1}$	$\beta_1$	$\beta_2$	$k_1=\dfrac{\delta}{\delta_1}$	$k_0=\dfrac{l_1}{l_0}$	Note
6	103	1,0	2,0	3,5	$\dfrac{0,1}{0,7}$	28°	46°—48°	0,5	0,61
7	35	1,2	2,2	3,6	$\dfrac{0,15}{0,7}$	31°	50°	0,55	0,6
8	101	1,6	2,5	3,7	$\dfrac{0,12}{0,8}$	30°—32°	50°—52°	0,64	0,65
9	106	1,75	3,2	5,5	$\dfrac{0,18}{1,2}$	32°—34°	56°—58°	0,55	0,68
10	104	1,85	2,8	4,0	$\dfrac{0,1}{0,6}$	34°	—	0,66	0,73
11	105	2,2	3,3	5,5	$\dfrac{0,15}{0,8}$	34°	54°	0,67	0,73
12	107	4,0	6	11	$\dfrac{0,2}{0,8}$	42°	60°	0,67	0,70

Table No. 8; $\alpha = 55^\circ$.

No. in order	Sample No.	“$\delta$”	“$\delta_1$”	“$s$”	$\dfrac{h}{S_1}$	$\beta_1$	$\beta_2$	$k_1=\dfrac{\delta}{\delta_1}$	$k_0=\dfrac{l_1}{l_0}$	Note
1	146	0,13	0,29	0,4	—	26°—28°	40°—42°	0,45	0,45	Chip of the second layer.
2	148	0,25	0,6	0,9	$\dfrac{0,1}{0,8}$	22°—24°	40°—42°	0,42	0,44	Chip of the second layer.
3	147	0,3	0,7	1,0	—	24°—26°	42°	0,43	0,47
4	71	0,32	0,65	1,0	—	32°—34°	50°	0,49	—	$\beta_2$ not quite clear.
5	9	0,4	0,95	1,3	—	24°—26°	42°	0,43	—
6	79	0,4	—	2,0	—	28°—30°	48°	0,47	—
7	150	0,6	1,4	1,9	$\dfrac{0,1}{0,8}$	26°	44°	0,43	—
8	151	0,6	1,4	1,8	$\dfrac{0,1}{0,7}$	26°—28°	44°	0,43	0,46
9	149	0,78	1,6	2,5	$\dfrac{0,15}{1,0}$	30°	52°—54°	0,49	0,48	Large values of $\beta_1$ and $\beta_2$ are explained by the abnormal structure of the chip.
10	153	0,8	2,0	2,8	$\dfrac{0,2}{1,2}$	24°	42°	0,4	—	Large values of $\beta_1$ and $\beta_2$ are explained by the abnormal structure of the chip.
11	97	0,8	2,15	2,9	$\dfrac{0,2}{1,3}$	22°—24°	40°	0,37	—
12	26	1,1	—	3,0	—	26°—28°	48°	—	0,53
13	86	1,15	1,8	3,2	$\dfrac{0,1}{0,7}$	30°—32°	52°	0,63	—

No. in order	Specimen No.	“δ”	“δ₁”	“S”	\(h/S_1\)	\(\delta_1\)	\(\beta_2\)	\(\kappa_1=\delta/\delta_1\)	\(\kappa_0=l_1/l_0\)	Note
14	4	1,2	2,6	2,9	—	28°	42°	0,46	—	Beginning of the chip
15	25	1,35	2,1	3,6	\(0,2/1,2\)	28°—30°	44°	0,64	0,57
16	21	1,45	2,3	3,8	\(0,2/1,2\)	28°—30°	46°—48°	0,63	0,56
17	24	1,8	2,9	5,0	\(0,3/1,4\)	30°	48°—50°	0,62	0,57
18	20	2,0	3,0	4,5	\(0,25/1,3\)	32°—34°	50°	0,66	0,53
19	108	2,0	3,4	4,7	\(0,2/1,0\)	32°	50°—52°	0,6	0,56
20	1	2,1	3,3	4,8	\(0,15/1,2\)	30°	48°—50°	0,64	0,53
21	19	2,3	3,3	5,2	\(0,25/1,3\)	34°	—	0,7	0,6	Chip of the second layer.
22	166	2,5	3,75	5,5	\(0,4/1,4\)	36°	50°	0,67	0,67
23	2	2,6	3,75	5,8	—	32°	50°	0,69	0,61
24	109	2,65	4,5	6,5	\(0,25/1,5\)	36°	54°	0,59	0,58	Chip of the second layer.
25	3	3,2	4,6	5,5	\(0,2/1,2\)	40°—42°	56°	0,67	0,66
26	23	4,1	5,12	10,8	\(0,3/1,5\)	42°	64°	0,8	0,68

Table No. 9; \(\alpha=65^\circ\).

No. in order	Specimen No.	“δ”	“δ₁”	“S”	\(h/S_1\)	\(\beta_1\)	\(\beta_2\)	\(\kappa_1=\delta/\delta_1\)	\(\kappa_0=l_1/l_2\)	Note
1	154	0,2	0,55	1,2	\(0,1/0,7\)	—	42°	0,17	0,38
2	118	0,25	0,9	1,4	\(0,15/1,5\)	14°—16°	30°	0,28	—
3	87	0,26	—	1,8	—	16°	27°	—	—	The chip divides into elements.
4	159	0,3	—	3,3	—	12°—16°	34°	—	—	”
5	156	0,4	—	2	—	18°	38°—40°	—	—	”
6	75	0,5	1,5	2,8	—	18°—20°	34°	0,33	0,33
7	157	0,5	—	2,2	—	20°	44°—46°	—	0,23	Divides into elements.
8	158	0,55	—	1,6	—	26°—28°	50°	—	0,29	”

No. in order	Specimen No.	“$\delta$”	“$\delta_1$”	“$S$”	$\dfrac{h}{S_1}$	$\beta_1$	$\beta_2$	$K_1=\dfrac{e}{\delta_1}$	$K_0=\dfrac{l_1}{l_2}$	Note
9	164	0,55	1,0	1,7	$\dfrac{0,1}{1,5}$	26°—28°	44°—46°	0,55	—
10	155	0,55	—	1,9	—	—	40°	—	0,32	”
11	38	0,6	—	3,5	—	18°—16°	34°—36°	—	0,27	”
12	67	0,6	—	3,8	—	18°—20°	36°—38°	—	—	”
13	69	0,87	2,3	3,5	—	24°—26°	40°	0,38	—
14	90	0,95	2,3	2,9	—	26°	44°	0,41	0,40
15	31	1,6	—	5,6	$\dfrac{0,4}{3,2}$	26°	44°—46°	—	—
16	110	1,6	3,2	5,0	$\dfrac{0,4}{2}$	26°—28°	48°	0,5	0,4
17	113	1,85	3,4	5,5	$\dfrac{0,5}{2,5}$	30°	48°—50°	0,54	0,46	Chip broken.
18	111	2,1	3,5	6,2	$\dfrac{0,7}{3}$	28°	46°—48°	0,6	0,51	$\beta_2$ not entirely clear.
19	137	2,1	2,9	4,5	$\dfrac{0,3}{1,7}$	32°—36°	54°	0,72	—
20	18	2,15	3,9	5,8	$\dfrac{0,5}{3}$	28°	48°	0,55	0,52
21	114	2,7	4,5	6,5	$\dfrac{0,5}{2}$	34°—36°	52°	0,6	0,64
22	115	3,7	5,3	9,0	$\dfrac{0,6}{2,2}$	34°—36°	52°—54°	0,7	0,78
23	165	4,0	—	7,5	$\dfrac{0.15}{0,9}$	36°—38°	52°—54°	—	—	Beginning of chip
24	17	4,3	6,4	9,0	$\dfrac{0,7}{2,5}$	42°	58°—60°	0,67	0,62

In these tables, as in the following ones, all the measurement results obtained by us, whatever they may be, are included without exception.

To clarify the character of the change in the quantities $K_1$, $\beta_1$, and $\beta_2$ with the change in “$\delta$,” graphs were constructed: from the data of Table No. 7—drawings Nos. 6–8; from Table No. 8—drawings Nos. 10–12; and from Table No. 9—drawings Nos. 14–16.

From the tables presented (Nos. 7–9), as well as from the corresponding graphs, it is evident that, as the thickness of the layer being removed increases, the angles $\beta_1$ and $\beta_2$ steadily increase; at the same time, there is a very considerable difference between their smallest and largest values. The greater the magnitude of the cutting angle, the smaller the absolute value of the angles $\beta_1$ and $\beta_2$. The difference between these angles is approximately constant and approximately equal to:

$$ \beta_2-\beta_1=[18^\circ—20^\circ]. $$

In order to verify the significant increase of the angles $\beta_1$ and $\beta_2$ with an increase in “$\delta$,” it is only necessary to compare with one another the photographs previously presented by us, figs. Nos. 2–5 and Nos. 7–11. For greater

for clarity they are arranged in order of increasing thickness of the layer being removed, “$\delta$,” and for each of them a diagram has been drawn up on which the angles $\beta_1$ and $\beta_2$ of interest to us are marked (see Figs. 2a–5a, 8a–11a). The photographs clearly show an increase of the angles with increasing “$\delta$.”

From Tables Nos. 7–9 it is evident that with increasing “$\delta$” the quantity “$s$” increases, but at the same time the second increases more slowly than the first, and thus the ratio

\[ \frac{\delta}{s} \]

increases. According to Prof. Time’s scheme (Fig. 2)

\[ \frac{\delta}{s} = \operatorname{tg}\,\beta_1 . \]

Thus, even if one accepts the scheme of Fig. 2, it is nevertheless possible to draw the conclusion that, with an increase in the thickness of the layer, the angle $\beta_1$ must inevitably increase. According to our scheme (Fig. 17)

\[ \operatorname{tg}\,\beta_1 = \frac{\delta}{s - BC}. \]

Here, for thin chips (for small “$\delta$”), $BC \cong 0$; for thick chips, however, it has some finite value that increases with increasing “$\delta$”; therefore, according to this scheme, the angle $\beta_1$ will grow more rapidly than according to the scheme of Fig. 2. Thus, even without resorting to measurement of $\beta_1$, it would have been possible to predict in advance the character of its change with increasing thickness of the layer being removed.

From the tables and graphs it is evident that $K_1$ (the quantity inverse to the chip transverse-thickening coefficient) also increases with increasing “$\delta$.”

Thus, a parallel increase is observed in three quantities: $\beta_1$, $\beta_2$, and $K_1$ with increasing “$\delta$.” The values of $\beta_1$, $\beta_2$, $K_1$, as we see, do not fall exactly on a curve, but have more or less noticeable deviations, which is quite natural in view of the complexity of the process; nevertheless, despite the deviations, sometimes fairly considerable ones, the general tendency of these quantities to increase with increasing “$\delta$” is still clear. The deviations are especially noticeable at small “$\delta$”; approximately at $\delta > 1$ mm they are very insignificant.

From the tables (Nos. 7–9) and also from the graphs it may be observed that, in the case of a deviation of some value of $K_1$ from the curve of mean values $[K_1 = f(\delta)]$, the values $\beta_1$ and $\beta_2$ for this same chip almost always give a deviation of the same sign from their curves $[\beta_1 = f'(\delta)$ and $\beta_2 = f''(\delta)]$. This is especially clearly noticeable if one takes several specimens with identical values of “$\delta$.” Thus, in Table No. 8, for specimens Nos. 9 and 79; Nos. 149 and 153; Nos. 86 and 4; Nos. 147 and 71; and in Table No. 9 for specimens Nos. 111 and 137, Nos. 75 and 164. In all these cases, with increasing $K_1$ the values of $\beta_1$ and $\beta_2$ increase at constant “$\delta$,” and the thought involuntarily arises of a mutual connection between the quantities $K_1$, $\beta_1$, and $\beta_2$. Such a connection must undoubtedly exist, since if, at a constant value of “$\delta$,” we for some reason obtain an increase of “$\delta_1$” (i.e., a decrease of $K_1$), then this will entail an increase of “$s$,” i.e., a decrease of $\beta_1$.

This is not difficult to show analytically. If, for simplicity, we take the case of a thin chip, then we shall have (see Fig. No. 18):

\[ \delta_1 = AC \cdot \sin(\alpha + \beta_1) \]
\[ \delta = AC \cdot \sin \beta_1 \]
\[ K_1 = \frac{\delta}{\delta_1} = \frac{\sin \beta_1}{\sin(\alpha + \beta_1)} . \]

For \(\alpha = \mathrm{const}\), an increase of \(K_1 = \dfrac{\delta}{\delta_1}\) is associated with an increase of \(\beta_1\), and conversely. Experience has shown, however, that with a change in \(\beta_1\), \(\beta_2\) also changes. The change in the magnitude \(K_1\), of course, by virtue of the nature of chip deformation, is associated with a change in the coefficient of longitudinal shrinkage \(K_0\).

Drawing No. 18.

In general, the visible change in the form of the metal during chip formation proceeds in three directions: in the direction of longitudinal shrinkage \((l_1 < l_0)\), transverse thickening \((\delta < \delta_1)\), and transverse widening \((b_1 > b_0)\). If there were no phenomenon of bending (curling) of the chip and the deformation in all directions proceeded absolutely uniformly, then, assuming the approximate constancy of the specific weight of the metal, we would have the equality:

\[ l_0 \cdot \delta \cdot b_0 = l_1 \cdot \delta_1 \cdot b_1 \]

and since:

\[ \frac{l_1}{l_0} = K_0 \]

and

\[ \frac{\delta}{\delta_1} = K_1, \]

then:

\[ K_1 \cdot b_0 = K_0 \cdot b_1 . \]

Here, according to the preceding:

\[ \begin{aligned} l_0 &— \text{length of the removed layer} \\ l_1 &— \text{” of the resulting chip} \\ \delta &— \text{thickness of the removed layer} \\ \delta_1 &— \text{” of the removed chip} \\ b_0 &— \text{width of the removed layer} \\ b_1 &— \text{” of the removed chip.} \end{aligned} \]

Experience shows that always \(b_1 \gtrless b_0\) (see Tables Nos. 4–6), and therefore it must always be:

\[ K_1 \gtrless K_0, \]

however, in the tables we do not everywhere have confirmation of this relation. Thus Tables Nos. 8 and 9 for angles \(\alpha = 55^\circ\) and \(\alpha = 65^\circ\) in the majority of cases confirm it, while Table No. 7 gives, for \(\alpha = 45^\circ\), the opposite relation.

The reasons for this are as follows:

First of all, the length of the removed chip, when it is bent, measured along the side of its contact with the cutter, is always greater than along the opposite side (the inner side of the coil).

For our calculation of the coefficient of longitudinal shrinkage

\[ K_0=\frac{l_1}{l_0}, \]

the quantity \(l_1\) was always measured along the side of contact with the cutter, as is usually done. Thus the value of \(K_0\) obtained by us was somewhat greater than the true one; to obtain the latter, one should take as the length of the removed chip (\(l_1\)) the mean value between the lengths of the two sides of the chip. Further, as the cutting angle \(\alpha\) increases, the transverse widening of the chip increases (see Tables Nos. 4–6).

Owing to the first of the indicated causes we could obtain \(K_1 < K_0\); the second, acting in the opposite direction, gave, for large angles \(\alpha\), \(K_1 > K_0\).

Returning to the mutual relation between the changes in the quantities \(\beta_1\), \(\beta_2\), \(K_1\), and \(K_0\), we shall try to prove it by experiment. On examining the photographs in connection with drawings Nos. 17 and 18, one gets the impression that the magnitude of the angle \(\beta_1\) depends first of all on the relative thickening of the chip as a result of its longitudinal shrinkage, and hence on the quantities \(K_1\) and \(K_0\). Assuming that any given amount of shrinkage, and therefore of thickening, is produced as the result of the complex of external forces acting on the chip and depends on their magnitude and direction, we attempted deliberately to change them, which, with \(s\) and \(\alpha\) constant, should have changed in one way or another the quantities \(\beta_1\), \(\beta_2\), \(K_1\), and \(K_0\).

According to Prof. Savvin\(^{26}\), the use of lubricant in cutting, in the best case when linseed oil is used, reduced the cutting force by 30%, which undoubtedly resulted from a reduction of the frictional force between the chip and the cutter, i.e. one of the forces acting on the chip. Even if all the other forces retain their magnitude and direction, a considerable change nevertheless occurs in the magnitude and direction of the resultant. This circumstance was used by us and was intended to serve for clarifying the assumptions stated above.

Experiments with lubrication were carried out at a cutting angle \(\alpha = 55^\circ\). Boiled linseed oil was taken as the lubricating substance; because of its high viscosity it was expected to produce a considerable effect. (In practice it is not used for lubricating cutting tools, since, thickening, it fills the pores and channels in the machines. In our case it was quite suitable, in view of the short duration of the experiment.) The experiments were carried out in such a way that the cutting operation took place in an oil bath fixed on the table of the planer, into which both the piece of iron from which the chip was removed and the cutting part of the cutter were immersed. With respect to the setup, measurements, and preparation of sections, the procedure and methods remained as before. Part of the experiments was carried out specially to establish the dependence of the coefficient of longitudinal shrinkage on the thickness \(s\) of the chip; part, to obtain the quantities \(K_1\), \(\beta_1\), and \(\beta_2\). In the first case the value of \(s\) was determined by weighing the removed chips. The results are collected in Tables Nos. 10 and 11.

Table No. 10; $\alpha = 55^\circ$.

With lubrication.

No. in order	Sample No.	$\delta$	$l_0$	$l_1$	$K_0=\dfrac{l_1}{l_0}$	$b_0$	$b_1$	Note
1	72	0,07	90	49	0,54	6,2	6,2	Taken in an oil bath.
2	70	0,08	"	46,5	0,52	"	6,2	Taken in an oil bath.
3	57	0,10	"	56	0,62	"	6,21	Taken in an oil bath.
4	56	0,11	"	51	0,57	"	6,21	Taken in an oil bath.
5	59	0,13	"	54	0,6	"	6,2	Taken in an oil bath.
6	62	0,17	"	52	0,58	"	6,2	Taken in an oil bath.
7	163	0,17	81	53	0,65	"	—	Taken in an oil bath.
8	58	0,17	90	54	0,6	"	6,2	Taken in an oil bath.
9	55	0,18	"	54	0,6	"	6,21	Taken in an oil bath.
10	140	0,23	64,8	40	0,62	"	—	Taken in an oil bath.
11	162	0,24	81	49	0,61	"	—	Taken in an oil bath.
12	161	0,28	"	48	0,60	"	—	Taken in an oil bath.
13	160	0,29	"	50	0,62	"	—	Taken in an oil bath.
14	141	0,30	38,5	24	0,62	"	—	Taken in an oil bath.
15	144	0,38	83,5	48	0,58	"	—	Taken in an oil bath.
16	54	0,46	90	50	0,56	"	6,45	Taken in an oil bath.
17	143	0,60	54,0	28,0	0,52	"	—	Taken in an oil bath.
18	145	0,95	29,0	14,0	0,48	"	—	Taken in an oil bath.

Table No. 11; $\alpha = 55^\circ$.

With lubrication.

No. in order	Sample No.	$\delta$	$\delta_1$	$s$	$\dfrac{h}{s_1}$	$\beta_1$	$\beta_2$	$K_1=\dfrac{\delta}{\delta_1}$	$K_0=\dfrac{l_1}{l_0}$	Note
1	140	0,23	0,42	0,6	—	36°	52°—54°	0,55	0,62	Taken in an oil bath.
2	141	0,30	0,56	1,0	—	34°	48°—50°	0,54	0,62	Taken in an oil bath.
3	144	0,38	0,72	1,5	$\dfrac{0,1}{0,7}$	34°	52°	0,53	0,58	Taken in an oil bath.
4	143	0,6	1,13	2,0	$\dfrac{0,08}{0,5}$	33°	50°—52°	0,53	0,52	Taken in an oil bath.
5	145	0,95	1,75	2,6	$\dfrac{0,1}{0,7}$	34°	50°	0,54	0,48	Taken in an oil bath.

From Table No. 10, the graph in Fig. No. 19 has been constructed; from Table No. 11, Figs. No. 20, 21, 22. On these same drawings, for comparison, the dependencies for the case of chip removal without lubrication, at the same angle \(\alpha = 55^\circ\), taken from Figs. 9—12, are plotted with a dotted line.

Thickness of the removed layer “\(\delta\)” in mm.

Fig. No. 19.

Thickness of the removed layer “\(\delta\)” in mm.

Fig. No. 20.

Thickness of the removed layer “\(\delta\)” in mm.

Fig. No. 21.

Thickness of the removed layer “\(\delta\)” in mm.

Fig. No. 22.

From consideration of the tables and graphs, the following conclusions may be drawn. At the smallest values, \(\delta = 0.07\) mm, the shrinkage when lubrication is used is considerably less than without lubrication; the shrinkage coefficient is considerably greater. As the thickness of the layer being removed, \(\delta\), increases up to the value \(\delta = 0.2\) mm, an increase in the shrinkage coefficient \(K_0\) is observed, similar to what we had also in the absence of lubrication; with a further increase in \(\delta\), the shrinkage coefficient passes through a maximum and then begins to decrease, but nevertheless exceeds the values of \(K_0\) obtained in the absence of lubrication.

(dashed curve). When the layer thickness is increased to $\delta = 0.9\text{--}1$ mm, the shrinkage coefficient with lubrication becomes equal to that without lubrication, and subsequently the change in $K_0$ for both cases should follow one curve.

This character of the change in $K_0$ when lubricant is used is quite natural and is explained by the fact that at a layer thickness $\delta > 0.2$ mm the lubricant evidently began to be squeezed out from under the chip and to act only partially; at $\delta = 0.9$ mm its action was completely annulled. Such a conclusion seems at first glance incompatible with the known and now proven fact that the specific cutting pressure decreases with increasing layer thickness $\delta$. However, it must be borne in mind that the magnitude of the specific cutting pressure refers to a unit area of the transverse section of the chip being removed, and not to the area of contact of the chip with the cutter; of the magnitude and character of the change in $\delta$ of this latter area we know absolutely nothing, just as we do not know the distribution of pressure on the front face of the cutter at the place of chip contact. Therefore we must be satisfied with the explanation given, especially since all the points of curves No. 19, obtained from the experiments, without exception gave an exact outline and character of the curve without any significant deviations.

When lubricant is used, the character of the chip itself differs from that of a chip removed dry. In this case, in addition to less shrinkage, we also obtain less transverse widening, as can be seen from a comparison of Tables No. 10 and No. 5, and, what is especially characteristic, when lubricant is used the chip curls more strongly and has a coil of smaller diameter than without lubricant. We have given a photograph (Fig. No. 12), where, for comparison, two chips removed without lubricant and two of exactly the same layer thickness removed with lubricant are presented; we see a sharp difference in the diameters of the coils, and it must be said that this phenomenon is not accidental and that all the chips we removed had exactly the same character.

In the photograph with lubricant: No. 54 has $\delta = 0.46$ mm and No. 70 $\delta = 0.08$ mm. Without lubricant: No. 12—$\delta = 0.45$ mm and No. 14—$\delta = 0.07$ mm.

The decrease in the diameter of the coil with lubrication also indicates a decrease in the friction force between the chip and the cutter.

If we turn to Table No. 11 and to graphs No. 20–22, constructed from it, we shall see that, when lubricant is applied, with an increase in the shrinkage coefficient the values $K_1$, $\beta_1$, and $\beta_2$ increased rather considerably. On these same graphs the values $K_1$, $\beta_1$, and $\beta_2$ obtained without lubricant are plotted with dashed lines. From the comparison it is evident that the solid curves lie considerably higher than the dashed ones, and also that the character of the change in the quantities $K_0$, $K_1$, $\beta_1$, and $\beta_2$ is such that there can be no doubt of their mutual connection. With the application of lubricant the $K_0$ curve was located higher; according to the considerations given above, in the event of the existence of an interrelation the values $K_1$, $\beta_1$, and $\beta_2$ should also have increased. This increase is indeed observed; moreover, when the values of $K_0$ fall after $\delta = 0.2$ mm, a fall in the values $K_1$, $\beta_1$, and $\beta_2$ can also be observed.

Thus, cutting experiments with lubricant confirm the existence of a dependence between the magnitudes of longitudinal shrinkage, transverse thickening, and the angles $\beta_1$ and $\beta_2$. In addition, they indicate that the change in all these quantities depends on the change in the forces acting on the chip, in particular the friction force between the chip and the cutter.

Since there is a connection between \(K_0\) and the angles \(\beta_1\) and \(\beta_2\), from the change in chip shrinkage one can form an idea of the change in \(\beta_1\) and \(\beta_2\).

It may be assumed that, in materials of different hardness, other conditions being equal, different magnitudes of \(\beta_1\) and \(\beta_2\) will be obtained. According to the experiments of Prof. Sokolov\(^7\), for harder metals the shrinkage coefficient is greater; one may say that in this case the magnitudes of \(\beta_1\) and \(\beta_2\) will also be higher than for softer ones.

We made use of the relation between the change in the longitudinal shrinkage of the chip and the angles \(\beta_1\) and \(\beta_2\) in order to solve the question of the change of the angles \(\beta_1\) and \(\beta_2\) as a function of cutting speed. This question interested us for the reason that, in fact, in our photographs we could observe chips obtained at speeds differing from the normal one at which we carried out the experiments, and corresponding to the moment when the machine was stopped, i.e., when the speed, over some more or less short interval of time, falls to zero. Generally speaking, we see no possibility of stopping the machine instantaneously, and at the last moment the speed will always change to zero, passing through intermediate values; moreover this change will occur differently for different chip sections and different cutting angles: faster for large ones, in view of the greater cutting resistance that brakes the motion of the table, and more slowly for small ones. It was therefore necessary to ascertain how noticeably this might distort the picture obtained with respect to the magnitudes of the angles \(\beta_1\) and \(\beta_2\). To clarify this, it was sufficient to determine the change in the chip-shrinkage coefficient when the cutting speed changed.

For the experiment, a previously turned and smoothly faced iron blank was taken; on its surface, parallel to the axis, a groove 0.5 mm deep and 1.5 mm wide had been milled. The blank was mounted on a lathe, where, at different speeds, a chip was removed from it; on this chip the cut groove left marks dividing the chip into segments exactly corresponding to the length of the cutting path along the full circumference of the blank (i.e., for one revolution). By measuring the length of the chip segments obtained at different speeds and knowing the circumference of the blank (measured at the mean cutting diameter), it was possible to calculate the shrinkage coefficients. In order to have a strictly constant feed throughout the experiment, the work was carried out by the lead screw, with appropriately selected change gears. The cutter used was made of high-speed steel, capable of withstanding high cutting speeds without becoming dull; moreover, each series of experiments was carried out with a single setting of the cutter, so that the magnitudes of the cutting angles, setting, etc., remained constant. During the experiment the cutter, being fixed in the tool post, was touched up with a whetstone to maintain approximately the same sharpness of the blade. The experiments were carried out at a cutting angle \(\alpha = 55^\circ\). The first series was carried out with a chip of the following dimensions: depth of cut \(t = 3.25\) mm; feed \(s = 0.43\) mm/rev; chip width \(b = 6.5\) mm; chip section \(q = 1.4\ \text{mm}^2\). For this, a powerful lathe with 8 speeds was used. Unfortunately, we could use this machine only for a short time.

The second series of experiments was carried out with the same cutting angle and with chip dimensions: \(t = 2.5\) [[unclear: unit]]; \(S = 0.3\) mm/rev; \(b = 5\) [[unclear: unit]];

$q = 0.75$ mm². The third series with the same cutting angle and with chip dimensions: $t = 2.5$ mm; $S = 0.15$ mm/rev.; $b = 5$ mm; $q = 0.375$ mm².

For these two series we had to use a weaker machine, which could remove chips of the indicated dimensions only at four speeds.

From the results of the experiments, Tables Nos. 12, 13, and 14 and the graph in Fig. No. 23 were compiled.

Fig. No. 23.

Table No. 12.

Cutting angle $\alpha = 55^\circ$, depth of cut $t = 3.25$ mm, feed $S = 0.43$ mm/rev., chip width $b = 6.5$ mm, chip cross-section $q = 1.4$ mm².

No. in order	$l_0$ path length, mm	$l_1$ chip length, mm	$V$ cutting speed, m/min	$K_0 = \dfrac{l_1}{l_0}$	$K_0$ average
1	214.5	115.0	1.72	0.536	0.53
2	214.5	113.5	1.72	0.529	0.53
3	214.5	116.0	1.72	0.541	0.53
4	214.5	115.0	1.72	0.536	0.53
5	214.5	116.0	1.72	0.541	0.53
6	214.5	113.0	1.72	0.527	0.53
7	214.5	111.0	3.25	0.527	0.51
8	214.5	110.5	3.25	0.515	0.51
9	214.5	110.5	3.25	0.515	0.51
10	214.5	111.0	3.25	0.517	0.51
11	214.5	110.0	3.25	0.513	0.51
12	214.5	109.0	3.25	0.508	0.51
13	214.5	111.5	4.55	0.519	0.51
14	214.5	111.0	4.55	0.517	0.51
15	214.5	108.0	4.55	0.503	0.51
16	214.5	109.5	4.55	0.510	0.51
17	214.5	111.0	4.55	0.517	0.51
18	214.5	110.5	4.55	0.517	0.51
19	214.5	111.0	6.50	0.517	0.51
20	214.5	108.0	6.50	0.503	0.51
21	214.5	111.0	6.50	0.517	0.51
22	214.5	110.0	6.50	0.513	0.51
23	214.5	111.0	6.50	0.517	0.51
24	214.5	111.0	6.50	0.517	0.51
25	214.5	110.5	10.8	0.515	0.52
26	214.5	114.0	10.8	0.531	0.52
27	214.5	113.0	10.8	0.527	0.52
28	214.5	111.5	10.8	0.519	0.52
29	214.5	111.5	10.8	0.519	0.52
30	214.5	112.0	10.8	0.522	0.52

No. of trial	\(l_0\), path length in mm	\(l_1\), chip length in mm	\(V\), cutting speed, m/min	\(K_0=\dfrac{l_1}{l_0}\)	\(K_0\), average
31	214.5	116.5	20.75	0.543	0.54
32	”	117.0	”	0.545	0.54
33	”	115.0	”	0.536	0.54
34	”	117.5	”	0.547	0.54
35	”	117.0	”	0.545	0.54
36	”	116.5	”	0.543	0.54
37	214.5	118.0	31.5	0.550	0.56
38	”	120.0	”	0.559	0.56
39	”	120.0	”	0.559	0.56
40	”	121.0	”	0.564	0.56
41	”	119.0	”	0.555	0.56
42	”	119.5	”	0.557	0.56
43	214.5	129.0	49.7	0.601	0.59
44	”	130.0	”	0.605	0.59
45	”	125.0	”	0.582	0.59
46	”	127.0	”	0.591	0.59
47	”	129.0	”	0.601	0.59
48	”	125.5	”	0.585	0.59

Table No. 13.

\[ \alpha = 55^\circ;\qquad t = 2.5\ \text{mm};\qquad S = 0.3\ \text{mm/rev}; \]
\[ b = 5\ \text{mm};\qquad q = 0.75\ \text{mm}^2 \]

No. of trial	\(l_0\), mm	\(l_1\), mm	\(V\), m/min	\(K_0=\dfrac{l_1}{l_0}\)	\(K_0\), average
1	196.3	90.0	4.2	0.46	0.46
2	”	90.0	”	0.46	0.46
3	”	89.0	”	0.45	0.46
4	”	90.5	”	0.46	0.46
5	”	90.5	”	0.46	0.46
6	196.3	92.0	7.0	0.47	0.46
7	”	91.0	”	0.46	0.46
8	”	91.5	”	0.47	0.46
9	”	90.5	”	0.46	0.46
10	”	91.0	”	0.46	0.46
11	196.3	91.0	10.2	0.46	0.47
12	”	92.0	”	0.47	0.47
13	”	91.5	”	0.47	0.47
14	”	92.0	”	0.47	0.47
15	”	92.5	”	0.47	0.47
16	196.3	95.0	4.61	0.48	0.48
17	”	94.5	”	0.48	0.48
18	”	95.5	”	0.49	0.48
19	”	95.0	”	0.48	0.48
20	”	94.5	”	0.48	0.48

Table No. 14.

\[ \alpha = 55^\circ; \qquad t = 2.5\ \text{mm}; \qquad S = 0.15\ \text{mm/rev}; \]
\[ b = 5\ \text{mm}; \qquad q = 0.375\ \text{mm}^2 \]

Test No.	\(l_0\), mm	\(l_1\), mm	\(V\), m/min	\(K_0 = \dfrac{l_1}{l_0}\)	\(K_0\), average
1	196.3	79.0	4.2	0.40
2	”	80.0	”	0.41
3	”	80.0	”	0.41	0.41
4	”	80.0	”	0.41
5	”	80.0	”	0.41
6	196.3	82.0	7.0	0.42
7	”	79.5	”	0.41
8	”	82.5	”	0.42	0.42
9	”	82.5	”	0.42
10	”	82.5	”	0.42
11	196.3	85.0	10.2	0.43
12	”	84.0	”	0.43
13	”	84.0	”	0.43	0.43
14	”	84.0	”	0.43
15	”	84.5	”	0.43
16	196.3	88.0	14.6	0.45
17	”	89.0	”	0.45
18	”	89.0	”	0.45	0.45
19	”	88.5	”	0.45
20	”	89.0	”	0.45

As both the tables and the graph show, in all three series of experiments, with three different chip cross-sections, the dependence of the coefficient of longitudinal shrinkage on cutting speed retains a constant character. At first, as the cutting speed increases, the shrinkage coefficient decreases slightly (the shrinkage increases), and then begins to increase slowly. Within the range of speeds from zero to 15 meters per minute this change is so insignificant that, when the speed is varied within these limits, we shall not have any appreciable change in the angles $\beta_1$ and $\beta_2$. Thus it may be concluded that the dependences obtained by us for these angles will not be distorted and, within the limits of the speeds mentioned, will retain their form. At higher speeds the magnitudes of the angles $\beta_1$ and $\beta_2$ will change, increasing somewhat with increasing cutting speed. How this increase will proceed for different chip cross-sections is difficult to say without carrying out special experiments, for which, unfortunately, we had neither the time nor the opportunity.

We have some grounds for supposing (we conducted preliminary experiments, not included here) that, at other cutting angles, the dependence of shrinkage on speed will have a somewhat different character; the minimum of the shrinkage coefficient will no longer be located at $V = 5$ m/min., as we have here, but at another speed. Even there, however, within the limits of the speeds ordinarily used, the change in $K_o$ is very slight and cannot produce any appreciable change in the angles $\beta_1$ and $\beta_2$.

Incidentally, we observed that, when the speed is increased, the diameter of the chip curl always increases, regardless of whether the shrinkage decreases or increases at the same time. In Fig. No. 19 chips are presented which were removed at an ever-increasing speed.

It is difficult to say anything definite about the causes of the change in longitudinal shrinkage with change in speed without conducting special experiments; nevertheless, one may suggest that one of the causes affecting the change in chip shrinkage is the chip temperature, which changes with speed, and the different capacity of the metal being machined, at different temperatures, for “work-hardening during cutting” (first discovered by Herbert)¹¹), and consequently also its different viscosity and capacity to undergo deformation.

Here, incidentally, the question arises whether, in the experiments using lubrication at a constant cutting speed equal to 8 m/min., we did not obtain an increase in the chip-shrinkage coefficient because, when increasing its cross-section, we maintained its temperature approximately constant, i.e., lowered it in comparison with what it would have been if the cross-section had been increased without applying lubrication. In this case, the lowering of the temperature by the introduction of lubrication should have produced the same effect as a decrease in speed from 8 meters per minute to some smaller value on the diagram of drawing No. 23. However, comparison of this drawing with drawing No. 19 makes it possible to conclude that, whereas with a change in speed the shrinkage changes, within the range of low speeds (8–0 m/min.), very insignificantly, we obtained, by introducing lubrication, a very considerable change in the shrinkage coefficient at the same values of chip thickness; this can be attributed, as we have already done earlier, only to a change in the magnitude and direction of the forces acting on the chip.

Thus, as a result of all the experiments described so far with continuous chips, we may say that:

1. The longitudinal shrinkage of the chip, for a constant material, depends on the cutting angle, the thickness of the layer removed, the cutting speed, and the magnitude and direction of the forces acting on the chip.

2. The shear angle \(\beta_2\) and the angle \(\beta_1\), which bounds the lines of deformation in the chip, are by no means constant quantities. They vary very considerably with changes in the thickness of the layer removed, and less considerably with changes in the cutting angle, the quality of the material being machined, and the cutting speed. With all other conditions being equal, a change in the magnitude and direction of the acting forces also entails a change in the angles \(\beta_1\) and \(\beta_2\).

3. The change in the angles \(\beta_1\) and \(\beta_2\) is connected with the change in the longitudinal shrinkage and the transverse thickening of the chip.

Incidentally, it should be said that Prof. Time asserted the existence of a connection between the angles of sliding and the shrinkage of the chip, but in his opinion the primary cause of all changes in the chip was precisely the constant sliding angle \(\beta_1\), which separated the deformed chip from the completely undeformed metal of the specimen; the magnitude of the shrinkage coefficient also depended on the angle \(\beta_1\).

The fact that, with a change in the thickness of the layer removed, the shrinkage changes—(this change, in Time’s opinion, was very small)—was the result of a change in the angle \(\beta_1\), which in his opinion was so insignificant that it was lost even in comparison with the change in \(\beta_1\) as a function of the cutting angle \(\alpha\), which, according to his data, fluctuated only within \(10^\circ\).

We see, however, that the angles \(\beta_1\) and \(\beta_2\) fluctuate within very wide limits, and an especially significant change in them occurs when the chip thickness changes. Thus, for example, for a cutting angle \(\alpha = 63^\circ\), \(\beta_1\) changes from \(14^\circ\) to \(42^\circ\); \(\beta_2\) from \(27^\circ\) to \(55^\circ\); the limits, as we see, are very wide.

Time considers the cause of chip shrinkage to be the deformation of successive sliding (shearing) of layers at a certain angle \(\beta_1\).

Afanasiev, on the contrary, believes that the shrinkage of the chip has as its cause only the compression of the metal in front of the cutter. Subsequent investigators, in the majority of cases, give preference to the opinion of Prof. Time.

The relationship between the sliding angle and the shrinkage coefficient, taking into account the process of chip formation, was given by him in the following form:

\[ K_o=\frac{\sin \beta_1}{\sin(180^\circ-\alpha-\beta_1)} \]

This expression was derived by Prof. Time for the case of a chip divided into elements, but the author believed that, since in a continuous chip there are invisible planes of sliding directed at an angle \(\beta_1\) to the direction of motion of the cutter, this expression is also suitable for calculating the shrinkage coefficient in the case of a continuous chip. Using this formula, he determined the shrinkage for iron chips at cutting angles \(\alpha = 45^\circ\) and \(\alpha = 55^\circ\); in this case, as he himself later notes, the chip was completely continuous. The values of the shrinkage coefficient obtained from the formula closely agreed with direct measurement of the chip, and this seemed to be good proof that the expression he had given for the shrinkage coefficient corresponded to reality, even revealing the very process of chip formation by successive sliding at the angle \(\beta_1\), and as if

indicates that the principal and predominant cause of shrinkage is the process of successive shears (slidings) of the chip layers.

But we have shown that in a continuous chip there are shears only at the angle \(\beta_2\), and if the shrinkage were produced solely by shears, then in that case, substituting into the shrinkage formula, instead of the angle \(\beta_1\), the actual shear angles \(\beta_2\) obtained experimentally, we ought to obtain by calculation values more or less closely coinciding with the shrinkage values obtained from experiment.

We made such an attempt for the cutting angle \(\alpha = 55^\circ\). The shrinkage coefficients obtained by calculation from the formula are plotted on graph No. 9 as a dotted curve. We see that this curve runs parallel to the actual (solid) one, but much higher than it, i.e. the shrinkage coefficients obtained by calculation are too large, the shrinkage too small, much less than the actual one. Thus the shears in the chip cannot produce the entire magnitude of the existing shrinkage; in the process, apparently, other kinds of deformation also participate, among which, undoubtedly, compression deformation is present as well.

Prof. Time’s formula does not take this into account; therefore, when the actual sliding angles \(\beta_2\) are substituted into it, it gives unsatisfactory results.

If, however, we substitute into it the angles \(\beta_1\) obtained by us from experiment, then by calculation we obtain shrinkage values extremely close to the actual ones. This is quite natural, since in reality Prof. Time’s formula for chip shrinkage expresses only purely geometrical relations between the dimensions of the chip and of the metal layer being removed; it can be derived from the condition of constancy of the volume of metal passing into the chip, and in whatever direction the shears in the chip may occur—whether at the angle \(\beta_1\), as Prof. Time assumes, or at some other angle (\(\beta_2\)), even if shears are altogether absent—it will give results corresponding to reality if the angle \(\beta_1\), which bounds the chip and is formed by the direction of motion of the cutter and the straight line running, in the simplest case, from the cutter tip to the point where the machined surface passes into the chip, is correctly determined. That is why the expressions for chip shrinkage of Prof. Time and Prof. Afanas’ev coincide completely, despite the fact that they see the causes of shrinkage as entirely different phenomena.

In addition to sliding and compression deformations, the bending deformation of the chip is undoubtedly also present. As proof, we present a photograph taken from cross sections of two chips of different section (see Fig. No. 13). Both one and the other give a clearly expressed trapezoidal section. The smaller of the parallel sides of the trapezoid is the side of the chip adjacent to the cutter. It is perfectly straight. The larger one, located precisely on the concave side of the chip, has a certain convexity. Thus, the width of the chip has a variable magnitude, increasing toward the side of the chip located on the inner part of the turn.

Such a section, having arisen from a rectangular one, indicates that here, besides all other deformations, as a consequence of the various acting forces, bending deformations took place, as a result of which the part of the chip located above the imaginary neutral line received a transverse widening. With pure bending we would have had, below the neutral line, a decrease in width, as a result of pro-

of longitudinal elongation; but in the present case, besides bending, there is also a considerable deformation of longitudinal compression, which eliminates the stretching of the chip layers adjacent to the tool that arises as a result of bending, and thus, as it were, carries the neutral line beyond the limits of the section, leaving in the bent beam only a compressed region with a variable degree of compression. If there were no bending, the cross-section of the chip would have to present a different picture. According to Prof. Time’s theory of chip formation, which assumes that the chip curls not as a result of its bending, but only as a result of compression deformations when the tip of the cutter penetrates the next element of the chip and the element subsequently shears off, the chip would apparently have to have an entirely different section, with the greatest widening on the side adjacent to the cutter.

Note. After we have shown the presence of bending in the chip, one may, returning to Tables Nos. 7, 8 and to drawings Nos. 5–12, say that the sharp fluctuations obtained at $\alpha = 45^\circ$ and $\alpha = 55^\circ$ in the values of $\beta_1$ and $\beta_2$ in thin chips, accompanied by similar fluctuations of $K_1$ and $K_0$, are probably explained by the fact that thin chips, for various reasons, resist bending less uniformly than thick ones; this causes a change in the frictional force of the chip on the cutter and the associated change in the quantities $\beta_1$, $\beta_2$, $K_1$, and $K_0$. The fluctuations of the same quantities at $\alpha = 65^\circ$ (see drawings Nos. 13–16) are explained, in addition, by the fact that some of the thin chips were divided into elements; in this case, as we shall indicate later, fluctuations in the values of $\beta_1$ and $\beta_2$ are obtained. In drawings 13–16 the chips divided into elements are indicated by double circles.

In concluding the section on continuous chips, we shall touch upon the question of the formation of irregularities (burrs) on the side of the chip adjacent to the cutter and on the surface under the cutter. This phenomenon extends to all types of chips, but in continuous chips it is most clearly expressed.

In analyzing the structure of the chip, it was pointed out that, as they approach the plane of the cutter, the shear lines in the chip lose their rectilinearity, bending in the direction opposite to the motion of the chip along the cutter. This curvature is the more pronounced, the greater the thickness of the layer being removed. A peculiar lagging of the metal adjacent to the cutter behind the rest of the chip is obtained in its motion along the front face of the cutter. A quite probable cause of the formation of such a “stagnation region” is the considerable friction between the chip and the cutter. The “stagnation region” is often supplemented by burrs on the chip, the formation of which was observed in almost all the cases we examined during the experiments. On the surface under the cutter there are also burrs, as has already been indicated.

Usachev[^8] was the first to draw attention to this phenomenon and to attempt to explain it. In his opinion, burrs on the chip and on the surface under the cutter are formed only in the case when there is on the cutter a built-up edge consisting of particles of metal compressed and welded to the tip of the cutter.

The phenomenon of the built-up edge has been known in the literature since the time of Taylor[^27], who believed that the chip is not cut by the blade of the cutter but is torn away ahead of it, creating irregularities on the surface under the cutter, which are cut off by the cutter blade in the form of small particles of metal, compressed and welded under the influence of great pressure and heat to the front face of the cutter, forming a built-up edge or, as it is called, an “edge appendage.” Rosenhain and Sturney⁴ hold the opinion that the built-up edge is part of the chip being removed; Gerbert³ considers that here there is a part of the metal also from the surface under the cutter. Usachev does not—

expresses no clear opinion as to the nature of the built-up edge, confining himself to the not entirely comprehensible assertion that “the metal of the built-up edge is compressed, belongs neither to the chip nor to the workpiece, and moves together with the cutter during cutting.”

As was said, the built-up edge, in Usachev’s opinion, is the cause of the formation of serrations both on the chip and on the cut surface. According to Usachev, the process of their formation is as follows: “The end of the built-up edge above the cutter blade has a rounded form and therefore cannot cut the metal as cleanly as the cutter blade cuts it. The rounded end of the built-up edge compresses the metal, carries it along with itself without tearing it away from the workpiece, and, when a large layer of this metal—serving, as it were, to supplement the built-up edge—has accumulated, it ruptures. One part of the layer remains on the workpiece, forming a serration; the other is removed with the chip, forming on it the same kind of serration, only more compressed.”

We have a number of objections to such an explanation. First of all, in the course of almost a hundred experiments we very rarely had a built-up edge on the cutter, whereas in almost all cases serrations were observed both on the chip and on the cut surface. Further, if there were a connection between the formation of the one and the other serrations in the form in which Usachev presents it, i.e., if one supplemented the other, then the number of serrations on the chip and on the cut surface ought to be the same, and, because of shrinkage, the serrations on the chip would be considerably shorter than those on the cut surface. From consideration of the photographs we present, one can see a relation between the two that is quite the reverse: in the overwhelming majority of cases the serrations on the chip are much longer than those on the cut surface (see Figs. Nos. 4, 5, 6, 10, 11); moreover, cases are not rare in which, with complete absence of any irregularities on the chip, sharply expressed serrations are observed on the cut surface.

Nor can one agree with Klopstock,^10) who attributes the formation of irregularities on the cut surface solely to vibration of the cutter during separation of the chip elements, and considers that the number of serrations is thus equal to the number of chip elements; this in no way can explain the phenomenon in the case of a completely continuous chip.

We have come to the conviction that there is no direct connection between the serrations on the chip and on the cut surface, except only that in the formation of both, a certain role is played by a crack propagating in front of the cutter tip, going somewhat forward and into the depth of the material being machined. Usually in a continuous chip it is very small and becomes noticeable only with very careful grinding and sufficient magnification; moreover, as we shall show below, it has a periodicity of formation connected with the formation of serrations on the cut surface, so that there are moments when it is absent. In Fig. No. 6 this crack is clearly distinguishable; it is evident that it is very insignificant and cannot be identified with the phenomenon of tearing-away of the chip, which, according to Taylor’s assumptions, should create favorable conditions for the work of the cutter blade itself, freeing it from contact with the metal, friction, and excessive heating.

The principal cause of the formation of serrations on the chip is the force of friction between the chip and the cutter, which creates a “stagnation zone.” However small the crack propagating ahead of the tip may be, it nevertheless condi-

it causes the inevitability of at least slight tears in the chip being separated. As a result of considerable friction, the layer of metal adjoining the front face of the cutter tends to be held back on it, lagging behind all the rest of the chip; owing, however, to the tendency of the whole chip—and especially of that part of it which has already passed out of contact with the cutter—to move forward and to separate from the part held back by friction, the tears in the chip deepen, small irregularities settle down and are transformed into burrs. The part of the metal held on the cutter by friction, having a lower velocity of motion, possibly even equal to zero, creates a depression in the material lying ahead which, continually increasing, finally overcomes the force of friction on the cutter and compels the next portion of the “stagnation region,” with the last burr, to move, to slip off the front face of the cutter, freeing space for the formation of the next burr. This slipping may occur either instantaneously or by gradual displacement. In the first case, a more or less step-like movement of the chip along the cutter is possible, which may be the cause of the periodic fall and rise of the frictional force on the cutter and of the total cutting pressure even with a continuous iron chip, as is shown by special diagrams 23). In the second case, the “stagnation region” and the last burr are divided into a number of small parts, successively slipping off the cutter. Such a case, a rather rare one, is presented in Fig. No. 14. Here, moreover, the crack in front of the cutter tip, mentioned earlier, is clearly visible.

But an entirely different course of the process of chip movement along the cutter is also possible. The force holding the part of the chip adhering to the cutter is sometimes so great that the increasing depression is not able to displace the burr formed in the “stagnation region” from the front face. The continuing formation and movement of the chip then produces, on the one hand, a further propagation of the tear between the chip and the burr; on the other hand, it increases the crack running in front of the end of the cutter, which in the end joins with the tear and thus separates the burr that has adhered to the cutter from the moving chip. Under the influence of high temperature and extremely considerable work-hardening, the burr is, as it were, welded to the front face of the cutter, acquires great hardness, and takes an active part in the further formation of the chip, replacing the cutter edge and protecting it from destruction.

The moment preceding the separation of the burr from the chip and its welding to the cutter is recorded in Fig. No. 15. Here, as we see, the tear between the chip and the last burr has gone very far, and the line of further separation of the burr from the chip has already been marked out.

Thus the burr separated from the chip is transformed into a build-up on the cutter (an edge appendage), and Fig. No. 15, as well as Figs. No. 14 and No. 16, clearly show that before its formation it belonged to the chip, constituting one whole with it.

The question arises: can a burr, separated from the chip and adhering to the cutter, perform the function of the latter when cutting the same material of which it consists? This can best be answered by comparing two works by Prof. Herbert. In one of them 26), the author, investigating by means of his pendulum apparatus the hardness of hardened cutters, with the aim of obtaining the best conditions of heat treatment,

of machining of the tool, found that under the most favorable conditions the hardness of hardened high-speed steel \([C = 0.7\%;\ Cr = 4\%;\ Va = 1\%;\ W = 18\%;\ Si = 0.2\%;\ Mn = 0.2\%]\), within the temperature range \(100^\circ\)—\(560^\circ\), is on average 55 (Herbert hardness number). In another work³, in studying with the same instrument the hardness at various places of the chip being separated, with the aim of investigating the process of cold-working of the metal during cutting, it was found that, whereas the initial hardness of the metal was 19.9, the hardness in the chip was 32–38, and the hardness in the burr adjacent to the front face of the cutter reached high limits—53.4—and thus was equal to the hardness of the best cutting tool. A diagram of the chip, with the hardness numbers marked on it at the tested places, borrowed from the above-mentioned work, is given by us in Fig. No. 16.

In the case of the formation of a built-up edge, cutting proceeds under more favorable conditions, since this entails a decrease in the cutting angle without diminishing the durability and strength of the cutter. During operation the built-up edge is gradually worn away or is torn from the front face by shaking or impact (for example, during planing). It may then be replaced by one of a series of subsequent burrs in the “stagnation region,” etc.

Thus we have shown that an indispensable condition for the possibility of formation of a built-up edge on the cutter is the presence of burrs on the chip, but not conversely, as Usachev assumes. The built-up edge is one of the burrs and, consequently, is formed from metal belonging to the chip.

Rosenhain and Styrney⁴ found that the “stagnation region,” or, as they call it, the “zone of deep deformations,” occurs in cutting bronze only at cutting angles \(\alpha > 60^\circ\), and the line separating this zone from the rest of the chip makes a constant angle of \(60^\circ\) with the direction of motion of the cutter and an angle of \(90^\circ\) with the direction of shear in the chip, which in turn determines the constancy of the shear angle in the chip, which must be equal to \(30^\circ\).

All these propositions are not confirmed in the case of iron chips. At all cutting angles beginning with \(45^\circ\) we had a more or less clearly expressed “stagnation region” in the chip; the constancy of the shear angle \(\beta_2\) has already been refuted by us in due course; moreover, the line separating the “stagnation region” from the chip, in our experiments, makes no constant angle either with the direction of motion of the cutter or with the direction of shear in the chip.

Turning to the burrs on the under-cutter surface, it must first of all be said that the principal cause of their formation is the crack advancing ahead of the cutter tip. Having a direction not only forward, but also into the depth of the workpiece, it can extend only up to a certain limiting magnitude, depending on the dimensions of the chip and the strength properties of the material being machined. At a certain moment, when its magnitude reaches the limiting value, its further propagation ceases; the cutter tip, cutting off the upper side of the crack (which passes into the chip), thus forms a sharp burr on the under-cutter surface. With further motion, the formation of a new crack begins; as it propagates, it reaches the limiting value, and so on. The process is repeated periodically at approximately equal intervals, dependent on the dimensions of the chip. In drawing No. 24 are schematically shown

successive stages of the process of burr formation on the undercut surface: “a”—the beginning of crack formation, “b”—its deepening, “c”—attainment of the limiting magnitude, “d”—formation of a burr and the beginning of formation of a new crack. The separate stages of the process can also be seen in the photographs (Figs. Nos. 14, 10, 11, and 17).

Drawing No. 24.

From Tables Nos. 7–9 it can be seen that the mean distance between burrs, “$S$,” and the depth of the burrs, $h$, increase with an increase in the thickness of the layer being removed, “$\delta$,” and with an increase in the cutting angle, “$\alpha$”; i.e., the nonuniformity of the machined surface increases.

2. Chip divided into elements.

In describing the structure of the chip obtained at a cutting angle $\alpha = 65^\circ$, with small values of the thickness of the layer being removed, we have already encountered a chip which differs in appearance from a continuous chip and consists of separate elements, small joints, nevertheless connected with one another.

At cutting angles $\alpha = 75^\circ$ and $\alpha = 85^\circ$, with small values of “$\delta$,” we obtained exactly the same chips. Fig. No. 18 gives an example of such a chip at a cutting angle $\alpha = 85^\circ$ and a thickness of the layer being removed $\delta = 0.1$ mm. In its structure it in no way differs from the chip of Fig. No. 7 and likewise consists of elements connected with one another. This type is commonly called a shear chip. It corresponds most closely to the conception of chip formation given in his theory by Prof. Time, but it is only a special case and in our work covers only a narrow region characterized by small values of “$\delta$” and large values of “$\alpha$.”

In this type, as was already said earlier, separation of the elements occurs in the direction of the angle $\beta_1$; in each separate element, however, deformation proceeds in the direction of the angle $\beta_2$. In photograph No. 18, both the one and the other deformation are extremely clearly visible. Thus, in the formation of an element we have the same process as in a continuous chip; then, when the element reaches a certain limiting size, shearing occurs in the direction of the angle $\beta_1$. The latter phenomenon was observed by Prof. Time and was extended by him to all types of chips; however, deformations in each separate element in the direction of the angle $\beta_2$ were not observed and did not find expression in any of the theories of chip formation that use as their basis the results of his observations (Zvorykin, Pogorzelsky, Afanasiev).

The deformations in such a chip are extremely large. The longitudinal shrinkage here depends already on two causes: on the one hand, on compression of the metal and shears within the elements; on the other hand, on relative-

of displacement of individual elements relative to one another. The two deformations are fundamentally different and, evidently, have different causes at their basis, since the first occurs in the direction of the angle \(\beta_2\), and, as we have already shown, the propagation of shears then goes in this direction, opposite to the motion of the tool, namely from the point where the machined surface passes into the chip, toward the tip of the tool; the second, however, is in the direction of the angle \(\beta_1\), and propagates forward from the tip of the tool.

We have already shown that the formula proposed by Prof. Time for the coefficient of longitudinal shrinkage in the case of continuous chips does not take into account the influence of compression of the chip, takes into account only its transverse thickening, and gives results that do not correspond to reality when the actual shear angles \(\beta_2\) are substituted into it. In the case of a shear chip consisting of separate elements connected with one another, for which this formula was in fact derived, it is entirely unsuitable, even if secondary deformations such as transverse widening, bending, etc., are not taken into account. Even under these conditions the expression for the shrinkage coefficient

\[ K_0=\frac{l_1}{l_0}=\frac{\sin\beta_1}{\sin(180^\circ-\alpha-\beta_1)} \]

can give results corresponding to reality only in the case where, after deformation, each separate element has the form of a parallelogram or a trapezoid with two parallel sides separated from one another by the same distance \((l)\) as before deformation. Diagram “a” of Drawing No. 25 gives the form of an element of a shear chip after the compression preceding shear, in which

Drawing No. 25.

the shrinkage would be expressed by the formula given above. In reality, however, during the period preceding shear, shears occur in the element in the direction of the angle \(\beta_2\), so that the ideal form of the element will be as in diagram “b” of that drawing. Each point of the element moves in the direction of the side of the angle \(\beta_2\), and since \(\beta_2>\beta_1\) always, the actual length of the element \(l_1'\) will always be greater than \(l_1\) (according to diagram a). The photographs obtained fully correspond to our diagram.

Thus, two causes producing the phenomenon of shrinkage act in opposite directions: sliding at the angle \(\beta_1\) (separation of the elements) shortens the chip, while sliding at the angle \(\beta_2\) (within each element) reduces this shortening, in the sense that without it the shortening would be considerably greater. In reality the phenomenon is further complicated by the fact that, at the beginning of the formation of each element, the tool removes a chip of variable (ever-increasing) thickness, and it may be assumed that in this process the magnitudes are also distorted

angles $\beta_1$ and $\beta_2$, on which, among other things, the variation of their values may depend; moreover, the separation of the elements often takes place at first along a horizontal line, and only then at the angle $\beta_1$, as may be observed in Fig. No. 20. In the latter case an appreciable role is played by the crack propagating ahead of the cutter tip, which here runs for the most part horizontally and, at the moment preceding the shearing-off of the element, attains a considerable size.

At cutting angles $\alpha = 75^\circ$ and $\alpha = 85^\circ$, with an increase in the thickness of the layer being removed, the individual elements of stepped chip lose their connection with one another and, after shearing in the direction of the side of angle $\beta_1$, come off the front face of the cutter in the form of separate pieces of metal. Otherwise the process proceeds as before. Fig. No. 21 may serve as an illustration of this. The chip was taken at $\delta = 0.65$ mm and $\alpha = 85^\circ$; the final moment of formation of the element preceding shearing is recorded. The preceding element is absent, since it has completely separated. It is clearly visible both how the direction of the shear lines in the element is at angle $\beta_2$, and the angle $\beta_1$ of inclination of the lines limiting the deformations in the chip. It is also visible here that from the cutter tip a horizontal crack begins to propagate, which in the further course of the process must separate the formed element.

With a further increase in the thickness of the layer being removed, “$\delta$,” and with the same cutting angles ($\alpha = 75^\circ$ and $\alpha = 85^\circ$), we obtained a transition to an entirely different type of chip. Fig. No. 22 gives an idea of the chip taken at $\delta = 1.6$ mm and $\alpha = 86^\circ$. Here we see that the shearing deformations are no longer so sharply expressed as in all the preceding chips, and particular attention is drawn by an extremely large crack running forward horizontally from the cutter tip. This crack is sometimes of monstrous dimensions in comparison with the depth of cut and gives the chip a peculiar appearance. Such a case is shown in Fig. No. 23 ($\delta = 2.0$ mm, $\alpha = 85^\circ$), and we see that the crack here exceeds the depth of cut by approximately 6–7 times. A definite impression is created that the chip is not sheared off but torn away by the cutter; therefore this type may be called a chip of tearing. Here, although not as clearly as in the earlier cases considered, the directions of the angles $\beta_1$ and $\beta_2$ are nevertheless noticeable.

At still more considerable values of “$\delta$” and the same cutting angles, the chip of tearing acquires still more characteristic features. In Fig. No. 24 (“$\delta$” $= 2.2$ mm; $\alpha = 75^\circ$) we have such a chip. First of all, here there is no longer a sharp boundary between the deformations in the chip and the rest of the metal of the specimen; in other words, there is no pronounced expression of the angle $\beta_1$, and the considerable deformations in the chip gradually disappear. Moreover, here the shear lines do not have a rectilinear character, but are clearly expressed, almost regular arcs of concentric circles. The crack in front of the cutter tip is also of great magnitude.

The same gradual transition of deformations from the chip into the specimen and the curved shear lines are also seen in photograph No. 25 (“$\delta$” $= 1.7$ mm; $\alpha = 75^\circ$). In all the photographs of tearing chips considered earlier, one can likewise note the transition of rectilinear shear lines into curvilinear ones.

In conclusion to the consideration of the photographs, we present an interesting photograph of a chip representing a transitional stage from the shearing type to the tearing-chip type. See Fig. No. 26 (“$\delta$” $= 1.1$ mm; $\alpha = 85^\circ$).

Here the first element has acquired the form characteristic of shear chips, while the second shows signs of a tear chip. From this photograph one may conclude how considerable the deformations are in such a chip. We see that, inside the chip on the side of the tool, clearly visible cracks spread, which in some places form voids; on the side opposite the tool, it is not difficult to notice that significant portions of the outer surface of the specimen, as a result of deformation, have passed into the interior of the chip.

In general, if one traces the character of formation, beginning with a continuous chip and ending with a tear chip, it becomes clear that, while in a continuous chip we have a fully established regular process, in a shear chip this process acquires a certain periodicity without losing its regularity, whereas in a tear chip neither regularity nor constancy in the course of the process can be observed: alongside small elements approaching in form the elements of shear, in the same tear chip we had elements of monstrous size, with a large deformed volume of metal, formed as a result of the propagation before the point of the tool of a crack many times exceeding the value of the depth of cut “$\delta$”; alongside the presence of shear lines, there were significant tears, cracks, crumplings, as a result of which the outer parts of the chip are pressed inward, etc. All this, on the one hand, shows how the process becomes more complicated with an increase in the cutting angle “$\alpha$”; on the other hand, it serves as a sufficiently good illustration of the proposition now proven, that with an increase in the cutting angle the force required for the formation of chips of constant dimensions increases. We see that, in this case, both the degree of deformation and the deformed volume inevitably increase.

Cutting angle $\alpha = 75^\circ$

Angle $\beta_2$
Thickness of the removed layer “$\delta$” in mm.
Chart No. 26.

Angle $\beta_1$
Thickness of the removed layer “$\delta$” in mm.
Chart No. 27.

Measurements of the angles $\beta_1$ and $\beta_2$ at cutting angles $\alpha = 75^\circ$ and $\alpha = 85^\circ$, collected by us in Tables No. 15 and No. 16, confirm the increase in the volume of metal deformed before the tool, since we obtain here smaller angles $\beta_1$ than for the same values of “$\delta$” at $\alpha = 75^\circ$. From these same tables it is evident that, at large cutting angles, the values of the angles $\beta_1$ and $\beta_2$ undergo, owing to the above-mentioned periodicity of the process (in the case of a shear chip) and nonuniformity (in the case of a tear chip), rather considerable fluctuations. In some cases, however, it is altogether impossible to determine the values of these angles, either

because of their indistinctness and the gradual transition of the deformation from the chip into the specimen, or because of the presence, instead of the rectilinear shear lines described above, of curvilinear ones. Nevertheless, as may be seen both from Tables Nos. 15 and 16 and from Drawings Nos. 26–27 and Nos. 28–29, compiled from them, one may conclude that here also, as in the case of continuous chips, with an increase in the thickness of the layer removed, \(\delta\), the angles \(\beta_1\) and \(\beta_2\) increase noticeably (as is also indicated by the increase in the ratio \(\dfrac{\delta}{S}\)).

Cutting angle \(\alpha = 85^\circ\)

Angle \(\beta_2\)

Thickness of the removed layer “\(\delta\)” in mm.
Drawing No. 28.

Angle \(\beta_1\)

Thickness of the removed layer “\(\delta\)” in mm.
Drawing No. 29.

Table No. 15. \(\alpha = 75^\circ\).

No. in order	Specimen No.	\(\delta\)	\(S\)	\(\beta_1\)	\(\beta_2\)	Note
1	78	0.25	1.6	\(19^\circ\)—\(20^\circ\)	\(26^\circ\)
2	68	0.4	2.9	\(16^\circ\)	\(30^\circ\)
3	8	0.4	2	\(22^\circ\)—\(24^\circ\)	\(42^\circ\)—\(44^\circ\)
4	10	0.55	3	\(18^\circ\)—\(20^\circ\)	\(36^\circ\)—\(38^\circ\)
5	81	0.7	3.1	\(20^\circ\)—\(22^\circ\)	\(38^\circ\)—\(40^\circ\)
6	16	0.62	3.5	\(18^\circ\)—\(20^\circ\)	\(35^\circ\)—\(36^\circ\)
7	63	0.7	2.5	—	\(44^\circ\)—\(46^\circ\)	Angle \(\beta_1\) indistinct
8	83	0.9	3.6	\(20^\circ\)	\(34^\circ\)—\(36^\circ\)	Beginning of chip
9	39	0.9	3.5	\(26^\circ\)—\(28^\circ\)	\(44^\circ\)—\(46^\circ\)	Angle \(\beta_1\) not reliable
10	33	1.0	4.0	\(32^\circ\)—\(34^\circ\)	\(52^\circ\)—\(54^\circ\)
11	5	1.35	6.5	—	—	Curvilinear shear lines
12	66	1.6	7.0	—	—	Same
13	6	1.7	5.5	—	—	Same
14	61	2.1	7.5	\(28^\circ\)—\(30^\circ\)	\(52^\circ\)
15	60	2.2	10.0	—	—	Curvilinear shear lines
16	7	4.0	6.5	—	—	Beginning of chip
17	27	4.2	11.0	—	—

Table No. 16. \(\alpha = 85^\circ\).

No. in order	Sample No.	\(\delta\)	\(S\)	\(\beta_1\)	\(\beta_2\)	Note
1	88	0.1	0.7	\(11^\circ\)	\(28^\circ\)—\(30^\circ\)
2	89	0.13	1.3	\(10^\circ\)—\(12^\circ\)	\(26^\circ\)
3	93	0.65	3.5	\(16^\circ\)	\(32^\circ\)—\(34^\circ\)
4	96	0.65	4.5	\(17^\circ\)	\(38^\circ\)
5	134	1.1	6.5	—	\(46^\circ\)—\(48^\circ\)	Angle \(\beta_1\) is unclear
6	131	1.6	9.5	\(22^\circ\)—\(24^\circ\)	\(42^\circ\)
7	117	2.0	18.0	\(28^\circ\)	\(48^\circ\)
8	130	2.3	6.3	—	—	Curvilinear shear lines
9	119	2.4	7.0	—	—	Same
10	133	3.4	18.0	—	—	Same
11	135	3.4	8.8	—	\(\sim 50^\circ\)
12	116	4.0	10.0	—	—	Curvilinear shear lines

In the graph of Fig. No. 30 and Fig. No. 31 we have assembled the dependences of the angles \(\beta_1\) and \(\beta_2\) on the thickness of the layer being removed for all angles \(\alpha\). The curves

Thickness of the removed layer “\(\delta\)” in mm.
Fig. No. 30.

Thickness of the removed layer “\(\delta\)” in mm.
Fig. No. 31.

for all \(\alpha\) run almost parallel; the angles \(\beta_1\) and \(\beta_2\) increase with increasing “\(\delta\)” and with decreasing \(\alpha\). At the same time, the change of the angles \(\beta_1\) and \(\beta_2\) with changing “\(\delta\)” is much more considerable than with changing “\(\alpha\)”.

In describing our experiments, we compared their results with the conclusions of previously conducted work and noted the discrepancies thereby obtained. The most significant point is that we did not obtain confirmation of the established opinion concerning the constancy of the shear angle in the chip. Quite apart from the fact that the extreme opinions on this question (Hankins, Rosenhain, and Styrney), which assume the absolute invariability of the shear angle, were not confirmed, the experiment carried out also did not agree with the more moderate views (Time, Afanas’ev, Zvorykin), which speak of the invariability of the shear angle within the limits of a constant cutting angle. It may even be said that we obtained results directly opposite to this. Indeed, the graphs in Figs. Nos. 30 and 31 not only testify to the absence of absolute constancy, but also indicate that one may rather accept an approximate independence of the angles \(\beta_1\) and \(\beta_2\) from the cutting angle “\(\alpha\)” than from the thickness of the layer being removed “\(\delta\),” with a change in which these angles vary extremely strongly.

If we turn to the theoretical studies of the question known to us, we shall see that they all arrive at the conclusion that the shear angle in the chip is constant and that it depends only on the cutting angle, i.e., at results that do not correspond to reality. The best-known theory among us, that of Prof. Zvorykin, presented in the technical literature alongside the conclusions of Prof. Time, is no exception here.

No less important is the fact that not one of the theories or experimental works sees a difference between the shear angle in a continuous chip \((\beta_2)\) and the angle at which separation of the elements in a shear chip \((\beta_1)\) takes place, which in a continuous chip only bounds the shear lines. Likewise absent is the conception that in a chip consisting of separate elements, the formation of the latter occurs by means of internal shears at the angle \(\beta_2\). Only Usachev expressed the correct view of the phenomenon, but his opinion remained unaccounted for in subsequent works.

Thus, the currently existing conception of the process of chip formation is not confirmed by experiment in its essential details.

Deformations in the Machined Surface.

There are indications in the literature that the action of the cutter is not limited only to deformation of the chip being removed, but also extends into the workpiece, both in front of the cutter tip and beneath the cutter, in the already machined surface. Time\(^2\) asserted that only the volume of metal belonging to the chip and enclosed within the angle \(\beta\) \((\beta = 180^\circ - \alpha - \beta_1)\) is deformed, which for this reason was called the “angle of action.” This opinion was refuted by Usachev\(^8\), who observed the spread of deformation in iron and steel beyond the limits of the angle \(\beta\), forward in front of the cutter and into the depth of the metal beneath the cutter; this, however, related only to the external lateral surface of the specimen, which, having been ground and having before machining a mirror-bright surface, after the passage of the cutter became covered with ripples, grew dull, and in places had undergone plastic deformation. The same was indicated by Zadneprovsky\(^9\) with respect to lead and by Prof. Coker\(^5\) with respect to brass and nitrocellulose.

However, all this referred to the outer lateral surface of the specimen, could have been produced by lateral bulging of the material, and could not serve as a criterion for the spread of residual deformations inside the specimen.

If we make use of the results of Herbert’s work¹¹), in which he measured hardness at various places on the removed chip, we can obtain indications on the question of interest to us. In Fig. No. 16 there is shown an image borrowed from the cited work, which we have already used once. From this photograph it is clearly evident that, far ahead of the cutter, as well as beneath the cutter, there is a considerable increase in hardness as compared with its original value for the untreated material. The increase in hardness is indeed less significant than in the chip already removed, but it is nevertheless present and may be a consequence only of the cold-working of the metal that occurred during machining as a result of the action of the cutter. In the indicated photograph the results are somewhat distorted and obscured because the deformations from the previous pass of the cutter had not been eliminated beforehand; this is indicated both by the condition of the upper surface of the specimen and by the increased hardness beneath it. It is possible that for this reason the author, in his article, does not point to the increase in hardness in the machined surface described by us. In any case, the photograph cited shows that the spread of deformation ahead of and beneath the cutter takes place not only on the lateral surface but also within the thickness of the workpiece being machined.

We set ourselves the aim of determining the magnitude of the spread of plastic deformation ahead of the cutter and the depth of its spread beneath the cutter, at various cutting angles, as a function of the thickness of the layer being removed. For this purpose we used the etching already described above by Fry’s method, in its second stage, i.e., when deformations are etched in the main metal of the specimen. A portion of a microsection was taken, etched in the corresponding manner, and subjected to measurement at tenfold magnification.

Draw. No. 32.

The picture obtained after etching is presented in Figs. No. 27 and No. 28. The characteristic darkening indicates the presence of plastically deformed metal. Such darkening, as is evident, is distributed—

extends far beyond the limits of the angle \(\beta\) in front of the cutter and deep under the cutter in the already machined surface. From such polished sections the thickness of the work-hardened layer \(\delta_2\) and the distance \(S_2\) from the tip of the cutter to the extreme point of the deformed region in front of the cutter were measured. The results of the measurements are collected in Tables Nos. 17—21. The dependence of the thickness of the work-hardened layer on the depth of cut “\(\delta\)” is graphically presented in Drawing No. 32. The designations used in the tables are clear from the diagram in Drawing No. 33.

Drawing No. 33.

Table No. 17

\[ \alpha = 45^\circ \]

No. in order	Specimen No.	\(\delta\)	\(\delta_2\)	\(S_2\)
1	74	0,35	0,3	1,5
2	73	0,8	0,5	2,6
3	36	1,0	0,6	3,5
4	101	1,6	0,9	5,0
5	106	1,75	1,8	3,5
6	105	2,2	1,2	6,8
7	107	4,0	3,8	16,5

Table No. 18

\[ \alpha = 55^\circ \]

No. in order	Specimen No.	\(\delta\)	\(\delta_2\)	\(S_2\)
1	79	0,4	0,25	1,5
2	75	0,5	0,7	2,7
3	4	1,2	0,5	1,9
4	24	1,8	1,5	6,0
5	1	2,1	1,0	3,5
6	166	2,5	1,7	8,0
7	2	2,6	1,8	—
8	3	3,2	2,25	8,5
9	23	4,1	3,5	12,5

Table No. 19

\[ \alpha = 65^\circ \]

No. in order	Specimen No.	\(\delta\)	\(\delta_2\)	\(S_2\)
1	67	0,6	0,8	4,2
2	113	1,85	1,7	7,0
3	137	2,1	1,8	8,5
4	114	2,7	2,1	9,0
5	17	4,3	3,5	14,0

Table No. 20

\[ \alpha = 75^\circ \]

No. in order	Specimen No.	\(\delta\)	\(\delta_2\)	\(S_2\)
1	63	0,7	0,9	4,5
2	33	1,0	1,0	5,5
3	5	1,35	1,5	8,0
4	66	1,6	1,3	7,0
5	61	2,1	2,5	9,5
6	27	4,2	4,0	—

Table No. 21

\[ \alpha = 85^\circ \]

No. in order	Specimen No.	\(\delta\)	\(\delta_2\)	\(S_2\)
1	93	0.65	0.75	3.6
2	131	1.6	1.9	11.5
3	130	2.3	3.0	12.5
4	119	2.4	2.5	10.0
5	133	3.4	4.2	23.5
6	135	3.4	3.5	11.0

From the tables and graphs presented it is evident that the thickness of the work-hardened layer, \(\delta_2\), increases with increasing depth of cut and with increasing cutting angle, and, for large chips, reaches a very considerable magnitude, exceeding in its value, at large cutting angles, even the depth of cut. In exactly the same way, only still more rapidly, the volume of work-hardened metal located ahead of the cutter and of the chip being removed increases with increasing \(\delta\) and \(\alpha\).

Thus, the deformation ahead of the cutter extends far beyond the limits of the angle \(\beta\), and the “angle of action,” as limiting the spread of the cutter’s action upon the metal, loses its specific significance.

The question touched upon is of interest with respect to the propagation of deformations under the cutter in the machined surface, for the reason that in the most recent technical literature there are definite indications that preliminary mechanical treatment considerably lowers the fatigue limit of the metal.

At the present time the fatigue limit is gaining ever greater recognition as a criterion of the suitability of a metal for one or another practical application. Its lowering as a result of preliminary mechanical treatment by cutting is usually explained by the fact that more or less significant scratches and cracks remain on the machined surface, which, during the subsequent service of the article under repeated alternating loads, give rise to fatigue cracks. We present a table characterizing the change in the fatigue limit as a function of preliminary treatment\({}^{29}\) for steel with 0.49% C.

Surface condition	Fatigue limit, kg/cm\(^2\)	Decrease, in %, relative to finely polished
Finely polished	3500	—
Standard	3400	2.9
Ground	3200	8.6
Cleanly turned	3000	14.3
Roughly turned	2900	17.2

On the other hand, there are indications that work hardening of the metal likewise lowers the fatigue limit rather strongly.

It may be supposed that the lowering of the fatigue limit after treatment by cutting is a consequence not only of the presence of remaining cracks, but also of the presence of a surface more or less

less significant work-hardened layer of metal, which appeared as a result of preliminary machining and differs in its mechanical qualities from the base metal. That the work-hardened layer really does differ in its properties from the base metal is evident from the photograph No. 16 given earlier. Whereas the hardness (according to Herbert) of the base metal is 19.9, the hardness in the work-hardened layer reaches 28.4. Along with the change in hardness, naturally, the other mechanical properties also change.

We carried out the following experiment.

From a surface previously planed with a large depth of cut, three specimens for tensile testing were prepared, measuring \(120 \times 6 \times 0.7\) mm. Exactly the same three specimens were made from the same piece of iron, but from a surface not work-hardened by cutting. Both were prepared in exactly the same way: they were carefully cut out with a hacksaw, then filed with a personal file and polished with emery paper. Thus the influence of work-hardening during preparation should have affected both sets equally, and the difference in mechanical properties revealed in the subsequent test could be attributed only to the preliminary work-hardening during planing.

The tensile tests gave approximately the same value of ultimate resistance for both sets of specimens, but different values of relative elongation. The results are collected in Table No. 22.

Table No. 22.

No. in order	Characteristic	Ultimate strength, kg/mm² \(K_z\)	Relative elongation in %, measured over a length of 10 mm: division 1	Relative elongation in %, measured over a length of 10 mm: division 2	Relative elongation in %, measured over a length of 10 mm: division 3	Relative elongation in %, measured over a length of 10 mm: division 4	Relative elongation in %, measured over a length of 10 mm: division 5	Relative elongation in %, measured over a length of 10 mm: division 6	Relative elongation in %, measured over a length of 10 mm: division 7	Relative elongation in %, measured over a length of 10 mm: division 8	Total relative elongation over 80 mm	Note
1	Not deformed.	35.0	11	12	12.5	14	22.5	49	12	7	17.5%	The division at which rupture occurred is underlined.
2	Not deformed.	34.0	10	16	16	13.5	13.5	19	45	21	19.3%	The division at which rupture occurred is underlined.
3	Not deformed.	33.8	24	17	14	12.5	14	20	49	21	21.3%	The division at which rupture occurred is underlined.
4	Work-hardened by planing.	34.2	52.5	12	8	5.5	5.5	4.5	4	4	12.0%	The division at which rupture occurred is underlined.
5	Work-hardened by planing.	40.0	0	0	1	2	4	8	16	45	9.5%	The division at which rupture occurred is underlined.
6	Work-hardened by planing.	39.0	2	1	0.5	1	1.5	0	4	27	4.6%	The division at which rupture occurred is underlined.

From the table it is evident that specimens prepared from a layer previously work-hardened by cutting gave a much lower value of relative elongation than the others; moreover, this elongation is almost entirely concentrated at the place of rupture, while in the other divisions of the specimen it is very insignificant. In the non-work-hardened specimens the total elongation is much greater and is distributed more uniformly over the entire length of the specimen.

For comparison we give two diagrams (see drawing No. 34) taken during tension: one (A) for a specimen not work-hardened by cutting (No. 3 in Table No. 22), the other (B) for a specimen work-hardened by cutting (No. 6 in Table 22).

Thus the influence of work-hardening by cutting on the mechanical properties of the specimens is beyond doubt. It need only be noted that even the un-work-hardened specimens showed rather low values of relative elongation, in comparison with those required for normal iron, which is in all probability explained partly by the fact that, owing to the small thickness of the specimens (0.7 mm), the effect of the treatment in preparing the specimens made itself felt. Two more specimens of the same iron were prepared, with a cross-section of \(2.5 \times 6\) mm, which before rupture were subjected to annealing. They gave a normal value of relative elongation: 32.5% and 35%.

Drawing No. 34.

Thus, rough machining by cutting creates on the machined surface a work-hardened layer which differs markedly in its mechanical qualities.

Concluding this exposition, we shall briefly summarize the results obtained:

Brief summary.

I. In the cutting of a viscous material, all chips are divided according to their appearance into continuous chips and chips divided into elements. The latter, in turn, are divided into chips of sliding and chips of tearing-off.

II. The longitudinal shrinkage of the chip changes with changes in the thickness of the layer being removed, the cutting angle, the cutting speed, and the magnitude and direction of the forces acting on the chip. Shrinkage increases with an increase in the cutting angle and with a decrease in the thickness of the layer removed. The curve of the dependence of shrinkage on speed has a maximum. With the use of lubrication, shrinkage decreases.

III. The cause of longitudinal shrinkage is not only successive shears in the chip, but also its compression and, possibly, bending, the presence of which in the chip is beyond doubt.

IV. The formation of a continuous chip occurs by successive shears of the metal at a definite angle \((\beta_2)\) to the direction of motion of the cutter. The direction of the shears does not coincide with the direction going from the tip of the cutter to the point at which the machined surface passes into the chip. The shears occur simultaneously in many parallel planes.

V. The formation of separate elements of a chip divided into elements occurs similarly to the formation of a continuous chip, i.e. by shears of the metal at an angle \(\beta_2\). The division of the chip into elements occurs in the direction from the tip of the cutter to the point at which the machined surface passes into the chip, at a definite angle \((\beta_1)\) to the direction of motion of the cutter. In this direction (at the angle \(\beta_1\)) no deformation occurs in a continuous chip.

VI. Both the angle \(\beta_1\) and \(\beta_2\) are by no means constant quantities. They change very considerably with a change in the thickness of the layer being removed, and less considerably with a change in the cutting angle and cutting speed. With all other conditions constant, a change in the magnitude and direction of the forces acting on the chip causes a change in the angles \(\beta_1\) and \(\beta_2\). The change in the angles \(\beta_1\) and \(\beta_2\) is connected with, and proceeds in parallel with, the change in the coefficient of longitudinal shrinkage of the chip.

VII. The angle \(\beta_2\) is always greater than the angle \(\beta_1\), and the difference between them remains approximately constant.

VIII. The cause of the formation of irregularities (serrations) on the side of contact of the chip with the cutter and on the machined surface is the considerable force of friction between the chip and the cutter and the presence of a crack propagating ahead of the cutter tip.

IX. The irregularities on the machined surface are the greater, the larger the cutting angle and the thickness of the layer being removed.

X. An indispensable condition for the possibility of the formation of a built-up edge on the cutter (edge accretion) is the presence of serrations on the chip, one of which, owing to the considerable friction between the chip and the cutter, tears away from the chip, is welded to the cutter, and turns into a built-up edge. Thus the built-up edge is formed from metal belonging to the chip.

XI. Plastic deformation takes place not only within the limits of the “angle of action” \(\beta\), but extends far ahead of the cutter and beneath the cutter into the depth of the material being machined. On the machined surface there is created a deformed layer of metal, differing in its mechanical properties from the parent metal. The depth to which the deformation extends under the cutter and ahead of the cutter increases with an increase in the cutting angle and in the thickness of the layer being removed.

XII. With an increase in the cutting angle, both the degree of deformation of the metal and the volume of metal deformed ahead of the cutter increase.

All the conclusions relate to the case of iron chips.

In conclusion, we express our deep gratitude for assistance and useful advice to Professor T. I. Tikhonov, as well as to all who in one way or another contributed to the execution of our work.

The experiments were carried out in the laboratory of metal technology and metallography of the Siberian Technological Institute.

Literature referred to in the work

1. Experiences sur la resistance utile... par Coquillhat, Paris 1848–1849.
2. I. Time. Resistance of metals and wood to cutting. St. Petersburg, 1870.
3. Afanas’ev. — Determination of the force and its work in the formation of chips. Proceedings of the St. Petersburg Technological Institute. 1883–4, p. 265.
4. “Mémoir sur le rabotage de métaux” par M. Tresca. 1873.
5. K. A. Zvorykin, prof. — The work and force necessary for separating metallic chips. 1893.
6. Pogorzhel’skii. — Bulletin of the Polytechnic Society. 1907, p. 179.
7. Sokolov. — On the form of roughing cutters. Proceedings of the Don Polytechnic Institute. 1914, p. 1.
8. Usachev. — Phenomena occurring in the cutting of metals. Proceedings of the Petrograd Polytechnic Institute. 1925, p. 259.
9. Zadneprovskii. — Deformation and resistance of metals to planing. Proceedings of the Don Polytechnic Institute. 1918.
10. Klopstock. — Untersuchung der Dreharbeit. 1926. Klopstock. Werkstattstechnik, 1923.
11. Herbert. — E. G. Maschinenbau. 1927 — p. 991.
12. W. Rosenhain and Styrney. — The Proceedings of the Instit. of Mech. Ing. 1925. No. 2, p. 141.
13. Kirpichev. — Optical study of elastic deformations. Bulletin of the Society of Technologists. 1913, p. 1.
14. Coker C. G. — Proc. I. Mech. Eng. 1922, p. 567.
15. Coker C. G. — Proc. I. Mech. Eng. 1925, p. 357.

1. Zaitsev.—Optical Method for the Study of Stresses. 1927, p. 133.
2. Kurgein.—Osterr. Wochenschr. f. öffentliche Baudienst. Sept. 1905.
3. Ad. Fry. Stahl und Eisen. 1921 No. 32.
4. I. D. Jevons.—The Journal of Iron and Steel Inst. 1925. V. CXI. E. W. Fell. Carnegie Schol. Met. 16. (1927). S—101/29 Stahl und Eisen. 1928. S—1926. Kraftwirkung im Stahl.
5. Dobrovidov and Rosenberg. Detection of Plastic Deformations in Iron. Proceedings of the STI, vol. 48 (1928).
6. Arshiv für das Eisenhüttenwesen 1928 S—511. Die Wichtigsten Eigenschaften und die Theorie der Fliessfiguren Von Itaro Takaba und Katumi Okuda.
7. Savvin—Cutting of Metals and Tools, p. 4. 1923.
8. Rejtö. Einige Prinzipien der theoretischen mechanischer Technologie der Metalle. 1927. S—441.
9. Kronenberg. Grundzuge der Zerspanungslehre. 1927. S.
10. Hankins.—Proced. Inst. Mech. Eng. 1923.
11. Savvin.—Cooling of the Cutter.
12. Taylor.—The Art of Cutting Metals, p. 49. 1909.
13. Herbert. Engineering. 1927. Orga-Information. 1928. No. 5 p. 142. Cutting temperature.
14. N.T.K. N.K.P.S. Issue No. 68. Methodology of Mechanical Testing of Metals. 1928, p. 62.

“Izv. Siberian Technological Institute”
1929, vol. 51, issue IV

Responsible editor
Prof. M. I. Evdokimov-Rokotovsky

Misprints and Omissions

in the article by A. M. Rosenberg

“Experimental Study of the Process of Formation of Metal Chips”
(in Izvestiya STI).

Printed	Should read
p. 33. 15th line from the top: “…changes of ‘$\delta$’ we completely…”	“…changes with a change in the magnitude of ‘$\delta$’ we completely…”
p. 35. Under drawing No. 23 OMITTED:	Cutting speed $V$ in meters per minute.
Beside the drawing OMITTED:	$$K_0=\frac{I_1}{I_0}$$
p. 36. 5th line from the bottom, column IV: 4.61	14.61
p. 52. Beside drawing No. 32: thickness of the work-hardened layer “$\delta_2$”	thickness of the work-hardened layer “$\delta_2$” in mm.

To the article by Eng. A. M. Rozenberg.

Fig. 1. × 100

Fig. 2. × 19
(Specimen No. 9; $\alpha = 55^\circ$; $\delta = 0.4$ mm.).

Fig. 2-a.

To the article by Eng. A. M. Rozenberg.

Fig. 3. × 20

(No. 153; $\alpha = 55^\circ$; $\delta = 0.8$).

Fig. 3a.

For the article by Engineer A. M. Rozenberg.

Fig. 4. × 8
(No. 1; $\alpha = 55^\circ$; $\delta = 2.1$ mm.)

Fig. 4-a.

To the article by Eng. A. M. Rozenberg.

Fig. 5.
\[ (\text{No. }3;\ \alpha = 55^\circ;\ \delta = 3.2\ \text{mm}.) \]
\[ \times 8 \]

Fig. 5-a.

To the article by Eng. A. M. Rozenberg.

Fig. 6. × 8

(No. 86; $\alpha = 55^\circ$; $\delta = 1.15$ mm).

Fig. 7. × 19

(No. 87; $\alpha = 65^\circ$; $\delta = 0.26$ mm).

To the article by Eng. A. M. Rozenberg.

Fig. 8. × 12
\((\text{No. }38;\ \alpha = 65^\circ;\ \delta = 0{,}6\ \text{mm})\).

Fig. 8a.

\(\beta_2 = 35^\circ\)
\(\beta_1 = 17^\circ\)

To the article by engineer A. M. Rosenberg.

Fig. 9. × 12
(No. 90; \( \alpha = 65^\circ \); \( \delta = 0.95 \) mm.)

Fig. 9-a.

To the article by Eng. A. M. Rosenberg.

Fig. 10

\[ (\text{No. }114;\ \alpha = 65^\circ,\ \delta = 2.7\ \text{mm}.) \]

Fig. 10-a.

To the article by Eng. A. M. Rozenberg.

[Figure 11-a.]

\[ \beta_2 = 59^\circ,\qquad \beta_3 = 42^\circ \]

Fig. 11. (No. 17; \(\alpha = 65^\circ\); \(\delta = 4.3\) mm.) \(\times 6\).

To the article by Eng. A. M. Rozenberg.

Fig. 12.

Fig. 13.

Fig. 4.

To the article by Eng. A. M. Rozenberg.

Fig. 15. × 60

Fig. 14. (No. 23; $\alpha = 55^\circ$; $\delta = 4.1$ mm). × 20

To the article by Engineer A. M. Rosenberg.

Fig. 16-a.

Fig. 16-b.

Fig. 17. $\times 22$

For the article by Eng. A. M. Rozenberg.

Fig. 18. × 22

(No. 88; $\alpha = 85^\circ$; $\delta = 0.1$ mm.)

Fig. 19. × 0.8

To the article by Eng. A. M. Rosenberg.

Fig. 20. × 12
(\(No.\ 68;\ \alpha = 75^\circ;\ \delta = 0.4\ \mathrm{mm}\).)

Fig. 21. × 18
(\(No.\ 96;\ \alpha = 85^\circ;\ \delta = 0.65\ \mathrm{mm}\).)

To the article by Eng. A. M. Rosenberg.

Fig. 22.
(No. 131; \(a = 85^\circ\); \(\delta = 1.6\) mm).
\(\times 8\)

Fig. 23.
(No. 117; \(a = 85^\circ\); \(\delta = 2\) mm).
\(\times 5\)

To the article by Engineer A. M. Rosenberg.

Fig. 24.
(No. 60; \(\alpha = 75^\circ\), \(\delta = 2.2\) mm). \(\times 8\)

Fig. 25.
(No. 6; \(\alpha = 75^\circ\), \(\delta = 1.7\) mm). \(\times 8\)

To the article by Eng. A. M. Rosenberg.

Fig. 27.
(No. 3; $\alpha = 55^\circ$; $\delta = 3.2$ mm).
$\times 2.5$

Fig. 28.
(No. 23; $\alpha = 55^\circ$; $\delta = 4.1$ mm).
$\times 2$

Fig. 26.
(No. 134; $\alpha = 85^\circ$; $\delta = 1.1$ mm).
$\times 12$