Deformation of Metals During Rolling

Deformation of Metals during Rolling

BY

I .YA. TARNOVSKII, A.A.POZDEYEV andV.B.LYASHKOV

TRANSLATED BY

M DE 0.TOLLEMACHE

TRANSLATION EDITED BY

A. SHUTT

P E R G A M O N P R E S S OXFORD · LONDON · E D I N B U R G H NEW YORK

PARIS · FRANKFURT

Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. 1

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 Pergamon Press Inc., 122 East 55th St., New York 22, N.Y.

Pergamon Press GmbH, Kaiserstrasse 75, Frankfurt-am-Main

First English edition 1965

Library of Congress Catalog Card No. 63-10070

This book has been translated under the auspices of the Department of Scientific and Industrial Research, British Iron and Steel Research Association. It is a translation

of the Russian book ,TJe<j)opMamiH MeTajina npH üpoicaTKe by M. Ά. TapHo6-CKHH, A. A. Πο3ΛββΒ and B. B. JIAIHKOB, published by

MeTanjiyprH3 aT, CBepanoBCK

1353

To the undying memory of

our beloved teacher PROFESSOR AKIM FILIPPOVICH GOLOVIN

Preface

THE present theory of longitudinal rolling makes wide use of the hypothesis of flat cross-sections. According to this, flat transverse-vertical cross-sections before deformation remain flat both in the zone of deformation and after rolling. This model of the rolling process has been used as the basis for various formulae and from it a number of general conclusions have been drawn.

This theory of rolling has been generally accepted until recently, but in the last few years many works have appeared dealing with the investigation of the inhomogeneity of deformation during rolling and with various aspects of its manifestation. These include in particular the zones of adhesion, diagrams of the velocities of the metal in the zone of deformation, etc. A discussion on I. M. Pavlov's theory of rigid ends which was organised by the journal "Stal" and took place during the years 1951-1953 revealed among research workers diametrically opposed points of view on the fundamental problems of the theory of rolling. These problems, which have not only scientific, but also great practical importance, especially for the rolling of billets and sections, can be solved only by investigating the general features of the great quantity of experimental data and by using the basic hypotheses of present-day mechanics of continuous media.

In the present book a large quantity of new, experimental data on the deformation of metal during rolling have been described and analysed; in our opinion these data go some way towards clearing up a number of disputed problems of rolling theory. Apart from the steady-state rolling process, consideration is also given to the conditions of deformation of the metal during the initial bite on the stock by the work rolls and the gradual transition to the steady state rolling process.

The book also gives theoretical solutions of individual problems based on the fundamental hypotheses of the mechanics of continuous media; in doing so experimental data have been used for the construction of a model of the deformation which is close to the real conditions

xi

XU PREFACE

of the rolling process, and also for the determination of the boundary conditions.

The data presented show that existing ideas and solutions of the basic problems of rolling theory (the conditions of the bite on the stock by the work rolls and of the established rolling process, the role of the frictional forces during rolling, formulae for the determination of the position of the neutral cross-section, of forward slip, of spread, of the stresses, etc.) do not give a complete picture of all the actual conditions of the rolling process, and are valid only as a first approximation, while in a number of cases, especially during the rolling of sufficiently thick stock, they must be considered useless. Further development of a rolling theory which takes into account the fundamental, real factors of the deformation of the metal, will bring about an improvement in technological rolling processes, and the introduction of a progressive technology.

The present book is the outcome of a series of investigations carried out under the direction of I. Ya. Tarnovskii. Chapter IV was written by I. Ya. Tarnovskii and V. N. Trubin, while the remaining chapters were written by I. Ya. Tarnovskii, A. A. Pozdeyev and V. B. Lyashkov.

The authors express their gratitude to the reviewer of the book, Professor T. M. Golubev, and to the editor, M. A. Zaikov, for their comments on the contents and the layout of the manuscript.

The Authors

CHAPTER I

The Hypothesis of Flat Cross-sections in Investigations of the Flow

of Metal during Rolling

1. THE HYPOTHESIS OF FLAT CROSS-SECTIONS AND THE

"THEORY OF RIGID ENDS"

For a very long time, the flow of metal during rolling has attracted the attention of research workers in the study of the rolling process, which is explained by the scientific and practical importance of the problem. On this subject investigators of the rolling process have expressed various opinions, and in many instances these expressions of opinion were not supported by theoretical or experimental bases. In the first quarter of the present century the opinion was current that the longitudinal velocities of the particles of metal in any given vertical cross-section of the zone of deformation are not identical (A. P. Vino-gradov,7 V. Ye. Grum-Grzhimailo,13 A. F. Rodzevich-Belevich41 and others).

Such a concept is tantamount to the assertion that different layers of the stock throughout its thickness are deformed differently on any given section of an element of the length of the zone of deformation.

In 1927, I. M. Pavlov propounded a new theory of the flow of metal during rolling,32 which subsequently acquired the name "the theory of rigid ends". According to the definition of the author of this theory, its essence is as follows: during an established rolling process the metal deformed within the roll gap is linked with the front and back ends of the stock, which at a given moment are not subjected to the direct operation of the force exerted by the rolls, so that during the study of the rolling process a start must be made with the principle of the interaction between the zone of deformation and the rigid ends. "This principle consists in the determination of the

1

2 DEFORMATION OF METALS DURING ROLLING

connection between the body being worked in a given mechanism as a whole and that part of it which at a given moment passes through the zone of deformation mentioned above, which is bounded in a known manner, , . .".3 2

This principle was sufficiently fruitful, since in essence it amounts to the formulation of the limiting conditions at the junctions between the zone of deformation and the rigid ends, whereby the limits of the extent of the zone of deformation need not necessarily coincide with the limits of the entry of the stock into the rolls and of its exit from them.

I. M. Pavlov formulated the "conditions at the entry and the exit", in accordance with which in the sections forming the limits of the junction between the zone of deformation and the rigid ends, the longitudinal velocities of the particles of metal are identical. The evident truth of these conditions seems to us indisputable and completely sound, but in the absence of identification of the limits of the extent of the plastic deformation of the metal with the geometric planes of the entry of the stock into the rolls and of its exit from them. During numerous investigations in the last two decades these conditions were successfully used, and up to the present time they have caused no objection of any sort.

Meanwhile during the course of the further development of the theory of rigid ends, a certain reassessment of the relationship of the flow of the metal in the zone of deformation from the rigid ends was permitted, which was expressed in the assertion that in any given transverse vertical section of the zone of deformation the longitudinal velocities of the displacement of the particles of metal are identical.

From this hypothesis it follows directly that along the whole length of the contact surface between the stock and the rolls slip takes place in the longitudinal direction, and also that the vertical deformation of the various layers of the thickness of the stock in the zone of deformation can be different only due to inhomogeneity of the transverse deformation, which very much restricts the inhomogeneity of the vertical deformation. This hypothesis, and likewise the conclusions which follow from it, brought a series of objections from investigators of the rolling process, and was repeatedly the subject of criticism in print.

A. F. Golovin considered that the longitudinal velocities of the particles of metal in any given transverse vertical section of the zone of deformation are different, and on the contact surface between the

FLAT CROSS-SECTIONS IN INVESTIGATION OF FLOW OF METAL 3

stock and the rolls, apart from the zones of slip which border on the boundaries of the entry and exit of the stock from the rolls, there is also a zone of adhesion, located in the central part of the arc of contact. Apart from this, A. F. Golovin also proposed an approximation of the equation for the determination of the extent of the zones of longitudinal slip and adhesion.8

N. A. Sobolevskii came to the conclusion that during rolling along the whole length of the contact surface between the stock and the rolls there is no slip,42 which was equivalent to the assertion that the vertical, and likewise the longitudinal, deformation of the various elements of the thickness of the stock over any given section of the length of the zone of deformation, and consequently the longitudinal velocities of the displacement of the particles of metal in any given vertical section of the zone of deformation, are also different.

Ye. V. Pal'mov during research into the flow of metal during rolling came to the conclusion that in the zone of deformation there are "sub-surface currents of metal", which likewise amounted to an assertion of the inhomogeneity of deformation of the metal during rolling, and inequality of the longitudinal velocities of the particles of metal in any given vertical section of the zone of deformation.35

The opinions set out above and the conclusions of the various investigators are based, for the most part, on representations of the kinetics of the rolling process.

A. I. Tselikov, when investigating the distribution of (specific) pressure along the contact surface between the stock being rolled and the rolls, came to the conclusion that on this contact surface the existence of an adhesion zone is possible.54 This conclusion on the basis of an analysis of the stress reaction between the rolled metal and the rolls was first investigated by A. I. Tselikov. He pointed out the nature of the relationship of the extent of the zone of adhesion to the parameters of the zone of deformation. Apart from this, in the works of A. I. Tselikov, graphs are given of the longitudinal velocities of the movement of the metal particles in various parts of the zone of deformation.55

Thus there is no unanimity in opinions on the kinetic and stress conditions of rolling, and concerning the basic problems of rolling theory, there were, and still are, differing opinions.

On the basis of experimental data obtained during the last 6 or 7 years, in Russian technical literature there has been wide discussion. The literature data published earlier, and also the numerous experi-


mental data set out in the present book, convincingly show that far from all the conclusions drawn from the theory of rigid ends may be accepted as valid.

The theory of rigid ends in the mechanical working of metals is to a certain extent analogous with the hypothesis of flat cross-sections, which has found extensive use in the study of the strength of materials. In accordance with this theory flat vertical cross-sections, during the bending of cylinders for instance, are not curved, but remain flat. This hypothesis is tantamount to the assumption that there are no grounds for the flexure of the flat vertical cross-sections during deformation, i.e. that there are no shear forces (pure flexure).

In a series of instances such an assumption should be considered a broad one. On the other hand more accurate methods of calculation of the theory of elasticity show that under certain conditions the use of the approximate theory of bending is permissible. Both these methods give extremely close results during the calculation of cylinders, if the ratio of the height of the cylinder to its diameter is greater than 5. Under other conditions the hypothesis of flat sections must be rejected, since it gives obviously false solutions, for instance during the deformation of a short cylinder, of an elastic parallelepiped, etc.

The condition of the equality of the longitudinal velocities of the displacement of the metal particles in any given vertical section of the zone of deformation, figuring in the theory of rigid ends, as one of the results of the basic condition of this theory, also means that the transverse vertical sections, which were flat before deformation in the rolling process, are not curved, i.e. they remain flat. In this sense it is possible to talk of the analogy between the theory of rigid ends in the mechanical working of metals, on the one hand, and the hypothesis of flat cross-sections in the study of the strength of materials on the other. Therefore the condition of the equality of the longitudinal velocities of the movement of the metal particles in any given vertical cross-section of the zone of deformation we shall in the subsequent text refer to conditionally as the "hypothesis of flat cross-sections" in the rolling theory.

It should be noted that in modern rolling theory almost all the conclusions which define the kinetic and stress conditions of rolling, are obtained by means of the hypothesis of flat cross-sections. Therefore it is natural to attempt to find the range of application of this hypothesis, similarly to the way in which this is done in the study of the resistance of material and in the theory of elasticity.

FLAT CROSS-SECTIONS IN INVESTIGATIONS OF FLOW OF METAL 5

It is known that a large number of solutions in the theory of rolling and upsetting are obtained for flat deformation. Let us consider the flat deformed state during upsetting between flat parallel plates and during rolling between smooth rolls. For analysis we shall use the differential equilibrium equations appropriate for a given flat medium

dax . drxv day drxy _ dx dy ' dy dx

and the relationship of the velocities of the shear deformation

(dux duy\ rxy = GYxy = G(— + — J,

where x, y are the co-ordinates of the point on the body under review;

ux and uy are the velocities of the point in the horizontal and vertical directions;

ax and ay are the normal stresses in the directions of these velocities;

xxy is the shear stress in the plane under review; and G is the plasticity modulus of the deformed material.

Thus during any real process of deformation there arise such stresses, the calculated value of which on substitution in the equilibrium equation will satisfy the latter identically. Under such circumstances the shear stresses are proportional to the shear velocities, and the plasticity modulus is a variable which may have a value between zero and infinity.

In the first instance (upsetting), if the hypothesis of flat cross-sections is correct, firstly shear stresses do not arise, and secondly the normal stresses ax änderbare independent of the relevant co-ordinates x and y. Since the application of the hypothesis of flat cross-sections means that ux is independent of y, and uy is independent of x, then

dux _ duy _ 0

dy dx therefore rxy = 0.

Thus in the equilibrium equations there remains only one component

ex dy DMR 2


whence it also follows that σχ is independent of x, and ay of y. In the actual process this condition, as a rule, is not fulfilled, and consequently, strictly speaking, the hypothesis of flat cross-sections is not valid.

Let us now consider the second instance, rolling between smooth rolls. Here the hypothesis of flat cross-sections amounts to the assertion that there is equality of the longitudinal velocities of the flow of metal in any given transverse vertical section of the zone of deformation. Under such circumstances no curvature of the transverse cross-sections takes place, but only the longitudinal horizontal cross-sections are curved. In fact

4 ^ = 0, 4^4=0. oy ox

In reality dujdy Φ 0 and is completely comparable with thevalue diiy/dx, in so far as the transverse cross-sections are curved during rolling. In consequence of the fact that the value of the shear stress is determined by the total of the relative shears along both axes,

'--*(£ + &) the calculated values of rxy obtained from the hypothesis of flat cross-sections will be considerably reduced.

Let us assume that the normal stresses are found exactly, and the shear stresses are determined from the hypothesis of flat cross-sections, i.e. understated. If the two sets of values are substituted in the equilibrium equations, these equations will not be satisfied.

Therefore the methods adopted in the theory of the mechanical working of metals for calculations by means of the hypothesis of flat cross-sections cannot give an accurate solution. But this does not by any means imply that under any given conditions of deformation it is impossible to use the hypothesis of flat cross-sections. Thus, just as in the strength of materials, under certain conditions these solutions will approach the true ones. In so far as at the present time accurate theoretical solutions applicable to different technical processes of the working of metals do not exist, then the results of the calculations based on approximate equations do not prove possible to compare with accurate theoretical equations. In this connection for verification of the approximate equations recourse should be made to experiment. By comparing the experimental data with the calculated, it is possible


to determine the sphere of application of the hypothesis of flat cross-sections.

Experiments show that the hypothesis of flat cross-sections is frequently not observed. On the contact surface, alongside the slip sections, there occurs a zone of adhesion of the metal to the rolls, and in the contact layers a zone of restricted deformation.

The longitudinal velocities of the contact surface are considerably different from the velocities of the central particles, i.e. considerable bending of the transverse sections takes place.

In certain instances, however, these phenomena are not clearly expressed, the zone of adhesion is compressed almost into a line coincident with the neutral cross-section, and the longitudinal velocities in any given vertical section of the zone of deformation are almost identical. Therefore, there is no foundation for completely rejecting the hypothesis of flat cross-sections.

The use of the hypothesis of flat cross-sections does not give rise to any objections in instances where the work done against the forces on the contact surface, brought about by external friction, appreciably exceeds the work done against internal forces (stresses).

As will be shown below, the whole of the work done in plastic deformation is expended in overcoming the internal and external forces. During upsetting between plane parallel plates or rolling between smooth rolls the external forces represent the forces of external friction. Since, in accordance with the circumstances of the mechanics of solid bodies, the total work (more precisely, its variation) determines the states of stresses and deformations, it may be asserted that the states of stresses and deformations are determined by the ratio of the amounts of work done in overcoming external forces and internal forces. These amounts of work can be calculated.

The work done in overcoming the force of external friction,

^τ = ffruTdS, V

where τ is the specific frictional force, directed along the surface of contact with the tool;

uc is the displacement in this direction; and S is the contact surface area.

As will be shown below the work done in overcoming internal stresses ABH= fjJTsrdV,

* V

2*


where rs is the shear stress; and Γ is the intensity of the shear stress.

By studying experimentally the distribution of deformation, it is possible to assess the value of the work done in overcoming the internal and external stresses. Experiments show that on increasing the ratio of the length of the zone of deformation to its thickness (llHmean), the proportion of the work done in overcoming external forces increases, and the supplementary work done in overcoming the internal stresses becomes relatively small, and it need not therefore be considered, although the deformation may not in fact be completely homogeneous, for instance in the rolling of thin sheet and strip. On the other hand in practical calculations in this instance use may be made of the hypothesis of flat cross-sections, since the distribution of the deformation is determined above all by the work done in overcoming the internal resistance of friction.

On studying the phenomena of slip and adhesion, and the inhomo-geneity of the deformation, it is possible to determine the conditions under which the work done in overcoming the forces of friction preponderates over all other factors, and thereby establish the zone of applicability of the hypothesis of flat cross-sections.

The hypothesis of flat cross-sections in the theory of mechanical working of metals, as in the strength of materials, has distinct limits of application; the accurate determination of these limits, and the production of new solutions within the limits of applicability of the hypothesis continues to remain one of the important problems of rolling theory.

Below, on the basis of experimental data, are shown the limits within which the hypothesis of flat cross-sections can give satisfactory results, although, understandably, the authors do not recommend them as absolute, and understand that this question should be studied in greater detail and accurately in the future.

In the approximate solutions of certain problems from time to time it is necessary to determine the distribution of the stresses or strains. For this purpose it is necessary to take into consideration the basic hypothesis of the theory of rigid ends. For instance the function for the strain should be chosen in such a way that on the boundary surfaces between the physical zone of plastic deformation and the rigid ends the longitudinal deformations are equalised. Consequently the condition of the equalisation of the velocities should be used in solutions as an abnormal limiting condition.


Let us consider further the question of the relationship between the concept of homogeneous (ideal) deformation and the hypothesis of flat cross-sections.

In the theory of rolling of metals, deformation without curvature of the transverse cross-section (hypothesis of flat cross-sections) is frequently identified with homogeneous deformation. In the future it will be necessary for us to use certain basic conditions, both of the hypothesis of flat cross-sections, and also of the theory of homogeneous deformation. Let us here consider this question in greater detail.

The special feature of homogeneous deformation, the concept of which in the theory of rolling of metals was first of all propounded by A. F. Golovin,8 consists in the fact that the surfaces of the deformed body are not curved, slip takes place over the whole contact surface, the deformations are proportional to the linear dimensions of the body, and the projections of the full velocity of a particle are linear functions of the relevant co-ordinates:

νχ = φ(χ); vy = φ(γ); vz = φ(ζ). The conditions of the existence of homogeneous deformation, as

propounded by I. Ya. Tarnovskii44 amount to the fact that homogeneous deformation is possible under a linear state of stress.

This state is ensured by the isotropicity of the body, identical absolute and relative reductions at any given point on the contact surface and the absence of rigid ends and frictional forces.

Homogeneous deformation and the theory of flat cross-sections are scientific abstractions. For creating conditions for homogeneous deformation a large measure of approximation may be used, the degree of which will depend on the experimental technique.

The achievement of homogeneous (ideal) deformation during rolling is in principle impossible, since the decrease in the forces of internal friction limits the possibility of the process taking place.

On the other hand, for the sake of simplicity, in the subsequent text we shall nevertheless retain the term "homogeneous deformation" for the instance of rolling within the limits of application of the hypothesis of flat cross-sections.

2. THE PRACTICAL AND THEORETICAL IMPORTANCE OF THE STUDY OF THE INHOMOGENEITY OF DEFORMATION DURING ROLLING

The development of processes of rolling metals creates an urgent necessity for the solution of a series of important technical questions.


In the sphere of rolling production, for instance, the following questions may be noted. In connection with the necessity for further stepping up schedules of reduction, in many instances it is important to know the conditions of the bite on the stock by the work rolls. The treatment of this question in present rolling theory is based on the assumption of homogeneous deformation, in so far as on the contact surfaces between the stock and the rolls the conditions are taken to be suitable only for the instance of contact slip along the whole surface of contact.

At the same time there are grounds for asserting that during partial filling of the gap between the rolls by the metal the slip friction develops only very slightly, the adhesion on the contact surfaces between the stock and the rolls is extremely limited, and on a great part of the contact surfaces static friction (friction at rest) takes place, and the metal of the stock adheres to the surface of the rolls.

It is a reality, therefore, that practice in many instances does not confirm the conditions of the bite on the stock by the rolls in that form in which they are explained in current rolling theory.

From rolling theory it is known that the initial bite on the stock by the rolls is only possible in the event that the angle of bite is less than the angle of friction, and after the metal has filled the roll gap, the rolling process can take place so long as the angle of bite is less than double the angle of friction. On this basis in rolling theory a series of important conclusions are drawn.

But experiments far from often confirm these assertions, and the conclusions which are drawn from them. It is enough to recall the experimental data of A. A. Presnyakov,36, 37 which in our opinion indisputably show the necessity for introducing substantial corrections into the theory of the bite on the stock by the rolls and of the conditions for establishing the rolling process. It seems to us that the working out of the conditions of the bite and the establishment of the rolling process should be based on observation of the actual state of stresses in, and deformation of, the metal.

In practice, during the filling of the roll gap by the metal the deformation is extremely inhomogeneous and is not a steady-state process, i.e. the state of stresses and deformations in any given area within the volume of the zone of deformation is dependent on the time during which the rolling process is not established, and adhesion on the contact surface is a consequence of the inhomogeneity of the deformation.


From this it follows that during a study of the conditions of the bite on the stock by the rolls and of the conditions of setting up the rolling process, it is necessary to take into account the inhomogeneity of the deformation, especially with the parameters of the zone of deformation which normally occur during rolling in blooming and section mills.

Further, let us briefly review the solution of questions of change in shape in present rolling theory. Here we are talking about calculation equations for spread, forward slip and backward slip. From literature sources on rolling we are familiar with numerous equations of various authors for calculating spread during rolling of stock with a rectangular cross-section between smooth rolls.

Naturally the equations which have appeared during the last 10-15 years are more accurate than previous equations for calculating spread, since in them is built up the relationship between spread and a larger number of rolling factors than in earlier equations, especially the relationship between spread and external friction.

But it must be noted that in all the published equations for calculating spread no account is taken of the inhomogeneity of deformation over a cross-section. Meantime every rolling mill operator knows, for instance, that spread of the stock is not the same throughout its thickness. Most frequently of all during rolling between smooth rolls the side surface of the stock bulges, acquiring the shape of a barrel, and the maximum spread is produced on the central horizontal layer in relation to height, and the minimum on the contact horizontal layers. Under other circumstances, however, where the parameters of the zone of deformation are different, especially during the first passes when rolling heavy ingots in a blooming mill, so-called double barrel formation occurs, when the maximum spread occurs in the contact horizontal layers, and the minimum in the central layers of the stock in relation to height. Sometimes no spread occurs at all in these central layers, or even has a negative value.

There is no doubt but that all these phenomena are the consequences of inhomogeneous deformation; not one of the equations for calculating spread during rolling, however, takes these important factors into account.

Moreover, in equations for calculating spread insufficient account is taken of the actual mechanism of transverse deformation. In reality, experiments show that during rolling of relatively thick stock, or more precisely under certain relationships of the parameters of the


zone of deformation which characterise section and billet rolling, spread on the contact surface of the stock takes place mainly due to the transition of the side surfaces into the contact surfaces, and not due to contact slip in the transverse direction. Or the contrary transverse adhesion occurs along almost the whole width of the rolled stock.

All this is the result of inhomogeneous deformation, and in the existing equations for calculating spread this is not taken into account in any sense at all. It is quite obvious that account must be taken of the inhomogeneity of transverse deformation in continuous relation to the vertical and longitudinal deformation.

The method of determining forward slip, which seems to many authors to be simple and clear, in reality suffers from grave disadvantages. Regardless of the fact that the equation for calculating forward slip is normally derived from the condition of constancy of a second volume of metal passing through any given vertical section of the zone of deformation, and this hypothesis remains in force during both homogeneous and also inhomogeneous deformation, the calculation of forward slip during section and billet rolling of course frequently gives false results. This is explained by the fact that forward slip is dependent on the position of the neutral cross-section, which is determined by the value of the neutral angle, and a certain equation for its determination is derived from the condition of homogeneous deformation.

It should generally be emphasised that during the derivation of an equation for the determination of the value of the angle of the neutral cross-section assumptions are made that there is no spread, and that the pressure remains the same over the width and length of the zone of deformation. The tangential forces on contact are always regarded as forces of sliding friction, and it even remains unnoticed that this is an assumption.

Experiments and calculations show that calculation of the change in the pressure along the contact surface, and also the calculation of the spread only slightly affects the calculated value of the angle of the neutral cross-section. But correct determination of the contact factional forces, which in the sphere of the zone of adhesion are subjected to another law than during sliding friction, has great importance.

Since the zone of adhesion is the result of inhomogeneity of deformation during rolling, then we come to the conclusion that the equations existing in rolling theory for the determination of the angle of the neutral cross-section are valid only for instance during rolling


when the inhomogeneity of deformation is small, for instance during the rolling of thin sheet. Under other parameters of the zone of deformation, on the other hand, when the zone of adhesion becomes considerable in extent, the equation for the determination of the angle of the neutral cross-section, obtained subject to the condition of slip along the whole length of the zone of deformation, gives unsatisfactory results. In this connection the calculation of forward slip, where inhomogeneity of deformation develops appreciably, likewise does not agree with experimental results.

Meanwhile under the working conditions of modern high-productivity continuous rolling mills there are increased demands for the calculation of forward slip. These demands have grown still further in connection with the development of longitudinal rolling of sections with varying cross-sections.

During the rolling of relatively thick stock, or more exactly at small values of the ratio of the length of the zone of deformation to the mean thickness of the stock, forward slip determined by experiment is considerably greater than the calculated value.

In order to bring the calculated value of forward slip into line with experimental data it is customary to take a higher value for the coefficient of external friction.

But by doing so, the coefficient of external friction, which has a limited physical sense, is turned into a correction coefficient, which determines the inhomogeneity of deformation and the presence of the zone of adhesion during rolling.

It seems to us that such a procedure for solving problems of forward slip during rolling cannot be considered satisfactory, since the concrete study of deformation of the metal is in this instance replaced by an artificial increase in a physical value which is already completely fixed for the given conditions, i.e. the coefficient of external friction. Hence it follows that the study and calculation of deformation has great importance for the determination of forward slip.

In connection with what has been mentioned above, we consider it necessary to dwell on the question of the experimental determination of the coefficient of external friction. The practical importance of a knowledge of the value of the coefficient of external friction for sundry technical calculations under varying conditions of the pressure rolling of metals is obvious and unquestionable. But in these experiments the coefficient of external friction is not obtained directly, but by means of its calculation on the basis of the equation for the angle of the neutral


cross-section, which is determined in turn from the experimental value of the forward slip.

It has also been pointed out above that certain familiar equations in rolling theory for the angle of the neutral cross-section are derived for the condition of slip along the whole of the contact surface, and do not take into account inhomogeneity of deformation and the presence of a zone of adhesion connected with the latter. Therefore during such experiments a sharp increase is normally produced in the coefficient of external friction with the decrease in the ratio of the length of the zone of deformation to its mean thickness, i.e. during the rolling of thick stock. Such results are presented in a series of investigations.8, 58

The actual coefficient of external sliding friction should not depend on the parameters of the zone of deformation, and the results obtained from the investigations, in our opinion, are the result of the fact that during analysis of the experimental data no account was taken of the inhomogeneity of deformation, which has great importance at small values of the ratio of the length of the zone of deformation to its mean thickness. On the other hand, in the event of rolling with a high ratio of the length of the zone of deformation to its mean thickness, for instance during the rolling of thin sheet, when the inhomogeneity of deformation and, consequently, the zone of adhesion are small, the determination of the coefficient of external friction from the experimental value of the forward slip should be considered completely reliable and correct. Below this question is reviewed in greater detail with quantitative analysis of sundry experimental data.

Finally let us note that it is especially necessary to consider the inhomogeneity of deformation during the rolling of heavy ingots in a blooming mill. In this instance the formation of internal and external defects is caused, basically, by the inhomogeneity of deformation under otherwise equal conditions. The difference in the signs of the state of stresses in appreciable parts of the volume of the zone of deformation can under definite conditions lead to the formation of both internal, and also external, cracks. Under such circumstances the internal cracks are in the majority of instances welded up during subsequent rolling, while the external cracks either increase the amount of work in cleaning the surface of the billet, or increase the quantity of scrap.

Therefore there seems to be every justification for a most thorough investigation of the inhomogeneity of deformation at extremely small ratios of the length of the zone of deformation to its mean height,


which is characteristic of the first passes during the rolling of heavy ingots in a blooming mill.

Questions of the inhomogeneity of deformation have exceptionally great importance during the rolling of metals and alloys with low ductility. It is known that one and the same metal or alloy, dependent on the diagram of the state of stresses, can in one set of instances display its plastic properties sufficiently well, while in other instances cracks and bursts will occur. In order to prevent the formation of defects, in practice a series of measures are adopted, for instance the use of very tight roll pass designs, etc., which in the final result bring about equalisation of the strains and of the stresses. Especially dangerous are those sections of the zone of deformation in which tensile stresses operate.

In order to tackle more successfully the defects produced during the mechanical working of metals and alloys of low malleability, it is necessary to study in detail the causes of the inhomogeneity of deformation and the factors which influence it, on the basis of which it is possible to work out measures for preventing the defects. It should be added to what has been said, that of great importance also are the chemical composition of the alloy, the presence of various harmful impurities even in small quantities, the structure of the ingot, the regimes of temperature and speed of rolling, and especially the temperature distribution over the cross-section of the ingot. All these factors must be reviewed in conjunction with the mechanical system of shaping the metal.

Even a brief enumeration of some of the problems of the theory and practice of rolling shows the importance of the study of the inhomogeneity of deformation during rolling.

Without exaggeration it may be said that the basic hypotheses of modern rolling theory are founded on the hypothesis of flat cross-sections, and can therefore be regarded as correct and suitable for practical calculation only in so far as, and in circumstances where, the hypothesis of flat cross-sections is more or less acceptable. Under other circumstances, for example, during billet and section rolling, the hypothesis of flat cross-sections cannot be regarded as acceptable, and the further development of rolling theory should be carried out with due account of the inhomogeneity of deformation during rolling.

CHAPTER II

The Theory of Inhomogeneous Deformation

3. THE INHOMOGENEITY OF DEFORMATION DURING ROLLING AND ITS MANIFESTATIONS

In the most common instance during rolling tri-axial deformation occurs, during which the maximum deformation is vertical. Therefore we shall start the review of the inhomogeneity of deformation with reduction.

In the various sections of the length of the zone of deformation the degree of reduction will be different. The gradual growth in the draft

FIG. 1. Diagram for the determination of the increasing draft during rolling.

of the stock is determined by the geometry of the tool. Thus during the rolling of stock between smooth rolls the increasing draft of the stock at any given point along the length of the zone of deformation may be written (Fig. 1) as

ΔΧΗ = ΔΗ- (Ηχ- Η,), 16

THEORY OF INHOMOGENEOUS DEFORMATION 17

but (/ - x)2 I2

* - * - ^ - £ ( · - τ Η " ( ' - τ ) . therefore

ΔΧΗ = ΔΗ 1 ('-τΠ (1)

Let us consider the vertical deformation of any given horizontal layer of an element, the initial thickness of which is equal to A0, and the final thickness hl9 and the increasing draft of this layer Zl* A, for which we introduce the concept of the "relative thickness of a layer of an element"; this represents the ratio of the thickness of a layer of an element under review to the thickness of the whole of the strip in a given vertical cross-section. Thus the relative thickness of a layer of an element before entry into the rolls, or more accurately before deformation, is equal to h0IH0, after exit from the rolls, or more accurately after deformation, to h1/H1, and in any given vertical cross-section of the zone of deformation to hJHx.

In the event of plane strain, i.e. in the absence of spread, the relative thickness of a layer of an element of the metal after the pass remains equal to the relative thickness of this layer before the pass

H0 H1

This follows from the law of constancy of the volume of the metal being deformed or from the "rule of closed contours" of I. M. Pavlov.33 In actual fact we shall rewrite this equation in the form

hg = H0

Ai # i '

From the equation it follows that the coefficient of reduction of any layer of an element is equal to the coefficient of reduction of the whole strip. In the absence of spread the coefficients of reduction are equal to the coefficients of elongation.

So far as the relative thickness of a layer of an element in any transverse vertical section of the zone of deformation is concerned, it remains constant only in the event of homogeneous deformation

A0 h1 hx

H0 H1 Hx


or

K Hx '

During inhomogeneous deformation the relative thickness of a layer of an element in the zone of deformation does not remain equal to the initial relative thickness, with the exception of the neutral cross-section,

K hx H0 Hx

or ho_ ]h_ A, Hx'

The experimental data which are set out below are evidence that in very many instances of rolling, especially in the rolling of sections and billets, inhomogeneous deformation occurs. In other instances, e.g. during the rolling of thin sheet, in practice there occur such parameters of the zone of deformation at which the deformation is sufficiently close to homogeneous.

During tri-axial deformation, i.e. in the presence of noticeable spread, apart from the vertical reduction it is also necessary to take into consideration the transverse deformation. The coefficient of the increasing spread of the whole of the strip

R B* i>o

and for a given section of an element of its width

R bx

b0

Let us introduce the concepts of local and cumulative deformations. The vertical and longitudinal deformations of any given element of the strip increase gradually as this element passes through the zone of deformation. Let us designate the dimensions of the element under consideration before deformation by h0, b0 and l0, after deformation by h1, b1, and lx, and at any given moment of deformation by hx, bx and lx.

Disregarding the curvatures of the elementary parallelepiped under review (Fig. 2), which are unavoidable during inhomogeneous


deformation, let us write the equation for the constancy of the volume of this parallelepiped

K b0 l0 = hx bx lx = hx bx l±. (2)

FIG. 2. Diagram of the deformation of an element of the stock during rolling.

The coefficients of deformation after the element under consideration has passed out of the zone of deformation are

η h, ' p ~ b0 ' A- l0 ' (3>

In any given position of the element under consideration in the zone of deformation these will be:

™ L ' Ρχ U ' x 7 ' Vx nx b0 l0

(3 a)

1 where — is the coefficient of reduction; η β is the coefficient of spread; and λ is the coefficient of elongation.


From eqn. (2) it is clear that a connection exists between the deformation coefficients:

In the event of plane strain β = βχ = /, therefore

n λ' Vx

By cumulative deformation should be understood the overall deformation of the element of the deformed body from the start of deformation, when the element had dimensions h0, b0 and /0, up to the moment of deformation at which the element has the dimensions hx, bx and lx. Consequently eqns. (3 a) express the coefficients of the cumulative deformation.

By local deformation should be understood the change in the cumulative deformation during the movement of the element in the longitudinal direction by a small value. For instance, let the vertical cumulative deformation of an element located at a distance x from the plane of entry of the strip into the rolls, equal 1/ηχ. After an infinitely small moment of time the element is displaced in the rolling direction by dx, and the cumulative vertical deformation becomes equal to 1/(ηχ + dx). Consequently, the local deformation is expressed in the form of the ratio ΙΙ(ηχ + dx): 1/ηχ. For the sake of brevity, we shall not, in future, draw attention to the fact that cumulative deformation is meant, and in the event of the use of the concept of local deformation this will be specified.

In so far as the value of the inhomogeneity of deformation can be studied only in contrast to homogeneous deformation, we shall derive a formula for the coefficient of homogeneous vertical deformation as a function of the longitudinal co-ordinate x (Fig. 1)

H0 H0

Hx H0 — AXH

Taking eqn. (1) into consideration and introducing the relevant co-ordinate

x


we obtain #o Ho Hx " H, + AH(l - q)2

or Ho_

HQ Hj. . . .

H'~ 1 + ( i r _ 1 ) ( 1 ~ i ) 2 ' During homogeneous deformation, for which Η0/Ηχ = hQjhx, this

equation is valid not only for every thickness of stock, but also for every horizontal layer of an element. During inhomogeneous deformation this equation is valid only for the whole thickness of the stock, and the vertical deformation of the various horizontal layers of an element will be different, with the exception of the instance where x = V, in which /' is the length of the zone of backward slip. Analysis of experimental data after an investigation of the inhomogeneity of vertical deformation consists of comparison of the experimental plots hQjhx for every horizontal layer of the strip with the calculated plots obtained from eqn. (5) (Fig. 3).

From the theory of the mechanical working of metals it is known that during the upsetting of cylinders and parallelepipeds between plane-parallel plates in the absence of frictional forces on the contact surfaces the deformation will be homogeneous,44 and the state of stresses of the body will be linear. In the presence of contact frictional forces, the state of stresses of the body will be three-dimensional, and the deformation of the body inhomogeneous.

Inhomogeneity of deformation is also possible, as I. Ya. Tarnovskii has shown, in the absence of contact frictional forces, but in the presence of external sections (rigid ends).45

V. I. Zalesskii and A. V. Puzanchikov21 showed that during upsetting between conical tools, even in the absence of barrelling, the deformation is inhomogeneously distributed.

In a very large number of instances of forging and rolling, the vertical deformation along the width and length of the contact surface between the deformed metal and the tool is inhomogeneous. Such is the case during elongation by forging between grooved tools, during hot stamping, during rolling in passes with varying reduction of the stock over its width and in a series of other instances. DMR 3

2 2 DEFORMATION OF METALS DURING ROLLING

In all these instances the inhomogeneity of deformation is the result not only of the presence of contact frictional forces, but also of the varying relative compression along the length and width of the contact surface, which is normally linked with the shape of the tool and the billet.

0 025 0-50 075 Wq FIG. 3. Calculated curves of the increasing homogeneous vertical

deformation.

During rolling through grooved passes the relative vertical reduction over the width of the stock is often varied, and during longitudinal rolling of batch sections the vertical deformation is also varied even along the length of the strip. It must be stressed that here, primarily, inhomogeneity of the relative, and not of the absolute, vertical deformation should be borne in mind. In fact, even during ideal upsetting, at any given point in the volume of a body not the absolute, but the relative, deformations are equal,44 which likewise follows from the basic determination of homogeneous upsetting, given by A. F. Golovin:8

dz __ dh

The expediency of a characteristic of the inhomogeneity of deformation as the difference of the relative deformations, and not of the absolute, will become apparent from the following example. In stock


being rolled it is possible to distinguish two layers of an element of differing thickness such that even during sharply marked inhomogene-ity of deformation their absolute changes in thickness can be identical. Only in the special instance, where all the elements of the metal, the deformations of which are equal during the investigation, have identical original dimensions, will the absolute changes in the dimensions of the various elements be in the same ratio to each other as heirt relative deformations.

It is evident from what has been set out above that inhomogeneity of deformation can be the result of various causes.

Let us here introduce the classification of the causes which give rise to unevenness of deformation, as proposed by I. M. Pavlov:34

1. Friction on the contact surfaces between the work tool and the body being deformed.

2. The special nature of the application of the deforming forces (curvature, etc.); the presence of residual stresses in the body.

3. The shape of the work tool. 4. The shape of the body before deformation in the relevant stage

under review. 5. Heterogeneity of the properties of the metal being worked

(chemical composition, structure, anisotropy, temperature, degree of work hardening, etc.).

6. The causes indicated above in one combination or another. Inhomogeneity of deformation brought about by the causes

enumerated in the third and fourth points of this classification, relate to numerous examples of rolling with identical reduction along the width of the stock. In the more general instance, for example during the rolling of sections with a varying cross-section along their length, the reduction will be different both along the width, and also along the length, of the stock. In all these processes inhomogeneity.of deformation is entirely apparent, and it may be judged from the shape of the stock before and after the pass. Therefore inhomogeneity of deformation brought about by these causes may logically be called "external".

As distinct from this, inhomogeneity of deformation brought about by the causes enumerated in the first and fifth points of the classification should be called "internal".

In so far as during the rolling of stock with a rectangular transverse section between smooth rolls the drafts and reductions are the same both over the width of the stock, and also along its length, then the external impression is created that the deformation is homogeneous, 3*


whereas in reality the various elements of the metal are variously deformed. Therefore the inhomogeneity of deformation may also logically in such circumstances be called internal.

Under practical rolling conditions internal and external inhomogeneity of deformation of course are manifest jointly, and therefore such a division should be considered arbitrary. The necessity for these concepts is linked with the fact that in literature on the rolling of metals, the deformation at which the reduction at any given point of the length and breadth of the stock is the same, are frequently termed homogeneous, which does not correspond to reality, since internal inhomogeneity still remains in this event.

Let us consider the basic phenomena of inhomogeneity of deformation.

Most frequently of all inhomogeneity of deformation during rolling is characterised by the diagram of the longitudinal velocities of the movement of the particles of the metal in any given vertical cross-section of the zone of deformation, or by the presence and the extent of the zone of adhesion. Less frequently, inhomogeneity of deformation is marked by the distribution of vertical deformation over the height, length and width of the zone of deformation. In the latter instance for the most part the inhomogeneity of vertical deformation produced after rolling has been studied.29'33 Inhomogeneity of the distribution of vertical deformation during the course of rolling has been studied (on stickers) mainly on the side surface.30'24

We are also familiar with works in which an investigation has been made of the distribution of deformation within the zone of deformation.29 '48 '53 Apart from this it is possible to indicate the investigation,11 in which the inhomogeneity of vertical deformation within the stock being rolled was studied during the course of the process without slowing down the rolls.

It seems to us that the most important characteristic of inhomogeneity of deformation is the distribution of the vertical deformation over the length, height and breadth of the zone of deformation, and all the other phenomena of inhomogeneity of deformation follow from this characteristic.

In actual fact, the different form of the diagrams of the horizontal velocities of the movement of the particles of the metal in any given vertical section of the zone of deformation, in the absence of spread, is directly connected with the distribution of the vertical deformation, since for any horizontal layer of an element the constancy of a second


volume of metal passing through the vertical section of the zone of deformation under review remains valid (Fig. 4)

vx hx bx = vx h1 bx = v0 K b0, whence at bx = bx = b0

h0 hl

where h and b are the thickness and width of any given horizontal layer of an element;

v is the horizontal velocity of the movement of this layer; and

the subscripts 0, 1 and x relate to the front and back ends of the stock and to the section, x9 respectively.

If the vertical deformation for all the horizontal layers of the metal is the same, then the longitudinal velocities of the movement of the particles of metal located in a given vertical section of the zone of deformation will also likewise be the same. In the converse instance the velocities in the same vertical section of the zone of deformation will not be the same.

FIG. 4. Longitudinal velocities of the particles of metal of a layer of an element.

It should be borne in mind that the vertical deformation expressed in the form of the coefficient of reduction 1/ηχ = h0/hx is increasing, and if in certain vertical sections of the zone of deformation the values of this coefficient and, consequently, of the longitudinal velocities of


the movement of all the horizontal layers of the elements of the stock being rolled become the same, then this does not in effect mean that the deformation is homogeneous along the whole extent of the length of the zone of deformation, and that in other vertical sections these coefficients will also be the same.

Thus, for instance, experiments show that between the entry cross-section of the zone of deformation and the neutral cross-section the difference in the vertical deformation and, consequently, the difference in the longitudinal velocities increases at the start, then decreases, and in the neutral section dies out, i.e. the values h0lhx and vx become the same in all layers of the thickness of the stock. Moreover, between the neutral cross-section and the cross-section in which plastic deformation ceases, there again occurs a growth in the difference in the values of the coefficients of vertical deformation and of the longitudinal velocities of the movement of the layers of the elements of the stock, and then their equalisation.

The presence of a zone of adhesion should be considered a phenomenon of inhomogeneity of deformation. It is interesting to note that this phenomenon is observed both during upsetting, and also during rolling. The relative extent of the zone of adhesion is dependent on the ratio of the longitudinal dimension to the thickness of the zone of deformation, i.e. the ratio of the diameter of a cylinder to its height during upsetting of a cylinder, or the ratio of the length of the zone of deformation to its mean height during rolling, and also on the coefficient of external friction, so that qualitatively these relationships are the same for rolling and upsetting. The zone of adhesion is the surface of a certain zone extending in depth (Fig. 5) and called the "zone of restricted deformation". In this zone, as experimental data show, in practice there is an absence of plastic deformation.

The barrelling which is normally observed during upsetting and rolling indicates the inhomogeneity of the distribution of the transverse deformation over the height of the body being deformed, which in its turn is the result of the inhomogeneity of the distribution of the vertical deformation over the height and width of the zone of deformation.

The stress conditions of deformation and the change in shape are closely connected one with another, and during theoretical analysis of the various processes of mechanical working of metals it is necessary to review them jointly.

In connection with the investigation of the inhomogeneity of deformation during rolling, great interest attaches to the question of "the


conditions on the contact surface", or of the value and distribution of the tangential frictional forces which operate on the contact surface of the zone of deformation during rolling. By deriving equations for

FIG. 5. Zones of adhesion and areas of restricted deformation (shaded areas) during upsetting (a) and rolling (b).

the calculation of pressure during rolling without account of the inhomogeneity of deformation and, consequently, without account of the presence of a zone of adhesion

τ = ρμ,

where τ is the frictional force per unit area; p is the normal pressure on the relevent area of the rolls; μ is the coefficient of friction.

On investigating the distribution of roll pressure along the length of the zone of deformation, A. I. Tselikov came to the conclusion that the growth in the frictional forces from the entry of the stock into the rolls, and from the exit of the stock from the rolls, in the direction towards the neutral cross-section takes place according to this law only in the sections of slip, while on the surface of the zone of adhesion the forces of friction are subject to another law. In Fig. 6 are shown diagrams of the distribution of the normal pressure and of the frictional forces per unit area along the arc of contact in accordance with the theory of A. I. Tselikov.55 Such a nature of the distribution of the contact frictional forces along the length of the zone of deformation


is confirmed by the investigations of A. A. Korolev.26 The experimental data set out below also confirm the correctness of the diagram

FIG. 6. Curves of the distribution of pressure and frictional forces per unit area along the arc of contact (data of A. I. Tselikov).

of the distribution of the contact frictional forces along the length of the zone of deformation, as propounded by A. I. Tselikov, which takes into account the zone of adhesion.

THEORY OF INHOMOGENEOUS DEFORMATION 2 9

4. THE PRESENT STATE OF THE STUDY OF THE INTERNAL

INHOMOGENEITY OF DEFORMATION DURING ROLLING

The gradual accumulation and circulation of experimental and theoretical data has made it possible for a series of investigators to arrive at the conclusion that there is an essential difference between the real process of rolling and the instance of homogeneous deformation.

During the last decade interest in questions connected with the inhomogeneity of deformation has grown considerably. This is explained, primarily, by the growth in the demands made by the rolling industry on the theory of rolling, by problems of the maximum utilisation of all the potentialities of rolling equipment, the stepping up of schedules of reductions, of the struggle against various forms of scrap during rolling, especially during the rolling of heavy ingots in blooming mills, during continuous rolling, etc. The solution of these problems is dependent on the depth of the study of the laws of deformation.

Let us pass to a review of experimental works. In the majority of investigations devoted to the study of the in

homogeneity of deformation during rolling, attention has principally been devoted not to direct study and determination of the deformation of various elements of the metal in various directions (vertical, longitudinal and transverse deformations), but primarily to the study of the presence and the extent of the zone of adhesion, and of curves of the longitudinal velocities of the particles of metal in various transverse-vertical cross-sections of the zone of deformation. At the same time in view of the appreciable experimental difficulties connected with the special features of the rolling process, the number of works completed in this direction up to the present time is still small.

The investigation of adhesion has been conducted in two basic directions. The first group of investigators studied movements along the side face of the stock; the second attempted to use methods which would permit the movement of the metal within the stock to be studied with varying degrees of accuracy.

Amongst works of the first group may be classified the investigation of O. G. Muzalevskii,30 who by means of a rapid cinematograph film observed the distribution of the velocities of flow on the side surface of metal being rolled in the zone of deformation, which simultaneously made it possible to determine the location and extent of the zone of


adhesion on the edges of the specimen. The author did not, however, investigate the effect of the basic factor which determines the extent of the zone of adhesion, namely the shape of the zone of deformation ////mean5 since the initial thickness of the specimens remained the same. As a result the "shape factor" varied inconsiderably, only in connection with the change in the reductions within the limits of 10-30 per cent.

For the determination of the zone of adhesion, M. L. Zaroshchinskii carried out experiments in the rolling of heated steel stock. Over the width of the stock vertical screws were inserted in the centre, while vertical graduation lines were drawn on the side surface of the stock.17,18 The method of experimental determination of the length of the zone of adhesion is not described by the author. According to the data of the author the relative extent of the zone of adhesion /adh// in his experiments varied within the limits of 0-26 to 0-53.

Amongst investigations of the second group may be classified the work of T. M. Golubev.11,12 The method applied by the author made it possible to determine, in essence, the limits of the spread of deformation, both on the side surface, and also within the volume of the stock. At the same time the author came to the conclusion that a section exists on the contact surface, where there is no slip.

E. Orowan31 also came to the conclusion that a zone of adhesion exists, by observing the curvature of vertical layers on a plasticine specimen cut along the middle of its width. It is certain that an investigation of this sort can give only a very approximate picture, even in the qualitative sense, of the rolling process.

Let us pass to a review of quantitative data concerning the extent of the zone of adhesion during rolling which are available in literature sources.

A. F. Golovin, using data obtained by him during the upsetting of cylinders, came to a series of original conclusions concerning the quantitative side of the problem under review. Above all he noticed that "under practical conditions of rolling slip is never set up in the presence of spread".8 This point of view is still held up to the present time by a series of authors.40

A. F. Golovin considered that the extent of the zone of adhesion during rolling, as during upsetting, increases with the increase in the thickness of the stock, and that the ratio between the zones of slip and adhesion is dependent on ////mean in a wide zone of deformation


and on BmeanIHmean in a long zone of deformation. Finally he proposed approximate equations for the determination of the length of the slip sections, and consequently, of the zone of adhesion, as the differences between the length of the zone of deformation and the sum of the zones of slip.8

Although investigations in latter years indeed permit the hypotheses expressed over twenty years ago by A. F. Golovin to be determined more precisely, the correctness of his basic ideas of the kinetics of the rolling process is indisputable.

N. A. Sobolevskii42 considered that the zone of adhesion extends along the whole of the contact surface during any rolling process which is taking place normally. At the present time the existence of a zone of adhesion has been confirmed by a series of investigations, but experimental data confirming complete adhesion during rolling still do not exist.

On the basis of an investigation of the distribution of specific pressure along the length of the arc of contact, A. I. Tselikov expressed the completely definite opinion that "during the rolling of thick stock with a small length of the arc of contact there is apparently no slip between the metal being rolled and the work rolls". At the same time "during the rolling of thin stock with a great length of the arc of contact there is undoubtedly slip".55

A. I. Tselikov came to conclusions of this sort during an investigation of the distribution of pressure along the length of the arc of contact. A. I. Tselikov55 also proposed approximate expressions for the determination of the length of the zone of adhesion: for hot rolling /adh = (0-5-2-0)tfmean; for cold rolling /adh = (0-3-1 0)#m e a n .

It must be supposed that the author did not intend, by means of these relationships, to give an exhaustive solution for the determination of the length of the zone of adhesion, but it is important to mention here that these approximate relationships, in our opinion, as a rule point to the fact that with the increase in the thickness of the stock, and under otherwise equal conditions, the length of the zone of adhesion increases. A. I. Tselikov also noticed the essential dependence of the extent of the zone of adhesion on the coefficient of external friction.

Ye. V. PaPmov35 points to the increase in the extent of the zone of adhesion in connection with the increase in the initial thickness of the stock, and, vice versa, to the increase in the sections of slip in accordance with the decrease in the thickness of the stock being rolled.


O. G. Muzalevskii,30 having investigated the real relationship of the extent and location of the zone of adhesion only to the coefficient of friction, connected with the physical nature of the metal being rolled, derives the following empirical relationships: for AlMg alloy at a temperature of 340-80°C the zone of adhesion amounts to 4dh ~ (0-3-0-6) //mean; for Al alloy D16 at a temperature of 250-340°C/ a d h * (0-15-0-30) i/mean.

In his works17 M. L. Zaroshchinskii came to the correct conclusion about the increase in the relative extent of the zone of adhesion with the increase in the thickness of the stock being rolled. But he considers that quantitatively its extent is dependent on the shape of the transverse section of the stock before the pass H0jB0, and is independent of the shape of the longitudinal-vertical projection of the zone of deformation / / i /m e a n . It is difficult to agree with the last, since jointly with the coefficient of friction the ratio of the length of the zone of deformation to its mean thickness is the basic factor which determines the relative length of the zone of adhesion.

F. Kasz and P. S. Varley,27 reviewing the process of inhomogeneous deformation during the upsetting of a cylinder between plane-parallel plates, came to the correct conclusions concerning the presence of zones of adhesion and restricted deformation. These conclusions were extended by them to the rolling process. But the authors had a onesided approach to the question. According to their opinion the extent of the zone of adhesion is dependent only on the coefficient of external friction. The dependence of frictional forces and, consequently, of the zone of adhesion, on the shape of the zone of deformation was altogether disregarded.

E. Orowan31 did not clearly formulate his view of the quantitative relationships between the zone of adhesion and the shape of the zone of deformation. None the less analysis of the theoretical section of his work shows that the increase in the thickness of the stock being rolled makes the presence of a zone of adhesion less probable, which contradicts the experimental data. The opinion expressed by E. Orowan is also adhered to by L. R. Underwood.49

A. A. Korolev26 also considers that with a change in the thickness of the stock at a fixed value of the coefficient of friction the extent of the zone of adhesion grows. The formula for the determination of the relative extent of the zone of adhesion is obtained by him on the basis of equations for the determination of the pressure along the length of the zone of deformation, derived on the assumption that the main


longitudinal stress is independent of the vertical co-ordinate, or in other words, internal transverse stresses in the vertical sections of the zone of deformation do not exist. It is true that with such a state of stresses only homogeneous deformation is possible, and from these equations it is difficult to produce characteristics of inhomogeneous deformation, in particular of the extent of the zone of adhesion. At all events the experimental data set out below do not confirm the conclusion of A. A. Korolev concerning the increase in the relative length of the zone of adhesion following a reduction in the thickness of the stock under otherwise equal conditions. The dependence of the zone of adhesion on the coefficient of external friction is, qualitatively, treated correctly by A. A. Korolev.

From what has been set down it is evident that in rolling theory up to the present time there are still no definitely formed views on the kinetics of the process of rolling. There exist two points of view on the question of the possibility of the formation of a zone of adhesion, as one of the phenomena of inhomogeneous deformation. Amongst the research workers who admit the existence of a zone of adhesion, there is no single opinion concerning the extent of the zone of adhesion in various instances of rolling; nor is there any single opinion on the question of the qualitative effect of certain factors on the extent of the zone of adhesion, for instance the thickness of the stock, the amount of reduction, etc.

While A. F. Golovin, A. I. Tselikov, Ye. V. Pal'mov and others consider that with the increase in the thickness of the stock under otherwise identical conditions, the length of the zone of adhesion is increased, A. A. Korolev, E. Orowan and others maintain the opposite opinion. Numerous experimental data, set out below, confirm the absolute correctness of the first group of research workers.

On the question of the dependence of the length of the zone of adhesion on the coefficient of external friction there exists a completely definite opinion to the effect that with the increase in the coefficient of external friction under otherwise equal conditions, the length of the zone of adhesion is increased at the expense of the corresponding reduction in the slip zones.

Let us further review the question of the longitudinal velocities of the movement of the particles of metal located in any given transverse-vertical section of the zone of deformation. The existence of a zone of adhesion and of zones of restricted deformation during rolling, which have been established by a series of investigations, is incompat-


ible with the condition of the hypothesis of flat cross-sections concerning equality of the longitudinal velocities in any given vertical section of the zone of deformation (with the exception of the neutral cross-section).

It has been noted above that the concept of dissimilar longitudinal velocities of the particles of metal in any given transverse-vertical section of the zone of deformation was expressed long ago by A. P. Vinogradov,7 V. Ye. Grum-Grzhimailo,13 A. F. Rodzevich-Belevich,41

A. F. Golovin8 and others. Inequality of the longitudinal velocities in a given vertical cross-

section of the zone of deformation has been obtained in recent years

FIG. 7. Distribution of the velocities of the metal being rolled over the height of its section in connection with the existence of a zone

of adhesion (data of A. I. Tselikov).


in the experimental investigations of T. M. Golubev,11'12 O. G. Mu-zalevskii30 and others.

These experiments, in essence, confirm the correctness of the diagrams of the horizontal velocities of the movement of the metal particles, as propounded by A. I. Tselikov (Fig. 7), according to which in the neutral cross-section all the particles of metal move with the same velocity. In the zone of adhesion, where slip between the metal and the roll is non-existent, the velocity of all the particles on the contact surface is likewise equal to the peripheral velocity of the roll. In the entry and exit zones the distribution of the velocities is presented by A. I. Tselikov as curves having the form of parabolas. Qualitatively these curves are confirmed by experimental data.

Apart from this, A. I. Tselikov hypothecates that extinction of the inhomogeneity of the longitudinal velocity of the movement of the metal takes place beyond the geometrical limits of the zone of deformation on account of the extensions of the upper and lower fibres of the section of the metal being rolled, and of compression in its central part, and that the overall limit of the zone of deformation is extended beyond the limits of its geometrical boundaries.

5. ON THE APPLICATION OF THE SAINT VENANT PRINCIPLE IN THE THEORY OF MECHANICAL WORKING OF METALS

In order to study the states of stresses and deformations during upsetting and rolling, in certain instances it is possible to apply the principle of the local nature of equipollent loads (Saint Venant principle). This principle states that, if forces operating on a relatively small area are replaced by an equivalent static system of forces on the same area, then for the new system a change takes place in the states of stresses and deformations only in the immediate vicinity of the applied load. At points distant from the application of the load, the effect of the redistribution of the forces is extremely small.

The Saint Venant principle is also called the principle of modification of the boundary conditions. In reality, a load distributed on a small area according to some complex law may be replaced statically by its equivalent simple system of forces and thereby the boundary conditions may be simplified. Hence it follows that in the theory of the mechanical working of metals the Saint Venant principle should have great importance. In reality, loads applied during upsetting and rolling are determined, as a rule, by a complex law, and therefore


exact satisfaction of the boundary conditions is complicated. This circumstance, that the boundary conditions may be satisfied in an integral form, considerably simplifies solution.

Various solutions in the theory of elasticity give good confirmation for this principle, and enable the limits of its applicability to be established.

The use of this principle in the theory of mechanical working of metals is more complicated, since the absence of strict theoretical solutions does not make it possible to name the conditions under which the Saint Venant principle will apply.

But the possibilities of an experimental investigation of the acceptable limits of the application of the Saint Venant principle are sufficiently wide even during the mechanical working of metals. By applying loads in different ways to the same areas, and by observing, for instance, the distribution of deformations, it is possible to clarify the conditions of the applicability of the modification of the boundary conditions. Below such experiments are presented, and the limits of the applicability of the Saint Venant principle are approximately established. In the subsequent text, by the method of application of a load we shall understand a diagram of the application of forces on a definite area, and also the form and orientation of the area itself.

By way of an example of the application of this principle the following may be given: during the upsetting and rolling of tall bodies, as will be established later on, the states of stresses and deformations are independent of the method of application of the load. During the solution of a series of problems it is here possible to dispense with a review of the work of the frictional forces on the contact surface, and it is also possible to replace the contact surface by a simpler one, for instance an arc, chord or parabola, and in many instances it is possible to consider the rolling process as a process of upsetting between inclined platens, without substantially changing the states of stresses and deformations.

But in order to make use of the Saint Venant principle, it is necessary to establish by experimental means the limits of its applicability.

On the basis of what has been set out above it is possible to classify bodies being deformed during upsetting and rolling into: (a) "high" (thick); (b) "moderately high" and (c) "thin". In doing so, the classification of a body under one of these groups is determined not by its absolute dimensions, but by the ratio of its height to the horizontal dimensions (in the instance of rolling, the ratio of the mean thickness to the length of the zone of deformation).


"High" (thick) bodies we shall call those to which the Saint Venant principle may be applied during their study; "thin" bodies we shall call those bodies, the deformed state of which may satisfactorily be described by the hypothesis of flat cross-sections.

The intermediate instance of deformation, for which neither the Saint Venant principle, nor the hypothesis of flat cross-sections, is true, we shall apply to the deformation of bodies "of moderate height". Below by experimental means the limits of these classes are established. Such a division, however, cannot be considered as absolutely accurate, in so far as the hypothesis of flat cross-sections and the Saint Venant principle are only conditionally observed. In addition any experimental data can be reliably employed only under conditions analogous to those of the experiment. Nevertheless, the concepts introduced prove to be useful and to facilitate the statement and theoretical solution of a series of problems.

6. VARIATIONAL METHODS IN THE THEORY OF THE MECHANICAL WORKING OF METALS

One of the important problems of the theory of the mechanical working of metals is the determination of the states of stresses and deformations. In order to fix a rational technical process, it is necessary to study the effect of various factors on the distribution of stresses and deformations.

But quite inadequate light has been thrown on this question of the theory of mechanical working. For the most part investigation has been made of the stresses on the contact surface (pressures), while the distribution of deformations has been studied in certain instances by methods which cannot make any claim to accuracy.

The majority of theoretical solutions have been obtained on the basis of the hypothesis of flat cross-sections, which is valid under definite, extremely limited, conditions. Extensive use of this hypothesis is apparently explained by the fact that the solution of the general differential equilibrium equations is extremely complicated, and in the majority of instances cannot be obtained in an abbreviated form. Approximate solutions of these equations are connected with fairly wide assumptions, while their numerical integration requires a long time.

It is therefore natural to seek more convenient and more reliable methods of solution of this problem. The use of approximate methods DME 4


of calculation, in fact of direct methods of variational computation, presents wide possibilities for the solution of the problem presented. As distinct from the theory of elasticity, by virtue of a series of special features of the process, the use of variational methods in the theory of the mechanical working of metals encounters specific difficulties.

For setting the variational problem there serves in one form or another the origin of the possible changes in the state of stresses (Castil-lian origin) or the origin of the possible changes in the state of deformations (Lagrangian origin).

These origins make it possible to determine the function, which in the real process acquires extreme values. After the function has been found, the problem can be solved with any degree of accuracy.

Recently variational methods of mechanics have found even greater application for the solution of a series of practical questions.

In the present work a review is given of the solution of the variational problem by direct methods for certain processes of the mechanical working of metals.

The basic special features of these processes are as follows: (1) the deformations are large; (2) the movement of the tool is fixed, therefore the movements on the boundary of the normal to the working surface of the tool do not vary; variation of the movement along the contact surface is possible; (3) on the contact surface frictional forces are set up; (4) inhomogeneous deformation takes place within the volume, and on the surface, of the body being worked; (5) in the general instance the structure of the body is heterogeneous, and indeed, the mechanical properties are also heterogeneous. Inhomogeneous distribution of the deformations and temperatures aggravates this phenomenon.

In consequence of the difficulties enumerated, use of the theory of small deformations is complicated, and the inhomogeneity of the medium does not permit the selection of an equilibrium equation; the boundary conditions are also extremely indeterminate, since the link between the frictional forces and the forces operating along the normal to the surface is unknown. By means of certain assumptions these difficulties may be overcome.

Firstly, it is possible to study the velocities of the deformations. Having determined them for a series of intermediate states, it is also possible to calculate approximately the larger deformations.

Secondly, the boundary conditions may be considerably simplified, if the problem is solved in permutations. The distribution of the defor-


mations on the contact surface between the metal and the tool, on the strength of the relative simplicity of the experiment, has been studied considerably more fully, which permits one or other set of boundary conditions to be adopted.

Thirdly, a large part of rolling processes is carried out in the hot state. Therefore, regardless of the great inhomogeneity of deformation, the body is more or less isotropic, since recrystallisation takes place simultaneously with work hardening. Under such conditions for simplification it must be assumed that the temperature of all points of the body is the same.

In consequence of the assumptions enumerated the basic conditions of the mechanics of solid bodies may be assumed. In the subsequent text we shall consider a solid isotropic body, which before deformation is in the natural, unstressed state. We shall also consider the medium to be incompressible, which may be considered as true for the processes of plastic deformation evolved. The divergence of he velocity of such a medium equals zero:

dvx dvy — dvz . "5— + T " + ~Έ~ εχ +■ £y + Ez = 0, ox oy oz

where vx, vy, vz are the components of the velocity of a point of the medium in the direction of the co-ordinate axes; and

£JC> £y> εζ are the corresponding strain rates. Apart from this, in the theory of the mechanical working of metals

in certain instances it is possible to use modification of the boundary conditions (Saint Venant principle), since frequently the total action on the contact surface is known.

We shall also consider that the deviatoric of the strain rates is proportional to the deviatoric of the stresses

De = ψΌσ, (6)

and that the function ψ is known for certain states of the medium. Below variational equations are derived which may be used for

solution of the problem set. Let us assume that the metal behaves as a viscous body; during the study of the state of deformations such an assumption, as experience has shown, gives satisfactory results, if certain conditions which are mentioned below are observed. In this instance ψ = η where η is the coefficient of viscosity. 4*


In accordance with the origin of the possible changes in the deformed state23 the rate of working of these external and internal forces on the possible movements around the equilibrium state is equal to zero:

jjj (<rx δεχ + <*y dky + ··· + τζχ δγ·ζχ) dV -

V *

~" J J V*adh δνχ + Fadh dvy + Zadh δνζ) dS = 0, (7)

where V is the volume of the body; and S is the surface, on which are set the forces Xadh, Yadh and Zadh.

We shall neglect the inertia and body forces, since they are negligible. On the basis of eqn. (6) we write:

σχ = σ + 2η εχ; oy = σ + 2η €y;

τζχ = ηγζχ-By substituting the values of the stresses obtained in eqn. (7), and

taking into account, firstly that the medium is incompressible, and consequently, the sum of the strain rates along the three axes is equal to zero, and secondly that the external forces do not vary, we derive the equation in the form

δ {nffftä + % + έ?) + itäy + #* + #*)] dv + W) = ° ' ^8> V where

d W = -j'j(X^dvx + Yadhdvy + Z,dhdvz)dS. 's*

The varied expression (8) represents the complete work done in, or rate of working of the, deformation. The forces, Xadh, Yadh and Zadh, represent the resistances to deformation arising during the application of the load, and directed in opposition to the forces operating.

It should be borne in mind that during upsetting between plane-parallel plates or rolling between smooth rolls the variation of the work done in, or rate of working of, the external normal forces is equal to zero, since the velocities of the tool do not vary (are set).

Thus from eqn. (8) it is evident that in the real process the work of deformation acquires an extremum. In those instances where the velocities on the contact surface do not vary, the last term of eqn. (8) disappears.


This is possible, for instance, in the event that on the whole of the surface of contact between the deformed body and the tool, 5Ί, a zone of adhesion occurs, and consequently δνχ = dvy = δνζ = 0. On the other hand on the surface of the body free from the tool, S2, the load is not applied, i.e. Xadh = Yadh = Zadh = 0.

Therefore

/ / (âdh 6VX + 7 a d h dvy + Z a d h δνζ) dS = / / (Xadh ÖVX + Yadh dvy + V Si

+ Zadh (5^) rfS + / / (Xadh δνχ + 7adh dvy + Zadh (5i;2) ^5 = 0. 's2

In this way the work of the reactions on the surface in this instance is equal to zero. Therefore here again the actual field of velocities corresponds to the minimum work of deformation, and may be written by eqn. (8), in which the value of W is 0.

If the body is in a state of flow, then, as is known from the mechanics of plastic bodies,23 the following relationship, which has been experimentally confirmed, is valid:

T2 = τ», (9) where rs is the yield stress in shear; and

T is the shear strain energy, determined by the expression

T = V£[(crx - O 2 + (σ, - σζ)2 + (σζ - σχ)2 + 6(τ*„ + r2yz + τ2

2Χ)]. The minimum of the full energy for any solid medium corresponds

to the real form of the equilibrium (equation)

0 3 = d(ffjndV - A\ = 0, (10)

where Π is the strain energy, unit volume, where Π = σχεχ + oyey + ··· + τζχγζχ.

The increment in the strain energy unit volume in a state of flow is

δΠ = δίή^+τ3Γ where emean is the bulk strain;

k is the bulk strain modulus; Γ is the yield criterion.

Γ =

+ 2 }'i[(ex - ey)2 + (e, - ez)2 + (ez - ex)2 + \(y2xy + y2

yx + γ2ζχ)] .


The bulk strain may in its turn be found from the equation ^mean = 3 / τ σ + 3 * 0 ,

where 0 is the change in temperature due to deformation; and a is the hydrostatic stress

= ax + ay + σζ

If the process takes place isothermally, then 0 = 0.

£ -„ 9 k2 a2

Then emean = 3 fc a

ok ak = 9&σ.

For an incompressible medium k = 0, consequently qka = 0. Neglecting the work of bulk compression, we derive eqn. (10) in the form

d3 = difffrsrdV - A\ = 0. (11)

In this equation the variation of the work of the external forces

dA = ff(XMdhdux + Y«hduy + Zadhduz)dS, (11a) s

where dux, duy and duz are infinitely small and continuous displacements, coincident with the boundary conditions and satisfying the condition of incompressibility

-^—dux + -r-bUy + -=— duz = 0. ox dy cz

Expression (11) is written subject to the fact that the inertial and body forces are neglected.23

During the use of eqn. (11) it should be taken into account that the displacements on the contact surface may be varied only along it.

In the instance under consideration the full energy is equal to the whole of the work of deformation, since the first term of eqn. (11) is the work done in changing the shape, while the second term is equal to the work done in overcoming resistance on the contact surface.

Therefore condition (11) may be formulated as: the real form of the equilibrium equation provides experimental values for the full work of deformation. This condition may also be used, if any given model


of the process permits the approximate work done in plastic deformation to be calculated. In such a form the condition of the minimum of work (done) was used by A. F. Golovin.8

It is also possible to obtain a variational equation for the determination of the stresses. For this the actual state of stresses is equated with all the possible states of stresses satisfying the equations of statics and the conditions on the surface. Thus the variation of the work done in actual displacements,

ÖA = / / / (ex δσχ + ey δσ, + ··· + γζχ δτζχ) dV. (12)

If the variation of the work of the external forces in actual displacements equals zero, then eqn. (12) can be derived in the form23

(εχ δσχ + eyday + ~- + yzx δτζχ) dV = 0. (13) ill·

The last equation is valid for every medium, regardless of its mechanical properties.

Thus it is possible to find the deformations and stresses by using eqns. (8), (11) or (13) in relation to the concrete conditions of the process. For the solution of these equations it is possible to use direct methods of variational calculation.

As is known, the idea of direct methods consists in the fact that the variational problem is considered to be the boundary problem for a certain problem to the extreme of the function of a finite number of the variables. Thereby the real function of the deformations or the stresses is approximated by a certain series with a finite number of terms:

m

εχ = Σ Cmq>m(x,y9z)9 etc. (14)

By substituting in the variational eqns. (11) or (13) the values of the stresses or deformations, it is possible to carry out integration of these equations, since the functions fm and φΜ are now fixed.

After this the value of the function 6A is determined only by the coefficients Cm

dA = d[F(CJ] = 0.


By varying A in relation to the coefficients, we obtain the necessary number of equations for the determination of these coefficients.

Consequently, the variational problem is reduced to the solution of a normal problem of the extremum of the function of a finite number of variables. By substituting the coefficients which have been found in this way into eqn. (14), we shall determine the deformations and stresses with a known degree of accuracy. The accuracy of the solution is all the greater, the greater is the number of terms of the series ntroduced into the variational equation.

Completing the limiting transition at n -> oo, if the limit exists and is a solution, we shall obtain functions of ux, uy and wz, or σχ, ay, ..., rrjc, which are an accurate solution of the variational problem under consideration.

The rapidity of the convergence of the solution to the real solution is dependent on the nature of the functions forming the series.14 It is expedient to determine the nature of these functions experimentally, and to seek for quantitative relationships, making use of variational equations.

If the convergence of the solution obtained with the real one cannot be proved theoretically, then the accuracy of the formula obtained may be verified by experiment in relation to the different factors themselves. Thus experiment is a reliable criterion of assessing the solution obtained.

The method proposed here for the solution is based on ideas of the finite differences method of Euler. This method, as is well known, consists in the fact that the permissible curves in a given variational problem approximate to broken curves composed of n links with the fixed abscissae of the vertices. The values of the functional are considered to be on these broken lines.28

On such broken lines the functional / \y(x)] is changed into a function of the ordinates yl9 y2, J>3, since the broken line is entirely determined by them (Fig. 8). Thus the functional is changed into a function of a finite number of variables yn. If this function / (yt, y2, • · · > yn-i) has an extreme, then the ordinates yx, y2,..., yn _i are determined from the condition

1L = 0 -fL-o -ii- = o


Passing to the limit at n -> oo5 we obtain a solution of the variational problem.

But in the majority of instances it is sufficient to limit oneself to a finite number of variables, and the solution thereby obtained will be approximate.

r° *

As

S/2

/ ,

X

FIG. 8. Diagram of the change of the curve of the broken line.

This method may conveniently be used for the solution of a series of problems of the theory of the mechanical working of metals. Dividing up the deformed volume into n zones, in each of these it is possible to choose the movement in the form of a linear function of the co-ordinates and any given parameters which determine the amount of deformation.

The variated function A is calculated as the sum of the integrals over the whole volume.

»A = δ£ ffffk (ak9 x, y9 z) dV = 0,

where ak is the parameter determining the size of the deformation. The coefficients, ak should be selected in such a way that on the boundary of the zones k-\ and k there will be continuity of the material. Using m such conditions and differentiating by n-m unknown variables, we obtain n-m equations, which is sufficient for the determination of all the coefficients au

ÖA 0, etc.


In this way the deformations in every zone of the deformed volume will be determined. Passing to the limit at n -> °o9 we obtain a solution of the variational problem.

But, as calculations show, in the majority of instances it is possible to restrict oneself to an approximate solution. Frequently, such a problem can be reduced to one-three equations with one-three unknown values, for instance the co-ordinates of the neutral cross-section and the size of the zone of adhesion. This considerably simplifies solution. By means of the variated parameters the co-ordinates of the neutral cross-sections and the size of the zone of adhesion will be established, while the value of the work (done) is determined by them. In this instance experiment permits the nature of the distribution of the deformations to be chosen, finding their values in relation to the chosen parameters. Thus, experiment helps us to divide up the basic values which determine the process of deformation.

Academician B. G. Galerkin has propounded a method of approximate integration, which makes it possible to calculate the deformations for an elasticmedium. Let us consider it in greater detail, since under certain conditions it may even be used for an incompressible medium.

The variational equation of Lagrange for a given medium may be derived in the form

- / /[(<** l+rxym + Txzn- A"adh) δ ux + s

+ (Ty2 I + ay ιη+ Tyzn— Yadh) δ uy + + (Tzx l+Tzym~G2n- Za d h) öuz] dS = 0, (15)

where ρ X, ρ Υ, ρ Ζ are mass forces; and /, m, n are directional cosines.

If we choose functions for the movements so that the continuities and the limiting conditions are simultaneously satisfied

0χ1+ Txy m T T K / ! = A"acih> ©tC,

then the second integral in eqn. (15) disappears. Thus it is possible to choose functions of ux, uy and uz, if the boundary conditions and the link between the deformations and the stresses are known.


If in addition we neglect the mass forces, then eqn. (15) acquires the form

- I _Îi^_ i ^l£2L 4- Aul. ~r{ dx ^ dy ^ dz

■)öuz dV=0. (16)

For an elastic medium this equation, in consequence of the arbitrariness of the variations öux, duy and 6uz breaks down into three equations:

V

so that the solution amounts to integration of these equations with their subsequent resolution in relation to the parameters which determine the value of the displacements.

If the medium is incompressible, the variations oux, öuy and buz are not arbitrary, but are linked with the condition

ox oy oz which complicates solution, but does not make it impossible.

To find the displacements in this instance it is possible to use eqn. (16). Finally, it is possible to obtain a solution, using the Lagrange equation in form

(15), and having used the proposed method of solution, by means of the division of the volume of the deformed body into parts. In actual fact eqn. (16) may be presented in the form

In this instance the body consists of n zones, in each of which displacements present themselves as definite functions of the co-ordinates and the parameter a{. This parameter and the function for the displacements should be chosen in such a way that on the boundaries of the zones continuity of the material is preserved. By using eqn. (17) and the equations of the link between the zones, it is possible to determine the parameters at, and thereby the displacements and the deformations in the medium under study.

This method is especially convenient in the instance where the boundary conditions may be satisfied in the integral form. Then the boundary conditions may be considerably simplified. This in its turn simplifies the corresponding choice of uX9 uy and uz.

When using eqns. (8), (11) and (13), it is recommended that extensive use is made of experimental data for the choice of the functions which approximate the displacements during the various simplifications, and


for the verification of the formulae obtained. Often, in order to carry out the solution to the correct result, it is necessary to use various simplifications. Experience makes it possible to judge their permissibility both directly, and also by verification of the formulae obtained. Accurate solution of eqns. (8), (11) and (13) by direct methods has no sense, since, firstly, the method of solution is itself approximate, secondly, the variational equations obtained are valid for idealised media, and the media studied by us differ from these, and thirdly, it is important to obtain a simple relationship which is convenient for practical calculations.

7. THE USE OF VARIATIONAL METHODS FOR THE CALCULATION OF THE DEFORMED STATE

Upsetting of a high cylinder. By way of examples of the use of the methods propounded we shall consider the solution of very simple problems of the theory of the mechanical working of metals.

Let us calculate the distribution of deformations during the upsetting of a high cylinder (d/h < 0-5). The boundary conditions on the displacements we shall take from experiment, and we shall also use the Saint Venant principle (modification of the boundary conditions). Experiments show that in this event the whole of the contact surface represents a zone of adhesion, while an increase in the contact surface takes place at the expense of the transition from the side surface. This latter circumstance is difficult to describe mathematically, therefore we shall neglect it. This will in fact be comprised in the use of the Saint Venant principle, since in accordance with this principle simplification of the boundary conditions is permissible.

In this connection by way of boundary conditions on the contact surface we shall assume that displacements do not take place along its length. It is known that during the upsetting of high cylinders double barelling takes place. Therefore the function of the displacements will be chosen in such a way that on the contact surface it will vanish, and at the same time it will satisfy the boundary conditions on the side surface.

In view of symmetry we shall consider half of the height of the cylinder. Let us define the function for a small amount of upsetting. For the determination of the displacements and deformations we shall use eqn. (11), where we shall reckon that in eqn. (11a)

δΑ = 0,


since in those sections of the surface to which the forces are applied, the displacements do not vary. Therefore eqn. (11) is written as:

S3 = δ fffrsrdV = 0, (18) V

where the yield criterion in the event of an incompressible medium is Γ = 2 ie2

r + ε,εφ + z% + 0-25y£ + 0-25 γ2φζ + 0-25y2

rz. (19) We shall choose the function of the displacements so as to satisfy the

boundary conditions indicated above (Fig. 9) π z 3π ζ ,ΛΛΧ

ur = atr cos^r-r- + a2 r cos -^-r-. (20)

z

A

-Q

f

*■ r0 1*

*

_ i ha

1\ h °ir J

FIG. 9. Diagram of the choice of the function for the radial displacement, ur, during the upsetting of high cylinders.

As may be seen from Fig. 9, the sum of the two terms of the right-hand part of the last equation satisfies the condition of double barrelling during the upsetting of high cylinders.

The parameters ax and a2 are not arbitrary, but are connected by the condition of incompressibility. We shall determine them, taking into account that the whole of the volume, displaced from the height,


goes into the formation of the barrel of the cylinder h

nr2Ah = f n[(R + ur)2 - R2]dz, 6

where ur is taken only for the side surface. Substituting for ur its value from eqn. (20), neglecting the second

order terms, integrating and solving the equation for a2, we obtain a2 = 3tfj — 2·355ε,

where A h

Let us determine er, εΨ, γτφ, γΓΖ, γζφ from the formulae dur

17 = _ ^ _ J V = 0

dr ΎψΖ = 0 ,

7rz = ■ du~

+ dur

63 =

er ■ ez (21>

and substitute their values in eqn. (19). The value of the parameter at we shall determine from eqn. (18)

« 3 - ί / / / . . « Κ - / / / 1 § - ( , . Γ > « ' - 0 . V V

If we carry out variation of the full energy, which is equivalent with one arbitrary parameter to differentiation in respect of the latter, we obtain

h R φ

{ r ->πζ ΛΛ t π ζ 3πζ - r πζ 3πζ

6tfi cos^-T^- + 14-1 εcos-r-rcos-^-=— 36ßi COS-T-T-COS-2h 2h 2h 2h 2h

21/3 L c o s ^ - - (2·355ε - 3a,) cos-3πζ

2h

A^ . Ί3πζ ΓΑ Ί3πζ ■ 42-4£Cos2 „ , + 54tfx cos2

2h 2h + 1-23(2! -77- sin2 ^ - + hr ^'-

πζ

+ 0-25 πα,χΥ . πζ 3πα2ν . 3πζ ——— sin — —— sin ^ t 2Α 2 A 2/2 2h

ft„ r2 . πζ . 3πζ ^^ ^ r2 . πζ . 3πζ + 8"7ε F-Sin 2/Γ5ιη^7Γ- 2 2 - 2 ^^ s m ΊΚ 5 ι η ΎΓ ~

ν 3τι ζ τ 3τίζ\ — 78-56-p-sin2 + 100^! -ρ- sin2 , \r άψ dr dz

= 0. (22)

THEORY OF INHOMOGENEOUS DEFOR MATION 51

Calculations show that in eqn. (22), as the co-ordinates change, the denominator changes considerably more slowly than the numerator. Therefore without making a large error, during integration it is possible to consider it as constant, and after integration it will vanish. As a result we obtain the expression

15a± R2h- 10·6ε R2 h + 0-154^ ^ - - 9·82ε ^ - + n n

R* + 12-6540!—- = 0. h

Solving the last equation for α±, we obtain

10-6 + 9-82-^-hr

"i= #Γ*. 15 + 12-654 4V-

/ r

The solution presented is valid for small deformations. The change in the radius vector of any point of the cylinder as a result of upsetting may be found by integration, considering the ratio R/h to be constant within the limits of the given upsetting. Taking into consideration that the changes in the radius vector dr is equal to the displacement wr, we obtain

dr I . πζ , 3πζ\ί dh = UN COS — - + ü2 COS —τ-τ— —

r X1 2h 2 2h J\ h where

ai = T> a2 = 3a[ - 2-355.

After integration we obtain ! rl (t πζ , 3 π ζ \ 1 h0 l o g 7 ^ = r c o s 2 ^ + a 2 C o s ^ r ) l o 8 - Ä 7 ·

On the basis of this equation the flow of metal was calculated for the upsetting of a cylinder of dimensions: H0 = 100 mm, D0 = 40 mm, Ηλ = 90mm, ΔΗ = 10mm. The experiment was carried out by V. N. Trubin. The results of the calculation are shown in Fig. 10, in which the solid lines represent the experimental radial displacements both on the side surface and also inside the cylinder, while the dashed lines show the calculated data obtained from the last equation. As


may be seen from Fig. 10, the calculated state of the deformations coincides sufficiently accurately with the experimental, which is evidence of the possibility of the use of the Lagrange origin for considerable deformations subject to the successful choice of the so-called "appropriate functions ", which describe sufficiently fully the boundary conditions in the deformation process under consideration.

45 z 40

30

20

10

l X

\

0

r v

\

15 20 FIG. 10. Comparison of the calculated and experimental states of

deformations during the upsetting of a high cylinder.

Compression of a long strip. Let us also calculate the distribution of the deformations during the compression of a long strip (state of plane strain). The ratio of the width to the height we shall take to be not more than 2-5 and not less than 0-5, i.e. we shall consider a body of moderate height. Experiments show that in this instance the following boundary conditions occur. On the predominating part of the contact surface adhesion takes place. Thereby the size of the contact surface is increased mainly at the expense of the transition of the side surface to it. The side surface becomes concave and barrelled, and can with a sufficient degree of accuracy be described as a parabolic curve.

In adopting the boundary conditions, we shall take the following into account: an increase in the contact surface can take place in the general instance, especially during the compression of low bodies, both at the expense of the transition of the side surface to the contact surface, and also due to slip on the contact surface. The latter may be considerably more easily described mathematically. We shall consider


that the increase in the contact surface takes place only as a result of slip, we shall not take the phenomenon of migration into consideration. Thereby we shall not make large errors, since the work done in migration and the work done in slip compensate each other. In actual fact, with the increase in migration the contact slip is reduced, and vice versa. Correspondingly there is also a change in the ratio of the work done in migration and in contact slip. Consequently the work done in migration may be conditionally replaced by the work done in slip. The correctness of this replacement can be confirmed or refuted only by experimental data. The work done in migration must be taken into account, since at a high ratio of ό/Α, this work represents a considerable part of the total work done in deformation. In reality, in bodies with the same volume, it is proportional to the increment in the width or the radius. Hence it is evident that for bodies with a high ratio of b\h or R/h, the work done in migration (or in slip) considerably increases, and to neglect it entails a large error.

Let us take horizontal displacements such as will satisfy the boundary conditions adopted (Fig. 11)

ux = αλ x '"fr + a2x.

bo

T T

Ux

a2b0 <*lbc

FIG. 11. Diagram of the choice of a function for the horizontal displacement, ux.

From the equation it is evident that the parameter a2 characterises the growth in the contact surface, and the sum of the parameters αγ + a2 the growth in the width of the stock at mid-height. The ratio of the parameters ax and a2 also indicates at any given moment to what extent DM 11 5


transition from the side surface to the contact surface, or slip, or both together have developed.

We find εχ = — ey and γχ Vxy

dux Λ y2\

__ 2ax x y y*y - ψ - ·

As in the example considered earlier, the parameters ax and a2 are not arbitrary, but are connected with the condition of constancy of the volume. Let us find them. From Fig. 11 it follows that:

bAh = f tfj bh + a2bh, a2 = ε — 0·666#!,

where Ah £ = —'

To discover the parameter ax we use eqn. (11), which for the problem under consideration may be written as:

b 03 = dff {2xs ye2

x + 0-25y2xy) dS + jrs dux = 0, (23)

s o where 2 v ε\ + 0-25 y2

xy is the intensity of the slip deformation for the state of plane strain in the event of incompressibility, while the last integral is the variation of the work done by the external forces. In reality, the work done by the external forces varies only with the horizontal displacement, since the vertical displacement is fixed by the movement of the tool. It is thereby assumed that the frictional forces are equal to the yield stress in shear. Such an assumption corresponds" to the real conditions, since during extensive development of adhesion the shear stresses are close to TS.51

Substituting in eqn. (23) the values of εΧ9 yxy and ux, and differentiating the sub-integral expressions by αλ, we obtain the variation of the full energy of the system

^ 1 ί ΐ - | Ι ) + ε - 0 · 6 6 6 α 1 | ί θ θ 3 3 - | Γ ) + 2 α ι/ ^ > ; \dxdy

\V(e + 0-084^ b

- [θ·333λί/.ν = 0,

x2 0-062562 2

)2 + - 7 ^ - « ϊ


where in the denominator of the first integral stands the mean value of the intensity of the shear strain at x = b\2 and y = A/2. This will not admit a large error during integration, since the numerator changes considerably more rapidly than the denominator, so that the latter may be replaced by its mean value. Let us transfer the last integral to the right-hand side, complete the integration, and square the left-hand side of the equation obtained. Then after several rearrangements we obtain:

[((M78A + 0-222 ητ-) - 0-00316 x 0-25 b2 - 0-0278 x 0-25-^-

-0-075 x 0·25Ζ>2£Ο! - 0-445 x 0-25b2 ε2 = 0.

From the last equation it follows that

« i -

0-55 ]/0-25B2 C ai = --Ä-+y—4^--Ä'

where

Λ = (θ·178 + 0-222-^-) - 000316 x 0-25|j - 0-0278 x 0 -25^

5 = -0075 x 0-254τ«; h2

C = -0-445 x 0-25 4r«2· h2

As in the first examples, by integration we shall determine the location of the points after completion of compression. Let x be the radius vector of the point and dx its change during a small amount of upsetting, then

dx = ux = αγ χ[ 1 — J T \ + 2 ■*·

Taking b\h to be constant within the limits of the given upsetting, and carrying out integration we obtain

"■»Ir-M'-SH] log£-, (24) where

„* ai „' a2 αλ = — , a2 = —

ε e


In order to verify this formula the following experiments were carried out: two lengths of stock sufficiently long to exclude longitudinal deformation, were compressed to a definite amount. The spread of the stock was measured along the centre of the width and along the contact surface, and for comparison their values were calculated according to formula (24). The results of the calculations and of the experiments are presented in Table 1.

TABLE 1. COMPARISON OF THE EXPERIMENTAL AND CALCULATED SPREAD IN THE STATE OF PLANE STRAIN

h0

(mm)

23-0 22-0

(mm)

20-2 18-6

bo

(mm)

54-8 2 2 0

bo h0

2-38 1-0

a\

0-50 0-72

a2

0-667 0-52

Spread after deformation (mm)

On contact surface

Experimental

60-2 24-5

Calculated

600 24-0

Along the centre of the height

Experimental

63-0 27-2

Calculated

63-5 27-2

As is evident from the data presented in Table 1, the experimental and calculated spread, both along the centre of the strip, and also along the contact surface, are sufficiently close to each other.

Upsetting of a cylinder of moderate height. In a similar manner to the preceding example, we calculate the distribution of deformations during the upsetting of a cylinder of moderate height, d\h > 0-5. The boundary conditions are adopted, as in the preceding example, replacing the work done in transition by the work done in slip. The shape of the cylinder after upsetting we shall consider to be parabolic. We shall take the radial displacements in the following form (Fig. 12):

ur = αγ r ί 1 - η-Α + a2r.

We calculate εΤ9 εφ, yrq) yr z , γφ2 from formulae (21)

er = εφ = a2 (1 - ητΑ + «2,

yrz = -Ια^-ττ, 7τφ = γφ.ζ = 0.


Variational eqn. (11) in the instance under consideration acquires the form:

03 = δ {flj 2TS ie2r +ε,€φ + ή, + 025yr

22 r d<p dr dz) +

V

+ ffrsdurdS = 0. (25)

In eqn. (25) S is the contact surface to which the forces are applied^ while the last integral is the variation of the work done by the external forces. In reality, the work done by the external forces can be varied

2

1

T

|- ' . * 1 fv

1 I I »

m 1 !

Y+1 ^

\

\Qfro

FIG. 12. Diagram of the choice of a function for the radial displacement, ur, during the upsetting of cylinders of "moderate" height.

only in relation to the displacement wr, since on the contact surface the vertical displacement is fixed by the movement of the tool and therefore does not vary.

In eqn. (25) variation should be carried out in relation to the independent parameter. The condition of constancy of the volume permits the parameter a2 to be expressed by ax

a2 = 0·5ε — 0*666«!. Let us substitute in eqn. (25) the values of the deformation sr, ...,

yr2, firstly carry out variation under the sign of the integral in relation to the parameter at, and then complete integration on the expression obtained. Thereby, as previously, we shall replace the denominator


in the integral of the volume by its mean value. After several rearrangements we obtain

r2 a\ 0-267 + 0-167 w-

|0·222>·ΐ/θ·75ε

Solving this equation for at, we obtain

2 + 0·252ε fll + 00212α? + 00625^-^-

B ^ + C where

B =

C

00125ε

Κ' ^2 Λ*_ „2

rz V h 0-267 + 0 - 1 6 7 - ^ ) -

0-148 x 0·25ε2

0-00105 - 0-0031 - j -/r

(' 0-267 + 0167-

2 L2 -r - 000105 rz 0-0031 -=-/r

In a similar manner as in the first two examples, we obtain a formula for the determination of the radius-vector of a point after upsetting of a finite value

T 2

log- 4 -TF) + «2 l o 8 7^ · From this formula a calculation was made of the deformed state

of a cylinder of dimensions HQ = 40-3 mm and D0 — 40-0 mm upset to a height of Hx = 36-2 mm. The experiment was conducted by V. N. Trubin. In Fig. 13 the solid lines indicate the experimental position of thepoints after deformation, and thedashed lines the calculated. From this figure it is evident that in this instance also the calculated deformed state is sufficiently close to the experimental.

X w

l i ft"

l X A.

I" 0 5"" 10 15 20

FIG. 13. Comparison of the calculated and experimental states of deformations during the upsetting of a cylinder of "moderate"

height.


In conclusion, let us mention that on changing from small deformations to finite deformations no account was taken of the second order values. This remains correct, so long as the finite deformations do not exceed a certain value. Calculations show that these values should not be greater than 10-12 per cent. For a greater degree of upsetting calculation should be carried out by splitting up the whole of the upsetting into several stages, so that the value of each of these does not exceed 10 per cent.

From the solution of the three problems set out above, which illustrate the possibility of the use of variational methods for the determination of the deformed state, it is evident that these methods can be used to obtain equations which are required for technical calculations of the mechanical working of metals.

The authors intend to present a detailed and wide treatment of these questions in a special monograph.

CHAPTER III

An Investigation of the Internal Inhomogeneity of Deformation during the

Rolling of Stock of "Moderate" Thickness. Instance of Plane Strain

8. REVIEW OF METHODS OF INVESTIGATION OF THE INHOMOGENEITY OF DEFORMATION DURING ROLLING

Above mention has been made of the importance and practical necessity of studying the question of the distribution of deformations of the metal during rolling in the zone of deformation and in the vicinity of its geometric limits. It is natural, therefore, that a large number of experimental investigations in the theory of rolling have been devoted in fact to problems of the flow of the metal during rolling.

While the experimental study of the distribution of stresses in the sphere of the zone of deformation runs into considerable experimental difficulties, which cannot at the present time be considered to have been overcome, for the investigation of the flow of metal and the distribution of the deformations different authors have employed a fairly large number of methods.

One of the earliest investigations was the experiment carried out in 1883 by A. Hollenberg,53 who observed the flow of metal during the hot rolling of wrought iron stock. The experimental technique consisted in drilling holes in the stock, along the centre of its width, into which short rods of the same material were then forced. The heated stock was rolled along half its initial length, after which the mill was stopped and the stock was withdrawn from the work rolls. After cooling, the stock was sheared lengthwise alongside the short rods, so that it would be possible to see their shape after rolling. The author restricted himself to establishing only that the short rods were bent during the rolling process in the direction opposed to the movement of the stock.

60

INTERNAL INHOMOGENEITY OF DEFORMATION DURING ROLLING 61

N. Mets29 improved the method of A. Hollenberg, replacing the smooth, short rods by bolts, which made it possible to judge the distribution of the deformation over the height of the stock from the change in the pitch of the thread. In addition, on the side surfaces of the stock rectangular co-ordinate networks were applied. The width of the grooves was 1*0-20 mm, while the distance between them was changed in the various experiments from 1Ό to 7-0 mm.

In order to study the distribution of deformations in the transverse direction, bolts were inserted into the stock parallel to the axis of the work rolls, while graduation lines were applied to the contact surfaces at right angles to the direction of rolling. Lengths of stock heated to the required temperature were completely rolled, or partially so by stopping the work rolls and then withdrawing the stock from them. After cooling and shearing of the lengths of stock, a study was made of the flexure of the bolts, their deformation and the change in the pitch of the thread.

On the basis of the results of his experiments, N. Mets drew a series of interesting conclusions, which at the same time incorporate quantitative relationships concerning the flow of the metal and the distribution of the deformations.

But the method described did not make it possible to explain the distribution of deformations in the individual horizontal and vertical layers along the length of the zone of deformation in the centre of the width of the rolled stock, and consequently to determine the location and extent of the zones of adhesion and of restricted deformation.

The use of rods and bolts for the study of the inhomogeneity of deformation was later repeated on several occasions.17'18, 3 3»4 9·e t c ·

As experience has shown, a method based on the use of bolts or rods inserted into drilled out apertures cannot be considered as completely suitable for the study of the flow of metal during rolling, and, all the more so, of the distribution of the deformations. Observation of the flexure of the bolts or rods after rolling can give only an approximate qualitative description of the process and an entirely approximate quantitative assessment of the distribution of the deformations in the zone of deformation. The major disadvantage of the method consists in the break in the continuity of the metal being rolled, which cannot fail to express itself in the nature of the deformation of the bolts themselves, even when they are prepared from the same material as that of the rolled stock.


This relates especially to the longitudinal deformation (extension), which is normally greater than the transverse deformation. If gaps are formed between the rod or bolt and the stock in a longitudinal-vertical plane, then the deformation of the bolt or rod can take place not only as a result of reduction, but also as a result of the deformation of longitudinal flexure, which makes it impossible to carry out a sufficiently accurate analysis of the experimental data.

W. Trinks48 employed a somewhat original method for the investigation of the flow of metal during rolling. Lengths of lead stock were prepared from two longitudinal halves, on the side surfaces of which vertical graduation lines were marked with a cutting tool. In order to explain the flow of metal within the stock these graduation lines were painted in, after which the halves of the specimen were soldered together at various points, so that the graduation lines on the side surfaces and within the specimen coincided. The specimens were rolled, and the work rolls were stopped during rolling. The conclusions of the author were based on observation of the curvature of the graduation lines.

The method employed by Trinks represents an attempt to compare the flow of the metal during rolling on the side surface of the stock and along the centre of its width. But the method described above of securing the two halves of the lengths of lead stock can scarcely be considered as well chosen or capable of reliably preventing them from being forced apart during the process of deformation. From the diagrams of the deformed specimens based on the work of Trinks, which are presented by N. Mets in his work,29 it is evident that severance of the halves of the stock of this sort in reality took place during the experiments.

G. Unkel,49 when carrying out experiments in the rolling of aluminium specimens of dimensions 40 x 365 x 415 mm in a 500 mm mill, employed a somewhat different method. He sheared a specimen before rolling longitudinally along its width. On one of the halves in the plane of the intersection a co-ordinate network was applied at 10 mm intervals, formed by grooves 1-5 mm wide and 8 mm deep. Copper-aluminium alloy strips of the appropriate dimensions were pressed into the grooves. Both halves of the specimen were united by means of bolts. The specimen was heated to a temperature of 450°C and rolled to a thickness of 30 mm. Then the ends of the specimen were sheared off, and the plane of intersection with the co-ordinate network was cleaned and etched with hydrochloric acid, after which


the strips of copper-aluminium alloy had a brown colour, so that their flexure could be observed.

Here, as in the experiments with the rods or bolts, the metal likewise loses its homogeneity and continuity. This disadvantage is intensified by the weak fastening of the two halves by means of bolts. L. R. Underwood49 notes that in the experiments of G. Unkel the halves of the specimen had the tendency to part from each other, and consequently the deformation of the composite specimen is substantially different from the deformation of a continuous body. The distortion of the co-ordinate network was studied after rolling of the complete length of stock, which afforded no possibility of studying the sequence of the process of deformation between the work rolls.

Recently, a series of new investigations have been carried out, which have thrown light on the study of the deformation of metal during rolling. In these investigations for the most part use has been made of a method based on the application of graduation lines or co-ordinate networks on the contact and side surfaces of the stock. Thus, in the works of M. L. Zaroshchinskii17· 18 experimental data are given for the rolling of specimens prepared from mark 10 steel. Along the centre of the width of the specimens at a distance of 10 mm one from another bolts 4-5 mm in diameter were inserted; on the side faces vertical graduation lines or co-ordinate networks were applied. Such lines 1-0 mm deep were applied on the contact surfaces of specimens at a distance of 3-0 mm one from the other. The initial thickness of the specimens was adopted within the limits from 9-74 to 48-0 mm. Rolling was carried out in two-high mills with 360 and 219-4 mm dia. work rolls; the specimens were preliminarily heated to a temperature of 1100°C.

The nature and dispersal of the deformation were judged from the distortion of the bolts and the lines on the side faces; the extent of the zone of adhesion was established by means of measurement of the distance between the graduation lines on the contact surface of the zone of deformation on stock which had been brought to a standstill in the work rolls.

By analysing the results obtained, the author came to the conclusion that the extent of the zone of adhesion will grow with the increase in the initial height of the stock, where the width is the same.

An original method of investigation of the distribution of the longitudinal velocities in the zone of deformation during the rolling of specimens of aluminium alloys D16 and AMts was employed by


O. G. Muzalevskii,30 who was the first to use the method of a highspeed cinematograph camera (64 frames per sec). Experiments were conducted under both laboratory, and works, conditions.

For rolling in the laboratory mill with 250 mm dia. work rolls specimens of 25 x 90 mm cross-section and 150, 200 and 300 mm long were prepared. On the side surface of the specimens a co-ordinate network was applied, and the end surface of the barrel of the work rolls was provided with scales with divisions applied in a radial direction. The cinematograph apparatus was placed at the end face of the work rolls so that exposure of a photograph would produce an image of the side surface of the specimen with the co-ordinate grid in the zone of deformation and of the scale of the work rolls. Then by means of an instrument microscope investigations were made of the deformed co-ordinate network in each frame. On the basis of the results obtained graphs were constructed of the longitudinal velocities in the various vertical cross-sections. The extent of the zone of longitudinal adhesion was determined by measurement of the mutual movements of points on the surface of the stock (of the co-ordinate network) and on the end faces of the work rolls (radial divisions of the scale).

Study of the inhomogeneity of deformation by the method of application of graduation marks or co-ordinate networks to the contact and side surfaces of the stock, as is already evident from the description of the experiments of N. Mets, makes it possible to conduct fairly accurate measurements, and form definite quantitative assessments of the process of plastic deformation, especially in conjunction with the use of a high-speed cinematograph camera.

But the study of the flow of metal on the contact and side surfaces does not afford the possibility of judging the kinematics of the process within the zone of deformation. On the other hand, the picture on the side surface in consequence of the phenomenon of spread, accompanied in addition by the transfer of metal from the side surface to the contact surface, does not correspond to the nature of the flow of metal over the greater part of the width of the stock.

The application of relatively deep grid lines to the contact surface substantially changes the conditions of contact friction, at the same time lowering the accuracy of the measurements.

An extremely interesting method of studying the distribution of the velocities throughout the height of a given cross-section of the zone of deformation during rolling was developed by T. M. Golubev.11· 12

Employing a special device of original design, the author studied the


movement of the individual layers of metal on the side surface of lead specimens, 14 mm high and 7, 16 and 30 mm wide. Rolling was carried out between 80 mm dia. work rolls with polished, roughly machined and ragged surfaces at a peripheral velocity of the rolls of 17*5 mm/sec. The device made it possible to portray graphically the movement of points on the specimen taken at various heights in one plane of the cross-section.

Analysis of the experimental results made it possible to construct graphs of the distribution of the velocities on the side surface of the specimen during rolling with various reductions. The method described above also made it possible to determine the fact of the existence of a zone of longitudinal adhesion.

For the determination of the limits of the zone of adhesion on the contact surface T. M. Golubev developed two other methods. One of these consisted in rolling specimens of coloured plasticine between rolls of transparent material with marks along the generatrix, applied at a definite distance one from the other. Observations of the movements were carried out during the rolling process through the wall of the work roll. Later for this purpose the method of cinematography was used.

The second method consists in the irradiation with X-rays during the rolling process of a plasticine specimen with lead needles inserted into it. Rolling was carried out in a mill with a cantilever arrangement of the work rolls. The wooden work rolls, 138 mm in dia., were also fitted with lead needles directed around the radius of the work rolls. On the basis of measurements of the mutual movement of the needles inserted into the specimen and the work rolls, conclusions were drawn concerning the existence and the extent of the zone of adhesion and of the zones of slip during the rolling process.

Considerable importance attaches to the method developed by collaborators in the rolling laboratory of the Urals Institute of Ferrous Metallurgy, and subsequently improved upon by the authors. In accordance with this method specimens are built up from individual lead plates of a definite thickness, braised together by intermediate layers of Bud alloy, as a result of which a "layered" specimen is produced, so that the planes of division between the layers may be either horizontal or vertical. After compression or stoppage within the work rolls, the specimen is cut in such a way that in the plane of the cross-section the boundaries between the individual layers are visible, then the surface of the cross-section is etched and the thickness •of each layer is measured.


The method indicated permits an investigation of the distribution of the vertical deformations within the specimen in any vertical plane, parallel or perpendicular to the direction of rolling, and also in any given horizontal plane, and at the same time the continuity of the specimen is not destroyed.

Check experiments showed that the nature of the deformation differs neither in the qualitative, nor the quantitative, sense from the deformation of a continuous body of the same dimensions.

It should be kept in view that in certain instances, especially during the rolling of thick lengths of stock, lamination of the front end of the stock may take place with the formation of a "fan" (Fig. 71) in consequence of the fact that the shear stresses exceed the strength of the Bud alloy. In order to avoid this it is necessary preliminarily to forge the front end of the stock.

An endeavour to eliminate the disadvantages inherent in the foregoing methods, led us to the need to develop a somewhat different method. For the experiments use was made of specimens of mark AO aluminium (99-5 Al, impurities 0-5 per cent) and furnace copper (98-8 per cent Cu, 0Ό1 per cent Fe). The metal was preliminarily forged in the hot state under a hammer and then annealed, after which pairs of specimens of the necessary dimensions were prepared. On the milled faces of one of the halves or on one side face of each of the halves along the centre of the length of the specimen a co-ordinate network, 40-50 mm long, was applied with an accuracy of up to 0-01 mm, on a universal milling machine with a vertical shaft (Fig. 14). The grooves had a width and depth of 0*10-0· 15 mm. The horizontal grooves were applied at distances of 1-0-1-5-2*0 mm and the vertical

FIG. 14. Half of an experimental specimen prepared for welding.


at distances of 1-5-2-0 apart. The first horizontal groove was applied at the minimum possible distance from the contact surface (up to 0-2 mm).

On the butt-ends of the specimen and on the edges of the faces of the two halves of the specimen in contact with each other, along the whole length, excluding the central sections with the co-ordinate network, bevels were machined away, as shown in Fig. 14.

Both halves of the specimen were welded together so that the first linesof eachof thetwo grid networks (if they were applied to different halves) lay in one transverse cross-section. The welding was carried out by means of an oxy-acetylene flame. The bevels were filled with the same metal from which the specimen was prepared (Fig. 15 a). Only a small slit, within which was the co-ordinate network, remained unwelded.

FIG. 15. Chink along the centre of the width of a specimen after welding:

a—before rolling; b—after rolling.

The specimens which were prepared by this method, were free from the greater part of the disadvantages inherent in the methods set out above. In actual fact the continuity of the metal throughout the height excluded the possibility of the formation of a "fan", and the metals which were used for the preparation of the specimens eliminated the risk of damage to the edges, while constancy of the dimensions between the grooves was ensured by mechanical means. The use of a co-ordinate network, as in the method of G. Unkel, made it possible to study the distribution of the vertical and longitudinal deformations, both within and also on the side surface of the specimen.

In so far as the movement of the specimen through the work rolls was stopped at the moment when the co-ordinate network was in the


am mm

FIG. 16. Deformed co-ordinate networks on the side surface (a) and along the centre of the width of a specimen (b); material: aluminium; HQ x B0 = 9-65 x 47-6 mm; AH= 2-5 mm; work roll dia. 210 mm.

FIG. 17. Deformed co-ordinate networks on the side surface (a) and along the centre of the width of the specimen (b); material: aluminium; H0 x B0 = 14-5 x 59mm; AH= 5-0mm; work roll dia.

210 mm.


zone of deformation, this made it possible to study the inhomogeneity of the deformation during the course of the whole rolling process.

The strength of the weld, as well as the influence of the chink left unwelded on the transverse deformation of the compound specimens were verified by trial rolling. The experiments showed that at a reduction coefficient of 1-5 and an initial width of the specimen of 50 mm the difference in the amounts of absolute spread in the welded and unwelded sections varied within the limits of tenths of a millimetre. The chink scarcely opened up at all as a result of rolling (Fig. 15b). No ruptures of the weld seam were observed. The strength of the weld was not decreased even after it was planed flush with the contact· surface.

Comparison of the nature of the flexure of the vertical graduation line on the side surface and along the centre of the width of the specimen (Figs. 16-18) confirmed a substantial difference in the distribution of the deformations.

It must be considered a disadvantage of the method under consideration that during rolling there is partial transference of the metal within the chink to the contact surface, which could be detected at reduction coefficients of 1-8-2-0 in the contact layer itself, which is explained by the break in the continuity of the specimen in the transverse direction.

This phenomenon made it necessary in a series of instances to dispense with measurement of the reductions of the first (contact) layer with corresponding corrections in the construction of the graphs. For this reason the first horizontal graduation line was applied at the minimum possible distance from the contact surface.

In our opinion, which does not differ from the opinion expressed by S. I. Gubkin,14 the method of the co-ordinate network can give the most accurate qualitative impression of the distribution of the deformations throughout the whole volume of a deformed body. In actual fact, not only is it possible to assess the distortion of networks on the side and contact surfaces, but also in relation to the method of preparation of the specimens it is possible to apply co-ordinate networks to any given cross-section of the zone deformation. At the same time it is necessary to observe strict parallelism of both the vertical and the horizontal graduation lines, and the minimum possible width and depth of the lines. The distance between the individual grooves should be chosen in accordance with the aims pursued by a given experiment. DMR 6


FIG. 18. Deformed co-ordinate networks on the side surface (a) and along the centre of the width of the specimen (b); material: aluminium; H0x B0 = 24-81 x 77-9 mm; AH= 8-05 mm; work roll

dia. 210 mm.

9. EXPERIMENTAL DATA ON THE INTERNAL INHOMOGENEITY OF VERTICAL DEFORMATION DURING THE ROLLING OF STOCK

OF "MODERATE" THICKNESS

Comparison of the dimensions of the deformed co-ordinate networks applied along the centre of the width of the specimen with their initial dimensions makes it possible to construct graphs of the increasing relative reduction of any given horizontal layer of an element along the whole extent of the zone of deformation as a function of the


longitudinal co-ordinate of the vertical cross-section under consideration

For the reasons set out above, the thicknesses of the limiting horizontal layers of the specimens in the majority of instances cannot be calculated; in consequence, during the construction of the graphs consideration is given to the deformation of a specimen somewhat smaller in thickness, by comparison with the true thickness.

Before rolling the specimens were heated in a muffle furnace, the aluminium to a temperature of 450°C, and the copper to 900°C.

Holding of the specimens in the furnace at the temperature indicated, was carried out in relation to their dimensions for a period of 20-30 min. The heated specimens were rolled in a two-high laboratory reversing mill with polished steel work rolls of 210 mm dia., the rolling speed was 0Ό88 m/sec, and the capacity of the drive motor 28 kVA.

The dimensions of the aluminium and copper specimens before and after rolling are given in Tables 2 and 3, while the dimensions of the cells of the co-ordinate networks before and after rolling are given for certain specimens in Tables Φ-8. The numbering of the cells along the length of the zone of deformation is designated by n and over the width of the stock by m (Fig. 19).

In Figs. 20-23 are shown the experimental graphs of the increasing reduction of the elementary layers as a function of the length of the zone of deformation. The curve for the instance of homogeneous deformation6 is drawn in as a dashed line.

The graphs presented enable an assessment to be made of the internal inhomogeneity of the vertical deformation during rolling to be made, in the particular instance of stock of "moderate" height. Comparison of the experimental curves with the curve of increasing reductions during homogeneous deformation gives an impression of the degree of inhomogeneity of the deformation in any given instance of rolling.

Considering the rolling process to be symmetrical relative to the central horizontal plane, during the construction of the graphs we consider the distribution of the vertical deformations for only one of the halves of the height of the zone of deformation. 6*


TABLE 2. RESULTS OF ROLLING WELDED ALUMINIUM SPECIMENS

No. of specimen

4 5 6 7 8 9

10 11 14 15 16 18 19 20 21 22 24 24'

Ho mm

25-44 19-60 19-93 14-85 14-88 15-18 14-50 2016 19-51 9-65 9-65 9-62

24-91 24-22 24-52 24-81 19-65 19-65

Bo mm

78-36 49-50 4900 49-68 49-42 59-88 5900 61-34 6000 47-60 48-60 47-30 78-30 78-30 78-10 77-90 5905 5905

AH mm

7-24 8-10 5-61 3-45 6-21 5-80 5-00

1003 6-39 2-50 3-60 2-29 2-73 5-82 7-20 8-05 4-68 4-68

1 η

1-40 1-705 1-39 1-30 1-72 1-62 1-53 203 1-49 1-35 1-59 1-31 1-12 1-32 1-42 1-48 1-31 1-31

/ mm

27-80 29-20 24-30 19-05 25-55 24-70 22-95 32-90 25-92 16-20 19-50 15-55 16-95 24-70 26-80 29-10 22-20 22-10

H mm

21-820 15-550 17120 13-125 11-775 12-280 12-000 15-145 16-315 8-400 7-850 8-500

23-550 21-310 20-920 20-780 17-310 17-280

/ LT

•"mean

1-270 1-880 1-420 1-450 2-170 2-070 1-910 2-170 1-585 1-930 2-480 1-830 0-720 1160 1-280 1-400 1-280 1-280

4dh mm

14-00 9-60

11-40 8-30 8-20 9-40 9-80 8-80

12-90 6-60 4-60 6-30

1100 15-90 15-70 1600 12-10 12-20

/ 4dh

0-504 0-329 0-470 0-436 0-321 0-381 0-428 0-267 0-497 0-407 0-236 0-405 0-650 0-644 0-586 0-555 0-545 0-550

TABLE 3. RESULTS OF ROLLING WELDED COPPER SPECIMENS

NO. of specimen

1 2 3 4 5 6 7 9

10 11 12 14 15 16 17

# 0 mm

19-97 9-85

10-00 9-38

1010 14-54 13-05 14-73 18-60 18-44 18-46 23-46 23-99 19-44 22-42

Bo mm

38-90 49-45 50-00 39-15 42-55 42-60 58-15 49-30 59-65 59-30 49-15 49-80 40-50 49-50 49-73

AH mm

5-45 1-41 2-53 3 04 3-21 4-54 2-79 3-49 3-23 3-93 4-61 7-12 4-49 2-04 4-89

1 η

1-37 117 1-34 1-48 1-47 1-45 1-27 1-31 1-20 1-27 1-33 1-44 1-23 1-12 1-28

/ mm

23-95 12-20 16-32 17-93 18-40 21-85 17-15 19-15 18-45 20-36 2200 27-60 20-90 14-65 22-70

H mm

17-25 9-30 8-73 7-86 8-50

12-35 11-65 12-98 17-03 16-47 1615 19-90 21-48 18-42 19-97

/ ** mean

1-375 1-310 1-870 2-280 2-160 1-770 1-470 1-480 1080 1-235 1-362 1-390 0-975 0-795 1140

4dn mm

9-00 4-80 5-30 5-10 5-50 6-40 6-50 7-60 8-30 9-50 8-90 9-95

10-60 8-90

10-60

/ 4dh

0-376 0-394 0-325 0-284 0-299 0-368 0-379 0-394 0-450 0-467 0-405 0-361 0-507 0-608 0-470

TABLE 4. DIMENSIONS OF SECTIONS OF AN ELEMENT OF A SPECIMEN BEFORE AND AFTER DEFORMATION.

MATERIAL: ALUMINIUM, SPECIMEN 6. /

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MATERIAL: ALUMINIUM, SPECIMEN 20.

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»n

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Direction of rolling -► /


MATERIAL: ALUMINIUM, SPECIMEN 20. /

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MATERIAL: COPPER, SPECIMEN N O . 1. /

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MATERIAL: COPPER, SPECIMEN N O . 4.

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p

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286

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200 200 200 200 200 200 2 00 200 200 200 200 200 200 1-99 200


> r 2 ac o

I o m Z m O

o m Ti O 5« > δ Ö c So 2 o So o r r 2 o

•-a

FIG. 19. Numbering of the layers of the co-ordinate network.

8 10 12 n 16 18 20 22 2h x FIG. 20. Graph of the increasing vertical deformations of horizontal layers of an element for specimen No. 6 (Table 2); material: aluminium; curves numbered in accordance with the numbers of the

layers of the specimen. Arrow shows direction of rolling. 78

FIG. 21. Graph of the increasing vertical deformations of horizontal layers of an element for specimen No. 20 (Table 2); material: aluminium; curves numbered in accordance with the numbers of the

layers of the specimen. Arrow shows direction of rolling.

FIG. 22. Graph of the increasing vertical deformations of horizontal layers of an element for specimen No. 1 (Table 3); material: copper, curves numbered in accordance with the numbers of the layers of the

specimen. Arrow shows direction of rolling. 79


Let us turn to a consideration of the graphs. Primarily let us analyse the experimental graphs presented in Figs. 20-23. These graphs relate to the series of experiments embracing the instance of rolling, where the shape factor of the longitudinal vertical cross-section of the zone of deformation////mean < 1-5.

FIG. 23. Graph of the increasing vertical deformations of horizontal layers of an element for specimen No. 24 (Table 2); material: aluminium; curves numbered in accordance with the numbers of the

layers of the specimen. Arrow shows direction of rolling.

As is evident from the graphs, plastic deformation starts even in front of the geometric boundary of the entry of the stock into the zone of deformation. Even before entry the contact layer is thinned down, while the central layer, on the contrary, is somewhat increased in thickness. Thereafter the deformation of the contact layer continues to increase sharply, while the central layer starts to undergo reduction considerably later and more slowly.

After the contact layer has attained a definite degree of deformation, which is not subsequently increased, the central layer starts rapidly to be reduced in thickness, and at a certain point along the length of the zone of deformation the relative increasing reductions of the contact and central layers are equalised. After passing beyond this point, the central layer continues to be deformed fairly rapidly, while


the contact layer still remains constant in thickness. Only close to the exit cross-section does the contact layer once again start to be deformed, at the moment when the level of deformation of the central layer decreases.

It should be noted that, although in front of the exit cross-section and, apparently, beyond it a certain equalisation of the deformations takes place, on the other hand the central layer in the central part of the width of the stock continues to be deformed to a greater extent than the contact layer, which points to an analogy in the distribution of the deformation during compression and rolling. In addition, this fact is evidence of the error of the conclusions drawn by certain authors concerning the distribution of the vertical deformations during rolling on the basis of the deformations measured on the side surface of a specimen. In actual fact the phenomenon of the partial transfer of metal during the deformation process from the side surface to the contact surface noticeably decreases the thickness of the contact layers in the vicinity of the side surface.

The distribution of the deformations beyond the exit cross-section should not be considered as sufficiently reliable, since under the conditions described above for conducting the experiments, the reversing of the mill could somewhat change the real picture beyond the exit cross-section.

As is evident from the graphs, the curves of increasing deformations for the horizontal layers located between the contact and the central layers, adopt an intermediate position, and in combination with the central layers afford the possibility of assessing the distribution of the vertical deformations throughout the whole volume of the body which is under the influence of the tool (the work rolls).

Analysis of the experimental curves (Figs. 20-23) indicates the extremely considerable degree of inhomogeneity of the vertical deformations, which takes place during the rolling of stock with a moderate ratio of the length to the mean thickness of the zone of deformation.

The experimental graph in Fig. 24 illustrates the distribution of the vertical deformations during the rolling of stock, where //^mean > 1-5. From the graph it is evident that in these instances plastic deformation also starts in front of the entry cross-section, when the nature of the distribution of the deformations is the same as during the rolling of specimens with the ratio IlHmm < 1-5. The difference consists in the fact that the rates of increase in the deformations of the contact and


central layers after entry into the zone of deformation differ little from each other in this instance and at the same time from the curve of the increase in homogeneous deformation, which indicates a lesser degree of inhomogeneity of the vertical deformations in the instance under consideration.

FIG. 24. Graph of the increasing vertical deformations of horizontal layers of an element for specimen No. 5 (Table 3); material: copper; curves numbered in accordance with the numbers of the layers of the

specimen. Arrow shows direction of rolling.

Let us observe that the graphs of the increasing vertical deformations for the aluminium specimens indicate a greater degree of inhomogeneity of the deformation than for the copper specimens at an identical shape factor llHmean. This fact expresses the influence of the coefficient of external friction on the nature of the distribution of the vertical deformations, for the value of the external friction of copper is known to be lower than that of aluminium.


FIG. 25. Summarised graph of the change in the relative reductions of the elements of the layers along the length of the zone of deforma

tion: a—for stock of "moderate" thickness (0-8 < ////mean < 1*5); b—for thin

lengths of stock (////mean > 1·5)


Thus, the experiments show that the inhomogeneity of the deformation increases with the increase in the coefficient of friction and with the decrease in the shape factor.

In Fig. 25 is presented a summarised graph of the change in the relative reductions along the length of the zone of deformation during the rolling of stock of moderate thickness (0-8 < //#mean < 1-5) and thin stock (//^mean > 1*5).

Graphs similar to those of the relationships for the increasing relative reduction can be obtained also as a result of analysis of the change in the longitudinal dimensions of each element of the coordinate network in a given layer. But in consequence of the fact that the vertical lines of the co-ordinate network undergo considerably greater flexure than the horizontal lines, the accuracy of measurements of the elements in the longitudinal direction is low.

In Fig. 26 is presented a comparative graph of the increasing extensions and reductions for the contact and central horizontal layers of aluminium specimen No. 4 (Table 2), illustrating the hypothesis expressed.

FIG. 26. Comparative graph of the increasing relative elongations (solid lines) and reductions (dashed lines) of elements of the contact (1) and central (2) layers. Material: aluminium. Specimen No. 4

(Table 2). Arrow shows direction of rolling.


The study of the distribution of the vertical deformations in a deformed body made it possible to establish the limits of the zone of occurrence of plastic deformations in various instances of rolling of stock of "moderate" thickness.

The analysed results completely confirm the opinion of A. I. Tseli-köv concerning the zone of occurrence and nature of the plastic deformations outside the geometric limits of the zone of deformation. As we know, A. I. Tselikov considers that attentuation of the deformations takes place in the vicinity of the entry and exit cross-sections outside the zone of deformation "on account of the occurrence of extension in the limiting, upper and lower, fibres of the cross-section of metal being rolled, and of reduction in its central part".55 S. I. Gub-kin presents a similar system.15

Extension of the external, and reduction of the central, horizontal layers are connected with the presence of a rear external end of the stock and its reaction with the zone of deformation.

FIG. 27. Diagram of the deformation of a plasticine specimen according to E. Orowan.

Let us dwell further on a consideration of the diagrams of the distribution of the plastic deformations which are presented by many authors: O. G. Muzalevskii,30 E. Orowan,31 A. I. Kolpashnikov24

et al. One of the diagrams of this type, that of E. Orowan, is presented in Fig. 27. In essence this diagram contradicts the data of a series of experiments in later years,14,24*.33 since it follows from it that the real limits of the occurrence of the plastic deformations do not extend DMR 7


beyond the geometric limits, and that plastic deformation extends only along the diagonals of the zone of deformation.

The experiments reported in the present work show that, if in section I (Fig. 27), even though they do not have the shape of a sharp wedge, vertical and longitudinal deformations do not in fact exist, then this cannot be said of sections III and, evidently, IV which the above named authors consider not be to deformed.

Plastic deformation takes place in sections III and IV in fact extending beyond the geometric limits of the zone of deformation. The diagrams of O. G. Muzalevskii, E. Orowan and A. I. Kolpash-nikov take into account only the plastic deformations of reduction in the vertical, and extension in the horizontal, direction. In addition, as will be shown below, these diagrams correspond only to a certain partial instance of rolling, where //i/mean > 0-8.

In sections III and IV plastic deformation in fact takes place as a result of reduction in the longitudinal direction and thickening in the vertical direction, which is explained by the action of the reactive forces connected with the presence of rigid ends.

^ h I H FIG. 28. Diagram of the distribution of plastic deformations at

/ / # m e a n > 0 - 8 :

/—zones of restricted deformation; 77—zone of plastic deformation with reduction in the vertical, and extension in the longitudinal direction; III & IV—zones of plastic deformation with

reduction in the longitudinal, and thickening in the vertical, direction.

In Fig. 28 is presented the diagram of the distribution of plastic deformations during the rolling of stock of "moderate" thickness, constructed on the basis of the experimental data presented in the present work.


10. LOCATION AND EXTENT OF THE ZONE OF LONGITUDINAL ADHESION.

ZONE OF RESTRICTED DEFORMATION

The extent of the horizontal section on the graphs of the increasing relative reduction for the curve of the contact layer characterises the extent of the zone of longitudinal adhesion. In actual fact the dimensions show that both the vertical, and the horizontal, dimensions of the elements of the contact layer remain practically unchanged in this section, i.e. here there is an absence of vertical and horizontal deformation. Consequently there are no grounds for considering that slip can occur in the transverse direction, as certain authors do. In addition, this phenomenon can be explained only by the absence (in practice) of plastic deformation on the section under consideration in all directions, and consequently also by the absence of slip of the metal relative to the work rolls, both in the longitudinal, as well as in the transverse, direction.

The results are set out below of an investigation of the location and extent of the zone of adhesion in the direction of rolling, i.e. of the longitudinal adhesion. Specimens were used of sufficiently great width, so that the influence of the width of the stock on the location and extent of the zone of adhesion in the transverse direction was excluded.

Consideration of the experimental graphs (Figs. 20-24) permits the conclusion to be drawn that the extent, as well as the location, of the zone of longitudinal adhesion for the instance of plane strain at a given external friction coefficient are dependent mainly on the shape of the longitudinal vertical cross-section of the zone of deformation J/#mean · Thus on the graphs corresponding to the rolling of specimens with small ratios of //#mean> the zone of adhesion has a considerable extent and is located almost in the middle of the zone of deformation. During the rolling of stock with high ratios of //#mean the zone of adhesion is smaller and is displaced in the direction of the exit cross-section.

Let us bear in mind that during experiments in accordance with the method described above directly on the contact surface the extent of the zone of adhesion was not determined in the majority of instances. But for the calculation of the "ejected" layer the contours of the zone of restricted deformation, obtained on the measured layers of a specimen with "corrected dimensions", were extended to their intersection with the contact layer. In consequence the extent of the zone 7*


of adhesion was somewhat greater by comparison with the dimensions which were obtained from an analysis of the graphs. There are also grounds for maintaining that the dimensions of the zone of adhesion obtained in this way are somewhat smaller by comparison with its actual dimensions on the contact surface.

Let us note that the absolute extent of the zone of longitudinal adhesion does not give a clear impression of the inhomogeneity of the deformation. Only the relative value of the deformation can sufficiently fully characterise the role of the zone of adhesion. Therefore the quantitative characteristic of the extent of the zone of longitudinal adhesion is given in the subsequent text in the form of the ratio of the absolute length of the zone of adhesion to the length of the arc of contact 4dh//·

The experimental data on the absolute and relative extent of the zones of longitudinal adhesion during the rolling of the aluminium and copper specimens are presented in Tables 2 and 3, on the basis of which was constructed the diagram establishing the experimental relationship of the relative extent of the zone of adhesion as a function of the shape factor given in Fig. 29.

The shape factor ////mean increases with the increase in the diameter of the work rolls and in the draft, and decreases with the increase in the thickness of the stock being rolled.

Taking into account the ratios between the diameters of the work rolls and the thickness of the stock which exist under normal conditions of rolling production, and also the values of the characteristic drafts and reductions during sheet and section rolling, it may be considered that higher values of //i/mean> greater than 3, are characteristic for the rolling of thin sheet, while moderate values, approximately within the limits from 1 to 3, are characteristic of section rolling mills. This naturally does not mean that in individual instances of rolling the values of the shape factor cannot exceed the limits indicated above. So far as blooming mills are concerned, the most characteristic value of l/Hmean for these mills is less than 1, but this instance will be considered below in a special chapter.

From Fig. 29 it is evident that as the shape factor is decreased the relative extent of the zone of longitudinal adhesion increases both for the instance of rolling the aluminium, and also the copper, specimens; at higher values of the shape factor, both curves merge, asymptotically approaching the axis of the abscissae. This fact is evidence of the fact that during the rolling of thin stock along the greater part of the arc


of the bite slip takes place between the stock and the work rolls, and consequently deformation in this instance will be close to homogeneous. The conditions on the contact surface approximate to the postulated hypothesis of flat cross-sections.

J_odh 1

FIG. 29. Plot of the relative extent of the zone of adhesion in relation to the shape factor of a longitudinal vertical cross-section of the zone of deformation during the rolling of aluminium (1) and copper

(2) stock.

An especially sharp change in the extent of the zone of longitudinal adhesion may be observed in the range of values of llHmean from 1Ό to 30. The maximum values of the relative extent of the zone of longitudinal adhesion were obtained experimentally: for aluminium

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from different metals—aluminium and copper. Literature data and the results of our experiments in the determination of the external friction coefficient are evidence that the latter is higher during the

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0 001 002 003 0-OV 005 ΟΌ6 007 FIG. 30. Relative extent of the zone of adhesion during rolling as a

function of A H: D and HG : D.

rolling of aluminium than of copper. The inhomogeneity of the deformation, and consequently the extent of the zone of adhesion, increase with the increase in the coefficient of friction.

From literature data we likewise know that during the hot rolling of carbon steel the external friction coefficient is less than during the hot rolling of aluminium, but greater than during the hot rolling of copper. Hence the conclusion may be drawn that during the hot rolling of carbon steel the curve for the relative extent of the zone of adhesion LdJl will lie between the corresponding curves for aluminium and copper.

Finally, the horizontal sections on the experimental curves (Figs. 20-24) relating to layers situated below the contact layers, indicate the existence of zones of restricted working during rolling in


the same way as during upsetting. The results of the experiments permit an approximate determination of the boundaries of these zones.

The experimental data presented above show that the shape of the zones of restricted working in the longitudinal direction during rolling is different from that which has been presented hitherto in a series of works, for instance of O. G. Muzalevskii,30 A. I. Kolpashnikov24

and E. Orowan.31

The experiments conducted by I. Ya. Tarnovskii in co-operation with V. N. Trubin and Yu. I. Kolemagin, showed that the zones of restricted deformation have the shape of a droplet with an extended base line lying on the contact surface. The location of the zones of restricted deformation in the zone of deformation is determined by the position of their base lines, the zones of longitudinal adhesion. We shall revert to consideration of this question below, in connection with a consideration of the problem of the location of the neutral cross-section. Let us observe in conclusion that the determination of the extent of the longitudinal zone of adhesion during rolling according to the change in the vertical dimension of an element of a co-ordinate network on the side surface of a specimen cannot give satisfactory results in consequence of the distortions described above which are introduced by spread.

On the other hand determination of/adh// on the basis of the change in the longitudinal dimensions of an element on the side surface likewise cannot be considered as correct, in so far as the flow of the metal on the side surface differs essentially from its flow along the greater part of the width of the zone of deformation.

Failure to take into account this difference explains the relatively low values of lauh/l obtained in the experiments of O. G. Muzalevskii.30

It is interesting to compare the relative extent of the zone of adhesion during compression and rolling. In Fig. 31 experimental curves44 are presented for the relative extent of the zone of adhesion during the compression of cylinders between plane parallel plates

<4dh A d \

where dadh is the diameter of the zone of adhesion; dcon is the diameter of the contact surface of the cylinder at the

moment of compression under consideration;


H is the height of the cylinder; and d is the mean diameter of the cylinder.

From a comparison of these curves with the corresponding curves obtained during rolling (Fig. 29), it is evident that the laws governing the change in the relative extent of the zone of longitudinal adhesion

"adh

1-0 2Ό 3090 12 15 FIG. 31. Diagram for dadh : dcon as a function of d: H.

18 H

are qualitatively identical both during rolling, and also during the compression of cylinders.

In both the first and second instances the relative extent of the zone of adhesion decreases with the increase in the ratio of the longitudinal dimension of the zone of deformation to its height, and increases with the increase in the external friction coefficient. Both during rolling, and also during the compression of cylinders, the transition from full adhesion to practically complete slip takes place fairly rapidly with a comparatively small change in the shape factor. From the figures it is evident that during rolling between unlubricated rolls even at //#mean > 2*5-3*0 it may be considered that adhesion is almost nonexistent; during the compression of cylinders between dry, polished


surfaces at d\H < 9 slip is non-existent on the contact surface, while at djH > 18 adhesion is practically non-existent. Where the external friction coefficient is decreased, these values are correspondingly lowered.

Above a comparison has been given between the inhomogeneity of the deformation, or more precisely of its main manifestation, during rolling on the one hand, and during the compression of cylinders without external sections or rigid ends on the other. If we compare again the inhomogeneity of the deformation during rolling between works rolls and compression between plane parallel plates in the presence of rigid ends, then we obtain an even closer conformity, as will be considered below.

In conclusion it is necessary to dwell on treatises available in literature on the mechanical working of metals concerning the relationship between the zone of adhesion on the one hand and the shape factor of the zone of deformation and the external friction coefficient on the other. In relation to the dependence of the length of the zone of adhesion on the coefficient of friction all authors come to the same results, namely the greater the external friction coefficient, the greater is the zone of adhesion under otherwise equal conditions. In relation to the dependence of the length of the zone of adhesion on the shape of the zone of deformation, and indeed on the thickness of the stock, on the other hand, diametrically opposed viewpoints exist. A. I. Tseli-kov,55 A. F. Golovin8 and others consider that the extent of the zone of adhesion increases with the increase in the thickness of the stock under otherwise equal conditions, while E. Orowan,31 A. A. Korolev26

and others consider that the zone of adhesion decreases with the increase in the thickness of the stock under otherwise equal conditions. Thus, A. A. Korolev26 writes: "With the increase in the thickness of the stock being rolled, where oc is constant, the relative extent of the zone of adhesion is decreased, and at a certain ratio of the thickness of the stock to the length of the arc of the bite and at a definite value of the coefficient of friction the zones of adhesion completely disappear. This conclusion does not confirm the conclusion drawn earlier in this respect by A. I. Tselikov." A. A. Korolev came to this conclusion on the basis of analysis of equations produced by him with the use of a series of assumptions.

The experimental data set out above concerning the relationship between the length of the zone of adhesion and the shape factor of the zone of deformation convincingly confirm that with the increase in the


thickness of the stock and under otherwise equal conditions the zone of adhesion is increased.

It is appropriate to mention that likewise in the sphere of forging theory certain authors also draw general conclusions to the effect that with the increase in the ratio of the diameter of a cylinder to its height the relative extent of the zone of adhesion is increased, which also contradicts the experimental data.

11. COMPARISON BETWEEN THE INHOMOGENEITY OF THE VERTICAL

DEFORMATION ON THE SIDE SURFACE AND ALONG THE CENTRE

OF THE WIDTH OF A SPECIMEN

It has already been mentioned that in a large number of works the distribution of deformations has been investigated according to the distortions of a co-ordinate network applied to the side surface of a specimen. Let us show that such a method of investigation cannot be considered as sufficiently satisfactory, in so far as it affords no possibility of judging the vertical deformation within the predominant part of the volume of the zone of deformation.

S. I. Gubkin, in considering the inhomogeneous deformation during plastic compression, shows that in the central part of the deformed bulk compressive stresses arise, while in the peripheral sections tensile stresses occur. Such a distribution of the stresses within the bulk in practice conforms to the system of linear extension for the external zone (more accurately for the barrel zone). The tensile stresses on the side surface can in a series of instances be sufficiently large and give rise to the occurrence of cracks.15

It is impossible to agree with this, if we analyse our experimental data, both for compression and for rolling.

In fact, "the compression experiment" confirms that the state of stresses in the internal zones of a deformed body and on its side surface differ sharply. In actual fact, where there is homogeneity of the physical state of the metal, ruptures take place not in the central section, but on the side surface, so that cracks arise on the generatrix of the cylinder. This phenomenon can be explained only by a difference in the state of stresses. Tensile stresses along the external contour provide less favourable conditions for ductility, while compression stresses within the specimen afford more favourable conditions.

In so far as in a simple system of loading the stresses are un-equivocably linked with the deformations, it is understandable that


distribution of deformations on the side surface and in the centre of a specimen is also different.

By force of the analogy between the processes of compression and rolling it may be expected that tensile stresses will occur on the side surface of rolled stock and compressive stresses in the centre of its width. The presence of "rigid ends" during rolling aggravates the inhomogeneity of the distribution of the stresses in the bulk, giving rise to supplementary tensile stresses on the side surface.45

Let us pass to the presentation of experimental data confirming the difference in the distribution of the vertical deformation on the side surface and along the centre of the width of a specimen.

In Fig. 32 is presented a transverse cross-section of a fairly wide length of lead stock, braised together with Bud alloy from plates of a definite thickness. The stock was rolled between smooth rolls of

FIG. 32. Transverse section of a wide specimen after rolling (plane strain).

200 mm dia. in the two-high laboratory mill. After rolling, the stock was sheared across the layers, the surface of the cut was polished and etched with molybdate. The boundaries between the platelets are clearly visible in the form of thin, dark lines.

From the illustration it is clear that the thickness of each platelet is the same over the greater part of the width. A sharp change in the thicknesses is observed only in the vicinity of the side surfaces. It should be mentioned that with the increase in the width of the stock being rolled, under otherwise equal conditions the relative extent of the portion of the width with homogeneous distribution of the deformation increased.

Even from cursory consideration of the deformed co-ordinate networks applied to the side surfaces a and along the centre of the width b of specimens prepared from aluminium and copper (Figs. 16-18) it becomes obvious that there is a substantial difference in the distribution of the deformations on the side surface and along the centre of the width of the zone of deformation.


The vertical graduation lines on the side surface, as distinct from those in the centre of the width, are only slightly bent. The difference in the distortion of the co-ordinate networks can be explained by the considerable difference in the range of velocities within, and on the side surface of, a specimen.

In reality, the curvature of the graduation lines is brought about by the mutual total movements of the particles during the passage of the specimen through the zone of deformation, by the velocities of the plastic flow which are discussed below. To the extent that the velocities of the plastic dislocation and the values of the deformation are mutually linked, to the same extent the deformations on the side surface differ from the deformation of the internal sections.

Thus it has been shown that the distribution of the deformations on the side surface of a specimen cannot serve as a characteristic for the greater part of the deformed bulk.

12. VELOCITIES OF THE MOVEMENTS OF THE PARTICLES OF METAL

IN THE ZONE OF DEFORMATION DURING ROLLING

In the most general instance of bulk strain of stock during rolling the vector of the velocity of the movement of the particles of metal is located at will in space. The components of this vector in the directions of the co-ordinate axes we shall in future call respectively the vertical, longitudinal (the component in the direction of rolling) and the transverse (the component in the direction perpendicular to the direction of rolling) velocities. In the instance of plane strain, when the transverse deformation is non-existent or negligibly small, the transverse velocity is equal to zero, and consequently the vector of the velocity of any given particle of the metal lies in a vertical longitudinal plane. Let us note that here we are considering the instance of the rolling of stock with a rectangular cross-section between smooth work rolls. In other instances during rolling with inhomogeneous reduction of the stock over its width, transverse movements of the particles of metal are possible from some parts of a transverse cross-section to others, even without an overall change in the width of the stock, but such instances are not under consideration here.

Further let us consider the velocity of the movement of the shaping tool (of the work rolls). Since the work rolls rotate, then at any given point on the surface of the work rolls the peripheral velocity is directed along a tangent, and in the general instance this velocity is projected


onto the vertical and longitudinal axes of the co-ordinates. In the following text we shall call the projections of the peripheral velocity of the work rolls the vertical and longitudinal velocities of the work rolls. The transverse velocity of the work rolls equals zero.

During plastic shaping mutual movement of the particles of metal takes place, in consequence of which so-called plastic movements take place as distinct from rigid movements,5, 4 4 which occur during the movements of a solid body which is not deformed. Corresponding to these two forms of movements we must consider the velocity of rigid movement and the velocity of plastic movement. The actual velocity of the movements of the particles of metal in the zone of deformation is in fact the sum of the vectors of the velocities of the rigid and plastic movements.

Let us apply these facts to the instance of the rolling of stock with a rectangular transverse cross-section between smooth work rolls. During rolling with a very small amount of reduction it is possible within limits to produce only elastic deformation, and the dimensions of the stock after rolling are unchanged. In this instance the work rolls fulfil the role of delivery (transport) rollers, and there is no plastic deformation. Therefore the velocities of the particles of the metal in front of and behind the work rolls are identical and equal to the peripheral velocity of the work rolls (the elastic deformations are disregarded). For this instance, the almost complete absence of a vertical velocity of the tool in the zone of its contact with the metal is characteristic.

Further let us consider the instance of compression of stock between plane-parallel plates or between cylindrical tools or work rolls; for the problem under consideration here the question of the shape of the tool has no importance. Where there is vertical movement of the instrument, or more accurately where there is movement of its surface parallel to itself, the longitudinal velocity of the tool equals zero, but the vertical deformation in its turn brings about longitudinal and transverse deformations, i.e. longitudinal and transverse plastic movements of the particles of metal; relating these movements to time, we obtain longitudinal and transverse velocities of the plastic movements.

The rolling process may be represented in the form of a combination of the two instances considered above. Since points on the surface o the work rolls have vertical and longitudinal movements and correspondingly vertical and longitudinal velocities, then during rolling/the stock on the one hand moves forwards like a solid body which is not deformed, and on the other hand is reduced in height, and is conse-

INTERNAL INHOMOGENEITY OF DEFORMATION DURING ROLLING 9 9

quently extended and widened, as in the instance of compression. Therefore the velocity of the movement of the particles of metal in the zone of deformation should be considered as the total velocity of the rigid and plastic movements.

Since the movement of the stock, like that of a solid body which is not deformed, takes place only in the longitudinal direction, then the concept set out above of the total velocity of the particles of metal relates only to a horizontal velocity. So far as the vertical and transverse velocities of the particles of metal in the zone of deformation are concerned, they are connected only with plastic movements.

Further let us consider the question of the trajectories of the particles of metal in the zone of deformation. Unfortunately neither theoretical, nor experimental, trajectories of the particles of metal in the zone of deformation during rolling are so far known to us. Therefore let us turn to the instance of the ideal compression of a parallelepiped. The conditions of ideal compression were set out earlier, when it was shown that during the compression of a parallelepiped between plane-parallel plates under conditions of a linear state of stresses the trajectory of the particles of metal forms a flat hyperbola lying on the plane passing through a given point and the axis of compression.44

In reality such trajectories of the movement of the particles of metal in the zone of deformation during rolling do not occur, since it is impossible to conceive of rolling under a linear state of stresses, owing to the cylindrical shape of the work rolls, the presence of external friction forces, and the presence of rigid ends. In Fig. 33 is shown a

FIG. 33. Trajectories of the particles of metal in the zone of deformation during rolling (plane strain).


diagram of the trajectories of the particles, relative to the work rolls, in the instance of plane strain during rolling without a zone of adhesion. During bulk strain the trajectories of the particles in the zone of deformation, relative to the work rolls, represent not flat, but spatial, curves. In the sphere of the zone of restricted deformation, corresponding to which on the surface there is a zone of adhesion, and within which there is in practice an absence of plastic deformation, the metal does not move relative to the work rolls. On the side surface of the stock, where the process of transition of particles of metal from the side surfaces to the contact surfaces develops, the trajectories of the particles of metal have their own fundamental peculiarities.

But, in so far as in the problem under review here no qualitative theoretical solutions unfortunately exist, and graphs of the velocities are at the most obtained from experimental data, then for the review of diagrams of the velocities of the movement of the particles of metal in the zone of deformation during rolling it is possible as a first approximation to use a diagram of the trajectories of the movement of the particles of metal during homogeneous or ideal compression. This is all the more permissible due to the fact that during the change in shape of a trajectory the diagrams set out below of the velocities of the particles of metal in the zone of deformation remain valid. In addition, the use of trajectories which occur only during plane, homogeneous strain, is in no way equal to the assertion that ideal compression exists during rolling.

Primarily let us consider the velocities of the plastic movement. The velocity of the plastic movement, vp, is directed along the tangent to the trajectory of the plastic movement of a particle of metal in the zone of deformation. In the most general instance of bulk strain this velocity, vp, may be represented in the form of the spatial diagonal of a parallelepiped, the edges of which are the projections of vp onto the axes of the co-ordinates, and correspond to the vertical, longitudinal and transverse velocities of the plastic movement of the particles of metal in the zone of deformation.

In Fig. 34 a is presented a diagram of the generation of the vector of the full velocity of the movement of a particle of metal, O, which is in the zone of backward slip. In consequence of the plastic deformation the particle acquires the velocity vp, directed along the tangent to the trajectory of the plastic movement of the particle at a given moment. The projections of this velocity onto the axes of the co-ordinates are designated by vpy, vpx and vpz.


At the same time the particle O is also moved as a result of the movement of the stock by the work rolls along the direction of rolling like a solid body which is not deformed. For the determination of the velocity of this rigid movement it must be borne in view that the horizontal

Direction of rolling

FIG. 34. Velocity of the plastic movement of the metal; a in the zone of backward slip; b in the zone of forward slip (bulk strain).

velocity of the work rolls changes along the arc of contact between the stock and the work rolls, but the velocity of the rigid movement must be the same for all the particles of the stock, while during the rigid movement the longitudinal velocities of the work rolls and of the stock are identical. Such equality of the longitudinal velocities of the work rolls and the stock occurs in the neutral cross-section. Let us note that with the existence of a zone of adhesion the longitudinal velocities of the work rolls and the particles of metal located along the whole contact surface of the zone of adhesion are identical. But under such conditions there is not equality of the longitudinal velocities of the DMR 8


particles of metal located in any given vertical section of the zone of deformation. Such equality of the velocities of the work rolls and of all the particles of metal in a vertical cross-section of the zone of deformation occurs only in the neutral cross-section. Therefore the longitudinal velocity of the rigid movement of the stock must be taken to be equal to the longitudinal velocity of the work rolls in the neutral cross-section, or

vy = vr = vwrcosy,

where vwr is the peripheral velocity of the work rolls; and γ is the angle of the neutral cross-section.

In order to substantiate the value of the longitudinal velocity of the rigid movement the following reasons may also be adduced. It is known that the neutral cross-section divides the zone of deformation into zones of backward and forward slip, and if by backward and forward slip we understand the relative difference between the mean longitudinal velocity of the stock in a given cross-section and the longitudinal velocity of the work rolls, then the value of the backward slip increases in the direction from the neutral cross-section towards the plane of entry of the stock into the work rolls, while the value of the forward slip rises in the direction from the neutral cross-section to the exit plane of the stock from the work rolls. If we consider the longitudinal plastic movement to be opposed to the direction of rolling of an element which is in the zone of backward slip and located at a distance x from the neutral cross-section (Fig. 35), one can satisfy oneself that at a certain rotation of the work rolls this movement is determined by the deformation of all the metal which is between the cross-section under consideration and the neutral cross-section (see the shaded area in Fig. 35). Hence it follows that the difference in the longitudinal velocities of any given layer of the metal in the various cross-sections of the zone of deformation are determined solely by the plastic movements,

In the neutral cross-section there is no longitudinal movement between the stock and the work rolls, and the longitudinal velocity of the movement of the stock amounts solely to the velocity of the rigid movement.

In this connection consideration should also be given to the question of the velocities of the front and rear rigid ends. Regardless whether the actual boundaries of the occurrence of plastic deformation coincide with the geometric boundaries of the entry and exit of the stock


to and from the work rolls, it may be asserted that on the boundaries of the zone of plastic deformation, if the sections of restricted deformation are not taken into consideration, the longitudinal velocities

FIG. 35. Diagram for the determination of the movement of a particle of the material in the zone of backward slip.

are equalised, and these velocities subsequently remain constant for the rear and front rigid ends respectively.

The velocity of the rear rigid end is equal to the difference between the velocity of the rigid movement vwrcosy and the longitudinal velocity of the plastic movement at the junction between the rigid end and the zone of plastic deformation, which is the result of the increasing plastic deformation between the neutral cross-section and the rear rigid end. In other words, the velocity of the rear rigid end is equal to the total longitudinal velocity of the metal on the rear boundary of the zone of plastic deformation. The same applies also to the front rigid end of the stock, but in this instance the velocity of the rigid movement is added to the longitudinal velocity of the plastic movement.

Reverting to a consideration of Fig. 34, let us note that the vector OA represents the velocity of the rigid movement

vr = vwr cosy.

The total longitudinal velocity of the movement of the particle of metal equals

vx = vr - vpx. 8*


The full velocity of the movement of the particle of metal, v, in the zone of backward slip is represented by the sum of the vectors vx, vpx and vpy, expressed in Fig. 34a by the vector OD.

By analogy we can consider the velocity of the movement of a particle of metal, v, which is in the zone of forward slip. In this instance (Fig. 34 b) the total longitudinal velocity is expressed as

vx = vr + vpx.

The full velocity, v (the vector O D), represents the sum of the vectors vx, vpz and vpy.

In the absence of transverse deformation these systems are simplified, since vp2 = O (Fig. 34).

In the neutral cross-section the longitudinal velocity of the plastic movement equals zero, therefore the total longitudinal velocity equals the velocity of the rigid movement

vx = vr = νγ.

Let us consider the experimental graphs of the longitudinal velocities of the particles of metal located in various vertical cross-sections of the zone of deformation. During the rolling of welded composite specimens with horizontal weld surfaces, of the lead specimens welded with Bud alloy, for instance, or of composite specimens with vertical weld surfaces and with a co-ordinate network in the zone of deformation, by means of accurate measurements it is possible to find the thicknesses of these layers or of all the cells of the co-ordinate network passing through the vertical cross-section of the zone of deformation under consideration. Let hx be the thickness of an element of a layer of metal in the vertical cross-section of the zone of deformation under consideration, vx be the absolute longitudinal velocity for the same layer of metal and in the same vertical cross-section, and hv be the thickness of the same layer of metal in the neutral cross-section. Then on the basis of the constancy of the volume of the deformed metal for the instance of plane strain, in the absence of spread or neglecting it, we have

vx = vy^-. (26)

It has already been shown above that in the neutral cross-section the longitudinal velocity of the metal during rolling is equal to the velocity of the rigid movement (νγ = vr), and that for all the layers of the metal this velocity is the same in a given pass. Therefore for


each layer the absolute longitudinal velocity, vX9 is proportional to the relative change in the thickness of the layer between the neutral cross-section and the vertical cross-section of the zone of deformation under consideration, while for the zone of forward slip this will be the normal expression of the reduction, and for the zone of backward slip the reciprocal of it.

On the basis of the experimental data obtained from the rolling of specimens consisting of lead strips brazed with Bud alloy, graphs of the longitudinal velocities of the particles of metal in various transverse-vertical cross-sections of the zone of deformation were constructed. The method of construction of such graphs consists in the following. At the time of rolling the drive of the mill was switched off and the work rolls were halted together with the incompletely rolled stock; after the removal of the stock from the work rolls it was sheared along the centre of its width to produce a longitudinal-vertical cross-section. After polishing of the surface of the section and treatment with

FIG. 36. Longitudinal-vertical cross-section of a lead specimen with intermediate layers of Bud alloy.

molybdate, the boundaries of the division of the layers of metal were clearly revealed. Along this surface a number of vertical graduation lines were applied, each of which was located at a definite distance from any given zero reference, for instance, from the plane of entry of the stock into the work rolls (vertical graduation lines 7, 2, 3, etc. in Fig. 36). Along each graduation line measurements were made of the thickness of the elementary layers.

Since the absolute velocity of the movement of the particles of metal is equal to the sum if the velocities of the rigid and plastic


movements, construction of the graphs is then carried out in the following manner. First of all a diagram is made of the velocities of the rigid movement; since in a given pass for any given layer of the stock vy is constant, then the diagram represents a rectangle. It is convenient to take νγ equal to unity. The thickness of the stock in the neutral cross-section is known from the experiments, and since the relative thicknesses in the neutral cross-section coincide with the original thicknesses, then consequently the values of the hy for each layer are known from experiment. The thicknesses of the layers hx in a given transverse-vertical cross-section of the zone of deformation are also known from experiment. On the basis of these data, employing eqn. (26), it is possible to find the absolute velocity of the movement, vx, for each layer of the metal in the transverse-vertical cross-section of the zone of deformation under consideration. On the other hand the difference between the velocities of the rigid ends and absolute movements of the metal represents the velocity of the plastic movement. A series of experiments were analysed according to this method; part of them are presented below.

In Fig. 37 a is presented a diagram of a longitudinal-vertical cross-section of an incompletely rolled length of stock with indication of the numbering of the layers of the stock and of the vertical graduation lines coinciding with the transverse-vertical cross-sections of the zone of deformation under consideration in the subsequent text. The thickness of the specimen before the pass was 19-56 mm and after it 10-46 mm; the draft was 9-10 mm and the length of the zone of deformation 30-1 mm. Rolling was carried out between 200 mm dia. work rolls.

In the diagram are shown the diagrams of the velocities in the transverse-vertical cross-section 2-2, located at a distance of 6 mm from the plane of entry of the stock into the work rolls. It is evident that this plane is in the zone of backward slip. In this diagram the figure AB CD is the diagram of the velocities of the rigid movement ιγ, ABEF the diagram of the velocities of the absolute movements of the particles of metal and CEFD the diagram of the velocities of the plastic movements of the particles of metal.

In the diagram in Fig. 37 b are shown the diagrams of the velocities in a transverse-vertical cross-section b-b located at a distance of 5 mm from the outlet plane of the stock from the work rolls. It is obvious that this plane is in the zone of forward slip. In this diagram the figure A'B'C'D' likewise represents the diagram of the velocities


of the rigid movement, C'D'E' the diagram of the plastic movements and ÄB'C'E' the diagram of the absolute velocities of the movements of the particles of metal.

FIG. 37. Diagrams of the velocities in the zone of deformation during rolling.

In Fig. 38a is shown the diagram of a longitudinal-vertical cross-section of an incompletely rolled specimen. Its thickness before rolling was H0 = 19Ό8 mm and after rolling Hi = 12*36 mm, the diameter of the work rolls 200 mm, and the draft Δ H = 6-72. The length of


the zone of deformation, / = 26 mm. In Fig. 38 b are presented the diagrams of the velocities in a transverse-vertical cross-section of the zone of deformation, located at a distance of 5-5 mm from the entry

(

a

l·

2

2

s

? s

y

0

2\z: -Vr

F I G . 38. Diagrams of the velocities in the zone of deformation during rolling.

into the work rolls, i.e. in the zone of backward slip, while in Fig. 38 c the same is shown for a transverse-vertical cross-section located at a distance of 5Ό mm from the exit plane, i.e. in the zone of forward slip.

In Fig. 39 a is presented the diagram of a longitudinal-vertical cross-section of an incompletely rolled specimen with dimensions H0 = 19-10 mm, Hx = 15-20 mm, AH = 3-90 mm, D = 200 mm, / = 19-8 mm. In Fig. 39b are presented the diagrams of the velocities for a transverse-vertical cross-section in the zone of backward slip, where the figure AB CD is the diagram of the velocities of the rigid


movements, ABEF the diagram of the absolute velocities of the movements and CEFD the diagram of the velocities of the plastic movements. In Fig. 39 are shown the corresponding diagrams of the velocities in a transverse-vertical cross-section located in the zone of forward slip.


In Fig. 40 ό are presented the diagrams of the velocities in transverse-vertical cross-sections of the zone of deformation, located in the zones of forward and backward slip of an incompletely rolled specimen with an original thickness of 9-90 mm, where the draft was 5*56 mm and the length of the zone of deformation 23-6 mm.

It must be mentioned that if the transverse-vertical cross-section, in which the diagrams of the longitudinal velocities are under con-


sideration, intersects the zone of adhesion, and consequently the zone of restricted deformation also, then in the contact layer the velocity of the absolute movement of the particles of metal should be equal to


the velocity of the rigid movement, if for this purpose the change in the cosine of the central angle during the transition from the cross-section under consideration to the neutral cross-section is not taken into account, while the velocity of the plastic movement is equal to zero. Only in the first of the diagrams of the velocities considered above has an absence of plastic movement on the contact layer been obtained. But this does not in any sense mean that in all the remaining instances the transverse-vertical cross-sections under consideration did not pass through the zone of adhesion. It is a question that the end sections of the zone of restricted deformation represent a very thin contact layer, while the experimental values of the longitudinal velocities are obtained from the deformation of elementary layers of 1-2 mm thickness. Naturally by measurement of the thickness of each


layer of the stock the mean velocity within the limits of a given layer is determined. Under these conditions the existence of a zone of adhesion is somewhat attenuated, but in the experiments under consideration the relevant aim was not the detection of a zone of adhesion.

The diagrams obtained by experimental means of the absolute longitudinal velocities for various instances of rolling and in various transverse-vertical cross-sections of the zone of deformation fully agree qualitatively with the diagrams produced earlier by A. I. Tselikov (Fig. 7). The quantitative difference between the absolute longitudinal velocities in one and the same vertical cross-section of the zone of deformation in the horizontal plane of symmetry, on the one hand, and on the contact surface on the other, is considerably less than it is normally represented.11'30

The inhomogeneity of the absolute longitudinal velocities of the movement of the particles of metal in any given transverse-vertical cross-section of the zone of deformation can be characterised by the relationship

Vxc ~~ Vxs (Λ VX S\ /T7\

ex= = 1 - — , (27) LXC \ VXCI

where vxc is the absolute longitudinal velocity at mid-height of transverse-vertical cross-section of the zone of deformation under consideration; and

vxs is the same on the contact surface in the same cross-section. The values of ex for the diagrams of the velocities according to the

operational data are shown in Table 10.

TABLE 10. INHOMOGENEITY OF THE ABSOLUTE LONGITUDINAL VELOCITIES OF THE CENTRAL AND CONTACT LAYERS OF THE METAL DURING ROLLING

Thickness of specimen before

rolling (mm)

1910 19-08 19-56 9-90

Reduction during rolling

Absolute (mm)

3-90 6-72 910 5-56

Relative (%) 20-4 35-8 46-5 56-2

Inhomogeneity of longitudinal velocities

(%)

In zone of backward slip

-5-3 - 9 0 -7-9 -9-0

In zone of forward slip

4-7 5-5 6-5 2-8


The inhomogeneity of the absolute longitudinal velocities inadequately characterises the inhomogeneity of the deformation of the stock. The latter is more fully characterised by the difference in the longitudinal velocities of the plastic movements.

It is appropriate to mention that between the difference between the absolute longitudinal velocities at mid-height and on the contact surface, and the difference between the longitudinal velocities of the plastic movements in the centre of the height and on the contact surface there is the following connection: for the zone of backward slip

vxs - vxc = vpc - vps (28)

and for the zone of forward slip

Vxc ~ Vxs = Vpc - Vps9 (29)

where vpc is the longitudinal velocity of the plastic movement at mid-height of a vertical cross-section of the zone of deformation under consideration; and

vps is the same for the contact layer in the same vertical cross-section of the zone of deformation.

Since this difference in the longitudinal velocities is caused only by the plastic movements, then the inhomogeneity of the velocities can be determined by the relationship

£p = ivîw = T / 1 _ i v 1 \ (30)

where the minus sign relates to the zone of backward slip and the plus sign to the zone of forward slip, which also takes into account the difference in the directions of the velocities of the plastic movement in the zones of backward and forward slip.

Let us observe that the maximum possible value of the relative difference in the velocities of the plastic movement is ± 100 per cent; this value is attained on the contact surface of all vertical cross-sections passing through the zone of adhesion. Since velocities are not obtained from the experiments but thicknesses of the different layers on each vertical cross-section of the zone of deformation, then we express eqns. (27) and (30) as the ratio of the layers of an element.

Starting from the condition of constancy of the second volumes for the various layers of an element in the zone of deformation, where B


*-xs

Vys =

' J CC

Vyc =

0 h rixs

v *°L 0 h nyS

v hs*. 0 h

nxc v - ^ riyC

is constant, we write: for the contact layer

u (31)

(32)

and for the central layer u

(33)

(34)

Here the subscripts of the thickness of the layers of an element in the zone of deformation coincide with the subscripts of the velocities of the corresponding layers.

By substitution of the values found into eqn. (27) we obtain

hxc

As has been mentioned above, the inhomogeneity of the deformation is most completely characterised by the difference in the velocities of the plastic dislocations expressed by eqn. (30).

Substituting into eqn. (35) the value of the velocities from eqns. (28) and (29) we obtain

± vxs - vxc χ 1 0 0 ^ Vy ~ Vxc

the minus sign relates to the zone of backward slip and the plus sign to the zone of forward slip, and we omit the second subscript with νγ9 since vyc = vys.

In the last equation, replacing the ratio of the velocities by the ratio of the thicknesses, on the basis of eqns. (31)—(34) after simple rearrangements we have

ε„ = + -τ τ— ■ (37) nOc _ nOc h h "vc ,lxc


100

UJU

u

0-50 \

100 j

Direction of rolling ^

* ι 8\ ί2\ ^.^J_ ' _——'

* > \

■^ —V

\ € p \

% 7ÖT 75T^ I I i

j . ^ L ^

FIG. 41. Diagrams of εχ and ερ for specimen No. 20 (Table 2); material: aluminium.

too

0-50

-0-50

-100

Direction of rolling ►

*ι Ί 'fi r ^ * · * ^ i 4—■— Γ \ \

£ / \

\

v^ ■fS, ^ ^ΤΊ

JZf -~L I I T

7 -L · Η

FIG. 42. Diagrams of εχ and ερ for specimen No. 1 (Table 3); material: copper.


By arranging experimental data on the change in the thickness of each horizontal layer of an element of the stock along the length of the zone of deformation, it is possible by means of eqns. (35) and (37) to construct diagrams expressing the inhomogeneity both of the absolute velocities of the movement of the particles of metal, and also of the velocities of the plastic movements in any given transverse-vertical cross-section of the zone of deformation. In Fig. 41 are presented the corresponding diagrams constructed on the basis of experimental data obtained during the rolling of aluminium specimen No. 20 and in Fig. 42 for the rolling of copper specimen No. 1. The dimensions of the specimens are given in Tables 2 and 3.

As is evident from the illustrations, the curves of the inhomogeneity of the absolute velocities of the movement of the particles, εχ, are close to the zero line, which indicates the small relative difference in the absolute longitudinal velocities of the movements of the particles of metal in every transverse-vertical cross-section of the zone of deformation.

On the contrary, from the curves of the velocities of the plastic movements it is evident that the relative difference in the longitudinal velocities of the plastic movements of the particles of metal in any given vertical cross-section of the zone of deformation is great, and in the zone of adhesion attains ±100 per cent.

CHAPTER IV

The Inhomogeneity of the Increasing Bulk Deformation during Rolling

Between Plain Rolls

13. METHOD OF CONDUCTING THE EXPERIMENTS

In order to carry out the experiments to study the inhomogeneity of deformation, the already familiar method of co-ordinate grids was used. This method makes it possible to study the deformation of the stock not only on its surface, but also within the stock. The co-ordinate grid can be located literally in any given cross-section of the stock, i.e. parallel, perpendicular, and at any angle to the direction of rolling. For this purpose lead stock is divided up by intersection along the planes in which the deformation is to be studied. On the plane of intersection the cutting tool on a graduating machine is used to apply a co-ordinate grid with an accuracy of up to 0*01 mm. Then the parts of the stock are brazed together with Bud alloy. The alloy is applied to the whole surface of the intersection, including also the area to which the co-ordinate grid is applied. In this way, therefore, a solid specimen is obtained, in reality with a thin film of about 0· 1 mm of Bud alloy in the plane of intersection, but it should be taken into account that the Bud alloy has plastic properties close to the properties of lead, and the layer of it is very thin.

The rolled stock is heated to the melting temperature of the Bud alloy, after which the stock is easily separated along the surfaces of the intersection. The Bud alloy is carefully removed from those areas to which the co-ordinate grid was applied; after cooling of the stock it is possible to carry out measurements of the grid.

A co-ordinate grid was also successfully applied to the contact surfaces.

After rolling lengths of stock with an initial thickness and width of about 20 mm, and with drafts of less than 6 mm, the grid is clearly

116

INCREASING BULK DEFORMATION DURING ROLLING 117

visible; during rolling with greater drafts the grid is partly washed out, and measurement of it is complicated (Fig. 43).

In the experiments described a co-ordinate grid was applied on the lower, A9 and the upper, B, contact surfaces and the horizontal-

FIG. 43. Photograph of the contact surface of specimens: a — H0 =20-6 mm; B0 = 14-8 mm; ΔΗ = 4-8 mm (specimen No. 33); b — Ho = 19-5 mm; B0 = 19*5 mm; AH= 6-2 mm (specimen No. 60).

longitudinal plane of intersection C of the stock at mid-height (Fig. 44). Before the experiment check measurement was carried out of the grid in two mutually perpendicular directions, as shown by the arrows Z. The rolling of the lengths of stock was carried out in such a way as to produce an incompletely rolled length of stock with the zone of deformation within the limits of the co-ordinate grid (Fig. 45). DMR 9


The co-ordinate grid was measured on an instrument microscope with an accuracy up to 001 mm. Each cell of the deformed grid was measured in two directions—in the direction of rolling (length of the

6rid B

6rid A Grid C FIG. 44. Two halves of a specimen with co-ordinate grids before

brazing.

cell, /) and in the perpendicular direction (width of cell, b). If the box had dimensions before deformation of/0 and b0 then after deformation its dimensions had changed and became correspondingly lx and bx. According to the measurements obtained the value of the elongation factor and of the spread factor for a given cell of the grid can be determined from the equations:

λ -k '0


Deformation along the third axis, the compression factor, is easily determined, starting from the law of constancy of the volume

— = βχλχ· (38) Vx

In this way may be determined the three factors of deformation, characterising the bulk strain of an element of the metal having an infinitely small height.

This method of determination of the coefficients of the deformations is suitable for any location of the plane of intersection relative to the rolling direction. Thus, if the plane of intersection is formed by a longitudinal-vertical cross-section, then according to the measurements of the co-ordinate grid on an incompletely rolled length of stock, for an element of the metal located in any given area of the co-ordinate network, it is possible to determine the increasing elongation and reduction factors, and on the basis of these data it is easy to find the increasing spread factor, taking into account that the element of the metal under review has an infinitely small width. It is possible to proceed in the same way in the instance where the section under consideration is perpendicular to the direction of rolling, etc.

FIG. 45. Experimental specimen—incompletely rolled length.

Since real deformations are expressed as natural logarithms of the relative change in the linear dimensions in a given direction, and the displaced volumes in each direction are proportional to the logarithms of the coefficients of deformation, then in future we shall employ the law of constancy of the volume, written in the following form:

log^ + logft^log-J-. (39) Ίχ

9*


This makes it the more desirable, that the ordinates of the curves of deformations in this instance are additive, which simplifies analysis of the experimental data.

14. EXPERIMENTAL INVESTIGATION OF THE INHOMOGENEITY OF BULK

STRAIN

The investigation of the inhomogeneity of bulk strain was carried out in a 200 mm laboratory mill. Lengths of lead stock were rolled in plain, steel work rolls of 210 mm dia. Before carrying out each series of experiments the work rolls were carefully cleaned off with alcohol and lead plated. The lengths of stock investigated were rolled simultaneously with additional lengths of lead stock, which were passed into the work rolls somewhat earlier than the lengths under investigation, in order to eliminate possible free play in the moving couplings, and in order to have a stable rolling process from the moment of delivery of the specimen under investigation into the work rolls. Since in the rolling process it was necessary to produce incompletely rolled lengths of stock, the mill was stopped, the lower work roll was removed by means of a tapered screwdown mechanism, and the stock was withdrawn from the work rolls.

In all, three series of experiments were conducted. In order to obtain more objective and reliable data, in each series of experiments three identical lengths of stock were rolled. The initial thickness of the stock was 20 mm, and the initial width of the stock was 15 mm for the first series, 40 mm for the second, and 60 mm for the third, while the draft was 4-5 mm.

In this way a study was made of the inhomogeneity of bulk strain and the relationship between this inhomogeneity and the width of the stock. In all three series of experiments co-ordinate grids were applied to the specimens on the lower A, and upper B, contact surfaces, and on a horizontal-longitudinal plane C, at the mid-height of the stock (Fig. 44).

The overall length of the co-ordinate grids C, was chosen somewhat greater than (by 15 mm) the length of the zone of deformation, so that it would be possible to study the deformation before the entry, and after the exit, of the metal into and out of the work rolls; the length of the elementary cells of the grid was in all instances constant and equal to 2 mm.


The co-ordinate network was applied over the whole of the stock; the width of the elementary cells was chosen at 1/10 of the width of the stock.

Upper contact surface View along arrow Z

i iimiYYYiwwKX b\b

fin

Plane of intersection

ower contact surface FIG. 46. Part of a length of rolled stock with co-ordinate grid, and

diagram of the distribution of the elementary layers.

In Fig. 44 the elementary planes are indicated by the Roman numerals, and the numbers of the cells in each elementary layer by the Arabic numerals. The distribution of the sections is given in Fig. 46.

First Series of Experiments

In the first series of experiments, three lead specimens of dimensions H0 x i?o = 20 x 15 mm were rolled; the reduction for the pass was 22-7-24 per cent. The data of the control dimensions of the co-ordinate grids of specimen No. 33, rolled with a reduction of 23-3 per cent, and data of the dimensions of the same grids after deformation are compiled in Tables 11-13. The horizontal figures in these tables indicate the length of the cells after deformation, and the vertical the width. The control dimensions for the width of the cells are given on the left of the numbering of the planes, and for the length of the cells of the co-ordinate grid at the foot under the numbering of the sections.

From the data of the dimensions, curves were constructed for the increasing relative amounts of spread, and the increasing reductions of the elements of the stock a{a'\ b(b'\ c{c'\ d(d% e(ef) and / ( / ' ) ,

z l

e'

f

\f

e

O

o

o

o

O

s ON

<*

ID

X

IX

VIII

VII

VI

V

IV

III

II

I

Z —>

1

o

2 01

o

2 00 o

200

2 0 0

1 200

TABLE 11. CO-ORDINATE GRID

2

o

2 0 0

2 0 0

2 0 0

r-

200

2 200

3

|2·01

201

200

00

2 0 0

3

4

o

201

200

201

OO

2 0 0

4 2 00|2 00j

5

o iTi

201

201

201

00

202

5 200|

1 6

o

2 0 2

202

202

OO

2 02

6 2 00

7

o

2 0 4

203

203

2 02

7 200

1 8

o

207

o

207 o

207

208

8 201

A OF SPECIMEN N O . 33 (LOWER

1 9 1 1 1 1 © 1 ~ 2 1 6 1

1 1 1 1 1 1 214

1 2· 14 1 1 1 1 1 1 ■ 5 1 -214 1 1 1

9 2 0 0

10

2 18

218

218

2·Γδ

10

11

2-17

218

218

r» <* 2 · Γ δ

1 ii 2001200

12

1 ^

2·Γδ

2-17

218

00

2·Γδ

12

1 13

2·Γδ

ο 2·Γδ

ο 2·Γδ

00

2 Ϊ δ

13

1 14

2 1 9

219

218

00

2·Γδ

14

15

218

218

219

οο

2·Γδ

15

C0NTAC1

16

en

2-20

2 18

218

2-21

16

17

m

2-27

2-26

2-26

' SURFACE

1 ι§

rn

2 31

2 3 0

2-29

OO I 00

2·22|2·30

17 2 00,2 00|2 00j2 00| 2 00|2 00|

18

11 9

m

2-33

2-33

2-33

2-33

19 |

AFTER ROLLING)

20

m

2-34

0

2-32 0 «0

231

2-30

20 2 00|2 00 |200

21

m wn

233

0

2-32 0

2 31

00

2-32

21 200

22

22 |

23

23 |

24

24

1 25

25

X

IX

VIII

VII

VI

V

IV

III

II

I

Direction of rolling Note 1. Control dimensions of the grid to deformation indicated by arrows.

2. Beginning of zone of deformation indicated by dotted line.

z l

α'

b'

b

a

© as ©

o

O

©

o

o T—1

o

O i—1

o

?

X

IX

VIII

VII

VI

V

IV

III

II

I

Z « — >

1

2 0 0

o

200 o in

2 0 0

in

2 0 0

1 200

TABLE 12. CO-ORDINATE GRID B OF SPECIMEN No. 33 (LOWER CONTACT SURFACE AFTER ROLLING)

2

5 2 0 0

o

2 0 0 55

200

in

201

2 200

3

o

2 0 0

o

2 0 0 o

1-99

2 0 0

3 199

4

ON

2 0 2

o

201 o

202

200

4 200

5

o in

201

o

201

2 0 0

2 0 0

5 200

6

ON

202

o

201 o

2 0 2

2 0 4

6

7

o

2 0 4

o

205 o

2 0 4

2 0 6

7 199 |200

8 1 1 1

2 1 0

1 1 1 1 1

2 1 2

2 1 3 1 1 1 1 1 1 | o

u 2 1 6 1

1 1 8

200|

9 | 10

o

221

o *n 2-21

o

2-20

o

2-20

9 200

o

2-21

o

2-21

221

o

218

10

11

o in

2 2 4

o

2Γ9

12

0 in

2 1 9

0 «n

2-21 in «r>

2·20|2-21

o

2-21

0 in

219

11 | 12 200 |200 |2 00

13

O

2 1 9

O

2 2 0 in

2 2 0

0

2 1 9

13 200

14

0 in

2-23

ON

2-22 0 in

221

00

2-21

14 200

15

0 «0

2-23

0

2-21 1 -

2-21

2-25

15 200

16 | 17

0

2-23

0 «0

2-25

2-24

2 2 4

16 |2 00

0

2-25

0

2-25 0

2-25

0 «n

2-29

1 17 200

18

0

2-33

0 •n

231

2-32

0

2-32

18 200

19 | 20

0 «n

2-31

0 *n

2-33 0

2-32

1 ©

2 31

19

0 in

2 3 0

0

2-32 in

2-29

0 «n

2-3l

20 |200;2 00

21

0 in

2-29

0

2-30

2-30

0 in

2-32

21 200

22

22

23

23

24

24

25

1 25

1 1 1

X

IX

VIII

VII

VI

V

IV

III

II

I

Note 1. Control dimensions of the grid to deformation indicated by arrows, 2. Beginning of zone of deformation indicated by dotted line.

H

c

d'

d

c

0 0

"?

<o

Ό

ON

o

O

O T—4

c o

X

IX

VIII

VII

VI

V

IV

in

II

I

Z ""*"

1

1 200

TABLE 13. CO-ORDINATE GRID C OF SPECIMEN N O . 33 (LOWER

2

2 200

3 1

3

4 00

m

2 0 0 o i n

200 in

1200

2 0 0 m

2 0 0

2 0 0 o m

200 o in

2 0 0 o

2 0 0 r* CO

200 4

2 0 0 | 2 0 0

6 oo m

1-99 o

1 99

1-99

2 0 0

1-99 ON

1-99 in

1-99

1-99 o «n

1-99 co

2 0 0 5

200

6 oo in

1 99 o "?

1-99

199 in

1-99

199 ON

1-99 o in

1 99 o m

1 99 in

1-99 CO

1-98 6

200

7 VO

1 99 o

2 0 0 CM in

2 0 0

199 in

2 0 0 o in

200

8 ^ 1 vo '

201

1 9 i—1 vo

201 O H H

2 Ό 0 2 0 2 1 ^1

«n ' 2-01

1 "*■ uo

202 O i l ro

2Ό1<2·02 c o 1 m '

201 o 1 >n ■

2Ό1 O 1 O |

2-00,2-01 O l M l in \ > o '

1·99|2Ό1 *n

2 0 0

1-99 7

2 0 0

2-01

201 8

200

in

2 0 2

202

10 VO

2 0 4 ro

2 0 4 in **?

2 0 4 in

2 0 4 oo

2 0 4 "3-

204 r o | τΤ

m m

2-02'2-04 CO

203 co in

203

202

11 ON m

209 m in

209 oo in

209 00 in

2 0 8 o

2 0 8 in

208 00

209 r f I oo in m

2-06J2-lb -3-in

205

205 9 | 10

200 |200

in

2 1 2

2-12

12 o

2-Γδ 00 in

216 o VO

2 Ϊ 5 CO vo

2 1 4 in vo

2 1 4 VO

215 VO

216 vo

217 ON in

2-Γδ

2-20

1 13 r o VO

2-23 0 VO

2-23 vo

2-21 VO

2-21 0 0 VO

221 VO

2-20 VO VO

2-22 VO

2-23 VO

2 2 6

2-28 11 | 12 | 13

14 CO vo

2 3 0 c o VO

2-28 VO

2-28 ON VO

2-28

2 2 6

2-27 0

2-28 VO

2-28

15 00 VO

231 vo

2 31 ON

vo

2 3 0

2-31

2-30 co

2-31 CO

2 31 VO

2 31 <N I d VO v o

2-29|2-33

2-30 14

|200|2 00j2 00;200

2-35 15

200

CONTACT SURFACE

16 O

2 31 m VO

2-32

2 31

2-30 00

2-31 m

231

2-31 ON vo

2-32 VO

2-33

2-33 16

2 0 0

1 17

231 VO VO

2 31 0 t^

2-31 in

2-30 00

231 « 0

2 31 in

2-31 ON VO

2-33 VO

2-32

2-33

117 2 0 0

18 ro

2-31 VO

2 31

2 31

19 r o

2 3 0

\FTER ROLLING)

20 ro

231 VO 1 VO

\o 0

2·30|2·31

2-31 r- 1 vo

2-30|2-31

231 231 VO I «O

2-32J2-32 r o t ^

2-32

in

2-33 O 1 ON

r- vo

2-32|2-33 vo

2-33

2-33 18

200

ro VO

2-32

2-33 19

|200

CN

2 31

2-30 00

2-31 »0

2-32

2-31 00 VO

2-33 vo

2-33

2-33 20

200

1 21

21

22

22

23

23

24

—

24

25

25

X

IX

VIII

VII

VI

V

IV

III

II

I

Direction of rolling Note 1. Control dimensions of the grid to deformation indicated by arrows.

2. Beginning of zone of deformation indicated by dotted line.


having infinitely small height. By this means a review was made of the deformation of the layers located along the centre of the width of the stock, and of the layers situated at a distance of one cell from the edge of the width of the stock, in three planes. The deformation of the edge particles along the width of the stock was not considered, since, firstly under a microscope it is somewhat difficult to carry out measurements on the edges of the stock, and secondly on the contact surfaces transition of metal takes place from the side surface to the contact surface (Fig. 43), which gives rise to essential peculiarities of the deformation of the edge sections. The deformation curves for layers a and a' and b and b\ etc., are located symmetrically, so that the data of the dimensions for both layers were averaged, after which curves were constructed from the averaged values.

LogX

8 6 ί 2 0 2 f 6 8 10 12 U 16 18 20 22 21 261 Direction of rolling

^. FIG. 47. Curves of increasing elongation factors for elementary

layers located on the contact surface: 1—along the centre of the width of the stock; 2—close to the edge of the width

of the stock. Arrow shows direction of rolling.

On the basis of the experimental data for the six contact surfaces of the three specimens curve 1 (Fig. 47) was constructed for the increasing elongation factor for the elementary layers b and / (Fig. 46), located along the centre of the width of the stock on the contact surface. It should be noted that there is good agreement between the relative elongation for all three specimens, which is confirmed by the small degree of scatter of the experimental plots (Fig. 47), regardless of the


fact that the specimens had different drafts—4-6, 4-8 and 5-0 mm. Thus, for the horizontal section of the curve the absolute variations in the length of the grid cells amounts to 2-195 ± 0Ό15 mm, i.e. the variations in the increase in the length of the cells equal:

ΔΙ = 0-195 ± 0-015 mm, which amounts to

0015 1ΛΛ nn x 100 = 7-7 per cent.

As is evident from curve 1 (Fig. 47), the deformation of the surface layer starts before the entry of the stock into the geometrical zone of deformation (section Α' Β' of curve i), as has already been noted earlier during the description of the experiments to study the increasing vertical deformation.

Deformation starts at a distance of 7-5 mm, or 0-33 / in front of the entry cross-section of the stock into the work rolls. At the moment of entry of the stock into the work rolls the elongation of the surface layer attains a considerable value of about 40 per cent of logAl5 where λ1 is the coefficient of elongation of a grid cell on the contact surface during entry into the work rolls.

The increase in the elongation factor is continued even after the entry of the stock into the work rolls; the corresponding section of curve 1 is Β' C. Consequently, on this relatively small section (1-8 mm, or 0-08 /) slip takes place between the metal and the work rolls in the direction opposed to the rolling direction.

Further increase in the elongation factor is restricted, and on the horizontal section CD' of curve 1 there is no increase in the elongation factor,* i.e. longitudinal adhesion occurs between the metal and the work rolls. Let us observe that the section of adhesion merges with the entry cross-section of the metal into the work rolls and has considerable length.

/adh = 15-5 mm or - ^ -=0-69 .

On the section D' E' in front of the exit plane of the stock from the work rolls the increase in the elongation factor is continued, and here slip takes place between the stock and the work rolls in the rolling direction. It is important to note that this section of slip is about three times as long as the section of slip at the start of the zone of


deformation. Beyond the exit plane from the work rolls no deformation of the metal was established in these experiments.

A similar curve was constructed for the elementary layers a, a' (Fig. 46), located at a distance of one grid section from the edge of the width of the stock. From the experimental plots obtained for the six contact layers of the three specimens an average curve, 2, was drawn (Fig. 47), which almost completely repeats the course of curve 7, but as distinct from the latter has a somewhat smaller length of the zone of adhesion.

As the measurements showed (Tables 11-13), no spread of the elementary grid cells was detected, and an increase in the width of the contact surface took place exclusively as a result of the transition of metal from the side surface to the contact surface. Consequently, for all the elementary sections of the width of the stock, except those on the edge, the spread factor equals unity.

Consequently logß* = 0 and log(l/iyx) = l o g ^ , i.e. curves 1 and 2 for \ο%λχ are also curves for log(l/^x).

The absence of spread of the elementary sections of the contact layer of the stock provides grounds for the assertion that in the sections of adhesion there is neither longitudinal nor transverse slip of the metal, i.e. there is an absence of elongation and spread, and consequently there is also an absence of vertical deformation in a layer of a certain thickness.

These experimental data convincingly show that under specific rolling conditions in the zone of deformation there exists such a zone as is frequently called the zone of restricted deformation, in which there is in practice an absence of plastic deformation of the metal. In this respect there is complete similarity between the processes of rolling and compression.44 At the same time the prevailing opinion exists that with the existence of a zone of longitudinal adhesion in this section vertical and transverse deformation occur, and longitudinal deformation is non-existent. The experimental data presented above refute this opinion.

Deformation of the stock in the horizontal-longitudinal plane of intersection, C, (Fig. 44) differs sharply from the deformation of the contact layer (Fig. 48). All the sections of the width of the stock in the zone of deformation as distinct from the contact surface have strongly developed spread. In Table 13 are shown the dimensions of the co-ordinate grid of the plane considered here. From these data were constructed the curves of the increasing elonga-


FIG. 48. Photograph of the co-ordinate grid in the intersection of a specimen after rolling. Grid C in Fig. 44.

tion factors of the elements c and d (Fig. 46): curve 3 for element d and curve 4 for element c (Fig. 49). In the given instance there is no similarity in the structure of the curves for the three specimens and

I—I—I ίο9λ*\—■—I—I—I—I—I—I 1—I—I—Π—I—I


FIG. 49. Curves of increasing elongation factors for elementary layers located on the contact surface (based on Fig. 47):

7—along the centre of the width of the stock; 2—close to the edge of the width of the stock and for the plane of the horizontal-longitudinal intersection; 3—for the elementary layer located in the centre of the width of the stock;

4—for the layer located close to the edge of the width of the stock; stock 20 X 15mm; reduction about 23 per cent. Arrow shows direction of rolling.

their subsequent averaging, as took place for curves 1 and 2, since, firstly within the stock the nature of the deformation is more stable than on the contact surface, and secondly the deviations occurring in the values of the deformation are less than on the contact surface, and they do not have such an important value, as during the determination of the extent of the section of adhesion on the contact surface.

Here on the other hand, for comparison, are shown curves 1 and 2 from Fig. 47 for the increasing elongation factors of the contact layer of the stock.


Curves 3 and 4, which indicate the increase in the elongation of the layer of the metal at the mid-height of the stock, differ sharply from the corresponding curves 1 and 2 for the contact layers. The deformation of the stock starts at about the same point at which it started on the contact surface, but in the section AB the elongation has a negative sign (not elongation, but compression), i.e. considerable shortening of the grid cells occurred. Without doubt, at this point the deformation of the stock was substantially affected by the rear rigid end of the stock.

In section BC an increase in elongation occurs, but as distinct from the contact layers, at the moment of entry into the work rolls the elongation factor is very small. In the section CD the deformation increases at first at an increasing and then a decreasing rate, but has not the slightest tendency to remain constant, as occurred on the contact surface. From 8 mm in front of the exit of the metal from the work rolls the elongation is restricted and thereafter remains constant. There is also no deformation beyond the exit from the work rolls. The absence of an increase in elongation in the section DE must be explained by the action of the front rigid end, since at mid-height under the parameters of the zone of deformation considered here the rigid end of the stock prohibits elongation.

On the basis of curves 7, 2, 3 and 4, which express the change in the elongations in the individual sections of a transverse cross-section of the stock, assessments may be made of the vertical and horizontal distribution of the absolute longitudinal velocities in the zone of deformation in any given transverse-vertical cross-section of that zone.

Thus, from the law of constancy of a second volume v0 ω0 = vx ωχ (40)

it follows that vx _ ω0 =

v0 ωχ

where v0 is the rate of entry of the stock into the work rolls; vx is the absolute longitudinal velocity of an element of the

metal in a given transverse-vertical cross-section; ω0 is the area of the transverse cross-section of the element

before deformation; and cox is the area of the transverse-vertical cross-section under

consideration.


The ratio of the absolute vertical velocities of the two elements lying in the transverse-vertical cross-section of the zone of deformation under consideration is equal to the ratio of the elongation factors

Employing this ratio for analysis of the curves of the increasing elongation factors, it may be said that in every vertical section of the zone of deformation the difference in the longitudinal velocities over the width of the stock is relatively small. This is evident from a comparison of curve 1 with curve 2 and of curve 3 with curve 4 (Fig. 49). The ordinates of the corresponding curves for the same abscissae differ considerably from one another.

On the contrary, the difference in the longitudinal velocities over the height of a cross-section at the start and end of the zone of deformation attains a high value. This is evident from a comparison of curve 1 with curve 3 and of curve 2 with curve 4.

On the basis of the experimental data, in addition curves were constructed for the increasing spread factors (Fig. 50) for the elementary layers c and d (Fig. 46). The spread of a central elementary layer (curve 5) is considerably greater than the spread of an elementary layer located close to the edge of the width of the stock (curve 6). Such a difference over the width of the stock, as we have already seen, was not observed in the relative elongations.

Let us pass to consideration of the distribution of the increasing reduction factors over the length of the zone of deformation. On the contact surfaces there was no spread of the elementary grid sections, therefore

For the horizontal-longitudinal plane C it is a different matter. On the basis of curves 3 and 5 and curves 4 and 6 for the elementary layers d and c, curves were constructed of the increasing reductions. From eqn. (39) it is evident that for the construction of curves of \og(lfyx) it is sufficient to add the ordinates of curves 3 and 5 or 4 and 6 for the corresponding values of the abscissae, and to transfer the values obtained from the addition to a new graph within the same axes (Fig. 51). In this figure are shown for comparison the curves of the increasing reduction factors for all the elementary layers considered


I r 22-5-

2 0 2 1 6 8 10 12 H 16 18 20 22 21 26L Direction of rolling

_►

FIG. 50. Curves of the increasing spread factor for the horizontal-longitudinal intersection:

a—stock 20 X 15 mm; b—stock 20 X 40 mm; c—stock 20 X 60 mm; reduction about 20 per cent. Solid line for the elementary layer located in the centre of the width (in the text—curve 5); broken line for the elementary layer located close to the edge of the width of the stock (in the text—curve 6). Arrow shows direction

of rolling.

Log 7

6 12 0 2 6 8 10 12 n 16 18 20 22 21 26 L Direction of rolling

FIG. 51. Curves of the increasing reduction factor located on the contact surface (based on Fig. 47):

/—in the centre of the width of the stock; 2—close to the edge of the width of the stock and for the horizontal-vertical intersection of the stock; 7—for the elementary layer located in the centre of the width; 8—locadet close to the edge of the width; stock 20 X 15 mm, reduction about 23 per cent. Arrow shows

direction of rolling


above; the dash-dotted line, 9, is the curve for homogeneous deformation of the stock.

From comparison of the curves it is evident that in the centre of the stock a high degree of deformation occurs; from the centre to the surface the deformations die away, both in the horizontal, and to an even greater extent in the vertical, direction.

FIG. 52. Change in a co-ordinate grid during the compression of a cylinder.

We draw attention to the fact that the distribution of the deformation in a transverse-vertical cross-section of the stock during rolling is highly reminiscent of the picture of the distribution of deformations during compression (Fig. 52).44 This once again confirms the conformity which exists between rolling between plain work rolls and compression between plane parallel plates.

Second Series of Experiments

In the second series of experiments three lead specimens of 20 x 40 mm were rolled between work rolls of 210 mm dia. with a reduction of 20-7-21-3 per cent.

The rolling conditions, and the number and distribution of the co-ordinate grids on the lengths of stock did not differ from those of the preceding experiments, with the exception of the difference in the width of the stock.

The curves of increasing relative deformations along the three axes were constructed for the same elementary layers a, b, c and d (Fig. 46) as in the first series of experiments.

From the data of measurements on the six contact surfaces the average curve, 1, of the increasing elongation factors was constructed for the elementary layer b, located in the centre of the width of the stock (Fig. 53). Similarly curve 2 was constructed for elementary layer a, located along the edge of the width of the stock. The nature


of the increase in the elongation factors along the length of the zone of deformation for the elementary layer located along the centre of the width of the stock and the increase in the initial width of the stock were almost unchanged.

A"

log λχ, —im

Ό06\

Λ ' ι ix

UJftrr ί*Τ

ΟΌίψ-

WA

)Ρΰ2> ff

C ν ο

V Ο Y-t-Q^^Q

y Β"

1 Α ό /A ν | fv

topfi ladh--H-5

6 4 2 0 2

1*20-8 J L

O V A O ^ y J l

D" Αέ

uc Ί^

o ^ = ΛνχΡ

6 8 10 12 H 16 18 20 22 24 2d1


FIG. 53. Change in the increasing elongation factors for elementary layers located on the contact surface:

/—in the centre of the width of the stock; 2—close to the edge of the width; stock 2 0 x 4 0 mm, reduction about 21 per cent. Arrow shows direction of

rolling.

It is true that the relative length of the section of adhesion was somewhat increased:

/ a d h =15-2mm or -^L = 0-73, *adh

/

whereas for lengths of stock with an initial width of 15 mm this ratio was 0-69.

The position was quite different for the elementary layer a, located close to the edge of the width of the stock. Curve 2 has no clearly marked section of adhesion. Although on section C" D" the increase in the elongation factor takes place very slowly, it does still occur. This is evidence;of the fact that in the given section there is partial slip of the metal of the stock in relation to the work rolls in the longitudinal direction. The latter is caused by the fact that with the DUE 10


increase in the width of the stock the action of the rigid ends of the stock on the zone of deformation is heightened, and in fact the extension of the edges is increased. Apart from this the section A" B" of curve 2 is longer than the relevant section of curve 1. Consequently. the extreme sections of the stock relative to its width start to be deformed in front of the zone of deformation under the action of the tensile stresses somewhat earlier than those in the centre. At the moment of entry of any given cross-section into the work rolls the elongation factor over the whole width of the contact surface has an identical value, i.e. the elongations of the contact layer are equalised at this moment.

Spread of the cells of the co-ordinate grids on the contact surface in the zone of deformation, as in the first series of experiments, was practically non-existent; spread of the stock took place as a result of the transfer of metal from the side surface to the contact surface.

Thus, with an increase in the initial width of the stock up to 40 mm it was found that on the contact surface:

1. the extent of the zone of adhesion was somewhat increased; 2. in the central part of the width of the stock there is complete

adhesion of the metal to the work rolls both in the longitudinal, and in the transverse, direction; and

3. close to the edge of the width of the stock there is considerable longitudinal slip of the metal of the stock relative to the work rolls.

From the data of the dimensions in the horizontal-longitudinal plane of intersection of the stock curve 3 was constructed for the increasing elongation factors for the elementary layer d, located in the centre of the width of the stock, and the similar curve 4 for the elementary layer c, located close to the edge of the width of the stock (Fig. 54). As distinct from the corresponding curves of the first series of experiments (Fig. 49) the section CD of curves 3 and 4 in Fig. 54 has a greater rate of increase in the elongation factor. The section DE, in which the increase in the relative elongations is restricted in front of the plane of exit of the stock from the work rolls, became considerably shorter and equalled 5 mm, while the relative elongation in the zone of deformation increased, although the reductions were somewhat less than in the first series of experiments.

Therefore . * l Q g A " _ 1.41 log*, " ' '

where the subscripts I and II indicate the series of experiments.


This is completely natural, since it is known that the greater the initial thickness of the stock, the less is its spread and the greater the proportion of metal, displaced from the height, which will flow in the

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Direciion of rolling

FIG. 54. Curves of increasing elongation factors for elementary layers located on the contact surface (based on Fig. 53):

/—in the centre of the width of the stock; 2—close to the edge of the stock and for the horizontal-longitudinal plane of intersection of the stock; 3—for the elementary layer located in the centre of the width; 4—for the layer located close to the edge of the width; stock 20 X 40 mm reduction about 21 per cent.

Arrow shows direction of rolling.

longitudinal direction. In fact, from the curves of the increasing relative degrees of spread for the horizontal-longitudinal plane of intersection it is evident that the spread was considerably diminished by comparison with the spread in the first series of experiments (Fig. 50).

The appearance of the curves has substantially changed. Curve 5b (Fig. 50) for the elementary layer d, located along the centre of the width of the stock, and curve 6b for the elementary layer C, located close to the edge of the width of the stock, practically coincide. Consequently, the relative spread at mid-height of the stock is evenly distributed over almost the whole of its width.

Let us return to Fig. 54, in which is given the combined graph of the curves of increasing elongation factors for all the elementary layers under consideration. As is evident from this graph, the overall 10*


picture of the distribution of the relative elongations between the various sections of the stock is the same as in the first series of experiments.

As will be shown below, the neutral cross-section is located at the point where all the curves of the elongations intersect, and consequently the absolute longitudinal velocities of the particles of the metal become equal.

From Fig. 54 it is evident that over the width of the stock there is also inhomogeneity in the distribution of the relative elongations and longitudinal velocities, but to a considerably lesser degree than in relation to height.

Close to the plane of exit of the stock from the work rolls the relative elongations and longitudinal velocities are equalised, over both the width and the height of the stock.

The spread factor of the elementary layers on the contact surface, as has been mentioned above, equals unity, so that the curves of the increasing elongation factors are also curves of the increasing reduc-

8 6 2 0 2 4 6 8 10 12 U Direction of rolling

16 18 20 22 2ϊί

FIG. 55. Curves of increasing reduction factors for elementary-layers located on the contact surface (based on Fig. 53):

7—in the centre of the width of the stock; 2—close to the edge of the stock and for the horizontal-longitudinal intersection of the stock; 7—for the •elementary layer, located in the middle of the width; 8 — for the elementary layer located close to the edge of the width; stock 20 x 40 mm, reduction about 21 percent: 9—for homogeneous deformation. Arrow shows direction

of rolling.


tion factors. The relevant curves, 7 and #, for the horizontal-longitudinal intersection were constructed with the use of curves 3 and 4 for the increasing elongation factors, and of curves 5 and 6 for the increasing spread factors, and are presented in Fig. 55. In the same figure are given the curves of the increasing reduction factors for all the elementary layers considered above, and curve 9 for homogeneous deformation. From a comparison of these curves it is evident that in the zone of deformation negligible inhomogeneity of the distribution of the reductions over the width occurs, but marked inhomogeneity over the height of the stock.

Under such conditions deformation proceeds so that at the start and at the end of the zone of deformation there is very marked reduction of the surface layers of the stock, at the same time that the layers of the stock at mid-height are only slightly deformed. In the central part of the length of the zone of deformation the reverse picture may be observed, namely: there is intensive deformation of the layers at mid-height of the stock, while on the contact surface adhesion of the metal to the work rolls occurs, and the formation of a zone of restricted deformation.

Third Series of Experiments

In the third series of experiments the width of the lengths of stock was increased to 60 mm, but in other respects there was no difference between this series and the preceding experiments; the reduction of the stock during rolling was 21-5-21*8 per cent. The curves of the increasing relative deformations were likewise constructed for the elementary layers a, b, c and d (Fig. 46).

From the data of measurements on the six contact surfaces of the three specimens the average curve /, of the increasing elongation factors was constructed for the elementary layer b, located along the centre of the width of the stock. The similar curve 2, was constructed for the elementary layer a, located close to the edge of the width of the stock (Fig. 56). The section Ä B' of curve / is 50 per cent shorter than the corresponding section of curve 2. Consequently, the surface layer of the stock in front of the entry into the work rolls is inhomogeneously deformed over the width of the stock. The extreme sections of the stock in relation to width, under the action of the tensile stresses, start to be deformed considerably earlier than the central sections.


logXx

D"\

Ladh = 13'5

1 = 21-3

n

^ Ί

2 4 6 8 10 12 U 16 18 20 22 24 26 L

Direction of rolling ►

FIG. 56. Curves of increasing elongation factors for elementary layers located on the contact surface:

7—in the centre of the width of the stock; 2—close to the edge of the width of the stock; stock 20 x 60 mm, reduction about 21·5 percent. Arrow shows

direction of rolling.

The section of adhesion CD' on curve / has scarcely changed with the increase in the width of the stock (/adh =15-5 mm, or /adh// = 0-728.

Curve 2 for the elementary layer located close to the edge of the width of the stock, as in the second series of experiments, has no clearly marked section of adhesion. On the section CD" of length 13-5 mm or 0 6 3 / the increasing elongation factor does indeed increase, but very slowly, so that on this section partial slip of the metal of the stock occurs relative to the work rolls in the longitudinal direction.

Spread of the sections of the co-ordinate grid on the contact surface within the limits of the zone of deformation did not in practice exist, so that the curves of the increasing elongation factor are at the same time curves of the increasing reduction factor. The curves of the increasing spread factor for the horizontal-longitudinal plane of intersection of the stock for the third series of experiments are drawn in on the same graph (Fig. 50) for the sake of comparison, just as for the first two series of experiments. The increasing elongation factors for the horizontal-longitudinal intersection of the stock are presented


in Fig. 57. In this diagram are also shown the corresponding curves for the contact surface.

The curves of the increasing elongation factor (Fig. 57) and the curves of the increasing spread factor (Fig. 50) for the horizontal-


FIG. 57. Curves of increasing relative elongation factors for elementary layers located on the contact surface:

1—in the centre of the width of he stock; 2—close to the edge of the width of the stock and for the horizontal-longitudinal intersection of the stock; 3—for the elementary layer located in the centre of the width of the stock; 4 — for the layer located close to the edge of the width; stock 20 X 60 mm,

reduction about 21 ·5 per cent. Arrow shows direction of rolling.

longitudinal plane of intersection of the stock do not differ in character in any way from the corresponding curves of the second series of experiments. It is true that the value of the relative elongations has somewhat increased, while the relative spread has decreased, which should have been expected on increasing the initial thickness of the stock. From Fig. 57 it is further evident that the distribution of the relative elongations over the width and height of the stock has a different character. While the relative elongations are more or less homogeneously distributed over the width of the stock in a vertical-transverse cross-section, over the height of the stock they are distributed extremely inhomogeneously. The same may be expected of the distribution of the longitudinal velocities in the same cross-section.


In Fig. 58 is given the combined graph of the curves of the increasing reduction factors for all the elementary layers under review. In this same graph is drawn in curve 9 for homogeneous deformation. From the diagram it is evident that the reduction factors are distributed extremely inhomogeneously over the height of the stock. Deformation

Γ ~ Γ Ί toHl I I I I I I I 1 1 I I I I


FIG. 58. Curves of increasing reduction factors for elementary layers located on the contact surface:

7 —in the centre of the width of the stock; 2—close to the edge of the width of the stock and for the horizontal-longitudinal plane of intersection; 7—for the elementary layer located in the centre of the width of the stock; 8—for the layer located close to the edge of the width; stock 20 χ 60 mm, reduction about 21·5 per cent; P—for homogeneous deformation. Arrow shows direction

of rolling.

takes place in such a way that a rapid increase in the reduction factor for the mid-height of the stock corresponds to a slow increase in the factor for the surface layer, and vice versa. Thus in the centre part of the length of the zone of deformation the mid-height layers are deformed at a high rate, while at the start and end of the length of the zone of deformation the surface layers of the stock are deformed at a high rate.

Experiment in Rolling with the Use of a Lubricant. As we know, the presence of a lubricant on the surface of the stock or on the surface of the work rolls substantially changes the value and distribution of the contact frictional forces, the relationship between elongation and


spread during rolling, and consequently also the whole process of the flow of the metal. In the experiments in rolling a specimen with a lubricant a lead specimen was selected of the following dimensions: thickness 20-1 mm, width 19-7 mm, length 150 mm. Rolling was carried out with a draft of 5-4 mm or a reduction of 26-8 per cent.

Rolling was conducted in the same work rolls as previously, but a layer of lubricating grease was applied to the surface of the stock. The stock was solid and had co-ordinate grids only on the contact surfaces. From the data of the measurement of the two contact surfaces curves were constructed of the increasing elongations for the elementary layer along the centre of the width of the stock (curve 1) and for the elementary layer located close to the edge of the width of the stock, curve 2 (Fig. 59). In this same graph are constructed curves 3 and 4 of the increasing relative degrees of spread for the same elementary layers respectively.

6 ϊ 2 0 2 * 6 8 10 12 n 16 18 20 22 2t 26 ί Direction of rolling

►

FIG. 59. Curves of the increasing elongation factors a and spread factors b for an elementary layer of the contact surface during rolling

with a lubricant: 1 and 3—in the centre of the width of the stock; 2 and 4—close to the edge of

the width of the stock. Arrow shows direction of rolling.

Section AB of curves 1 and 2 are evidence of the fact that, as in the experiments described above, the deformation of the surface layer of the stock starts even in front of the plane of entry of the stock into the zone of deformation. At this stage the deformation is homogeneously distributed over the whole width of the stock. Since the original dimensions of the stock and the reduction in the present experiment


were almost the same as in the first series of experiments described above, so the curves obtained are compared with the curves of the first series of experiments. As is evident from a comparison of the curves, lubrication of the stock produced scarcely any changes in the preliminary deformation of the surface layer of the stock in front of the plane of entry into the work rolls.

In our opinion, this is explained by the fact that the preliminary deformation of the surface layers of the stock (their extension) is the consequence not of the presence of contact frictional forces, but of the stabilising action of the rear rigid end of the stock.

Beyond the line of entry of the stock into the work rolls the distribution of the increasing elongations along the length of the zone of deformation is substantially changed by comparison with the rolling of the same specimen without a lubricant. There is no section of adhesion, there is no marked slip of the metal of the stock relative to the work rolls, either at the start, or at the end of the zone of deformation, such as occurred in all the preceding experiments. The increase in the relative elongations in the given instance takes place almost homogeneously along the whole length of the zone of deformation. Over the width of the stock the relative elongations are also distributed more or less homogeneously, as is evident from a comparison of curves 1 and 2.

The presence of a lubricant also brought a change in the distribution of the relative spread in the surface layer of the stock. Whereas during rolling without a lubricant the width of the cells of the co-ordinate grids on the contact surface remained unchanged, and the spread of the stock took place as a result of the transition of metal from the side surface to the contact surface, in the present experiment all the cells of the co-ordinate grids had considerable spread. Over the width of the stock the amounts of spread were distributed somewhat inhomogeneously, for the elementary layer located along the centre of the width of the stock (curve 3, Fig. 59) had somewhat less spread than the elementary plane located close to the edge of the width of the stock (curve 4, Fig. 59). The increasing factors of spread have the maximum rate of increase in the first half of the zone of deformation, and it is basically here that the spread of the stock occurs, which is apparently connected with the restriction in the reduction along the length of the zone of deformation.

On the basis of curves 1 and 3 and curves 2 and 4 were constructed the curves of the increasing reduction factors—curve 5 for the elemen-


tary layer located along the centre of the width of the stock, and curve 6 for the elementary layer located close to the edge of the width of the stock. Both curves are given in one graph (Fig. 60), in which curve 7 has on the other hand been drawn for homogeneous deformation. From the diagram it is evident that the presence of a lubricant led to a more homogeneous increase in the increasing reduction factors along the length of the zone of deformation, which is clearly visible from comparison of the given curves with curves 1 and 2 of Fig. 51, and also with curve 7 for homogeneous deformation in Fig. 60.


FIG. 60. Curves of increasing reduction factors for elementary layers located on the contact surface during rolling with a lubri

cant: 5—in the centre of the width of the stock; 6—close to the edge of the width of the stock; 7—for homogeneous deformation. Arrow shows direction of rolling.

Thus, lubrication of the contact surface of the stock before rolling substantially changes the nature of the deformation, making it more homogeneous.

Experiment in Rolling with a Long Zone of Deformation. In all the preceding experiments the ratio of the length of the zone of deformation to the mean thickness of the zone was within the limits of 1-16-1-36. In order to explain what changes in the distribution of


deformations within the sphere of the zone of deformation are produced by a considerable increase in the length of the zone of deformation, one further experiment was carried out, without lubrication.

The initial dimensions chosen for the stock were: thickness 13-5 mm, width 14-5 mm, length 150 mm. Rolling was carried out with a reduction of 5-9 mm or 43-7 per cent. The ratio of the length of the zone of deformation to the mean thickness of the zone was therefore 2-36.

Rolling was carried out in the same work rolls as in the preceding experiments. The stock consisted of two halves with a horizontal-longitudinal intersection at mid-height of the stock, as in the first three series of experiments. Joining of the stock in the plane of intersection was obtained by brazing. In the plane of intersection and on both the contact surfaces a co-ordinate grid was applied. From data of the measurements were constructed the curves of the increasing deformation factors along the three axes for the elementary layers a, b, c and d (Fig. 46).

From the data of the measurements on the contact surfaces curve 1 was constructed (Fig. 61) for the increasing elongation factors. The

6 8 10 12 n 16 18 20 22 2<t 2d 28 L

Direction of rolling . FIG. 61. The increasing relative degrees of spread and elongations

on the contact surface. Arrow shows direction of rolling.

experimental plots for element b, located along the centre of the width of the stock, and for element a, located close to the edge of the width of the stock, lay very close together, so that for both elements a common curve was drawn, 1. Similarly curve 2 was constructed for the increasing relative degrees of spread. The nature of the curves is


almost the same as in the instance of rolling of stock with a lubricant. There is no zone of adhesion. Along the whole length of the specimen longitudinal and transverse slip of the metal of the stock occurs relative to the work rolls.

From data of the measurements in the transverse-horizontal plane of intersection curves were constructed of the increasing elongation factor and of the increasing spread factor, 5, for the element d (Fig. 46) and also corresponding curves 4 and 6 for the element c (Fig. 62).


FIG. 62. Curves of the increasing elongation factors a and spread factors b for an elementary layer located in the horizontal-longitu

dinal plane of intersection: 3 and 5 = in the centre of the width; 4 and 6—close to the edge of the width of

the stock. Arrow shows direction of rolling.

From a comparison of these curves with the corresponding curves, for instance of the first series of experiments (Figs. 49 and 50) it is evident that the increase in the increasing elongation factors takes place more homogeneously along the whole length of the zone of deformation, and they are also distributed more homogeneously over the width of the stock. The spread in the horizontal-longitudinal plane of intersection is less markedly developed than in the lengths of stock of the first series of experiments, but it nevertheless considerably exceeds the relative spread of the elementary layers on the contact surface. At the same time the spread of the elementary layer located along the centre of the width of the stock is greater than that of the elementary layer located close to the edge of the width of the stock, which was observed in the other experiments with lower ratios of the length of the zone of deformation to its mean thickness.


In Fig. 63 is given the summarised graph of the curves of the increasing elongation factors for all the elementary layers under consideration. From the figure it is evident that along almost the whole length of the

Direction of rolling ^.

FIG . 63. Composite graph of the increasing elongation factors: 1 —for the contact surface; 3 and 4—for the plane of intersection. Arrow shows

direction of rolling.

zone of deformation the elongation factors in the vertical-transverse cross-section of the stock are distributed homogeneously both over the width and over the height of a transverse cross-section of the stock. There is no doubt but that the distribution of the longitudinal velocities in the vertical-transverse cross-section will also be almost homogeneous with the exception of the small section at the start of the zone of deformation.

Finally, in Fig. 64 is given the summary graph of the curves for the increasing reduction factors for all the elementary layers under review—curve 7 for the elementary layers a and b, located on the contact surface, curve 8 for the elementary layer d, located in the plane of intersection in the centre of the width of the stock, and 9 for the elementary layer c, located in the same plane close to the edge of the width of the stock, and in the same figure is given curve 10 for homogeneous deformation. As is evident from the figure the increase in the reduction factors along the length of the zone of deformation is more homogeneous, and the concomitant distribution of the reductions in the vertical-transverse sections of the stock also became more homogeneous, both over the width, and also over the height of the stock, which is evident from a comparison of the given curves


with the curves in Fig. 51 and with curve 10 for homogeneous deformation.

Thus, the increase in the length of the zone of deformation, or more accurately in ////mean? in the given instance showed the same influence on the distribution of the deformations within the sphere of the zone

t 2 0 2 4 6 8 10 12 H 16 18 20 22 24 26 I Direction of rolling

>-FIG. 64. Composite graph of the increasing reduction factors: 7—for the contact surface; 8 and 9—for the plane of intersection; 10—for

homogeneous deformation. Arrow shows direction of rolling.

of deformation as the lubricant, namely an increase in the increasing deformation factors, and their distribution in the vertical-transverse cross-sections of the stock in the zone of deformation over the height and width of the stock became more homogeneous.

The experimental data set out above concerning the inhomogeneity of the bulk strain during rolling convincingly confirmed the opinion expressed earlier to the effect that the inhomogeneity of the deformation in any given direction is closely linked and is mutually bound up with the inhomogeneity of deformations in other directions.45

The experimental data refute the widely extended opinions that the inhomogeneity of the vertical deformation during rolling is completely absorbed by the inhomogeneity of the transverse deformation, while


the longitudinal deformation remains seemingly homogeneous (hypothesis of flat cross-sections). On the contrary, these data convince us that under definite rolling conditions, namely at small values of the shape factor (//i/mean) and at a high external friction coefficient sharply marked inhomogeneity exists not only in the vertical and transverse directions, but also in the longitudinal.

The three first series of experiments described above, which are distinguished from one another by the considerable change in the initial width of the stock, convincingly show that the initial width of the stock under otherwise equal conditions has little influence on the course of the curves of the increasing elongations along the length of the zone of deformation. This confirms the opinion that the basic parameter which determines the degree of inhomogeneity of the deformation at a given value of the external friction coefficient, is the ratio of the length of the zone of deformation to its mean height. On the other hand this refutes the opinion expressed earlier concerning the relationship between the zone of adhesion, and consequently of the inhomogeneity of the deformation also in general on the one hand, and the ratio of the thickness to the width of the stock on the other.11^17

In actual fact, in the experiments reviewed this ratio varied within the limits of 0-75-3-0, which did not show a marked influence on the relative extent of the zone of adhesion, and consequently on the inhomogeneity of the deformation also.

The experimental data likewise convince us that the kinematics of the rolling process between plain work rolls is similar to the kinematics of compression. The distribution of the deformations in the vertical, transverse and longitudinal directions does not differ in principle from such during compression. Naturally, rolling has its special features, distinguishing it from compression, and they should be taken into account during a review of the various facets of this or that technological process.

The inhomogeneity of the deformation during rolling is the consequence of the inhomogeneous distribution of the stresses within the sphere to the zone of plastic deformation. Theoretical solution of the basic problems of the theory of rolling require combined consideration of the states of stresses and deformations.

CHAPTER V

Experimental Investigation of the Inhomogeneity of Deformation during

Compression and the Rolling of "High" Bodies

15. DUCTILITY OF METALS

The ductility of metals is not a constant value, but is dependent on the conditions of deformation. During mechanical working these conditions are: the composition of the alloy or the type of material, the temperature of the metal being worked, the rate of deformation, the structure of the metal, the state of stresses, etc. With a change in these factors, there is a noticeable change in the ductility. It is known, for instance, that brittle marble becomes ductile under conditions of the sharply pronounced state of bulk strain of compression.

Another example of the dependence of ductility on the conditions of deformation may be given by the following: as a test of ductility, and also to reveal external defects, compression tests are frequently carried out. The test consists in compressing cylinders with a certain ratio DQJHQ until cracks appear, and the amount of deformation up to the appearance of cracks affords the possibility of assessing the quality of the metal.

In order to establish to what extent the result of a ductility test is dependent on friction on the contact surface, i.e. on the state of stresses during compression, experiments were carried out on the compression of cylinders of 10 mm dia. x 20 mm; the cylinders were of lead with the addition of antimony (5 and 10 per cent, in order to reduce the ductility to a certain extent), and also of silumin.

The compression was carried out with rough machined, dry polished, and lubricated and polished tools with small degrees of deformation DMR 11 149


until cracks appeared on the side surface; the appearance of cracks was detected by means of a microscope at a magnification of x 100.

The results of the experiments (the mean of 10 measurements) are presented in Table 14. From the table it is evident that the greatest

TABLE 14. THE DEGREE OF DEFORMATION DURING THE COMPRESSION OF CYLINDRICAL SPECIMENS UP TO THE APPEARANCE OF CRACKS IN RELATION TO THE SURFACE

STATE OF THE TOOLS (MEAN DATA FROM 10 EXPERIMENTS) INITIAL DIMENSIONS OF THE SPECIMENS 20 mm x 10 mm dia.

Chemical composition of alloy

95%Pb;5%Sb 90%Pb;10%Sb 88-5% Al; 11-0% Si; 1 0-5%Mg;0-l%Fe;0-l%Cu }

Degree of deformation until appearance of cracks during compression,

% between tools

Polished and lubricated

77-00 68-26

47-73

Polished, dry

64-73

54-79 37-71

Rough machined, dry

59-81 38-88

26-53

ductility is possessed by cylinders, on which compression is carried out under conditions close to the conditions of homogeneous compression, i.e. between polished and lubricated tools. Lower ductility is shown by cylinders compressed between dry, polished tools; on cylinders compressed between rough machined tools, the occurrence of cracks was observed at even earlier stages of compression. The first allowed 50-100 per cent greater compression than the last.

In all instances the occurrence of the cracks was observed on the side surface of the specimen. The first series of experiments with the polished and lubricated tools were characterised by the fact that deformation proceeds homogeneously under conditions close to a linear state of stresses. Convexity of the side surface (a barrel) scarcely arises. With the increase in the contact frictional forces convexity increases.

In so far as all other factors remained constant, the ductility of the metal in the zone of the side surface should in this instance be related to the diagram of the state of stresses, determined by the boundary conditions. As S. I. Gubkin14 remarks, there are grounds for considering that in the external layers of a cylinder tangential, tensile stresses are set up, and that they are manifested the more rapidly, the earlier a barrel is formed. In reality, the layers forming the barrel may be represented in the form of a tube with a varying wall thickness stress-

COMPRESSION AND ROLLING OF "HIGH" BODIES 151

ing the deformed volume (Fig. 65). Unfortunately there is no theoretical solution to this problem, and in the meanwhile we are obliged to satisfy ourselves with a merely qualitative explanation.

FIG. 65. Diagram of the extension of the external layers of a cylinder during compression.

The tensile stresses lower the ductility of the volume of the metal ABC and bring about the occurrence of cracks along the generatrix of cylindrical specimens. During the rolling of sheets the tensile stresses give rise to cracks on the edges; during the rolling of blooms cracks sometimes occur in the zone of maximum spread on the contact layers. The unfavourable system of the state of stresses may even lead to internal bursts, which sometimes takes place during the rolling of heavy ingots in a blooming mill. The causes of the occurrence of bursts are reviewed in greater detail below.

16. EXPERIMENTS IN THE COMPRESSION OF " H I G H " CYLINDERS

Recently in rolling literature the problem of the conditions of deformation of heavy ingots in a blooming mill has been considered.1,2° This question has great practical and theoretical importance. During the rolling of heavy ingots in primary mills, external and internal cracks sometimes occur in the ingots.

The same metal, on the other hand, shows adequate ductility under other relative dimensions of the zone of deformation. It is natural to suggest that during the rolling of "high" bodies stresses occur in them, which are unfavourably expressed in the plastic properties of the metal. Great importance also attaches to the fact that the plastic properties of the central part of the transverse cross-section of the ingot are considerably lower than those of its peripheral part. It would be rational, by changing the parameters of the zone of deformation, 11*


to change the system of the state of stresses, in order to increase the ductility of the metal being worked. The experiments carried out by us permit certain conclusions to be drawn concerning the nature of the state of stresses and concerning the relationship between the distribution of the deformations and stresses and the entirely natural factors.

Let us consider initially the compression of "high" bodies between plane-parallel plates, as a simpler process and therefore more readily subjected to study. The investigation of such processes and a comparison of them with rolling likewise permits an assessment of the limits of the applicability of the Saint Venant principle, which has great importance for the theoretical solution.

Above a theoretical solution has been given for the determination of the state of deformations during the compression of high cylinders. Let us present here additional data, which permit assessment not only of the state of the deformations, but also that of the stresses.

Observations of the compression of heavy ingots (according to works records) in powerful, hydraulic presses show that at low ratios of the diameter of the ingot to its height a so-called double barrel is formed. At a sufficiently low relative diameter of the ingot (DjH ^ ^ 0-3-0-4) the barrels are concentrated in areas located close to the contact surfaces, while the cylinder forming in the central part remains vertical (Fig. 66). But plastic deformation takes place even in the volume ABCD; one may convince oneself of this by observing the flaking off of scale during compression.

For an approximate assessment of the inhomogeneity of deformation the following experiment was initiated. A lead cylinder was compressed between rough machined steel tools. The initial dimensions of the cylinder were: diameter 30 mm, height 100 mm, D0IH0 = 0-3. The outline and dimensions of the specimen after compression are shown in Fig. 66. From the Figure it is evident that the part of the bulk at mid-height has a cylindrical shape. The mean, relative change in the height of the cylinder is Η0\Η1 = 100/85-6 = 1-17.

The mean diameter of the cylinder after compression, corresponding to homogeneous deformation, found from the condition of constancy of the volume, would be 32-4 mm.

From the relative change in the diameter in the central part of the cylinder, ABCD (Fig. 66), it is possible to assess the states of deformations and stresses of this part of the volume. In our experiment the diameter of the central part of the cylinder after compression equalled


32-5 mm, so that DJDQ^ 1-082. Thus the relative change in the diameter of the central part of the cylinder practically coincides (a series of parallel experiments confirm this) with those during homogeneous compression of the whole cylinder. Consequently, in this part of the volume of the cylinder the deformation took place under conditions of a linear state of stresses, and there are no grounds for considering that in the volume ABCD the deformation took place other than during homogeneous compression. This was also confirmed by experiments in the compression of cylinders consisting of individual circular plates.

In the volume EFCB (Fig. 66) the deformation was inhomogeneo.us. Before compression grids were applied to the side surface of the

cylinder in the form of concentric circles of definite diameters, which were measured again after compression.

As a result it was established that the whole of the contact surface represents a zone of adhesion, while a small increase in the diameter of the contact surface takes place as a consequence of the transition of metal from the side surface to the contact surface. As we know,44,

the zone of restricted deformation corresponds to the zone of adhesion. Therefore it may be asserted that in the zone BEFC the deformations are inhomogeneously distributed. The presence of a barrel after compression is always the consequence of inhomogeneous deformation. Where the coefficient of external friction is decreased, the in-homogeneity of the deformation is diminished. For a more accurate study of the deformation special experiments were put in hand.

The deformations within the volume of the cylinder were studied in the following manner.

The cylinders were built up from round, lead plates of a definite thickness, which were welded together with Bud alloy and machined to the required dimensions. After deformation the specimens were cut across the diameter, the surface of the intersection was polished and etched with molybdic liquor. The intermediate layers of Bud alloy were revealed as thin, dark lines. The distances between them were measured by means of an instrument microscope. The thickness of the plates before and after deformation were compared, and the vertical deformation, εζ, of each layer was calculated. The actual experimental apparatus made it possible to determine the deformation only along the z axis, but in consequence of the symmetry of the specimen the deformations along the two other axes were also determined. In actual fact, within cylindrical co-ordinates, taking


into account the incompressibility of the medium, we have (Fig. 67):

du7 d ur ur

dz dr r (42)

where = εζ is the vertical strain; du, Ahi ~JT^~h7

-^r— = er is the radial strain; dr ur — = ew is the circumferential strain; r ψ

hi is the thickness of the /-platelet after a small amount of compression; and

Ahi is the change in height of the /-platelet.

FIG. 66. Compression of a high cylinder (D0/H0 = 0-3).

FIG. 67. Diagram of the deformation of an element during the compression of

a cylinder.


Thus, if the deformations along the z axis (vertical) are studied, then this will serve to determine the overall state of deformations. In order to obtain a more complete picture of the distribution of deformations, the ratio of the diameter to the height of the cylinder, and also the coefficient of external friction, were varied within wide limits during the experiments.

The experiments show that when the ratio of the diameter to the height of a cylinder falls below 0-5 (during compression between dry, unpolished plates), a double barrel is formed; if this ratio is reduced still further, the nature of the distribution of the deformations changes little. The most typical diagrams of the distribution of deformations are presented in Fig. 68.

a 1\ I 2

3 4 5

I 6 7 8 3 10 11 11 13 '* 15 16 17

\Zf /

V V \ \ Λ

} Jl l /

/ ;

ώ

1 1 I

1 1 1 1 J.

K* v\ 1 ^

1 \

£iz

\\ \\

\l\ ' / 1

1

/ 1 1 2

3 4

5 6 7 8

1 9 10 11 11 13 1* 15 16 n

\h \ \ \

M Ί I I i

Π l| I I II

11 1 ti<

μζ

h\i \ \ ! M

N / / / / I

£zz

FIG. 68. Diagrams of the distribution of deformations during the compression of "high" cylinders between dry, rough machined (dashed lines) and dry, polished (solid lines) tools in two stages:

a-H0= 101-72 mm; H x = 96-4 mm; A H = 508 mm; AH/H0 = 0050; b-H0 = 96-4 mm; Hx = 9200 mm; ΔΗ = 4-64; AHfH0 = 0048.

The deformations of each layer were measured at several points on the radius and compared with the mean (homogeneous) deformation for the whole of the cylinder.


From Figs. 68 and 66 it is evident that in the intermediate zone ABCD the deformations are in reality distributed homogeneously, corresponding to the mean for the whole of the cylinder, and do not differ from one another at various points on the radius.

In the vicinity of the contact surface there were no deformations, thus forming a zone of restricted deformation. Then they rose fairly rapidly, exceeded the mean deformation and on the boundary CB fell to the mean. At this point adhesion took place along the whole of the contact surface; the particles of metal did not move along the surface of the tool, which was also evident from the concentric lines applied to the contact surface before deformation. Compression was carried out in several stages with the application of new grids to the newly forming contact surface. Thus it may be considered as established that the increase in the contact surface takes place only as a result of the transition of metal from the side surface to the contact surface with subsequent adhesion.

It should be mentioned that the values eZi and εΖ2 on the contact surfaces of cylinders should equal zero, in so far as the z1 and z2 axes pass through the zone of adhesion and of restricted deformation. But from Fig. 68 it is evident that εΖί and εΖ2 are not always equal to zero on the contact surface. This is connected with the fact that the thickness of the contact surface layer exceeded the thickness of the zone of restricted deformation, and the mean value of εΖι or εΖ2 in this instance will already be greater than zero.

On the basis of what has been set out already, it is possible to present the conditions of deformation in the volume CBEF in the following form (Fig. 66). Due to the presence of constraining frictional forces on the contact surface deformation takes place under conditions of triaxial compression, so that the influence of the constraining forces is decreased in proportion to the distance from the contact surface. On the boundary CB the effect of these forces dies out, and there is a transition to a linear state of stresses, corresponding to homogeneous deformation.44

It is understandable that the limit of the effect of the frictional forces at fixed initial dimensions of the cylinder and coefficient of external friction change in proportion to the increase in the compression, since under these conditions there is a change in the ratio of the diameter to the height. The position of this boundary at any given moment of compression is dependent only on the coefficient of friction and the ratio of the dimensions, so that it is rational to


characterise the state of deformations not by the value of the compression, but by the indicated ratio of the diameter to the height of the cylinder. Experiments in the compression of high cylinders of various metals show that the distribution of deformations is little dependent on the type of material, while at the same time they convince us of the applicability of the Saint Venant principle of the modification of the limit conditions. "High" cylinders (DjH = 0-33) were compressed between rough machined, dry polished, and polished and lubricated, tools. The last were lubricated with machine oil.

As is evident from Fig. 68, the nature of the distribution of the deformations and their values in the zones remote from the point of application of load, i.e. from the contact surface, are not dependent on the coefficient of external friction, and in general on the system of application of the external load, in so far as the diagrams of the normal pressures on the contact surfaces in the presence of a lubricant and of slip along the whole surface were different to those in the absence of a lubricant and the presence of a marked zone of adhesion. By this means it was shown by experiment that it is possible to apply the Saint Venant principle to certain instances of the mechanical working of metals.

The deformations are distributed somewhat differently during the compression of "high" and " long" parallelepipeds. The experiments were carried out with lead, welded specimens of the following dimensions: width 16-7 mm, length 50 mm, thickness 50 mm or / : H: B = 3 -0 :30 : 1-0. The reduction chosen was 7 mm or 14 per cent.

Compression was carried out between dry polished, and rough machined, tools. A co-ordinate grid was applied to the contact surface. According to the changes in the dimensions of the cells of the co-ordinate grid it was possible to assess the value and shape of the zone of adhesion. The results of the experiments are presented by the graphs in Fig. 69. The experiments permit the following conclusions to be drawn. On the contact surface there exists a zone of adhesion, the dimensions of which increase with the coefficient of external friction. Close to the contact surface is located a zone of restricted deformation. There is no clearly expressed zone of homogeneous deformation, which is explained by the comparatively great length of the specimens, and in consequence of this by the existence of a complex diagram of the state of stresses.


Section along A A

FIG. 69. Diagrams of the distribution of deformations during the compression of "high" parallelepipeds between dry polished (solid lines) and rough machined (dashed lines) tools. Longitudinal cross-

section along the plane of symmetry: b0= 16-7 mm; /0 = 50-2 mm; //„ = 501 mm; Hx = 42-8 mm; AH = 7-3 mm;

AHjH0= 00146.

17. COMPRESSION OF " H I G H " BODIES WITH EXTERNAL ZONES (RIGID ENDS)

During the compression of "high" bodies without external zones no tensile stresses arise in the internal zones, in which a linear state of stresses or a state close to this occurs, and the plastic properties of these zones are not diminished.

The study of the states of stresses and deformations during compression with the existence of external zones is somewhat complex. But, by comparing these states with the states of stresses and deforma-


tions during compression without external zones, it is possible to reveal the qualitative relationship between the stresses and deformations and the most essential factors of working.

The influence of the external zones on the process of deformation is described in the works of I. M. Pavlov,32,34 A. I. Tselikov and V. V. Smirnov,57 I. Ya. Tarnovskii44 and others.

In the works of A. I. Tselikov and V. V. Smirnov it was shown that the force required for the compression of a parallelepiped in the presence of external zones is greater than for the compression of the same parallelepiped under otherwise equal conditions, but in the absence of external zones. The difference in the forces required for compression in the first and second instances is dependent on the ratio of the length of the zone of deformation to its height, and is a decreasing function of this ratio.

Thus, the degree of influence of the external zones (rigid ends) on the forces of compression is dependent on the shape of the zone of deformation.

In order to study the kinematics of the process in those instances where the ratio of the length of the zone of deformation to its height is low, special experiments were conducted. As we know, during the rolling of heavy ingots in a blooming mill, during the first passes the ratio of the length of the zone of deformation to the mean thickness of the ingot is less than unity and normally exceeds 0-5.

In so far as during the experiments it was necessary to ensure conditions of deformation close to those in a blooming mill, the following dimensions were taken for the specimens: thickness and width, both 50Ό mm, length of the zone of deformation, equal to the width of the tools, 14-4 and 6-7 mm.

The overall length of the specimens was made relatively great, about 200-230 mm, in order to ensure the corresponding boundary conditions

GxiVy^xy, etC. |.v==00 = 0 .

The conditions indicated ensure lack of deformability and adequate rigidity of the external zones (rigid ends). In this way it is possible to be certain that the real limits of the zone of deformation are located within the specimen. If the specimen is insufficiently long, the deformations can affect the whole of its volume.

The lead stock for these experiments was welded with Bud alloy from plates of a definite thickness with the plates located perpendicular


to the direction of compression. After compression the stock was cut in the longitudinal direction so that the plane of intersection a-a coincided with the vertical plane of symmetry, and the plane b-b was located at a distance of 8 mm from the side surface (Fig. 70).

\ \ \

31-0

I I

! ** 5:

-31-6-

ΓΨΓ~\ I

\ \ m. -210-

5? II

FIG. 70. Compression of a length of stock with external zones.

The deformations were also determined on the side surface. After polishing and etching on the surface of intersection the boundaries between the plates were revealed. According to the thickness of a plate before and after the experiment, its relative vertical deformation was determined

h0i - hu ezt = Aoi

where / is the number of the corresponding plate. The longitudinal and transverse deformations were not measured,

since the results obtained were sufficient to explain the diagram of the state of stresses.


Apart from this, the transverse deformation under the conditions of the experiment was small, so that it may be considered that εζ = — ey.

It is interesting to note the following. Regardless of the fact that the plates were fairly rigidly welded together, the reaction of the metal being deformed with the external ends was so great that, if special measures were not taken, then a "fan" is formed during compression (Fig. 71). Apparently in the external zones high shear and tensile stresses arise, the total action of which tears one plate from another.

FIG. 71. "Fan" during the compression of stock with external zones.

Therefore, in the preparation of the specimens special measures were taken to ensure sufficient rigidity and compactness of the external zones. This simple example shows to what extent there is an intense reaction between the external zones and the zone of deformation.

The distribution of the vertical deformations is represented in Figs. 72 and 73; here for shortness only half the zone of deformation is given, and the other, which is symmetrical, is omitted. The numbering of the plates was carried out from the contact surface towards the centre.

For the contact plate, regardless of the presence of a zone of adhesion, sz Φ 0, which is explained by its great thickness (see above).

Comparison of the deformations in the two longitudinal cross-sections a-a along the centre of the stock and b-b close to the edge of the stock shows that they differ little from each other (Fig. 73). This special feature permits the tensor of the stresses to be considered as independent of the transverse co-ordinate. By solving the problem theoretically, in this instance it is possible to consider the state of plane strain. On the side surface and close to it the deformations are distributed differently. This is understandable, since the conditions of deformation on the side surface differ sharply from the conditions of deformation


within the bulk. Therefore it is impossible to judge the flow of the metal in the zone of deformation by studying the distribution of deformations on the side surface only.

σ Ζη b Ζη C Ζη

FIG. 72. Diagrams of the distribution of the vertical deformations during compression under a press of "high" parallelepipeds with rigid ends between dry, rough machined (dashed lines) and dry, polished (solid lines) tools. Longitudinal cross-section along the

plane of symmetry: « - / = 14-4mm, A HjH0= 0185; 6 - / = 14-4 mm, ΔΉ\Ήϋ = 0155;

c _ / = 6-7 mm, AHjH0= 0-155.

On a considerable part of the side surface the vertical deformations are homogeneously distributed and do not exceed the mean in value.

The horizontal layers of the metal, located close to the contact surface, undergo the maximum vertical deformation. Consequently no zone of restricted deformation is formed in the vicinity of the side surface.

Special experiments in the compression of similar specimens show that there is no slip on the contact surface; almost the whole contact surface represents a zone of adhesion, so that the layers which are closest to it form the zone of restricted deformation. The existence of a zone of adhesion is evidenced by the fact that the cells of the co-ordinate grid applied to the contact surface did not change their dimensions. An exception is provided only by those sections located in the corners of the contact surface. Here the size of the cells increased considerably, which points to a high degree of slip in these corners.


But the area of these sections was very small, which allows us to neglect them during theoretical investigations.

Let us compare the distribution of the deformations during compression without external zones (Fig. 69) and with external zones (Figs. 72 and 73) and at identical ratios of the length of the zone of deformation to its thickness. As is evident from Figs. 72 and 73, the zone of restricted deformation has a thickness of less than 5 mm, and the

Q Z b Z c Z

FIG. 73. Diagrams of the distribution of the vertical deformations during compression under a press of "high" parallelepipeds with rigid ends between dry, rough machined tools. The solid lines indicate the longitudinal cross-section along the plane of symmetry a-o, and the dashed lines that close to the edge of the length of stock

(plane b-b in Fig. 70): a-l= 14-4 mm, ΔΗ\Η*= 0185; b-l = 14-4 mm, ΔΗ\Ή*= 0-155;

c-l= 6-7 mm, Δ HjH0= 0155.

reduction of the contact plate 1 relates only to the bottom layer of this plate. The layers closely adjacent to the contact surface were not deformed, since a zone of adhesion was discovered on the contact surface. With a width of the tools of 6*7 mm there was very high reduction of plate 2 and the minimum reduction of plates 4 and 5. A similar picture was also observed with a width of the tools of 14-4 mm, but with the greater length of the zone of deformation the high reductions penetrate deeper; on further increasing the ratio of the length of the zone of deformation one may convince oneself that the greatest reduction in thickness takes place in the internal elements of the specimen.14 So far as the spread is concerned, as might have been expected,


this is most extensively developed in the vicinity of the contact surface. There is practically no spread in the central sections.

Comparison shows that the presence of external zones is essentially expressed in the distribution of the deformations. The part at mid-height, during compression with external zones, is deformed considerably less, which points to a change in the distribution of the stresses; close to the contact surface, apparently, the frictional forces predominate over other factors. The influence of the external zones is most noticeably expressed in the sections of the specimen at mid-height.

According to the deformation of each plate it is possible to judge the actual size and shape of the zone of deformation. In Fig. 70 the limits of the zone of deformation are indicated by dashed lines. As is evident from this drawing, the deformations in the sections adjacent to the contact surface extend to a shorter distance than in the central sections.

18. STATE OF STRESSES DURING THE COMPRESSION OF " H I G H " BODIES WITH EXTERNAL ZONES

Analysis of the experiments permits the following conclusions to be drawn concerning the state of stresses. During compression without external zones, in the central part of a "high" parallelepiped a state of stresses occurs which is close to linear. The presence of external zones considerably changes the distribution of stresses, which may be qualitatively described.

From the experiments it is evident that the layers, at mid-height are deformed over a great length. Apparently, the full force of compression in the central layers operates on a larger area, i.e. the mean vertical stress in the central layers is considerably less than on the contact surface. Since during the compression of "high" bodies the mean pressure on the contact surface is scarcely greater than yield point, while the area of the deformed zone in the centre is considerably greater than the area of contact, then it is natural to assume that at a certain distance from the contact surface the vertical stresses become less than the yield point. But the central layers are deformed (Figs. 72 and 73), so that the plasticity equation is observed. This is possible only when the longitudinal stress σ3 becomes positive, i.e. tensile

ffi + σ 3 = l'15<rs. (43)


Thus in the layers of the zone of deformation at mid-height a system of the state of stresses with differing signs occurs—a vertical compressive stress σ1 and a horizontal tensile stress, σ3.

In Fig. 74 is a rough representation of the diagram of the state of stresses: zones I and II are deformed under conditions of triaxial compression; zone III is deformed under the action of stresses of different signs. It is frequently asserted that deformation does not

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V

/

i

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L

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FIG. 74. Provisional diagram of the state of stresses in the zone of deformation at low values of //if and high values of b/l; OIBD is the diagram of the vertical compressive stresses; ECD the diagram of the longitudinal compressive stresses; and OEF the diagram of the

longitudinal tensile stresses.

penetrate into the central layers of an ingot, but said that at low ratios of the length of the zone of deformation to its mean thickness a linear state of stresses occurs in the central layers of an ingot. The experiments reviewed refute both the first and the second opinion. It is difficult to concede that with such a complex distribution of the deformations in any given zone a linear state of stresses could arise.

In so far as beyond the geometric limits of the zone of deformation the internal sections are noticeably deformed, it is important to consider the diagram of the state of stresses at this point. So far there is no theoretical solution for this problem, so that we shall attempt, as in the first instance also, to give a qualitative description of the diagram of the state of stresses beyond the limits of the geometrical DUE 12


boundaries of the zone of deformation. As has already been shown, during the compression of "high" prismatic bodies without external zones two barrels are formed, located symmetrically relative to the central axis of the horizontal plane. The diameter of the central part of the height is noticeably less than the diameter of the barrel. The same double barrel formation takes place also during the compression of parallelepipeds. It is natural to assume that the tendency to double barrel formation is preserved also during the compression of high bodies with external zones. But the external zones prevent the formation of a double barrel and strive to homogenize the longitudinal deformation at the junctions of the external zones with the zone of deformation. In consequence one may expect the occurrence of tensile stresses in those sections of a high body, in which the longitudinal deformation is minimal during compression without rigid ends, i.e. at the mid-height of the body. These tensile stresses also reduce the ductility of the metal and can be the cause of the occurrence of internal cracks even before the entry of the relevant volume of the stock into the geometrical zone of deformation. If defects have also arisen, their subsequent development continues in the zone of deformation under the action of the tensile stresses, after the corresponding "delivery" of the metal.

The tensile stresses lower the ductility of the internal layers of the metal both before its entry into the zone of deformation, and also within the zone of deformation, which is also a cause of the occurrence of internal defects—cracks.

Of great importance also is the structure of the ingot subjected to forging or rolling. Even small defects in the central part of the ingot, under the action of the longitudinal tensile stresses, will be increased and opened up, while under the action of the longitudinal compressive stresses such defects can also be welded up.

19. INVESTIGATION OF LOCAL VERTICAL DEFORMATIONS DURING MODELLING OF THE ROLLING OF HEAVY INGOTS IN A BLOOMING MILL

One of the important special features of an established process of rolling as distinct from compression in the presence of rigid ends is its stationary nature. Therefore a comparison between the deformation during rolling and during compression is, strictly speaking, possible only for local deformations, i.e. deformations which are completed within the course of a short time. For compression these will be small


deformations, and for rolling the deformation of a particle of metal during its movement to a position sufficiently close to some initial position.

If we draw a series of transverse-vertical cross-section in the zone of deformation during rolling, which are located at sufficiently short distances from one another, then the local deformation of any given elementary volume may be determined from the changes in the dimensions of this element during its movement from one position to the other.

For the investigation of the local deformations the following experiments were initiated. Packs of lead plates, brazed with Bud alloy, were rolled in a two-high mill with a work roll diameter of 113 mm; the rolls had a rough, polished surface. The ratios between the dimensions of the specimens, the reductions and the diameter of the work rolls were chosen such as they frequently are during the rolling of blooms. The mill was stopped when the specimen was between the work rolls. The photograph of a longitudinal-vertical cross-section of a specimen is shown in Fig. 75. The layers of Bud alloy after etching with molybdic fluid are revealed in the form of dark lines.

FIG. 75. Photograph of the longitudinal-vertical cross-section of a lead sticker, consisting of plates brazed with Bud alloy.

In order that the results of the experiments could be compared with compression, the increase in the thickness of each plate in a given transverse cross-section was calculated relative to the next nearest preceding transverse cross-section. The relative increase in the thickness 12*


0 1 2

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Nv

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FIG. 76..Local deformations in transverse-vertical cross-sections of a specimen H0 x B0 = 41-9 x 42-2 mm, rolled with a reduction of

7-1 mm between work rolls of 11*3 mm dia.: 1 — in the section a-a along the vertical-longitudinal plane of symmetry; 2 — on the side surface; 3 —in the section b-b along the vertical-longitudinal

plane, located at 5 mm from the side surface.

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FIG. 77. Local deformations in transverse-vertical cross-sections of a specimen H0 x B0 = 64-9 x 62-0 mm, rolled with a reduction of

11-2 mm between work rolls of 11-3 mm dia.: 7 —in the cross-section a-a along the vertical-longitudinal plane of symmetry;

2—idem, 3 mm from the side surface.


corresponded to the deformation during the rotation of the work rolls through a small angle. In consequence of the stationary nature of the process the thickness of the plate in the preceding cross-section z-1 is the initial for the following cross-section i, so that in this instance the methods of determination of the deformations during compression and rolling were identical, which permitted a comparison of the results. On the basis of the calculated deformations graphs were constructed (Figs. 76 and 77). Along the co-ordinate axis is placed the number of the plate, and along the abscissa the value of the deformation

_ Aht Ez ~~ hi '

where Ay is the increase in thickness during passage from the cross-section i-\ to the cross-section i; and

hi is the thickness of the plate in the transverse cross-section /. In view of the symmetry, on the graphs the deformations are given

only for the upper half of the specimen. Comparison of the deformations in the cross-sections a-a along the

vertical-longitudinal plane of symmetry, on the side surface and at 5 mm from the side surface (cross-section b-b) shows (Figs. 76 and 77) that on the side surface the deformations are distributed differently to those in the predominant part of the volume. Just as during compression the contact layers on a certain section of the length of the zone of deformation are not deformed and form a zone of restricted deformation. Then in the intermediate layers the vertical deformations reach a maximum value and die away towards the centre of the specimen, becoming less than the mean for the given cross-section. Further, the measurements showed that the actual zone of deformation is considerably greater than the geometric zone, especially along the horizontal plane of symmetry.

From the experiments it is evident that the distribution of deformations and, consequently of the stresses during the rolling of "high" bodies is qualitatively in accordance with those during compression.. In reality, even in this or that instance adhesion takes place on the contact surface, and then the deformations increase to exceed the mean, and in the vicinity of the horizontal plane of symmetry become considerably less than the mean.

Thus, the nature of the distribution of the deformations during the rolling of "high" bodies agrees with the distribution of deformations during compression with rigid ends.


20. INVESTIGATION OF THE INCREASING BULK STRAIN DURING MODELLING OF THE ROLLING OF HEAVY INGOTS IN A BLOOMING MILL*

With the aim of modelling the rolling of ingots in a blooming mill the parameters of the zone of deformation in the laboratory experiments were adopted so that they corresponded to the rolling of heavy ingots in a blooming mill. Specimens of 60 x 60 mm transverse cross-section were rolled between 100 mm dia. work rolls. The drafts were chosen from 6 to 10 mm and the reductions from 10Ό to 16-7 per cent. During the conduct of the experiments the method described in Chapters III and IV was used. The lead specimen consisted of sections brazed together with Bud alloy along longitudinal-vertical and horizontal planes of intersection. A co-ordinate grid was applied by means of an indexing machine both on the plane of intersection, and also on

FIG. 78. Diagram of the location of the elements of a transverse cross-section of the stock being rolled.

* Yu. I. Kolemagin took part in the conduct of these experiments.


the contact surfaces. The increasing deformations in two directions were determined from the changes in the dimensions of the cells of the co-ordinate grid; the deformation in the third direction was determined from the condition of constancy of the volume. In the preceding chapter this method was described in greater detail.

In Fig. 78 is given the diagram of the location of the individual elements of a transverse section of the stock (A, B, C, D, etc.), the increasing bulk strain of which is considered below. As is evident from this diagram, there is a horizontal plane of intersection III—III, passing through the midheight of the stock (the horizontal plane of symmetry), two horizontal planes of intersection II—II and IV-IV, located at a distance of 5 mm from the contact surfaces, and one longitudinal-vertical surface VI-VI, passing through the centre of the width of the stock (the vertical plane of symmetry).

The location of the co-ordinate grids and the initial dimensions of their cells on the various surfaces are shown in Fig. 79.

In Fig. 80 are given the curves of the increasing vertical, longitudinal and transverse deformations of the element A, located on the contact surface at a distance of 7 mm from the side surface of the stock after rolling with a reduction of 8 mm or 13-3 per cent. From the figure it is evident that even an element located fairly close to the edge of the width has a large section of adhesion (horizontal sections of the curves in Fig. 80). The plastic deformation of the element A starts considerably earlier than the entry of the metal into the work rolls. In addition, the sections of the curves of the increasing reduction and elongation factors up to the entry plane into the work rolls show that the deformation of the element A up to its entry into the work rolls is very great and amounts to about 75 per cent of the total deformation of this element per pass. In the zone of exit of the metal from the work rolls the maxima of the curves are attained, which are characteristic above all for the contact layers and the adjacent layers. It is obvious that these maxima of the curves are connected with the homogenization of the longitudinal deformations of the various elements of the transverse cross-section of the stock beyond the line of exit of the metal from the work rolls in connection with the presence of a front rigid end of the stock. In Fig. 81 is shown a photograph of the deformed co-ordinate grid applied to the contact surface I-I.

Let us consider the deformation of the elements F, G, I (E), located in the horizontal plane III—III (Fig. 78). The curves of the increasing vertical, transverse and longitudinal deformations of these elements

Vl-VI

ui-m

FIG. 79. Location of the co-ordinate grids on various surfaces.

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FIG. 80. Curves of the increasing coefficients of deformation of element A. Arrow shows direction of rolling.

172


FIG. 81. Deformation of the co-ordinate network located on the contact surface.

are presented in Figs. 82 and 85. From the figures it is evident that the limit of the real zone of plastic deformation over the whole width of the horizontal layer at mid-height of the metal is located at a great distance from the geometric plane of entry of the metal into the work rolls. The distance from the initial boundary of the actual zone of plastic deformation to the plane of entry of the metal into the work rolls amounts to 50-60 per cent of the geometrical zone of deformation, while the vertical and longitudinal deformations up to the entry into the work rolls have the same sign as in the whole of the zone of deformation, i.e. an increase in the length and a reduction in the thickness.

Of importance are the experimental data on the transverse deformation of the elements located in the horizontal plane III—III. While the elements Fand G, located in the central part of the width of the stock, acquired positive spread (Figs. 82 and 83), the elements E and /, located close to the side surface, acquired negative spread (Figs. 84 and 85). Thus, the absence of spread observed during the rolling of heavy ingots in a blooming mill (or extremely small changes in the width of the stock both in the direction of an increase and also that of a decrease) at mid-height in many passes is in practice the result of the fact that the central part of the width has positive spread, and the edge sections negative spread, so that the total spread of the whole horizontal layer is very close to zero. Negative spread (reduction in width) of the elements E and / and extremely low positive spread of

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FIG. 82. Curves of the increasing coefficients of deformation of element F. Arrow shows direction of rolling.

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FIG. 83. Curves of the increasing coefficients of deformation of element G. Arrow shows direction of rolling.

174

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FIG. 84. Curves of the increasing coefficients of deformation of element /. Arrow shows direction of rolling.


FIG. 85. Curves of the increasing coefficients of deformation of element E. Arrow shows direction of rolling.

175


the elements F and G confirms the existence of longitudinal tensile stresses in the central layers of the thickness of the stock. In Fig. 86 is shown a photograph of the deformed co-ordinate grid applied to the plane III-III (Fig. 78).

FIG. 86. Photograph of the deformed co-ordinate grid applied to the horizontal plane of symmetry, III-III (Figs. 78, 79).

Further, let us consider the deformation of the elements B, C and D of the plane II—II, located at a distance of 5 mm (or about 17 per cent of half the width of the stock) from the contact surface. Element C is located in the centre of the width of the horizontal layer II—II; the centre of the element B is located at a distance of 10 mm from the side surface, or x/3 of half the width of the specimen; the centre of the element D is located at a distance of 5 mm from the side surface, or l /6 of half the width of the stock. In Fig. 87 are shown the curves of the increasing vertical, longitudinal and transverse deformations for the elements considered.

Let us note the special features of the deformation of the element of the metal II—II. The most important of them is the existence of large spread, especially for elements B and D, located close to the side surface. From Fig. 87 it is evident that the transverse deformation of these elements amounts to 35-40 per cent of the vertical deformation, while the remaining 60-65 per cent applies to the longitudinal deformation (logftc/logl/ife = 0-35-0-40; log^/logl/ifc = 0-65-0-60); there is neg-


ligibly small development of transverse deformation of the element C. In general the spread of these layers of the metal attains considerable

values, which is confirmed also by the works experimental data set out below, obtained during the rolling of an ingot in a blooming mill.

FIG. 87. Curves of the increasing deformations of elements B, C and D. Arrow shows direction of rolling.

Of importance is the nature of the deformation of the elements of the transverse section of the stock before entry into the work rolls. The element C located in the centre of the width of layer II—II undergoes longitudinal compression (negative elongation) and of vertical thickening even before entry into the work rolls (Fig. 87), which is not observed for the element D; the element B, located further from the side surface than element D undergoes deformation before entry into the work rolls, which is intermediate between the deformations of elements C and D.


In Fig. 88 is shown a photograph of the deformed co-ordinate grid located on the plane II—II (see Fig. 78).

Let us consider the deformation of the elements K, L, M and N> located in the longitudinal-vertical plane of symmetry VI-VI. The curves of the increasing vertical, longitudinal and transverse defor-

FIG. 88. Photograph of the deformed co-ordinate grid located on the plane II-II (Fig. 78).

mations of these elements are shown in Figs. 89 and 90. From the figures may be seen the change in the nature of the deformation before the entry of the metal into the work rolls. The element N, located in the centre of the mid-height of the specimen, is thinned and lengthened before entry into the work rolls; the nature of the deformation before entry into the work rolls of the other elements, located closer to the centre of the height, remains the same, which is confirmed by Fig. 82 for the element F. On the contrary, the deformation before entry into the work rolls of the metal, located in the layers adjacent to the contact ones (elements K, L and M) bears another character, namely: compression in the longitudinal direction and thickening in the vertical (Figs. 89 and 90). Such a change in the nature of the deformation is explained, in our opinion, by double barrel formation not only in the transverse, but also in the longitudinal direction, which is confirmed by the photograph of the deformed co-ordinate grid in the


longitudinal-vertical plane VI-VI (Fig. 91) and the homogenising action of the rear rigid end. In Fig. 90 is presented also the curve for the contact layer, the horizontal section of which indicates the presence of a zone of adhesion, and the length of this section gives its extent. The horizontal section of the sub-contact layer corresponds to the

FIG. 89. Curves of the increasing vertical deformation of various elements located in the longitudinal-vertical plane of symmetry: K—sub-contact surface; L, M—layers at 5 and 7 mm from the contact surface; N—at a quarter of the height of the stock. Arrow shows direction of rolling.

length of the zone of restricted deformation on the corresponding level. Up to the entry of the metal into the work rolls the layers of metal, located close to the contact surface, or in the zone of barrel formation and triaxial compression, undergo negative elongation (shortening), while the layers, located in the central part of the height of the stock, acquire positive elongation (lengthening). A similar conclusion may be drawn also for the vertical deformations. The


difference between the corresponding curves in Fig. 89 and 90 determines the transverse deformation (eqn. (38)).

The transverse deformation of plane VI after the exit of the metal from the work rolls amounts to 20 per cent of the vertical (logßxj logl/nx & 0*2), and the longitudinal to about 80 per cent (logA*/ J o g l / ^ ^ 0 - 8 ) .

Ü-060 — i 1 1 Elements//,/Τ,£,Λ7,Λ/

U


FIG. SO. Curves of the increasing longitudinal and transverse deformations of various elements located in the longitudinal-vertical

plane of symmetry: K— sub-contact surface; L, M—layers at 5 and 7 mm from the contact surface; TV—at mid-height of the stock. Here is shown the curve for the contact layer,

H (Fig. 92).

Let us compare with the results obtained the data of the experiments for the increasing deformations of the elements of the horizontal planes III and II, located in the mid-height of the stock (Figs. 82-85) and at 5 mm from the contact layer (Fig. 87). Regardless of the fact that these elements / and £, located close to the side surface, undergo negative transverse deformation, nevertheless the overall nature of the change in the transverse deformation over the width of this layer


remains the same as for the other parameters of the zone of deformation (Chapter IV), namely the transverse deformation is decreased on passing from the central element to the side elements.

FIG. 91. Photograph of the deformed co-ordinate grid located on the longitudinal-vertical plane of symmetry.

As distinct from the other planes the transverse deformation of the elements B and D in plane II—II, located closer to the edges of the width, is considerably greater than for the central element C It is possible that this is linked with the location of this element close to the zone of restricted deformation, which thins down towards the edges of the stock, in consequence of which the elements B and D undergo greater vertical, and consequently also greater transverse, deformation than the element C. This is confirmed by a comparison of the curves for the vertical deformation of the elements B and D on the one hand with the same curve for the element C on the other (Fig. 87).

All the operational data set out above relate to rolling of 60 x 60 mm specimens with a draft of 8 mm. The same experiments were conducted at drafts of 6 and 10 mm. For the sake of brevity all these operational data are not presented here, in so far as they in large measure confirm the material set out above. Let us restrict ourselves merely to a comparison of the curves of the increasing deformations at different drafts of the elements and subject to otherwise equal conditions. DMR 13


In Fig. 92 are to be found the curves of the increasing longitudinal deformations for the element H, located in the centre of the width of the contact surface (Fig. 78), at various drafts of the stock. As is evident from this figure, with an increase in the draft there is likewise

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FIG. 92. Curves of the increasing longitudinal deformations of the element H at various reductions of the stock. Arrow shows direction

of rolling.

an increase in the longitudinal deformation, which is completely natural, but the relative extent of the zone of adhesion is somewhat decreased with the increase in the draft in connection with the fact that under such conditions there is an increase in the ratio of the length of the zone of deformation to its mean thickness.

Also noteworthy is the fact that with an increase in the draft there is a decrease in the negative elongation (shortening) before the entry of the metal into the work rolls, which is also evidence of a certain decrease in the inhomogeneity of the deformation with the increase in the draft and in the ratio of the length of the zone of deformation to its mean thickness under otherwise equal conditions.


In connection with the fact that element H does not undergo transverse deformation, the curves for the increasing elongation factors are simultaneously also curves for the increasing vertical deformation. A photograph of the deformed co-ordinate grid, located on the contact surface of the specimen, to which a draft of 10 mm was given, is presented in Fig. 93.

FIG. 93. Photograph of the deformed co-ordinate grid located on the contact surface of the specimen. Draft 10 mm.

In Fig. 94 are presented the curves of the increasing vertical and transverse deformations of the element F, located in the centre of the plane III-III at various drafts of the stock. The curves of the increasing longitudinal deformations are determined by the difference between the relevant curves of the vertical and transverse deformations. From the figure it is evident that with an increase in the draft the deformations in all three directions are increased; the transverse deformation is positive at any of the drafts. The elongation of the element F before entry into the work rolls at any given draft of the stock is also positive. As distinct from the elements, located on the contact surface, in the zone of exit of the metal from the work rolls the maxima on the curves of the increasing reduction factors are very indistinctly marked.

In Fig. 95 are presented the curves for the increasing vertical and transverse deformations of element /, located in the plane III-III close to the side surface. As is evident from the figure, the transverse 13*

DEFORMATION OF METALS DURING ROLLING

08 0-6 04 02 Direction of rolling

FIG. 94. Curves of the increasing vertical (solid lines) and transverse (dashed lines) deformations of the element F at various reductions

of the stock. Arrow shows direction of rolling.



FIG. 95. Curves of the increasing vertical (solid lines) and transverse (dashed lines) deformations of the element / at various reductions of

the stock. Arrow shows direction of rolling.


OS Ού Μ 0-2

Diredion of rolling 02 -0·ϊ

FIG. 96. Curves of the increasing elongation factors for elements F and / at various reductions of the stock. Arrow shows direction of

rolling.

FIG. 97. Photograph of the deformed co-ordinate grid applied to plane 111—III (Fig. 78). Reduction of stock 10 mm.


deformation is negative, so that with the increase in the draft it approaches zero; which confirms the fact of the decrease in the longitudinal tensile stresses with the increase in the draft under otherwise equal conditions. This condition has great practical importance, since it convincingly shows that in order to deal with internal defects it is necessary to increase the draft in each pass.

It is important to compare the elongation factors of the elements F and /for corresponding drafts (Fig. 96); they were shown to be practically identical, while the curves for the increasing vertical deformation of the element F are located considerably higher than the corresponding curves for the element /. This is explained by the difference in the signs of the transverse deformation of the elements Fand /(Figs. 94 and 95).

In Fig. 97 is shown a photograph of the deformed co-ordinate grid, applied to the plane III—III, for a draft of the stock equal to 10 mm.

In Fig. 98 are to be found the diagrams of the increasing vertical, longitudinal and transverse deformations of the element L for various drafts of the stock. The maxima of the curves for the vertical and longitudinal deformations are expressed all the earlier the less the draft, which is explained by the increase in the ratio of the length of the zone of deformation to its mean thickness with the increase in the draft and consequently a certain decrease in the inhomogeneity of the deformation. The longitudinal deformation before the entry into the work rolls for any given draft is negative, so that thickening and spread of the element of the stock occurs. Let us note that the increasing reduction and elongation factors before the entry of the metal into the work rolls is the greater (by absolute values), the less is the draft, which is explained by the decrease in the ratio of the length of the zone of deformation to its mean thickness and the increase in the inhomogeneity of the deformation with the decrease in the draft.

Finally, let us consider the deformation of element D (Fig. 78) at various drafts. In Fig. 99 are presented the curves for the increasing vertical, longitudinal and transverse deformations of this element. Attention should be drawn to the fact that in this part of the transverse section of the stock high transverse deformation develops. But, as is evident from the figure, with the increase in the draft the transverse deformation increases considerably more slowly than the longitudinal. It is obvious that during a further increase in the draft the increase in the transverse deformation of the element D is restricted, and to this will correspond the transition from a double barrel shape to a single.


■\0080

FIG. 98. Curves of the increasing, vertical, longitudinal and transverse deformations of the element L at various reductions of the

stock. Arrow shows direction of rolling.



FIG. 99. Curves of the increasing, vertical, longitudinal and transverse deformations of the element D at various reductions of the

stock. Arrow shows direction of rolling.


Such a change is confirmed also by the curves for the transverse deformation of the element F (Fig. 94), located in the horizontal plane of symmetry, which indicate a considerable increase in the spread with the increase in the draft. Thus, on changing over from a draft of 8 mm to a draft of 10 mm, the transverse deformation of element Fis increased approximately by 100 per cent, and of the element D by only 20 per cent.

In Fig. 100 is shown a photograph of the deformed co-ordinate grid, located in the vertical plane VI-VI (Fig. 78), obtained at a draft equal to 10 mm.

FIG. 100. Photograph of the deformed co-ordinate grid located on the longitudinal-vertical plane of symmetry. Reduction 10 mm.

In conclusion let us observe that similar experiments were conducted during the hot rolling of welded specimens of aluminium and copper by the method described in Chapter III. The results of these experiments did not differ in practice from the experimental data described in the present section for the modelling of the deformation of heavy ingots in a blooming mill during the rolling of lead specimens.

21. WORKS EXPERIMENT IN THE ROLLING OF A STEEL INGOT IN A BLOOMING MILL

The experimental data set out in the preceding section show that the deformation of metal during the rolling of heavy ingots in a blooming mill is extremely inhomogeneous, and one of the phenomena of

COMPRESSION AND ROLUNG OF "HIGH" BODIES 191

this inhomogeneity is double barrel formation. For the study of the deformation of a steel ingot and to verify the possibility of application of the results of the laboratory experimental data for the rolling of different metals to the rolling of steel ingots in a blooming mill under works conditions, we carried out rolling of a 6-ton ingot of rimming steel in a 1150 mm blooming mill. All the dimensions of the ingot before rolling are shown in Fig. 101. Before rolling the ingot was heated

l·*-2100 »I

FIG. 101. Shape and dimensions of the steel ingot before rolling in the blooming mill.

to a temperature of 1200°C. In the first pass the draft was equal to 60 mm (relative to the thin end); the second and third passes were carried out without edging with drafts of 85 and 90 mm. In the second pass the work rolls were stopped at the time when rolling was still incomplete; after reversing, the ingot was withdrawn from the work rolls. In this way the natural shape of the zone of deformation was obtained. Similarly the natural shape of the zone of deformation was obtained in the third pass. After cooling, all the dimensions of the ingot were taken (Fig. 102). As might have been expected, in the zones


adjacent to the contact surfaces extensive spread took place, while in the central part in practice no spread occurred.

As is evident from the figure, on reducing the ingot by 60 mm at the thin end the spread of the central part of the height of the ingot

lAh=250

ΣΔΟ-88 ΣΔο=1 CC

Σ Ah-160 lAb=66

FIG. 102. Shape and dimensions of the incompletely rolled ingot.


equalled zero, while the maximum spread in the areas adjacent to the contact surfaces was 22 mm. The transverse cross-section B-B (Fig. 102) was reduced in two passes with a total draft of 160 mm. The maximum total absolute spread in the zones adjacent to the contact layers thus amounted to 66 mm, and in the zones of mid-height to only 8 mm. Finally the transverse cross-section C-C was reduced in three passes with a total draft of 250 mm. Under such conditions the maximum total absolute spread in the zones adjoining the contact surfaces was 88 mm, and in the zones at mid-height only 7 mm.

During the conduct of the laboratory experiments in the rolling of different metals, not only qualitative, but also quantitative results were obtained, which were about the same as during the rolling of the steel ingots in the blooming mill. This provides grounds to consider that the results of the modelling carried out under laboratory conditions of the rolling of heavy ingots in a blooming mill reveal laws which apply under production conditions.

Naturally, it would be wrong not to draw attention to the intrinsic factors which occur under production conditions, but are absent during modelling of the rolling process under laboratory conditions. Here it is necessary to have in view at least the two following factors. Firstly, the inhomogeneity of the structure and chemical composition of a steel ingot, and also the presence of defects, especially in the axial zone of an ingot, which leads to inhomogeneity of the mechanical properties in various parts of an ingot. There are grounds for the assumption that the axial zone of an ingot of killed steel has lower plastic properties than the peripheral zone. This has very great importance during the rolling of ingots. Therefore greater demands on the quality of a steel ingot are necessary and legitimate.

Secondly, the inhomogeneous temperature of the ingot both over its length, and also over a transverse cross-section (which is especially important) gives rise to a corresponding inhomogeneity of the mechanical properties of the steel. If the temperature of the surface layers of an ingot is greater than the temperature of the axial zone, then this intensifies the double barrel formation and the inhomogeneity of the deformation. On the contrary, if the temperature of the axial zone of an ingot is greater than the temperature of the peripheral zone, then this decreases the extent of the double barrel formation and can decrease the danger of the occurrence of internal bursts.


22. THE STATE OF STRESSES DURING THE ROLLING OF " H I G H " BODIES

The experimental data set out above, obtained during the modelling of the deformation of heavy ingots during rolling in a blooming mill, show that the state of stresses and deformations during rolling are the same as during compression, but in the presence of rigid ends. Therefore the state of stresses during rolling with double barrel formation may be described in the following manner. Almost the whole of the contact surface represents a zone of adhesion, forming the basis of the zone of restricted deformation I (Fig. 103). The upper and lower zones of restricted deformation have a small volume, since they are sufficiently thin. In these volumes in all directions compressive forces

FIG. 103. Diagram of the state of stresses during rolling in a blooming mill.

operate, whereby the ratio between the stresses is such that plastic deformation does not take place in these volumes. Beyond zones I are located zones II of more pronounced vertical and transverse deformation.

During the deformation of the metal in the transverse direction in this zone there occurs great convexity (a barrel); in the longitudinal direction convexity also occurs, but it is eliminated at the ends of the zone of plastic deformation under the influence of the homogenizing action of the front and rear rigid ends. In zone II the state of stresses is also described as a system of manifold compression, but the ratios between the compressive stresses are such that there is an absence of plastic deformation. The longitudinal compressive stresses, as we pass from the contact to the horizontal plane of symmetry, decrease and


on the boundaries between zones II and III become equal to zero. In zone III there is a state of stresses with differing signs—vertical compressive, and longitudinal tensile stresses. In relation to the transverse stresses it may be said that they are very small in their absolute value. In actual fact, it was shown above that the transverse deformation in zone III is extremely small, so that by neglecting it, we may consider that in this zone there is a state of plane strain. In this event the transverse stress equals half the sum of the other two stresses. Taking into consideration the difference in the signs of the components of this half total, we come to the conclusion that the transverse stress will be small. Let us note that the side dashed lines in Fig. 103 diagrammati-cally limit the zone of diffusion of the vertical compressive stresses, and not the boundaries of the zone of plastic deformation, which naturally is not one and the same thing.

The state of stresses of the metal described here for the rolling of heavy ingots in a blooming mill is in close and direct connection with the experimental data set out above in relation to the deformation of the metal. In actual fact, the full vertical force in any given horizontal cross-section is the same. But as we pass from the contact surface towards the horizontal plane of symmetry, the area onto which th e vertical force is applied is increased, and consequently the mean v alue of the vertical compressive stress is also diminished. Therefore at a certain boundary (in Fig. 103 the plane of division between zones II and III) the vertical stress becomes less than the yield point. Then the deformation should be concentrated only in zone II, but in connection with the presence of rigid ends the longitudinal deformations of all parts of the transverse cross-section of the stock, as I. M. Pavlov34

remarks, are identical. Therefore the metal of zone II, being lengthened, as it were compulsorily draws the metal of zone III along with it; the latter in its turn resists elongation and consequently decreases the elongation in zone II; the transverse deformation in zone II is consequently increased.

Thus the longitudinal tensile stresses are a consequence of the presence of the rigid ends. This is also confirmed by the data set out above concerning the compression of "high" bodies with and without rigid ends. In the latter instance tensile stresses are non-existent in the centre of the body, and in the former they occur (Fig. 74).

Even the elementary description of the state of stresses during rolling in a blooming mill presented here shows that this question is fairly complicated, and its further study is necessary.


23. THE LONGITUDINAL VELOCITIES OF THE FLOW OF THE METAL DURING ROLLING IN A BLOOMING MILL

On the basis of the experimental data it is possible to construct diagrams of the velocities of the flow of the particles of metal in the zone of deformation. As has been mentioned above, the velocity of any given layer in any given transverse-vertical cross-section of the zone of deformation may be calculated, if the dimensions of the transverse cross-section of the element under consideration in the given plane are known and also the absolute velocity of the particles in the neutral cross-section. From the condition of equality of the second volumes40

it follows that /

where vx is the longitudinal velocity at a given point; vy is the same in the neutral cross-section; lx is the length of a cell at the given point (zone); ly is the length of a cell in the neutral cross-section (zone).

By measuring the cells of the co-ordinate grid in the longitudinal vertical cross-section of the zone of deformation, it is possible to determine 10 and lx in any given section of the zone of deformation. The velocity in the neutral cross-section is taken to be unity, since we are interested not in the absolute velocities, but their relationships.

The location of the neutral cross-section may be determined from experimental data, starting from the fact that in this cross-section the increasing longitudinal deformations become equal, and consequently the absolute velocities, vX9 also become equal.

Since vY is also the velocity of the rigid movement, then the velocity of the plastic movement, determining the inhomogeneity of the deformation, may be calculated as the difference

vP = i'x - Vy

The absolute velocities in any given transverse-vertical cross-section of the zone of backward slip may also be determined from the equation

lx vx = ι·ο T > l0

where v0 is the longitudinal velocity up to the entry into the zone of plastic deformation, identical for all the elements of a transverse cross-section of the stock; and

10 is the initial length of the element under review.


Since vy is the velocity of the rigid movement, then it is more convenient to construct graphs for vx on the basis of an equation into which vy enters.

A lead specimen of dimensions H0 x B0 = 60-5 x 58-0 mm was rolled with a reduction of 6*2 mm, to which the corresponding ratio of the length of the zone of deformation to its thickness was 0-31; the angle of the bite was 20°. The velocities were determined for the longitudinal intersection along the vertical plane of symmetry. The method of preparation of the specimen has been described above. The system of numbering of the layers of the stock and of the transverse cross-section, in which the velocities were reviewed, is shown in Fig. 104.

The velocities of the particles of metal were calculated in the cross-sections xjl = 0-9; 0-7; 0*4 and 0-3 so that the origin of the coordinate system was located in the exit plane of the metal from the work rolls, and the positive direction of the x-axis was opposed to the movement of the metal.

The diagrams of the velocities are presented in Fig. 104, where the rectangle ABCD represents the diagram of the rigid body velocity νγ, and the figure DEFC the diagram of the velocities of the plastic movements. The vector sum of these two diagrams gives the diagram of the absolute velocity, vx, in the transverse-vertical cross-section of the zone of deformation under review.

From Fig. 104 it is evident that the absolute velocities of the particles of metal differed only slightly from each other, but the inhomogeneity of the velocities of the plastic movement was large. From the diagrams of the velocities it is evident that around the entry into the zone of deformation (cross-section 1-1) the velocities of the plastic movement are distributed in the same way as during the compression of a cylinder or a "high" parallelepiped, forming a characteristic double barrel. The same picture was observed also in the vicinity of the exit from the zone of deformation (cross-section 3-3).

From the graph it is further evident that at the actual exit (cross-section 4-4) the velocities of the flow of the metal become almost entirely equal.

Thus, the experiments confirm that equalisation of the longitudinal velocities takes place beyond the exit plane of the metal from the work rolls, and this takes place in consequence of the existence of a forward rigid end. The same may also be said of the deformation of the metal, DMR 14


1 2 3 t y

|

--2_

9

\10 f

77

77

I 1 ««—

^

^ 3 2 Γ — ·

7 3 Ί — I — *■

7

1-1

2-2

V?

FC

E

3-3

-Vy-

vr

CF

\

H

C F

0\£

FIG. 104. Longitudinal velocities of movement and velocities of plastic movement in the zone of deformation during rolling.


and also of the diagrams of the longitudinal velocities before the entry into the work rolls, but the zone of expansion of the plastic deformation before the entry of the metal into the work rolls is greater than behind its exit from the work rolls.

From the data set out above it follows that for the study of the states of stresses and deformations during rolling it is necessary to take into account the presence of rigid ends.

From the aspect of mechanics the rigid ends determine the boundary conditions for the theoretical study of the states of stresses and deformations.

The experiments convincingly confirm that the influence of the rigid ends is increased with the decrease in the ratio of the length of the zone of deformation to its mean thickness, and consequently during the rolling of heavy ingots in a blooming mill their role is very great.

Here only the so-called internal inhomogeneity is reviewed, since we are talking about the rolling of stock with a rectangular, transverse cross-section between plain rolls. It should be born in view that during rolling in shaped passes the influence of the rigid ends is increased.

14*

CHAPTER VI

The Use of Variational Principles for the Solution of Some Problems of Rolling Theory

24. THE USE OF INTEGRAL EQUATIONS OF THE WORK DONE IN DEFORMATION FOR THE DETERMINATION OF THE FORCES

The solution of problems of the mechanical working of metals with the use of plasticity theory methods encounters considerable mathematical difficulties. The assumptions normally employed do not allow a series of factors to be taken into account. The plane, and all the more so the triaxial, problem are not solved to the extent to which this is necessary for the theory and practice of the mechanical working of metals.

The need for such solutions makes it necessary to dispense with the accurate methods of plasticity theory, and by adopting a simplified model of the process, to search for a suitable, approximate solution.

In order to solve a series of problems of the theory of the mechanical working of metals the energy conditions will be used, which have been widely employed by A. F. Golovin.8,9

The method suggested here amounts to establishing integral equations, which connect the work done by an external load on an element of the deformed body, with the work which is done by the internal stresses in deformating this element. This makes it possible to determine the pressures on the contact surface of the body being worked.

We shall calculate the work done by the external load during the deformation of a given element, dependent on the nature of its deformation and co-ordinates, making use of the equality of the amounts of work done by the operative forces and the resistances to deformation, where an ideal plastic medium is considered. During compression between plain tools, or rolling (and indeed such instances are also

200

VARIATIONAL PRINCIPLES FOR SOLUTION OF SOME PROBLEMS 201

reviewed), the work required for deformation of the element equals (Fig. 105)

A= pAHAxAz, (44)

where p is the resistance of the metal to deformation (pressure); Ax, Az are the dimensions of the element under review in plan; Ah is the absolute deformation.

f-'

<*< r&h vl/ /

FIG. 105. Diagram for determination of the work of the external force during compression:

P—external force; Tx and Tz—forces applied to the element from the medium.

Then A = Ai + A2, (45)

where Ax is part of the whole work, determined only by the yield point as of the deformed body and the change in shape of the element, and not dependent on the reactions, external in relation to the elements under review. This part of the work, let us call it "ideal", is the same for all elements and equals

Ax = kAHAxAz. (46)

But the applied force should carry out additional work equal to the work done in overcoming the reactions on the element from the surrounding medium

A2 = A - Αλ.

Such a setting of the problem makes it possible to take into account a series of existing factors. Let us note that the corresponding integral


equations can be solved by the method of successive approximations or by some other method.

In accordance with what has been set out it is possible to calculate the work expended in the deformation of each element, having thereby solved the problem of the distribution of the pressures.

We shall show that results which are known in the theory of the mechanical working of metals can easily be obtained by employing the suggested method.

25. FORCES DURING COMPRESSION. PLANE PROBLEM

Let us consider compression between two plane-parallel plates (Fig. 106). Let us select the element ABCD; let us locate the co-ordinate axes, as shown in the sketch. Let us give a small amount of compression

v \ \ \ \ N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ ^

I L

^ ΔΧ

^ N \ ^ \ N \ \ \ \ \ \ \ ^ ^ FIG. 106. Compression. Plane problem.

A H. The symmetry of the problem makes it possible to calculate the work done in a quarter of the whole area of the cross-section.

Let us assume, as in the work of Ye. P. Unksov,50 that 1. the vertical cross-sections remain vertical after deformation, and

also flat (hypothesis of flat cross-sections); 2. the frictional forces on the contact surface are proportional to the

pressures, i.e. τ = p μ. The work done on the selected element per unit length in accordance

with eqn. (44) is determined A = pAxAH,

and its ideal part as At = kAHAx,

where k is the constrained yield point, equal to 1-15 as.


The work of the supplementary resistances AA, connected with the existence of external friction, may be calculated from the following considerations. The element ABCD, being deformed, displaces the volume OBAE to an amount equal to the change in the dimension of the element in the direction of the x axis and determined by the law of constancy of the volume

ΔΗΔχ As =

H

The force, operating on the element from the volume OB AC, is completely determined by the summation of the forces of contact friction from the edge of the body to the element, i.e.

. r , ΔΗΔχ r A2 = μ ΔΞJ pdx = μ ——J p

dx.

The whole work of deformation will equal X

A = ρΔΗΔχ = ΙιΔΗΔχ + μ —— / pdx, 0

Reducing to Δ ΗΔχ, differentiating by x and solving the differential equation and taking into account the boundary conditions (/>/*=o= k\ we obtain

p = k eH .

Taking into account that we chose the origin of the co-ordinates differently to Ye. P. Unksov, to obtain the formula of Ye. P. Unksov it is necessary in the last equation to replace H by H/2, x by bjl — x, after which we obtain the known formula of Ye. P. Unksov50

2/i / b \

p = keH ^2 "ΑΛ (47)

26. FORCES DURING THE COMPRESSION OF A CYLINDER

In the cylinder let us choose as small element, bounded by two concentric arcs (surfaces) and two radial planes (Fig. 107). Let us locate the axes of the co-ordinates, as shown in the figure; in view of the symmetry, we shall calculate only the work done on the upper half of the element.


FIG. 107. Compression of a cylinder.

The work done by the force p during the deformation of such an element will equal

A = pAx x xocAH= AL + A2.

The movement of the section ADEF on account of the deformation of the selected element equals

dxAH

During such movement the forces of friction on the contact surface ADEF do work. Let us take into account that these forces obstruct the movement of the element ABCD, and the forces £<r3 sin^x/2, on the contrary, lighten the deformation, pushing out the volume AFED in the direction of the x axis. Then the work of the supplementary


resistances during deformation of the element will equal

dxAH r dxAH r x 2 = μ Η' J pxocdx W~J 2a3^>Hdx> X X

whence p xocAHdx = k xocAH dx +

r r xAHdx Γ , (xAHdx Γ + μ — pxdx — a3Hdx.

H J H J (48) Let us assume, as Ye. P. Unksov50 does, that σ2 = <r3, and using the

plasticity equation in the form

or3 = p - k,

after which the desired equation is derived in the form r r

px = k x + -j— j p xdx — I (p — k) dx, X X

or after differentiation by x

p x + p = k — ' — p x + p - k, M

whence dp _ μ ~dx~= ~~WP

or _ μ

p = Qe Ht

Determining the integration constant from the boundary conditions p\x=r = k9

w e obtain

p = k eHr

In our symbols H' equals half the height of the cylinder. Replacing / / ' = Hj2, we obtain the equation of Ye. P. Unksov for the determination of the pressure50 during the compression of a cylinder

p = k e H ' 2 /. (49)

2 0 6 DEFORMATION OF METALS DURING ROLLING

27. FORCES DURING ROLLING. PLANE PROBLEM

(a) Zone of Backward Slip

Let us consider half the vertical cross-section of the zone of deformation OABC (Fig. 108). As the work rolls turn through a small angle Αφ, the element will acquire a reduction Δ H'x.

FIG. 108. Rolling. Plane problem.

During such reduction, in the zone of backward slip the element displaces the volume of metal OAEF to the extent

ΔΗ'Δχ Asx = HI

Just as above, account is taken of the displacement of the volume OAEF on account of the deformation only of the element under review, and not the total displacement of this element, equal to the absolute backward slip at the given point.

The forces operating from the volume OAEF on the element, equal .V X

Tx = μ j p dx — ί ρ tan 9^ dx.

VARIATIONAL PRINCIPLES FOR SOLUTION OF SOME PROBLEMS 2 0 7

In this equation account is taken of the horizontal components of the radial pressure. Then the work done in the deformation of a unit width of the element, will equal

X

A = ρΔΗ'χΔχ = kAH'xAx + μ -^- pdx -o

ΔΗ'χΔχ r jf,—J P tan <px dx, (50)

0

where the first term of the equation represents the "ideal" work done on a unit width of the element.

Following A. I. Tselikov, let us replace the arc of the periphery by the corresponding chord. Therefore

ΔΗ' \&ηφχ = — — ,

After replacement of the values found in eqn. (50) and cancelling AH'xAx, we obtain eqn. (50) in the form

m-^xfl P = k + TW— I pdx-—;— jjjr- I pdx, AH' 1_

HQ — X

ΔΗ' and after multiplication by H0 — —-— x and differentiating by x

dp μρ kAH' =

<*-Hi-**Lx l(Hi-*»Lx)~ '

Replacing H0 by H0j2 and AH'byA H/2, we obtain dp μρ kAH _ rfj" i/0 'ΔΗ + (H0 AH \ = ·

2 2/ X Z ' \ 2 ~ΊΓΧ)

The equation obtained is a linear differential equation.


The pressures p can be found from the formula

p = e-fnx)dx[c- fcpWef'^dx],

where the coefficients and the free term are designated respectively:

μ , „ kAH Ax) = -

2 AH 21

φ(χ) = 2/ H0 AH

Ί Γ 2/

Calculating the integrals, we obtain:

P = kAH 2μΙ C

H0 AH \U± , X) Δ H + 1

2 2/ The integration constant we find from the boundary conditions

= k =

C =

kAH 2μΙ

2μΙ AH

+ 1

Introducing the criterion of A. I. Tselikov δ = 2μ l/AH, and taking into account that Hx = H0 - (A H/l) x, we obtain the known formula of A. I. Tselikov for the zone of backward slip55

ρ = τ ( 0 _ 1 ) ( | L ) + 1 (51)

(b) Zone of Forward Slip Just as simply it is possible to obtain a formula for the zone of

forward slip. Placing the co-ordinate axes at the outlet from the zone of deformation, let us obtain the initial integral equation

P = k + -jj-\ pdx + jjj-j Pdx

After manipulations similar to those presented above, we obtain the equation of A. I. Tselikov55 for the zone of forward slip

ρ = τ 05 + ■>(£)' 1 (52)


As we see, both methods give identical solutions. In the first instance the equilibrium and plasticity equations were solved jointly. In the second, the law of equality of the work done by the operative forces and that of the resistances to deformation is the basis. Here the role of the equation of the state of the medium, evidently, fulfils the condition of additiveness of the action of the forces. This condition may be formulated as follows: the "ideal" work of deformation of an element is dependent of the amount of work transmitted by the element.

In reality, without this condition the problem would be more complicated to solve, since At would be difficult to determine.

There are grounds for asserting that such a condition is not too wide an assumption. The solution obtained below gives values of the specific pressures, closely corresponding to the experimental values.

Justification of the condition adopted is confirmed in an indirect manner by the agreements indicated above between our solution and the solutions of A. I. Tselikov and Ye. P. Unksov.

Certain difficulties arise during the determination of the work transmitted by the element. But in the simplest instances this work can be calculated.

Let us note that for the derivation of eqns. (47), (49), (51) and (52) the hypothesis of flat cross-sections was employed, in so far as it was assumed that the stresses are not dependent on the vertical co-ordinate, and consequently the flat cross-sections remain flat during deformation. Below it will be shown that the utilisation of integral equations for the work is possible also even in the instance of inhomogeneous deformation.

28. FORCES DURING COMPRESSION. PLANE PROBLEM IN THE ABSENCE OF CONTACT SLIP

Let us consider plane strain (Fig. 109 a). Let us state the relationship between the shear stress and the vertical co-ordinate, as E. P. Unksov51 does, in the form

r °s y

xy ]/3 H' let us consider that there is no slip on the contact surface. Let us assume that after deformation the horizontal cross-sections remain flat, and the horizontal movements of the particles are determined by the expression jji _ yi χ


This is equivalent to the hypothesis that the barrel has a parabolic shape. Below are reviewed the forces applied to a unit width of the deformed body.

y////////////A v//y///h//////////A///

a

♦y , p

>\y/Ä//A

V/////////////////A////////7///////////// FIG. 109. Diagram of plane strain in the absence of contact slip:

(a) first variant; (b) second variant.

Being deformed under the action of the force p Ax, the element produces displacement in the volume adjoining it. In accordance with the kinematic system adopted, there is no slip on the contact surface, and the stresses τχγ, preventing the deformation of the element, work in the planes parallel to the x axis. Their sum from x to b and from 0 to H amounts to the complete resistance to deformation.


In order to be deformed, the element should displace a layer of thickness dy relative to one another to an extent of (dexldy) Axy where ex is the strain along the x axis.

The total displacement equals

The work done by the shear stresses rxy in the corresponding movements is completely determined for each element by the expression

H b

where

ex =

D

I

0 x

d)

dux Ab H2 - y2

dx ~~~b W2

dex ab 2y dy b H2 '

T dx - °s b~X txydX ^ H

y-

The total work done on the element will equal

ρΔΗΔχ = —jâsAHAx + Ax 13 ml·-dx dy,

or, taking into account the signs of the stresses, and dividing by AH Ax, we obtain

173 + ^-bir4wjib-x)y2dy-

2 2 1'3 f3"s bH3 ΑΗΨ Χ} 3 '


If we take into account the parabolic shape of the barrel, then from the equation of the displaced volumes we find

Then

Introducing the values of E. P. Unksov:51

u h u a

we obtain the formula of Ye. P. Unksov51 in the absence of contact slip

2 a, = -jas

29. FORCES DURING COMPRESSION. VARIANT OF THE SOLUTION OF THE PLANE PROBLEM IN THE ABSENCE OF CONTACT SLIP

The experiments provide grounds for the assertion that during the compression of bodies of moderate height between rough machined tools, i.e. under conditions close to real ones, the zone of slip does not arise until the ratios D\H and B\H attain respectively 7-9 and 6-8.44

An increase in the contact surface takes place on account of migration from the side surface to the contact surface.

The particles transferred adhere, thereby causing the migration of further particles. The process continues without the occurrence of slip on the contact surface, until the ratios indicated above are attained.

The experiments to determine the inhomogeneity of the deformations showed that in the predominant part of the width of the body the deformations increase from the contact surface towards the horizontal plane of symmetry. The picture changes in the opposite sense only in the vicinity of the side surface (Fig. 69).

Having studied experimentally the kinematics of the plastic deformation of long lengths of stock, adopting a definite model of the process, and finding τ „ as a function of the co-ordinates, it is possible to calculate the work of deformation of each element and determine the

1 + — x

(53)

pressures.


If we assume the following system of the process (Fig. 109 b): (a) slip is non-existent; (b) the volume being deformed consists of six layers of equal thick

ness h (there is nothing against an increase in the number of layers); (c) the side surface is formed by six chords drawn on the correspond

ing curve of the barrel; (d) the displacements are determined in each layer by the expressions

layer 1 uxl = Ab^ h

x Ί

layer 2 Λ1 x Λ1 h — y u2x = Ab1j+Ab2—ji^-

layer 3 ux3 = {Ab1 +Ab2)j,

The origins of the co-ordinates we locate respectively at points O, A, B. In consequence of the symmetry it is possible to consider a quarter of the transverse cross-section.

The model adopted does not express all the special features of the process. Here, no account is taken of the transition of the side surface to the contact surface and of the curvature of the horizontal planes. But in it account is taken of the restricted deformation on the surface, the gradual increase in the deformations from the contact surface to the horizontal plane of symmetry, which is well confirmed by experiment. It may be suggested that the total work of the slips in such a model will be close to the work of the slips of the real process, and consequently the value of the calculated pressure will not differ appreciably from the real pressure.

The work done by the forcepAxis determined by the equation

pAHAx = kAHAx + fl%rH 2

— H 3

X V "

H 7" b

\dex2

dy +

+/[^1^/*"H^+/[%L^/ H

~3~

Ax j rxydx dy. (54)

Here the first, second and third integrals express the work of movement of the elements in the respective layers. DMR 15


Let us take into account that for the system of deformations adopted the movement in the direction of the y axis is not dependent on x.

Then duyjdx — 0 Differentiating the equations determining the movements in the

direction of the x and y axes, we calculate the values necessary to us subsequently:

layer 1 duxi _ _ _ Abx h - y m

layer 2

layer 3

Here yxy

dx "-1"1 ~'1 b dexi Abi dy b h '

Abi x 7xyl " ~~ΊΓΤ;

dux2 Abi Ab -j^-ex2- - e y 2 - - r + —

dex2 _ _ Ab2 m dy b h

Ab 2 x yxy2~ ~ ~ 7 T T ;

Abi +Ab2 eX3 - -b ;

4 ^ = 0; dy

Vxy3 = 0 .

is the shear strain dux d uy

h '

2(h-y) bh

Inserting the deformations determined in this way into the Saint Venant consistency equation, we satisfy them absolutely. We find the value of the shear stress only for the first and second layers, since the third integral in eqn. (54) equals zero.

In the state of flow we have23

^ y „ ; (55)


where Γ is the shear strain rate

r=2i\I2(De)\. (56)

For plane strain the second invariant of the deviatoric of the deformation tensor equals

I2(De)= -82x-0'25y2

xy. (57)

Inserting into eqn. (55) the values of the incoming values for the relevant layers, we express the shear stresses as a function of the co-ordinates: 1 1 TSX layer 1 rxy = ; yX2 + 4 ( A _ y)2

layer 2 rxy = .

\*+*{%'+>->)' For the determination of the pressure /?, it is necessary to compute

the integrals in eqn. (54). The equation obtained in this way is of very great length46 and is not

therefore given here. But the conclusion set out above from this equation has paramount importance, since it shows the possibility of the use of the suggested method for the determination of the forces during inhomogeneous deformation. In accordance with this formula, the graph of the pressure for the compression of lengths of stock with ratios bjH = 1; 3-3 and 10 was prepared (Fig. 110).

The nature of the distribution of the pressure and the values of the mean pressures correspond to existing experimental data and present-day impressions of the distribution of the pressure on the contact surface. Hence it follows that the method set out makes it possible to calculate relatively well the factors which are operative during the real process.

The great size of the formula obtained in accordance with the suggested method for the determination of the forces results from the endeavour to take into account the greatest possible number of the factors which are operative in the real process of deformation.

Below, during the derivation of formulae for the determination of the forces during the rolling of thick lengths of stock, it will be shown that by adopting this method, but employing certain simplifying assumptions, it is possible to obtain calculation formulae for the forces, which are completely suitable for practical use. 15*


Above, during the derivation of formulae for the determination of the specific pressures, the deformation was assumed to be that known earlier. But this method affords the possibility of finding out the

0 10 20 SO FIG. 110. Diagrams of pressure:

l—b == 120 mm, H = 120 mm; 2-b = 100 mm, H = 30 mm; 3—b = 300 mm, H = 30 mm.

distribution of the deformations. In fact, by carrying out variation of the deformations, we shall obtain differing pressures, and in fact differing values of the work required for deformation. Assuming that in the real process the work is the least of all the possible values, in order to discover the actual distribution of the deformations it is sufficient to solve the variational problem for the minimum of the work. Here it should be noted that, by describing approximately the distribution of the deformations by sufficiently simple functions, we apply the variational problem to the solution of the ordinary problem to the minimum for a finite number of variables (see Chapter II).

30. DETERMINATION OF THE LOCATION OF THE NEUTRAL CROSS-SECTION DURING THE ROLLING OF THICK LENGTHS OF STOCK AND THOSE OF

MODERATE HEIGHT

The generally adopted formulae for the determination of the location of the neutral cross-section are obtained by starting from the hypothesis of flat cross-sections. In all the conclusions it is assumed


that the whole of the supplementary work of deformation is carried out by the forces of external friction; the work completed within the volume and brought about by the inhomogeneity of the deformation, is completely disregarded. The location of the neutral cross-section was determined by us from experiments by several methods set out below. Comparison of the experimental data with the results of the calculations on the basis of the formula obtained from the conditions of equilibrium of the forces, but without taking account of the zone of adhesion, which is most widely current at the present time, reveals considerable discrepancies.

It should be noted that these discrepancies between the results of calculation and the experimental data are increased in proportion to the increase in the thickness of the stock, or in accordance with the terminology adopted, in proportion to the decrease in the shape factor or the ratio l/Hmean.

Undoubtedly it is also impossible to expect that a formula obtained without taking account of the essential peculiarity of the process of deformation during rolling—the inhomogeneity of the deformation— will be valid for the instance of rolling "thick" lengths of stock, when, as the experiments show, the inhomogeneity of the deformation is especially sharply marked.

The experimental data, presented above, show that during the rolling of thick lengths of stock, on the contact surface there is a strongly developed zone of adhesion, and proportionately to the decrease in the shape factor its relative extent is increased.

Our experiments, therefore, confirm the opinion of a series of authors,1 1 '1 7 5 5 who have noticed the great extent of the zone of adhesion during the rolling of blooms, for instance.

In consequence of the fact that the zone of adhesion in this instance occupies the predominant part of the contact surface, the work completed by the forces of external friction is relatively small. A large part of the work completed by the external normal forces during deformation, is expended in the internal displacements.

For the determination of the location of the neutral cross-section it is possible to use the following variational principle, which is not difficult to obtain from the origin of the possible changes in the state of deformations:

6A = 0, (58)

i.e. the variation of the full work is equal to zero.23


We shall carry out variation of the work, making use of the method suggested in Chapter II. Let us divide the deformed volume into two zones in front of, and behind, the neutral cross-section. In each zone let us set ourselves a definite system of deformation such that into the work equation the co-ordinate of the neutral cross-section enters as a parameter. We shall carry out variation of the work along this coordinate. In this way, the variational problem is carried out for the investigation to the extremum of the function of a single variable. The system suggested for the calculation of the work of deformation consists of the following. The work carried out by the external force above the element (Fig. 105) is made up of the work of the resistances, external relative to the element under review, and of the work done in the change in shape of the same element, independent of the quantity of transmitted work (see Section 24).

As shown below, variational methods permit the determination of the location of the neutral cross-section for various states of a deformed body.

31. DETERMINATION OF THE LOCATION OF THE NEUTRAL CROSS-SECTION DURING THE COMPRESSION OF WEDGE-SHAPED SPECIMENS BETWEEN INCLINED TOOLS WITH FULL ADHESION AND CONSTANT DRAFT ALONG

THE LENGTH

Let a wedge shaped specimen of length / be deformed between two inclined tools, moving vertically without rotation (Fig. 111).

As a result the wedge receives a draft uH, identical along its whole length. Let the draft uH be small. We shall consider the strain to be plane; the width of the wedge should correspondingly be relatively large. Let its contact surface represent a zone of adhesion.

Experimental data permit the following system of distribution of the displacements to be adopted:

for the zone of backward slip

« x l = 4 / ' f ( l - | r ) , (59)

for the zone of forward slip

ux2=Al"jr{l-^j, (60)

where ux is the horizontal movement of the particle; Hx is the current value of the height.


The other indications are shown in Fig. 111. From the continuity equation

dux du, +

we find dx dy

dux

0

where <p(x) is an arbitrary function.

FIG. 111. Model for derivation of a formula for the determination of the location of the neutral cross-section during the compression of a wedge between inclined tools, moving parallel to themselves.

If we determine this function from the condition Uy\y = 0 = 0 ,

this shows that the boundary conditions uy\y = Hx = uH are not completely satisfied. This has little influence on the solution, since for relatively high lengths of stock there is no necessity to satisfy precisely the boundary conditions.


The question of the possibility of modification of the boundary conditions, or the possibility of the application of the Saint Venant principle for the deformation of thick bodies, has been considered above.

Let us consider that the metal is in a state of flow, for which consequently eqns. (9) and (55-57) are applicable.

The functions determined in this way for the shear stresses rxy are complex and inconvenient for computation of the work done in deformation. Calculations of the shear stresses in accordance with the formulae presented above, show that the expression for their determination can be approximated with sufficient accuracy by means of the following functions:


•„-..£(£)'. (60 for the zone of forward slip

These expressions give certain deviations only in the layer adjoining the contact layer. But the deformations in this layer are small, so that such an assumption is not expressed in the calculated value of the work of deformation.

The work necessary for the deformation of an element, in accordance with eqn. (45) will equal

ΔΑ = ΔΑλ + AA2.

Being deformed under the action of the force p dx, the element y dx itself produces movement in the volume adjoining it. The stresses τχγ, obstructing the deformation of the element, are themselves resistances, external relative to the element. Summating them from x to /" or from x to /', we obtain the full resistance obstructing the deformation of the element.

During the deformation process each elementary layer of thickness dy (Fig. 112) slips by means of the selected element relatively to the layer adjacent to it to a certain extent. Let us compute it; the overall slip at a given point equals yxy dy (Fig. 112). The slip of one layer relative to another, taking place as a result of the deformation of the


element itself of length dx, equals

tyxy dx ■ dy dx.

^ •o

/////y/////y/////// — ax —

FIG. 112. Diagram for calculation of the slip of an adjacent elementary layer.

\

at.

*

By the adopted system of deformation the value of the movement in the direction of the y axis is only slightly dependent on x, so that it may be considered that

δγχν cex a ■ dy dx ^ -r-2- dx dy dx dv

or tyxy dx

δεχ dy (63)

In a series of instances, where this does not give rise to considerable errors, for simplification we shall make use of eqn. (63).

By carrying out integration of the resistance to movement from x to /", we obtain the work of the element expended on the slip of a single layer

T?dydxf Txy dx.

Integrating this expression between the limits from 0 to HX9 we compute the work transmitted by the element during deformation

AA2 = dx -^- / rxy dx dy.


The work necessary for the deformation of the element itself equals

AA1 = k uHdx,

where uH is the decrease in height of the element. Then for an element, taken to be in the zone of forward slip, the

work equation acquires the form

Δ A" = p uHdx = k uH dx + dx Γ\δεχ2 Γ „ J Vwi T~ dx dy.

and in the zone of backward slip

A A' = p uH dx = k uH dx + dx \ xl / r ' v dx dy.

Inserting into the last two equations the values of the strains

ex dux2

dx '

determined by differentiation of (59) and (60), and the values of the shear stresses from (61) and (62), after integration we obtain

Δ A" = p uH dx = k uH dx + — Δ I" dx +

+ τΑΙ τππ, dx> A A = p uHdx = k uH dx + —Δ V -^ dx +

(64)

k ( / / p - Z / i X / ' 2 - * 2 ) * (65)

where H0 is the height of the thick end of the wedge; Ηγ is the height of the thin end of the wedge; and

rs = fc/2 is the yield stress in shear of the metal. In view of the fact that Hx changes relatively slowly, in the zone of

forward slip we take Hx = Ηγ and in the zone of backward slip Hx = H0.


We find the overall work of deformation per unit width of the wedge, by adding the work of the elements in the zones of backward slip (64) and forward slip (65)

/" A = ΔΑ" dx + I ΔΑ dx = uHk\l + — /' + — /" +

J J L 12 uH 12 uH 0 0

1 ΔΙ" „., . 1 ΔΓ (H0- Hi) I'2

24 IH1 24 uH IH0 (66)

The values ΔΥ\ιιΗ and ΔΙ"\\ιΗ we find from the condition of constancy of the volume and of the parabolic distribution of the displacements

ΔΙ"_3__Γ_ QnH ΔΓ _ 3 /' " 2 H0 '

3 /" _, ΔΓ — —- and 2 Hi uH

where /' + /" = /. Inserting the values found into eqn. (66), we obtain

A = k uH , 1 I"2 1 (/ - I")2

l Λ 1 —I 8 H, 8 H0 64

3 (H0 - Ht) Γ 3 _ I HI

3 (H0 - / / . ) 64 IH0

(/ - I")3

The value of the work done in deformation is determined by the parameter /". If the work of deformation has a minimum, then

dA dl" = 0

or dA dl"

21'

+ -77

Ho 20 - I") 9_ (Hp - 7/,)

8 IH\ 9 (Ho-Ηλ)

I"2 +

(l - I")2 = 0. (67)

Taking the second derivative, we can convince ourselves that the work done in deformation does in reality have a minimum.

Equation (67) may be presented in the form of a quadratic equation

9(θ3 - Θ2 + Θ - 1) m2 + (1602 - 2Θ + 18) m - 7Θ - 9 = 0. (68)


In this equation /" _γ

m = , -/ «

AH

0 = 1 + 1 l , Λ» » (69) 1

/ '

/ 1 Zl//

Λ 2 '

where Δ H is the difference between the height of the thick and thin ends of the stock in the instance of compression of the wedge between inclined tools, and the draft of the stock in the instance of rolling;

/ is the length of the wedge in the instance of compression, and the length of the zone of deformation

ΔΗ during rolling; —— is the tangent of the angle of taper of the wedge and

the tangent of the bite angle, on replacing the arc by a chord, assuming tan <x ôc and oc = ΔΗ\1\ and

m = — is the ratio of the angle of the neutral cross-section to a the bite angle during rolling.

On the basis of eqn. (68) calculation was made of the values of m = /0*> //#mean)> from which the corresponding diagram was constructed (Fig. 113). As is evident from the diagram, m is a decreasing function of the bite angle, and also of the shape factor.

In the majority of practical instances of rolling of lengths of thick stock and those of moderate height, m varies within the limits of 0-4-0-5.

For comparison, on this diagram are drawn in the curves for various values of μ (see the dashed lines in Fig. 113), obtained by calculation in accordance with the known formula

-K'-f)' <™> where β is the angle of friction; approximately it may be taken that β*>μ·

From Fig. 113 it is evident that the values of w, found from eqn. (70), which has been obtained in accordance with the hypothesis of flat cross-sections, and consequently without taking into account the


inhomogeneity of the deformation and of the zone of adhesion, are considerably less than those obtained from eqn. (68).

For the compression of wedge specimens between inclined tools eqn. (68) gives good results, if the wedge is sufficiently high, and the zone of adhesion occupies the predominant part of the contact surface. For wedges with a high ratio of the length to the mean thickness, this formula becomes invalid, since on the contact surface slip occurs, which was not taken into account in the derivation of the formula.

FIG. 113. Diagram of the relationship between the location of the neutral cross-section m = / ( a , l!Hmcan)9 constructed from eqn. (68)

(solid lines) and from eqn. (70) (dashed lines).

Applying eqn. (68) to the instance of rolling, it should be kept in view that during its derivation, account was taken of the shape of the zone of deformation during rolling, but inadequate account was taken of the law of the distribution of compression along the length of the zone of deformation. In the following section this factor is taken into account with greater accuracy.


32. DETERMINATION OF THE LOCATION OF THE NEUTRAL CROSS-SECTION

DURING ROLLING WITH FULL ADHESION ON THE CONTACT SURFACE

One of the fundamental special features determining the deformations during rolling is the distribution of the velocities, vy9 on the contact surface, to which corresponds the distribution of the draft along the length of the zone of deformation.

The vertical velocities along the arc of the bite change from zero in the plane of exit to a maximum in the plane of entry of the metal into the work rolls. In consequence of the smallness of the angle φΧ9 this relationship is close to linear. Therefore the function for the horizontal displacements will also be different to that in the preceding section.

Let us consider the following simplified model of the rolling process. A wedge of length / is deformed by two inclined tools, which are rotated about the points A and A' through a small angle, φ (Fig. 114). Thereby the vertical displacements uH at the points A and A' equal

u 4 · — L " — /

FIG. 114. Model for the derivation of a formula for the determination of the location of the neutral cross-section during the compres

sion of a wedge between inclined tools with rotation.


zero, and at the points B and B' 2uH0, so that from A to B the reductions increase as a linear function.

In the following, in view of the symmetry of the wedge relative to the x axis, we shall consider only the top half of the wedge.

We consider that the rolling process can as a first approximation be replaced by such compression between inclined tools, since the distribution of the deformations is taken to be about the same as during rolling, when the work roll is rotated through a small angle άφ. The model employed by us is the same as the model of A. I. Tselikov for the determination of the specific pressures during rolling, when the arc of the circumference is replaced by a chord,55 and the distribution of the velocities over the height is taken to be parabolic.

Let us consider the deformation as plane, we shall not take hardening into account, and we neglect slip on the contact surfaces.

The velocities on the surface of the tool during rolling are determined by the equations:

vy = νκ$ϊηφχ

vx = r a c o s ^ ,

where vR is the peripheral speed of the work rolls; φ is the angle at the centre, corresponding to the point under

consideration. In the model presented we have replaced the function for the vertical

velocities by a rectilinear relationship

vy = νκφχ.

Since φχ is small, then such a replacement is justified. The horizontal component of the velocity of the tool in the model

adopted equals zero. During rolling this component changes slowly, imparting a translatory velocity to the body being rolled. Therefore the velocities, governed by this change in the horizontal component of the velocity, will be small. In this way, a certain difference in the boundary conditions should not be noticeably expressed in the distribution of the deformations.

The important feature is that rolling, as distinct from forging, is an established (steady-state) process. But we are considering the changes in the state of a body within a small interval of time, and therefore we can neglect the change in the parameters determining the deformation.

Better confirmation of the justification for the proposed model is an


experiment which permits the model to be considered as sufficiently close to the real process.

In our opinion, the most essential disadvantage of the model must be considered the exclusion of the interaction with the external zones. Evidently, the subsequent development of the model and the method permits us to take into account this important factor.

It should be noted that the experiments confirm the analogy between the states of stresses and deformations of "high" bodies during rolling and forging. Theoretically such a replacement is also justified, since the load during the rolling of "high" bodies is applied to a small area. In this way, the use of the boundary conditions is found to be in accordance with the principle of modification of the limiting conditions of Saint Venant.

The system of the distribution of the movements we find from the conditions of constancy of the volume, taking into account that the horizontal movements are described by a parabolic relationship (Fig. 114).

Let us find the function of the horizontal movement of points for the zone of forward slip.

The volume displaced over the height in the section from the neutral cross-section to the intersection with the x co-ordinate, equals

x2

I" x - — AVh = — w H o - T - + UHO i )x = uH0-

The whole of this volume is displaced in accordance with the parabolic curve of ux, which also limits the displacement of the points in the cross-section under review. Since the volume, displaced in the vertical direction, equals the volume displaced in the horizontal direction, then

where ux0 is the movement in the horizontal direction at y — 0. Since the movements in the horizontal direction are described by a

parabolic curve, we obtain x2

lit Λ

, IX — 3 uH0 ( y2 \ 2


It is just as simple to obtain ux for the zone of backward slip also

/" x + — u =Aiiî_2^ L· (72) Ux 2 HY V m i ' {U)

where ux is the horizontal movement of a particle; uH0 is the maximum value of the draft during the rotation of the

tools about the point A or A' through a small angle d<p; Hx is the current value of the height of the wedge; /" is the length of the zone of forward slip.

The values of the small reductions on the contact surface are described:

for the zone of forward slip I" — x uHx = uH0 - — ; (73)


UHX = UHO j . (74)

Such a distribution of the movements closely corresponds to the experimental data presented below.

We take the equations for the shear stresses rxy to be the same as in the preceding section:


*-*'f (·£)*: (75)


^ = T-f(ir)2· (76)

The equation for the work which must be carried out for the deformation of an element in the zone of forward slip is expressed

Hx I"

δγχ ΔΑ" = p uHx dx = k uHx dx + dx I J*y / τ"γ dx dy. (77)

DMR 16


After inserting into the last equation the appropriate values from eqns. (63), (73) and (75) and integrating we obtain

i 3 7 wH0 (/"2 ~ X2W ~ x) i ΔΑ = puHx ax = kuHx ax + -ry- k —— — ax +

lo iix I I ΔΗ{Γ'2 - x2)(l" x - ^

+ TT*-#f TTTTi '-dx. (78) 9 , uH0

3 2 H2X I" I1

The full work of deformation in the zone of forward slip we obtain by integrating eqn. (78) from 0 to /". Initially in the first term of the equation we replace uHx by its value from eqn. (73), Hx we consider to be equal to H1 for the zone of forward slip, which for high bodies with a small relative change in Hx is entirely permissible

l"2 5 «„o /"3 . 33 uH0 AH,„ ΐΗ0ΎΓ + Ί4Ιζ~ΗΓ i "64ÖK~~HT I2 4

Similarly for the work of deformation in the zone of backward slip we obtain

A' = A£S2.<2/"/' + I'2) + A i i ^ l ( 8 r _ 3Γ) -

3 k "»o,?**/3 (15/" - 41'). 640

The full work of deformation after replacement of /' by / — /" will equal

A = A' + A" = _/_ k (5l"3 + 3 / 3 - 7 / " 2 / - / 2 / " 5/"3\ 2 + 64/1 i/o + Hj

kAH /31 l"l3 - 69l2l"2 + 61 ll"3 - 19/"4 - 41* 33T*\'l "64ÖPI Hi ^r-jj"Ho.(79)

We examine the expression found at the extremum with respect to the parameter /", which at given values of /, H0, Hx and A H determines the value of the work

dA

Using the symbols of (69), after differentiating and corresponding rearrangements of eqn. (79) we obtain

[13203 - 13202 + 76Θ - 76] m3 + [15O02 - 330 + 183] m \ 2

- (20 + 138) m - 410 + 31 = 0. (80)


The relationship m = /(Θ) is shown graphically in Fig. 115. As is evident from the graph, in order to discover the neutral cross-

section during the rolling of thick bodies without particular errors it is possible to replace the curve of m = f(ß) by the linear relationship

m = 0-79 - 0-270. (81) ' 1

0-5

W

0-3

0-2

(H

n

*+S k ^

Ί-0 /·? W Π 1-6 7-0 2-2 0

FIG. 115. Relationship between m and Θ from eqn. (80).

Taking into account the symbols of (69), we obtain

m=f(—r, — ) = / ( * , 77 h \ l -"mean / \ " m e a n /

i.e. the required value m = V'\l is independent of the external friction coefficient, in so far as during its derivation it was assumed that adhesion takes place along the whole of the contact surface.

On the basis of eqns. (80) and (69) calculation was made of the values of m = f(oc, llHmean), on the basis of which the relevant diagram was constructed (Fig. 116).

In the same figure are also drawn in the lines for m calculated in accordance with eqn. (70) at various values of the external friction coefficient. As is evident from the figure, the difference between the values of m = γ/α, obtained from the condition of full adhesion, on the one hand, and of full slip on the contact surfaces, on the other, are very large.

From comparison of the magnitudes of these values, obtained during constant reduction along the length of the zone of deformation (Fig. 113) and during varying reduction (Fig. 116), it is evident that in the second instance they acquire somewhat larger values. This is explained by the fact that during homogeneous distribution of the 16*


reductions, the work expended in internal movements per unit length in the zone of entry of the stock into the work rolls is greater than in the zone of exit.

FIG. 116. Diagram of the relationship between the location of the neutral cross-section m = / ( a , l/Hmeaa)f constructed from eqn. (80)

(solid lines) and from eqn. (70) (dashed lines).

Below will be presented experimental data, confirming the reality and practical applicability of the solution obtained in the present section for the location of the neutral cross-section in those instances of rolling, where the zone of adhesion occupies the predominant part of the contact surface.

Taking into account that


from eqn. (69) we obtain (AH H0\

In Fig. 117 are presented the curves for m = f(A H/D; H0ID); from consideration of the curves it is evident that m = γ/oc increases sharply with the increase in the thickness of the stock at a given diameter of the work rolls.

FIG. 117. Diagram of the relationship between the location of the neutral cross-section and the reduction and thickness of the stock

during rolling m = f(AH/D, H0/D).

33. DETERMINATION OF THE LOCATION OF THE NEUTRAL CROSS-SECTION FOR A MEDIUM WHICH IS IN A STATE OF FLOW

As has been shown in section 6, for a medium which is in a state of flow, the variational principle expressed in the form of eqns. (11) and (12) is valid.

Let us consider a wedge, compressed by parallel moving plates (Fig. 118). In view of the symmetry, we shall consider only the top


half of the wedge. If the ratio ////mean is sufficiently small (llHmean < 2), then the predominant part of the contact surface represents a zone of adhesion. We shall consider that the adhesion extends over the whole of the contact surface, so that along it movements are impossible, i.e.

dux = duz = 0.

The vertical movements uy are fixed and therefore also do not vary (duy = 0). Then eqn. (11) is written

v

If the width of the wedge is great, the deformation may be considered to be plane, and integration can be carried out over the area (along the x and y axes). Then

dtfrsrdS = 0. S

We represent the deformations in the following manner (Fig. 118). Let the wedge undergo a small reduction 2w0. We shall consider

that the horizontal movements ux are distributed in accordance with

FIG. 118. Model for derivation of a formula for the determination of the location of the neutral cross-section during the compression

of a wedge which is in a state of flow.


a parabolic function. Then from the condition of constancy of the volume it follows that

_ 2 u u0 x — — ux0 rlx

and _ 3 u0

UXO"2~H;X'

But since the movements are distributed over the height in accordance with a parabolic function, then the maximum values will be on the x axis; on the contact surface they decrease to zero. Therefore

ux = ux0(\ - _ j = T - g - * ( i - — In view of the fact that the height in the zone of deformation changes

considerably more slowly than*, we assume that in the zone of forward slip

2 H, \ m "*2 = 4 " Ξ Μ Ι --ΠΓ (83)

and in the zone of backward slip

^ - T T H 1 - ^ ) · (84)

We find the deformations in accordance with the formulae dux dux duy /0_.

ε* = 17' ε> = -ε» γχ> = IF +17· (85)

The vertical movements may be determined from the condition εχ = —By.. Since

then du

_ ÖUy €y ~~dy~'

Uy = "J ~Jx y +ψ^' The arbitrary function <p (x) may be found from the condition

Uy\y=-.0= 0 .

Inserting into (85) the values of the movements from eqns. (83) and (84), we obtain:


for the zone of forward slip 3 w0 _ 3 w0 / yM . w0 xy .

-τ^Γΐ1--^]' 7xy2=-3ΊΪ;ΊΓ[ (86)

and for the zone of backward slip

Z / / 0 \ / / 0 / Ή 0 /7 0 l--kh y„x--3^-S-. (87)

Knowing εχ and y^ , it is possible to find the shear strain rate in the zone of backward slip and that of forward slip. In fact

Since we are considering an incompressible medium and a state of plane strain, then e2 = 0, yyz = 0, γ2Χ = 0, εχ = — sy and

r = 2 V d + 0.25y5y. (88) Inserting into equation (88) the values of the deformations from

eqns. (86) and (87) ,we obtain: for the zone of forward slip


îl/('-ii+4f Let us simplify these equations, employing the approximate formula

fa2 + z2 = a + — , 2a

and replacing 11 —ττγ\by its mean value, equal to 2/3. Then for the zone of forward slip we obtain

''->%(>-% + !*£) m


Γ - 3Ü2./1 _ Z l + λΞΐΐί) (90)


We find the whole work of deformation during compression of the wedge

HX l" H0V

A = / / rs Γ dS = / / τζ Γ2 dx dy + ff TS J \ dx dy. (91) S 0 0 0 0

Here, for simplification of the final result, integration is carried out up to Hx and H0, which, in view of their relatively small change, does not noticeably affect the result. Inserting into eqn. (91) the values of the shear strain rate for the zone of forward slip (89) and of backward slip (90) and integrating we obtain

(2 1 I'3 2_ J_ Γ 3 \ ^ - 3 T S W O ^ 3 / + — - ^ - - 3 / + 1 2 my

or, replacing /' = / — /", •2 . . 1 „ ,fi„ . 1

A = 3T S W 0 3 + 12«? l ; + 12//? (92)

The value of the work done in deformation in the last equation is now determined only by the parameter /", and variation of the work should be carried out with respect to this parameter

dA _ ( / - /")» , I"2 _

Solving eqn. (87) relative to /", we obtain

__ / ] / I2 I2

02 - Γ + [/ (Θ2 - l)2 + 92 - 1 ' ( ^ here Θ is the ratio of the height of the thick edge of the wedge to that of the thin one.

This ratio characterises the angle of slope of the wedge, on the one hand, and the shape of the vertical-longitudinal cross-section of the wedge, on the other. In actual fact, the ratio Θ may be presented in the following form:

Ο = 1/Γ= 1 +1/Γ' or

ΔΗ - 1 + - ^ -

Hm

1 ΔΗ' 2 I


Replacing Δ H\l = tan* « a, we obtain (X

e = l + 1 /

Hm

1

In the instance of rolling « is the bite angle. After simple rearrangements, eqn. (93) can be presented in the following form:

or

/" m = -r =

1

AM =

/ 1 + Θ '

1

2 + 1 1

\ i^mej

(94)

FIG. 119. Diagram of the relationship of the location of the neutral cross-section m = / (a , I/Hmeaa) during the compression of a wedge, which is in a state of flow, constructed from eqn. (94) (solid lines)

and from eqn. (70) (dashed lines).

VARIATIONAL PRINCIPLES FOR SOLUTION OF SOME PROBLMES 239

From the last equation values were calculated for m = f(oc, ////mean) on the basis of which the relevant diagram was constructed (Fig. 119). For comparison, on this diagram are also drawn in the curves for m9 constructed from eqn. (70).

As is evident from eqn. (94), at very small values of oc (where a -> 0), and also at small values of //#mean > at a given value of oc, m approaches 0*5. In the instance of rolling m — γ/oc.

Let us recall to mind that eqn. (94) is derived from the condition of full adhesion on the contact surfaces, and therefore it is suitable only for instances where //#mean < 2Ό and with dry friction. From eqn. (94) we find values of m = γ/oc for certain instances of rolling. The results of the calculation are presented in the form of Table 15.

TABLE 15. CALCULATED DATA FOR m = γ/oc [from eqn. (94)] FOR VARIOUS INSTANCES OF ROLLING

D

mm 1000 1000 750 750 400 400

Ho

mm 500 500 300 300 50 50

ΔΗ

mm 50 100 40 80 10 20

I

mm 158 223-5 122-4 173-0 44-7 63-3

-"mean

mm 475 450 280 260 45-0 400

is "mean /

0-333 0-497 0-438 0-666 0-992 1-58

" = ZH radians

0-316 0-448 0-327 0-462 0-224 0-316

m = — oc

0-475 0-445 0-465 0-423 0-445 0-375

On the basis of the data presented in the table, it is evident that for various instances of rolling billets, and also during section rolling, the ratio ////mean is normally less than 2-2*5, and the zone of adhesion is very large.

34. DETERMINATION OF THE LOCATION OF THE NEUTRAL CROSS-SECTION DURING ROLLING. INSTANCE OF THE SIMULTANEOUS OCCURRENCE OF

ZONES OF ADHESION AND OF SLIP

In the preceding three sections are given three variants of the derivation of an equation for the determination of the location of the neutral cross-section, or more accurately for the determination of m = /"// = γ/oc. In all these derivations, as a limiting condition the complete absence of mutual movements on the contact surfaces was


assumed or, which is the same thing, the existence of complete adhesion along the whole of the contact surface. In the instance of rolling between smooth work rolls the process of deformation diiTers from the instances reviewed of compression of wedge-shaped stock, firstly by reason of its stationary nature, and secondly by reason of the pressure of the reductions along the length of the zone of deformation.

The latter difference can give rise to a substantial change in the location of the neutral cross-section during the rolling of wedge-shaped stock by comparison with compression with parallel movement of the tools. Therefore the system of distribution of the reductions (Fig. 114) adopted for the derivation of eqn. (81) should be considered the closest to the rolling process. But even eqn. (81) is suitable, firstly in the presence of complete adhesion, and secondly on the occurrence of the system of movement of the particles of metal adopted for its derivation.

As experimental data show, during hot rolling, when there is no lubricant, at //i/mean < 2-5-2-0 the length of the zone of adhesion represents a considerable part of the whole length of the zone of deformation. This affords grounds for the use of eqn. (81) for the determination of m subject to the condition of the introduction into this equation of a correction, taking into account the presence of zones of slip in addition to the zone of adhesion.

So far as the kinematic system of the movement of the particles of metal is concerned, which was adopted for the derivation of eqn. (81), then as experimental data show, the formation of a single barrel takes place at ////mean > 0-8, while //i/mean < 0-8 double barrel formation takes place both in the transverse, and also in the longitudinal, direction of the zone of deformation. Let us note appropriately that eqn. (81) gives the maximum values of m, not greatly exceeding 0-5, while in fact during the rolling of very thick stock (at low values ofllHme&n), and consequently in the presence of double barrel formation, the value of m will be considerably greater than 0*5. But herewe are not considering the theoretical determination of the location of the neutral cross-section during double barrel formation.

In connection with what has been set out above, eqn. (81) can be employed for the determination of the location of the neutral cross-section in the simultaneous presence of zones of adhesion and slip only during single barrel formation, on the one hand, and the presence of a developed zone of adhesion on the other. To these conditions correspond instances of rolling, where 2-0 > //i/mean > 0-8.


Under practical conditions of rolling production, such a range of variation in the ratio of the length of the zone of deformation to its mean height takes place during section rolling.

At higher values of the ratio of the length of the zone of deformation to its mean thickness, when the relative extent of the zone of adhesion is small or if the zone of adhesion is completely non-existent (thin sheet and strip rolling), the formula is valid for the determination of the location of the neutral cross-section, which was obtained on the basis of the hypothesis of flat sections (70).

During the use of eqn. (81) the presence of certain sections of slip can be approximately calculated, assuming that for the sections of slip eqn. (70) is valid, and for the zone of adhesion eqn. (81). This somewhat lowers the values of m, calculated from eqn. (81) in accordance with the relative extent of the zone of slip. On this basis it is possible to write

y I = 7adh 4dh + ymean Anean ( 9 5 )

where yadh is the angle of the neutral cross-section calculated from the condition of adhesion of the stock to the work rolls along the whole contact surface and determined from eqn. (81);

7mean is the angle of the neutral cross-section, calculated from the condition of slip along the whole contact surface and determined from eqn. (70);

γ is the critical angle in the presence of a zone of adhesion of length /adh and of two zones of slip of total length 'mean? s o t h a t / a d h + 'mean = ' ·

After simple rearrangements, eqn. (95) can be presented in the form

m = m a d h ίψ + mm,.„ (\ - 1-ψ^ (96)

or after insertion of the values of the incoming values from eqns. (81) and (70) at β = μ5 we obtain

m = (0-79 - 0-270)-^p- + j(l - γ-)(ΐ - ^ψ) · (97)

On the basis of the experimental graph of /adh// = ^(////mean) (Fig. 29) are derived the following relationships valid for 0 < ///^mean < 30:


for aluminium

(«=)'--'*·(¥-™): for copper / / \ 2 7

' ' \ = _ 1 5 0 - ^ 1 - 0-6 " m e a n / *

Solving these equations in relation to /adh//, we obtain: for aluminium

'adh \ -Οπιβϊ

for copper / 120

/ ^2

9 'adh \ H.

I 15-0

In the denominator of both equations there is a coefficient equal to 12-0 for aluminium and 15-0 for copper. Let us denote it by C. It is apparent that the value of this coefficient for various metals and alloys is determined by the value of the external sliding friction. Let us take the mean values of the coefficients of sliding friction in accordance with our experimental data, presented above, for aluminium 0-25 and for copper 0-20.

Assuming that the relative extent of the arc of adhesion is directly proportional to the coefficient of sliding friction, we obtain

C - * .

where B is a certain constant, which is independent of the value of the coefficient of external friction.

Then for aluminium B = 0*25 x 12-0 = 3 Ό and for copper B = 0-20 x 150 = 30.

Inserting the values obtained into the preceding expressions, we obtain

-hr-) «adh

—μ* I 3 or

4dh

~r 3 - 1 3 \Hm

2 i

μ*


Then from eqn. (97) we have

m = (0-79 - 0-270) [3 - 1 ( 7 7 ^ ) A +

4("-£)('-44(τ6)1· <*> Let us express llHmm by 0 and a, or, which is the same thing by 0

and Δ HID 1 0 - 1 0 - 1

/ / m Ä ( 0 + l ) y (Θ+1) 1/—

Having inserted the value of ljHmtm into eqn. (98) we obtain

m = (0-79 - 0-270) 3 - 1 (6 - I)2

3 Λ2 μ* +

l - μ 5 3 - 1 Φ ~ I)2

3 Λ 2

(θ + l)2 £L

or m = (0-79 - 0-27Θ) 3 -

1 (Θ - \f 3<* + »·|#

Λ +

4-m 1 - i « s

1 (0 - l)2

3 <»+ ' ) *#

(99)

In this way, in the general instance m = φχ (0, <%, μ$). Taking into account that 0 = 9?20%, ////mean)? we obtain

JW \ -"mean /

Apart from this, it is known that α = φ4(ΑΗ/ΰ), therefore (ΔΗ

m (ΔΗ i \

(100)

(101)


Above it was shown that /

-"me

therefore it is possible to write

Ψβ (ΔΗ Η0\ { D ' D r

(ΔΗ H0 (102)

In this way, by the suggested formula it is possible to calculate the influence of all the basic factors determining the change in shape during rolling, the shape of the zone of deformation and the coefficient of external friction. This is related to the instance of rolling an ideally plastic body with identical mechanical properties at any given point.

010-fr FIG. 120. Diagram of the relationship between the location of the neutral cross-section and the thickness of the stock and reduction during rolling m = q)7(HolD9AH/D) at an external friction coefficient of 0-25 (solid lines) and 0-20 (dashed lines), constructed from eqn. (102). For comparison, the curves a are given, constructed

from eqn. (70).

In Fig. 120 are presented comparative curves of the relationship of the location of the neutral cross-section to the thickness of the stock and the reduction during rolling at a coefficient of external friction of 0-25 and 0-20, constructed from eqns. (102) and (70).


Figure 121 shows the curves of the relationship of the location of the neutral cross-section and the bite angle and the shape factor during rolling, with an external friction coefficient of 0-25 and 0*2 in accordance with formula (100) and (70).

I =oc FIG. 121. Diagram of the relationship of the location of the neutral cross-section and the bite angle and the shape factor during rolling m = <p3 (oc, l/Hmtan) at a friction coefficient of 0*25 (solid lines) and

0-20 (dashed lines): 7 —from eqn. (100) at ////mean = 0-5; 2—idem at ////mean = 1-5; 3—from

eqn. (70).

Analysis of these curves shows that during the rolling of thick lengths of stock, i.e. at high values of the ratio of the thickness of the stock to the diameter of the work rolls, the value m is considerable even at high bite angles, and vice versa during the rolling of thin strips with low ratios of the thickness to the diameter of the work rolls and also small bite angles, the curves approximate to the curve obtained with the formula of I. M. Pavlov at the same value of the coefficient of friction. Such an approximation takes place as a result of the gradual decrease in the relative extent of the zone of adhesion (ladh/l -> 0) and a corresponding increase in the relative extent of the zone of slip (/mean /-> 1'0) as the stock becomes thinner.

At the limit we obtain the instance of homogeneous deformation, when slip exists over the whole of the contact surface, i.e. when the DMR 17


hypothesis of flat cross-sections becomes valid, and it may be considered that τ = ρ μ5.

It should be noted that comparison of the experimental data with those calculated from eqn. (99) shows that in a series of instances, especially for copper, discrepancies are observed; the formula gives somewhat too low results.

35. EXPERIMENTAL DETERMINATION OF THE LOCATION OF THE NEUTRAL

CROSS-SECTION DURING ROLLING

For the development of the technical rolling process in continuous mills, the pass designing of sections of changing cross-section, for the determination of the forces occurring during rolling, it is necessary to know the location of the neutral cross-section.

Up to the present time in literature not a single method has been published, which makes it possible directly to determine more or less accurately by experiment the location of the neutral cross-section during rolling, having completed computations from experimental forward slip. Concerning the location of the neutral cross-section, assessment has been made either on the basis of experimental data concerning forward slip, or of calculations from qn. (70).

The need for experimental determination of the neutral cross-section is heightened by the fact that the results set out above of the theoretical investigation are considerably divergent from existing impressions of the relationship between the location of the neutral cross-section and the various factors, and all the more so concerning the limiting values of the angle of the neutral cross-section.

A. I. Tselikov,55 indicating the existence of a zone of adhesion during rolling, considers that in the zone of deformation there is one cross-section, the neutral one, in which all the particles of the metal being rolled have an identical longitudinal velocity, equal to the horizontal component of the velocity of the work rolls at a point corresponding to this cross-section. Employing the equation of constancy of the volume for the plane problem, we may write

vy = υ0(Η0ΙΗγ) or vY = v0(h0lhY),

where v0 is the velocity of the rear (entry) end of the stock; and Hy and hy respectively are the thicknesses of the stock and of

an elementary layer in the neutral cross-section.


From the expression presented it follows that constancy of the velocity in the neutral cross-section of the zone of deformation is observed at ho/hy = constant, or \\Ύ\Ύ = constant for any element of the thickness of the stock.

The graphs presented in Figs. 20-23, show that such a cross-section in the zone of deformation really exists, and to it corresponds the point of intersection of all the curves of the increasing deformations, whereby, apart from the curves constructed for all the horizontal layers, at this same point the curves for the instance of homogeneous deformation also in fact intersect.

0-05 OUb ~H-

FIG. 122. Experimental diagram for m as a function of AH/D; H0/D.

Regardless of the fact that deformation between neighbouring cross-section is inhomogeneous, the difference in the increasing deformations for various layers increases initially, is then evened out and dies away in the neutral cross-section. In this way, the results of the experiments presented afford the possibility to determine the location of the neutral cross-section, having completed computations from experimental forward slip by means of eqn. (70), obtained from the hypothesis of flat cross-sections.

In Fig. 122 are given the curves for m = /(AH/D, H0ID), obtained from experiments in the rolling of specimens, the dimensions of which are shown in Tables 2 and 3. These data show that during the rolling of thick stock the neutral cross-section is found close to the centre of the zone of deformation. 17*


The considerable discrepancy between our experimental data and existing impressions of the limiting values of the neutral cross-section, compiled as a result of employment of the known eqn. (70) for the determination of the angle of the critical cross-section without taking into account the shape of the zone of deformation, has instigated us to turn, for the sake of a check, to other experimental methods, which allow even an approximate determination of the location of the neutral cross-section.

One of the methods widely used for the study of the kinematics of the rolling process, is the method of vertical grids, which are applied to one half of a specimen cut in two or to its side surface. After a small amount of compression, in consequence of the inhomogeneous distribution of the deformations, initially the vertical grids are bent in both directions away from the neutral cross-section. But the grid, corresponding approximately to the neutral cross-section, remains vertical.

This method was laid down as the basis of the experimental investigation of the location of the neutral cross-section during compression between backing rings,44 and also of the influence of the front rigid end of the stock during rolling, carried out by B. P. Bakhtinov and M. M. Shternov.4 The method in a somewhat changed form was employed by us for the determination of the location of the neutral cross-section during rolling.

Two rectangular lengths of lead stock of identical dimensions were delivered in turn into the work rolls with a barrel diameter of 210 mm. Identical reductions were given. The mill was switched off when the zone of deformation was approximately in the middle of the length of stock, and then the work rolls were reversed. The sticker was sheared through along one side surface for the removal of the barrel forming. Then on one of the sheared lengths of stock in the section of the zone of deformation vertical grids were applied, after which both halves were brazed together with Bud alloy. While preserving the former setting of the work rolls, rolling of the specimen obtained was carried out in such a way that an additional turn was imparted to the work rolls through a very small angle, with the intention that the length of stock should move forward over a distance of 1-2 mm in the direction of exit. After this the stock was withdrawn from the rolls and taken apart. The location of the neutral cross-section was determined in the form of the ratio of the length of the zone of forward slip to the complete length of the zone of deformation /"// = γ/oc = m (Fig. 123).


The state of the surfaces of the work rolls and of the specimens in all the experiments was identical.

F I G . 123. Location of the neutral cross-section during rolling (/ /^„. .n = 1-32; m = y/α = 0-500).

The results of the series of similar experiments in the rolling of brazed lengths of stock are presented in Table 16.

By comparison of the data of Table 16 with the graphs (Figs. 20-23) sufficiently satisfactory agreement is revealed.

TABLE 16. EXPERIMENTAL DATA FOR m = γ/α DURING THE ROLLING OF LEAD SPECIMENS. WORK ROLL dia. 210 mm

Ho mm

55-40 40-40 34-70 3000 29-10 2500 24-60 18-00 9-00

Hi mm

3800 25-40 20-30 21-80 20-30 17-50 2110 11-80 6 0 0

AH mm

17-40 1500 14-40 8-20 8-80 7-50 3-50 6-20 300

/ mm

42-70 38-70 38-90 29-40 30-40 2800 19-20 25-50 17-35

fexp

mm

28-45 22-70 15-70 1300 16-40 1400 1000 8-50 2-35

/ mean

0-92 1-18 1-41 113 1-23 1-32 0-84 1-72 2-31

*exp '"exp ~j

0-665 0-587 0-404 0-442 0-540 0-500 0-520 0-330 0135


Similar data were obtained during the compression of wedge-shaped lead specimens both with and without rigid ends, between cast iron, inclined tools with a rough surface. The angles of inclination of the contact surfaces of the specimen before compression and of the tools were taken to be equal to a/2 = 0*125. Vertical grids were applied both on the side surface of the wedge-shaped specimen, and also on an intersection, such that the specimen consisted of two identical halves with the plane of intersection along the centre of the width of the specimen.

The identical nature of the processes of compression of a wedge-shaped specimen between inclined tools and rolling has been shown by I. Ya. Tarnovskii.44

TABLE 17. DATA ON THE COMPRESSION OF LEAD, WEDCE-SHAPED SPECIMENS

Ho mm

42-80 41-00 22-00 51-70 51-50 62-45 46-35 21-20 1400

# 1

mm

31-80 3010 10-60 40-70 40-70 5200 35-65 10-75 3-20

/ mm

53-30 52-70 54-70 51-00 51-40 48-00 49-80 50-45 50-50

AH mm

1100 10-90 11-40 11-00 10-80 10-45 10-70 10-45 10-80

•"mean mm

37-30 35-55 16-30 46-20 46-10 57-22 41-00 15-97 8-60

/ If

•"mean

1-43 1-48 3-36 1-10 1-15 0-84 1-22 3-16 5-87

'exp

mm

2100 14-40 20-30 22-80 22-80 21-45 19-20 14-40 11-40

"»exp

0-395 0-368 0-372 0-447 0-444 0-448 0-388 0-286 0-226

Remarks

| Compression [with rigid Jends

The results of the experiments are presented in Table 17. In Figs. 124 and 125 are shown photographs of the experimental specimens. From the analysis of the experimental data and a comparison of them with the calculated data, it is evident that during the rolling of thin lengths of stock eqn. (70) for the determination of the angle of the neutral cross-section gives good compatability at the determined value of the coefficient of external friction. In the instance of rolling of thick lengths of stock this was not observed.

This is explained by the presence of a zone of adhesion during the rolling of lengths of stock with a low ratio of the length of the zone of deformation to its mean thickness, which in a radical manner changes the limiting conditions on contact, making them far from the conditions for the derivation of eqn. (70).


From the experimental graph of m = y(AH\D, H0ID) (Fig. 122) it is evident that the value of the angle of the neutral cross-section increases with the increase in the ratio of the height of the stock to the diameter of the work rolls and with the decrease in the ratio of the draft to the diameter of the work rolls. Apart from this, comparison of the location of the experimental curves for aluminium and copper points to the substantial influence of the coefficient of external friction at low ratios of the height of the stock to the diameter of the work rolls

FIG. 124. Location of the neutral cross-section during the compression of a wedge (l/Hmean = 0-925; m = γ/oc = 0-66).

FIG. 125. Location of the neutral cross-section during the compression of a wedge (//i/mean = 3-16; m = γ/oc = 0-286).


and an increase in the influence of the coefficient of friction with the increase in its ratio even at high bite angles.

The similar relationship confirms that during the rolling of thick lengths of stock, of the type of blooms, regardless of the physical nature of the metal, a strongly developed zone of adhesion exists. In this instance the connection between the shear and normal forces is not the same, differing from that adopted during the derivation of eqn. (70).

In so far as the extent of the zone of adhesion is dependent directly and primarily on the shape of the zone of deformation, the question of the location of the neutral cross-section is likewise connected with the influence of this factor, which determines the change in shape during rolling.

It is completely obvious that during rolling in a blooming mill, in connection with the considerable value of the ratio of the height of the ingot to the diameter of the work rolls, values of m are sufficiently large.

The theoretical and experimental investigations which have been presented permit the assertion that the value of the angle of the neutral cross-section can differ considerably from those values which are determined from eqn. (70). In a series of instances the angle of the neutral cross-section can become greater than half the bite angle. These investigations show the relationship of the ratio m to the basic factors of the process. As is evident from the graphs (Fig. 121), with the increase in the bite angle the value of m decreases, and the shape factor exerts an influence in the same direction. At low values of the shape factor (r///mean < 1)> m normally exceeds 0-5.

As is known, the value of 0-5 is limiting for eqn. (70). This is understandable, in so far as this formula was obtained from the condition of equilibrium of an absolutely solid body. Undoubtedly, equilibrium equations remain true even for a deformed medium. But here use should be made of differential equilibrium equations. There is also no doubt about the fact that, if we project a system of forces applied to the surface of a deformed body, onto the axis of co-ordinates, then the sum of the projections along the three axes is equal to zero. Nevertheless it is impossible to use this condition, since in the instance of adhesion neither the directions, nor the value of the forces of friction are known. The distribution of the deformations, including also the location of the neutral cross-section, can here be determined from the condition of the^ minimum of work. In the presence of a developed


zone of adhesion the work done by the external forces, as has been shown above, need not be considered, since it does not vary. In this way, having calculated the work done by the internal stresses and minimizing it, it is possible to find the location of the neutral cross-section, completely neglecting the external forces, a part of which is unknown to us in the presence of adhesion.

At high values of the ratio of the length of the zone of deformation to its mean thickness, when the hypothesis of flat cross-sections becomes valid, and the inhomogeneity of the deformations is considerably decreased, on the greater part of the contact surface slip occurs. Under such conditions the forces of friction on the contact surface are easily determined (111), and the equilibrium equations as a whole can be employed. Therefore eqn. (70) becomes true at greater values of the ratio of the length of the zone of deformation to its mean thickness.

36. DETERMINATION OF THE FORCES DURING THE ROLLING OF THICK

STOCK

The method proposed in Section 32 enables us to calculate the rolling forces. In reality, the work of the external force is made up of the internal work of an element. The location of the neutral cross-section, determining the value of the work of the stresses, is given by eqn. (81).

Therefore the specific pressures in the zone of forward slip can be calculated, if eqn. (78) is divided by uHx dx,

Px 3 {Γ* -x*)(l" -x)uHQ

+ 16 /" HxluHx

( Γ 2 _ X2)^rx_^.yHUHo

+ 32 l"H2xl2uHx

After inserting uHx from (73) and reducing, we have

3 (/"2 - x2) 9 1 H - - ^ 16 I" Hx 32 /" HI I

(103)


For the zone of backward slip correspondingly we obtain

3 (/«-,*) 9 ( / ' 2 -* 2 ) ( / -*+4W p'x = k 1 + 16 l'Hx 32 V Hl(l" + x)l (104)

Calculations show that the third terms of eqns. (103) and (104) can be neglected as a result of their smallness.

Then the formulae for the determination of the specific pressures will be: for the zone of forward slip

3 (I"2 - x2) 1 +


px = k 1 +

or correspondingly: Px k

= 1 +

16

3 16

3 16 3

/"

{I'2

/'

(/"2

/" {I'2 -

Hx .

- x2) Hx

-x2) Hx

- A 2 )

k 16 Γ Hx

Expressing Hx by x, we obtain: for the zone of forward slip

4=1 + k

3 1 6 i»

I"2 - x2

Hi + -jril -X)

(105)


p'x , 3 I'2 - x2

k 16 /' H.+^r x)

(106)

The co-ordinate of the neutral cross-section enters into eqns. (106) and (105). Above has been set out a method of determining the neutral cross-section theoretically during complete adhesion and rotation of the tools. Equations (106) and (105) were obtained for the same conditions and are valid for "high" bodies, when the ratio of the length of the zone of deformation to its mean thickness is small.


Let us derive them in a form convenient for investigation, so that into these equations shall enter the value m, the method of determination of which is set out in section 32. For convenience we transfer the origin of the co-ordinates to the exit plane from the zone of deformation. Then (Fig. 114): in the zone of forward slip

Λ- = / " - χ';

in the zone of backward slip x = x' - I",

where x' is the longitudinal co-ordinate, reckoned from the exit plane. After simple rearrangements we obtain:


p"x i 3 -*-?- = 1 - 1

k 16

2m q — q2

m 1 1 /

η— + y<*(? - 1)


Ρχ Λ 3

k 16

H„

1 — 2m — 2m q — q2

(107)

(1 - / » )

Hms

-jocq (108)

In eqns. (107) and (108) q = x'jl. Let us construct the diagrams of the pressures for the following

instances:

(a)D = 700mm;#0 = 570mm; AH = 60mm; Hmeaa = H° + Hl =

= 540mm; / = 145mm;

From eqn. (69) we find

/ 145 AH H„ 540 = 0 2 6 9 ; - p = 0-414.

AH

= 1 + 1 ('

I 1 AH

\ 2 i

1-118.


Inserting into eqn. (81) the values found, we obtain m = 0-488. In order to construct the diagram of the pressures, we shall change

the relative co-ordinate q from zero to unity. Inserting the values of q, m, AHjl and //i/mean into eqns. (107) and

(108), we obtain the values of the pressures at the selected points. On the basis of these data the diagram of the distribution of pressure

over the length of the zone of deformation was constructed (Fig. 126a). (b) D = 1100 mm; H0 = 725 mm; Hl = 650 mm; AH = 75 mm;

/ = 194 mm. The diagram constructed from the data of the calculation is presented

in Fig. 1266. a p

K

102

10

γ

ί' M ("

i K

*

FIG. 126. Diagrams of the distribution of the pressure along the length of the zone of deformation, constructed from eqns. (107) and

(108).

As is evident from Fig. 126, the diagrams of the distribution of pressures have a certain break in the neutral cross-section, which is explained by the following circumstances. For the derivation of eqn. (78) for the work of the shear stresses, with the aim of simplifying the integration, in place of a variable value of the thickness of the zone of


deformation in any given cross-section of the zone, we adopted constant values of H0 for the zone of backward slip and of Hx for the zone of forward slip. By more accurate integration this break would be removed. It is understandable, therefore, that eqns. (107) and (108) will be all the more true, the less there is a change in the thickness of the stock over the length of the zone of deformation. Appropriately let us note that the equations were obtained from the condition of full adhesion, which is characteristic of the rolling of thick stock, and in these instances namely the thickness of the stock does not change sharply along the length of the zone of deformation, since the relative reductions are comparatively small.

37. CALCULATION OF FORWARD SLIP TAKING INTO ACCOUNT A ZONE OF ADHESION

Determination of the location of the neutral cross-section taking into account the real factors operating during the rolling process has great importance, especially for the calculation of forward slip. Account of the inhomogeneity of the deformation cannot substantially influence the equations (full or simplified) for the calculation of forward slip, since these equations are derived from constancy of a second volume of the metal, passing through any given cross-section of the zone of deformation and remaining constant even during inhomogeneous deformation. But into the equations for the calculation of forward slip there enters the value of the angle of the neutral cross-section, which is essentially dependent on the kinematics of deformation.

Apart from the generally known eqn. (70), suitable for instances of rolling in the absence of a zone of adhesion, eqn. (80) was derived above for the determination of the angle of the neutral cross-section in the instance of full adhesion along the whole length of the arc of contact between the stock and the work rolls, and also eqn. (99) in the instance of adhesion on a certain length of the contact surface and slip in the other sections.

For the calculation of the influence of the zone of adhesion on the forward slip it is necessary to insert into the calculation formula for the forward slip the value of the angle of the neutral cross-section from eqn. (99).

In Table 18 are presented experimental data for forward slip, obtained by the method of punch marks during the hot rolling of

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lengths of aluminium stock between 210 mm dia. work rolls. According to the experimental data set out in the following chapter, the coefficient of external friction during the hot rolling of aluminium is taken to be 0-25. Apart from experimental forward slip data, values are given for forward slip calculated with the use of eqns. (99) and (70), and also the deviations of the latter from the former. As is evident from the data presented in Table 18, the calculated forward slip, obtained from eqn. (99), is considerably closer to the experimental data than the forward slip calculated with the use of eqn. (70).

It should be noted that during the rolling of very thick stock, when the ratio of the length of the zone of deformation to its mean thickness is small, and the zone of adhesion is strongly developed, greater forward slip is produced, and determination in accordance with it of the value of the angle of the neutral cross-section shows that the latter is considerably greater than half the bite angle. Meanwhile according to the existing views on the present-day theory of rolling, during the rolling of very thick stock, for instance during the rolling of heavy ingots in a blooming mill, when greater drafts are used and greater bite angles, the values of the angle of the neutral cross-section and of the forward slip should be very small, which follows from eqn. (70). In fact, as the experimental data show, in these instances of rolling the forward slip is very large, by comparison with the elongation, and the angle of the neutral cross-section is greater than half the bite angle. Here, naturally, greater forward slip should be understood as a greater proportion of it to the overall elongation of the stock.

From what has been set out above it follows that the question of forward slip, which appears to be completely simple and finally to have been solved in the theory of rolling based on the hypothesis of flat cross-sections, is, under real conditions of billet and section rolling when the inhomogeneity of the deformation is considerable, far more complicated, being connected with the whole kinematic and force conditions of the rolling process and in need of further study.

CHAPTER VII

The Coefficient of External Friction and the Conditions of the Bite on the Stock by the

Work Rolls during Rolling

38. REVIEW OF METHODS OF DETERMINATION OF THE COEFFICIENT OF EXTERNAL FRICTION

When considering problems of external friction during the plastic working of metals, research workers normally reckon that on the contact surface only kinetic frictional forces occur, in fact the forces of sliding friction. This is entirely understandable, for until comparatively recently it was considered that the metal being deformed could only slide in relation to the tool. It is therefore assumed that the external friction is subject to Coulomb's Law, i.e. at every point on the surface of contact between the deformed body and the tool the specific force of friction is directly proportional to the normal pressure. And so most frequently of all it is considered that the coefficient of external friction is identical at all points on the contact surface.

Without dwelling for the time being on the correctness of such an integration of the external friction during rolling, let us observe that a knowledge of the value of the coefficient of external friction at the present stage of the development of the theory of the mechanical working of metals has great importance, as has already been noted above. But until now no reliable methods have existed for the direct measurement of the value of the external friction coefficient under conditions of the plastic working of metals.

The majority of existing methods are indirect, and are extensively linked with the degree of accordance of one calculation equation or another with the real conditions of the process. For the determination of the external friction coefficient, certain authors even employ empirical formulae, for instance, normally used for the calculation of spread

260

THE COEFFICIENT OF EXTERNAL FRICTION 261

during rolling. It is natural that the value of the external friction obtained by such a method does not inspire much confidence.

Below the most widely used methods are considered for the calculation of the friction coefficient during the plastic working of metals under conditions close to real compression and rolling.

In literature on the mechanical working of metals, for practical calculations values of the friction coefficient obtained during compression between tapered tools are often used and recommended.

The method of compression between tapered tools was first of all suggested by E. Siebel and A. Pomp22 for the construction of curves of the real stresses during compression; to determine the external friction coefficient this method was first employed by S. I. Gubkin in 1934.

The authors of the method considered that during the compression of a cylindrical specimen having tapered grooves on the end faces between tapered tools, such that the angle of taper, ρ, of the grooves and the tools was identical, a linear state of stresses is created.

On this basis it was asserted that if the generatrix of the cylinder remains vertical during compression, then tan@ = μ. But V. I. Zales-skii and A. V. Puzanchikov21 showed sufficiently convincingly that compression by the method of tapered tools takes place under conditions of a volumetric state of stresses, in connection with which the values of the external friction coefficient determined by this method are exaggerated.

In the research of I. M. Pavlov and P. S. Kostychev34 the external friction coefficient was determined during the compression of specimens between plane-parallel plates by means of a special mechanism developed by the authors of the research. At the time of compression the specimen was moved sideways by means of a special screw mechanism. In the process of compression measurements were made of the vertical pressure of the press and the resistance of the specimen being displaced, on the basis of which the external friction coefficient was determined. In this research a study was made of the "anisotropy " of friction, i.e. the relationship between the force and coefficient of friction on the one hand and the direction of displacement in those instances where the state of the surface changes in various directions in connection with the method of machining the tool.

For the determination of the coefficient of friction during rolling three basic methods are known at the present time:

(a) from the value of the experimental forward slip; DMR 18


(b) by means of the tongs mechanism of L. M. Pavlov; (c) from the maximum bite angle. Let us consider each of these methods briefly. (a) The coefficient of external friction is determined from the value

of the experimental forward slip measured by the centre-punch method, starting from the known theoretical relationships between the value of the forward slip and the angle of the neutral cross-section

C " 2 ^

and between the angle of the neutral cross-section and the coefficient of external friction (eqn. (70)).

There is no doubt that the accuracy of the determination of the external friction will be governed by the validity of the hypotheses forming the basis for the derivation of the equations set out above. As experience shows, the boundary conditions adopted for the derivation of the equation for the angle »of the neutral cross-section (70) are valid to a large extent for the rolling of thin stock, i.e. for instances close to homogeneous deformation, and are inapplicable for a large part of the contact surface during the rolling of thick stock (rolling in cogging, heavy section and medium section mills).

Very typical is the fact that many authors, by using the method under consideration, have obtained values for the external friction coefficient, which rise with the increase in the thickness of the stock being rolled.8'58 Below it will be shown on the basis of the experimental data that a similar relationship between the external friction coefficient and the thickness of the stock, obtained during analysis of the experimental data by means of eqn. (70), is explained by the occurrence of a zone of adhesion during the rolling of thick stock, by the change in the whole kinematics of the process, and consequently by considerable deviation from homogeneous deformation.

(b) By means of the tongs mechanism of I. M. Pavlov33 the external friction coefficient is determined at the moment when the stock is slowed down in the work rolls. By this means the braking force and the pressure of the metal on the work rolls are determined, and the value of the friction coefficient is calculated from the equation put forward by the author of the device

Q oc ^ = ypr + t a n T '


where Q is the braking force directed horizontally in relation to the rolling axis, measured with a dynamometer; and

Pz is the vertical pressure on the work roll determined from the readings of load cells placed under the screwdowns.

The equation set out above is derived from the hypothesis of the existence of sliding friction along the whole of the contact surface. It should be noted that the value of the external friction coefficient, obtained by means of the device of I. M. Pavlov, is determined at the moment of the start of skidding of the work rolls, i.e. under conditions of full slip between the metal of the stock and the work rolls. In this way, the conditions for conducting the experiment correspond in large measure to the assumptions made during the derivation of the formula.

As a result, data for the value of the external friction coefficient, obtained by means of the tongs mechanism, are more reliable than data obtained by the other methods.

(c) Determination of the external friction coefficient from the value of the angle of the bite based on the "condition of the natural bite"

tan oc S μ· (109)

This method makes it possible to determine the friction coefficient at the moment of the bite in the event that the last inequality is valid. But the derivation of this inequality, like the conditions of an established process of rolling (tanoc S 2μ), is made on the basis of assumptions also employed in the preceding methods, except that it is assumed that the cross-sections of the stock which were vertical before deformation remain vertical even during gradual filling of the roll gap by the metal.

Recently, in literature on rolling a large number of works have appeared which are devoted to definition of the applicability of eqn. (109). Thus experiments conducted by A. A. Presnyakov,36'38

I. L. Perlin and K. K. Goderzian39 place in doubt the validity of the generally known theory of the conditions of the bite and of the established rolling process.

Below it will be shown on the basis of experimental data that at the moment of the bite a zone of adhesion is formed, the dimensions of which are continuously changing right up to the moment of the formation of the front "rigid end" of the stock. Consequently it is impossible to maintain that during the process whereby the work rolls bite the stock and the roll gap is gradually filled by the metal, slip takes place along the whole contact surface, and that the value of 18*


the external friction coefficient, obtained by this method, is completely reliable.

39. DETERMINATION OF THE EXTERNAL FRICTION COEFFICIENT FROM THE EXPERIMENTAL FORWARD SLIP

In the present section are set out the results of work carried out with the aim of explaining the causes which exert an influence on the change in the value of the external friction coefficient obtained by calculation from the experimental forward slip. The calculation was carried out in the following way.

The forward slip was determined from the known equation by the method of centre punch indentations

e/, _ Lx — L0 ~ 7 '

where Lx is the distance between the centre punch on the rolled stock; and

L0 is the distance between the pop marks on the work rolls along an arc of the circumference.

Using the simplified equation for the determination of the forward slip8

S" = y 2 - ^ - , (110)

from the experimental forward slip we find the value of the angle of the neutral cross-section. It must be noted that the derivation of eqn. (110) for calculation of the forward slip is based on the equality of the second volumes passing through the vertical cross-sections of the zone of deformation, which is undoubtedly valid during an established rolling process.

It is true that for the derivation of the equation no account was taken of the occurrence during rolling of inequality of the velocities of the metal over the height of the stock in a given transverse vertical cross-section of the zone of deformation. In other words, in every cross-section a certain mean velocity is assumed, and thereby the inhomogeneity of the distribution of the velocities is artificially wiped out.

But such an "averaging" of the velocities scarcely introduces material errors during the calculation of forward slip, since during a steady-state process equality of the second volumes is true for any


expression of the longitudinal velocities in any given vertical cross-section. Therefore it should be considered that the decisive influence on the accuracy of calculation is the correctness of the determination of the position of the neutral cross-section, which at the present time is normally determined from eqn (70).

It is evident that solution of eqn. (110) for the relative angle of the neutral cross-section at a value of the forward slip which is known by experiment makes it possible to determine with sufficient accuracy the position of this cross-section.

The results of calculation from eqn. (110), as we know, differ considerably from the results obtained by means of the more lengthy eqn.

1 Hx 1/71 ΗΛ2 IH\~~

where λ is the coefficient of forward slip, which in its turn represents the ratio L^LQ = 1 + S".

The external friction coefficient was determined in our experiments after insertion of the experimentally determined value of the angle of the neutral cross-section into eqn. (70), when

oc2

4 ( T - y )

Analysis of the method set out above for calculating the external friction coefficient from the experimental forward slip shows that the accuracy and validity of the determination of the value of the external friction coefficient are dependent on the degree of reliability of eqn. (70), which, as we know, is derived from the condition of equilibrium of the forces during rolling. Spread is considered to be negligibly small,. and the pressure to be evenly distributed over the arc of the bite. If it is considered that the external friction coefficient is also taken to be constant, then in consequence it is considered that on the whole of the contact surface forces of Coulomb friction of identical value operate, i.e. τ = ρ μ = constant.

The last assumption in its very essence excludes the possibility of the occurrence of a zone of adhesion, i.e. of taking into account the inhomogeneity of deformation.

The presence of a zone of adhesion substantially changes contact friction. In reality, if we take into account that the metal being rolled


at any given point on the contact surface is displaced relative to the work roll, then the Coulomb Law will be valid. In the presence of a zone of adhesion movements of the metal and the tool in relation to each other do not take place, nevertheless the force of external friction has a certain limiting value, i.e. there is friction at rest or static friction.

B. V. Deryagin shows that friction at rest, dependent on the value of the external force, can acquire different values within the limits from zero to a certain maximum value, and at the same time has an entirely different direction in space. "Determinant only is the limiting value of the force of friction, equal to the minimum or threshold value of the external force, which is necessary to bring a body into a state of motion. By friction at rest, or static friction, we normally in fact understand this limiting or threshold value."16 For friction at rest, therefore, i.e. for the limiting or threshold value

τ=ρμ. ( I l l )

But in the absence of any relative movement on the contact surface, which in fact occurs in the zone of adhesion, this equation cannot be considered valid.

Certain authors consider25 that for the instance of homogeneous deformation the position of the neutral cross-section during rolling is substantially influenced by the inhomogeneous distribution of the pressures along the arc of the bite. But it can be shown that the influence of this factor is little expressed in the position of the neutral cross-section in the simple instance of rolling, even if it is considered from the evidence that there is no zone of adhesion.

In reality, if as a result of simplification of the derivations the concave curves on the theoretical diagram of the pressures (according to A. I. Tselikov) are replaced by straight lines, then the diagram will represent two trapezoids (Fig. 127a). We disregard work hardening. The equilibrium equation in this instance will be

pf I pH _j_ Ύ1" T' Γ χ Τ" Γ χ T 1 χ 1 χ.

Where the shape of the diagram of the pressures is trapezoidal, and in the absence of hardening, the pressure will be identical both in the zone of forward slip and in the zone of backward slip, so that the equipollent pressures are located at a different angle to that during homogeneous distribution of the pressure.


We shall derive an equation for the angle of the neutral cross-section, starting from the diagram of the pressure, shown in Fig. 127. The projections of the active forces onto the x axis will equal:

Px=pBR(oc-y)ocx; P'J = pBRyoc'J;

T'x = p B R(oc — γ) μοο$αχ;

T" = pBRy μοο$<χχ.

FIG. 127. Diagram for the derivation of the relationship between the angle of the neutral cross-section and the distribution of the pres

sures along the arc of the bite.

Assuming that c o s ^ = cos«*^ = 1, and setting up the equilibrium equation we have

(oc - γ)θί'χ + y(x"x + γμ = {a - γ) μ. (112)


We find odx from the equation R*x = xa + Ry,

where xa is the distance from the centroid to the largest side a of the trapezium (Fig. 1270).

From the equation for the centre of gravity

h a-2b x* = 3 a + b

Designating the maximum pressure in the neutral cross-section by p

n k, where n = —ψ*- 9 we obtain

_ R(oc -_y) nk + Ik _ R(oc - y) η + 2 X& — 3 nk + k 3 n + 1 '

Then /__<* — γ n + 2 __ ft « + 2 γ 2n + I

*x " —Γ" TTT + r " T 7ΓΓΤ + T n + l ' We find ft" from the equation

Aft" = Xb,

where xb is the distance from the centroid to the smallest side b of the trapezium:

_ h 2a + b *b~T a + b l

γ 2n + 1 ax = 3 n + 1 '

Inserting the values ft* and ft* into eqn. (112), after simple rearrangements we obtain

voc n — 1 . ft2«+ 2 — — + 2// V = ft μ - — . 3 /i + 1 ^ ' p 3 n + 1

Dividing this equation by 2μ, and taking into account that y/ft = m, we find

ft 2 72 + 2 1 ~ ΎμΎ'τΓΤΤ m — 2 ft 7 7 — 1

2μ 3(/i + 1)

Let us designate -z -—r ° 3 n + 1

THE COEFFICIENT OF EXTERNAL FRICTION

2 n + 2

269

«—I ~ 2μ . . = A> i, , ΊΛ = *> — = <5, then

3(« + 1) Λ

m = l__ δ_ 2 , B 1 + T

(113)

The values of A = 9Ί(«) and 5 = «^(n) are presented in Table 19, and in Fig. 128 is shown the graph of the function m = φ(η, δ).

n

A B

10

1-000 0000

TABLE 19. VALUES OF A =

20

0-890 0111

30

0-835 0167

4-0

0-800 0-200

φχ (n) AND B =

50

0-778 0-222

6 0

0-762 0-238

<Pi(n)

7-0

0-750 0-250

80

0-742 0-259

10-0

0-728 0-273

From the graph it is evident that at high values of δ, a change in n changes the ratio γ/oc = m only slightly.

Let us observe that high values of δ are characteristic for rolling thin stock, when the inhomogeneity of the distribution of the pressures is very great.

0·5Γ

FIG. 128. Graph of the plot of m as a function of n and δ.


Hence it follows that to take into account the inhomogeneity of the distribution of the pressure in the derivation of an equation for the angle of the neutral cross-section from equilibrium of the forces only slightly changes the value of the unknown quantity.

In the literature3 there are indications that, under normal conditions of rolling, forward slip, as calculated from any of the equations with the use of values of the angle of the neutral cross-section, obtained from eqn. (70), does not always correspond to the experimental data. Specially large discrepancies are observed during the rolling of thick stock.

In the preceding sections experimental data were presented, which confirm the existence of considerable inhomogeneity in the distribution of the deformations during the rolling of thick stock. One of the most characteristic phenomena of inhomogeneity of deformation is a zone of adhesion. From the experiments it was established that the relative extent of the zone of longitudinal adhesion increases with the decrease in the shape factor of the longitudinal vertical cross-section of the zone of deformation (////mean)> i.e. with the increase in the thickness of the stock being rolled.

In order to follow the change in the value of the external friction coefficient, as calculated from experimental forward slip with the use of eqn. (70), in connection with the change in the parameters of the zone of deformation graphs were constructed; along the abscissa were placed the values of the shape factor, and along the ordinate the values of the external friction coefficient obtained as a result of calculation.

The experiments in the determination of forward slip were carried out on a laboratory, two-high mill with steel work rolls of 210 mm dia. Three grids were applied along the generatrix of the work rolls, located at a distance of 165 mm from each other along the arc of the circumference.

Lengths of aluminium and copper stock were rolled in this mill. Lead specimens were rolled in a two-high mill with cast-iron work rolls of 130 mm dia., and the distance between the grids on these rolls was 100 mm. The rolling speed in the 210 mm mill was chosen as ΟΌ88 m/sec, and in the 130 mm,mill 0*211 m/sec.

Specimens of various dimensions were prepared from hot forged metal, with the exception of the lead specimens which were cold forged.

For the preparation of the specimens grade A 2 aluminium (0*33 percent Fe; 0-35 per cent Si) and electrolytic copper Ml (99-6 per


cent Cu; 0Ό03 per cent Pb; 0Ό02 per cent Fe; 0-002 per cent Ni) were used.

The experimental specimens were heated in a tubular, resistance furnace placed in the immediate vicinity of the mill before rolling. The heating temperatures selected were: 450°C for the aluminium and 900°C for the copper. The temperature in the furnace was maintained by means of a contact galvanometer. At the given temperature the stock was held in the furnace for 20-30 min dependent on the dimensions. Measurement of the temperature was carried out with a platinum thermocouple. The heated lengths of stock were delivered into the work rolls with tongs, without preliminary removal of the scale. The roll barrel was divided along its length into three sections, and in each section a definite metal was rolled. The surface of the work rolls in each of these sections were initially covered with copper or aluminium, by passing through the work rolls a large quantity of heated test lengths of stock of the metals enumerated above.

The forward slip of the hot rolled lengths of stock was determined, taking into account their linear contraction during cooling to room temperature. For this purpose the following equation was used

c „ _ L[-L S L '

where L[ = L± (1 + oc t) is the distance between the impressions on the hot rolled stock;

a is the coefficient of linear expansion or contraction of the metal of the stock; and

t is the temperature of the stock at the time of rolling. The values of the coefficient of linear expansion were taken from

the data of G. A. Kashchenko to be as follows, taking into account the heating temperature: for aluminium 23-1 x 10~6 and for copper 20-6 x 10"6.

It is necessary to indicate the comparatively small amount of study which has been given to the coefficients of linear expansion for the majority of metals at rolling temperatures, which must inevitably be expressed in the accuracy of the values of the forward slip as determined by the centre-punch method.

Let us turn to the presentation of the results of the experiments. In Fig. 129 are given the experimental graphs of the relationship between the external friction coefficients of aluminium, lead and


0-70 u

060

0-50

CHO

0-30

0-20

0-10

0 1 2 3 4 5 6 7 8 9π "mean

FIG. 129. Experimental graph for the external friction coefficient at various values of the shape factor.

copper on the one hand and the shape factor, constructed on the basis of the data given in Tables 20-22.

Naturally, there are no grounds for considering that the external friction coefficient is dependent on the shape factor, //i/mean. But, as has been shown above, the relative extent of the zone of adhesion at a given external friction coefficient is dependent on the shape factor. Since for the calculation of the external friction from the experimental forward slip eqn. (70) was used, for the derivation of which no account was taken of the presence of a zone of adhesion, then from the nature of the relationship between the external friction coefficient and the shape factor we may judge the limitations of the possible application of this equation. Let us stress that the graphs of the relationship between the external friction coefficient and the shape factor were constructed only with this aim, and not with the intention of recommending that excessively high values of μ should be taken at low values of the shape factor.

Analysis of the relationship between the external friction coefficient and the shape factor for all the metals investigated by us shows complete identity of the nature of the experimental curves.

On all the graphs the mean value of the external friction coefficient obtained by calculation from the experimental forward slip, rises sharply with the decrease in the shape factor (//#mean)> remaining almost constant at higher values of the shape factor.

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\ 0 \

o X

l a • V

Si *Al x>

• p o

X

y-v X ~

"cüT -Pb

μ-

μ-

Κ)

00

GO C

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Let us observe that the results of our experiments coincide with the results obtained by other research workers,858 in confirming that on increasing the thickness of the stock being rolled the external friction coefficient, as calculated from the experimental forward slip using eqn. (70), increases. Thus the graph of the relationship between the external friction coefficient and the shape factor presented in Fig. 130, which was constructed from the experimental data of A. F. Golovin8

(Table 23) and that in Fig. 131 for type St. 1 steel at 1000°C from the data of Yu. M. Chizhikov58 (Table 24), were in both instances obtained from the experimental forward slip.

It is characteristic that during the construction of the graph of the relationship between the external friction coefficient and the shape factor from the data of Yu. M. Chizhikov, the authors did not take into account the method of change in the value of the reduction (i.e. H0 = const, H1 = const, or Δ H = const), but all the experimental points were located on one curve, completely identical in its nature with the curve constructed from the data of A. F. Golovin.

As a result of examination of all the experimental curves it becomes evident that at high values of the shape factor the external friction coefficient remains almost constant. This is completely natural, since at such values of the shape factor, as has been shown in Chapter III, in practice over the whole length of the zone of deformation slip takes place on the contact surface, the deformation is close to homogeneous, and eqn. (111), which is essentially based on the derivation of eqn. (70), will be valid. In proportion to the decrease in the shape factor the relative extent of the zone of longitudinal adhesion is increased, and in this instance it is already impossible to consider that over the whole of the contact surface eqn. ( I l l ) is valid. Let us note appropriately that the opinion set out above is confirmed during comparison of the experimental curves of the relationship between the external friction coefficient and the relative extent of the zone of adhesion on the one hand and the shape factor on the other (Figs. 29 and 129), since an increase in the external friction coefficient, obtained by calculation from the experimental forward slip, is observed in the same range of values of the shape factor, in which there is a noticeable increase in the extent of the zone of adhesion.

On the basis of what has been set out, the conclusion may be drawn that eqn. (70) for the determination of the angle of the neutral cross-, section makes it possible to determine more or less accurately the


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values of the external friction coefficient only at high values of the shape factor, at which the deformation is close to homogeneous. As the shape factor decreases, there is an increase in the inhomogeneity of the deformation, which eqn. (70) entirely fails to take into account. Therefore at low values of the shape factor, for the determination of the external friction coefficient from experimental forward slip the equation will in reality give fictitious values of the external friction coefficient, which increase sharply with the increase in the thickness of the stock being deformed.

Experimental forward slip during the rolling of thick stock is considerably greater than the value which can be obtained by calculation, if it is determined from eqn. (70). Above experimental data have been presented which confirm this hypothesis. This is indeed understandable, in so far as the experiments show that the angle of the neutral cross-section will in these instances rise to half or more than half the angle of the bite. From eqn. (70) such values can be obtained, only by taking the friction coefficient to be close to infinity, and also where the angle of the bite approaches zero. And in reality, the experimental forward slip corresponds to the calculated, when the external friction coefficient is taken to be equal to 0-9-1Ό; otherwise during the rolling of thick stock the values of the friction coefficient calculated from the experimental forward slip considerably exceed unity. This last is one further confirmation of the fictitious nature of the values of the friction coefficient obtained from eqn. (70) under conditions of sharply marked inhomogeneity of the deformation.

Analysis of the experimental data presented in the present work is evidence to the effect that the main assumption during the derivation of the equation for the determination of the angle of the neutral cross-section is not constancy of the pressure along the arc of contact and not neglect of the presence of spread, but the absence of any account of the inhomogeneity of the deformation, i.e. of a zone of adhesion, and the use of the sliding friction equation for the whole length of the contact surface of the zone of deformation.

The problem of the change in the external friction coefficient along the arc of the bite for various instances of rolling, and all the more so in the presence of a zone of adhesion, has remained little studied up to the present time and has not been studied by the authors of the present work. But it should be observed that the inhomogeneity of the deformation observed during the rolling of thick stock, requires the employment of another method of determination of μ, divorced from the use 19*


of the assumption that p μ = constant, i.e. in other words that the deformation is homogeneous.

Certain authors consider that out of the large number of factors which exert an influence on the value of the external friction coefficient, decisive importance attaches to the velocity of the slip between the stock being rolled and the work rolls.

Indeed, T. M. Golubev and M. A. Zaikov10 came to a similar conclusion when considering forward slip and the external friction coefficient during rolling. By simplifying the equation of A. F. Golo-vin for the mean velocity of the slip along the arc of the bite, the authors obtained the following expression

_ v0 AH Vmean~ 3 ht

or, which is the same thing,

*Wan = -2Vo[— ~ l)· 0 1 4 )

This expression establishes a relationship between the mean slip velocity and the reduction at a given peripheral velocity of the work rolls, v0. If we consider eqn. (114), the velocity of the slip increases with the increase in the shape factor, since with the increase in the reduction factor the arc of the bite increases, and the mean thickness of the stock decreases. Here let us observe that the equation is derived from the assumption of complete slip of the metal over the whole extent of the contact surface, i.e. for the instance of homogeneous deformation.

The external friction coefficient was determined from the experimental forward slip, using eqn. (70) and taking into account spread from the data of I. M. Pavlov. Amongst sundry others the authors present experimental data tables and a graph of the relationship between the friction coefficient and the mean slip velocity (Fig. 132), obtained during the hot rolling of type St. 2 steel in a two-high mill with 79*6 mm dia. work rolls. The rolling speed was varied within fairly wide limits from 0Ό9 to 6 m/sec. From the graph it follows that the value of the friction coefficient rises with the decrease in the slip velocity. Analysis of the tables containing the experimental data,10

shows that the change in the slip velocity exerts a certain influence on the values obtained for the external friction coefficient. But this analysis to all appearances points to the fact that all the high values


of the friction coefficient (0-5-0-7) were obtained by the authors during the rolling of stock with a ratio !IHme3n < 1-3, and small values of the coefficient (0-2-0-3) at llHmtan <* 2-0.

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FIG. 132. Relationship between the friction coefficient of type St. 2 steel at 800°C and the rate of slip (from the data of T. M. Golubev

and M. A. Zaikov).

In order to judge the nature of the curves of the relationship between the friction coefficient and the shape factor at a constant slip velocity we carried out experiments, during the conduct of which the ratios between the peripheral velocity of the work rolls and the relative deformation were chosen so that the slip velocity, as calculated from eqn. (114), remained constant. In this way, the effect of the slip velocity was completely excluded.

The experiments were carried out between the same work rolls with grids which were used for the determination of the experimental forward slip during the rolling of aluminium and copper stock. The specimens were made of lead. The slip velocity was taken to be about


0-130 m/sec. At the same time the rolling speed was varied within the technical capacity of the mill from 0-106 to 2-3 m/sec.

The results of the measurements and calculations in accordance with the method set out earlier are presented in Table 25 and Fig. 133. As is evident from this graph, the curve of the relationship between the friction coefficient and the shape factor at constant slip velocity scarcely differs in character from those obtained earlier (Figs. 128-30), for the construction of which the value of the slip velocity in accordance with eqn. (114) varied within fairly wide limits.

0Ί

FIG. 133. Experimental graph for the external friction coefficient of lead at various values of the shape factor and a slip velocity,

Vs~ 0-138 m/sec.

The curve in Fig. 133 provides grounds for the assertion that the basic cause, which influenced the change in the value of the external friction coefficient in the experiments of T. M. Golubev and M. A. Zai-kov, was not the slip velocity, but the shape factor of the zone of deformation (the inhomogeneity of the deformation), the change in which at a given friction coefficient was accompanied by a change in the degree of inhomogeneity of the deformation, and consequently by a change in the reliability of the calculation formula (70).

It is indisputable that the change in the slip velocity of the metal relative to the work rolls on the contact surface should in some measure change the value of the external friction coefficient. Nevertheless the slip velocity cannot be the decisive factor; in confirmation of this we have the identical nature of the curves obtained at constant (Fig. 132) and varying (Figs. 128-30) slip velocities.

The same eqn. (114) is derived for the instance of homogeneous deformation, and therefore its use cannot do otherwise than be

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reflected in the reliability of the values of the slip velocities in the presence of a markedly developed zone of adhesion.

40. LITERATURE DATA ON THE CONDITIONS OF THE BITE ON THE STOCK

BY THE WORK ROLLS

In modern rolling theory it is considered that the so-called natural bite on the stock by the work rolls, i.e. the bite on the stock by the work rolls without enforced entry into the roll gap, is possible only in the instance where

oc < ß, where oc is the bite angle;

ß is the angle of friction. At any given moment of filling of the roll gap by the metal (Fig. 134)

the condition of the realisation of the process is φ<β, (115)

where φ is the angle between the resultant of the normal pressure and the vertical.

FIG. 134. Diagram of the filling of the roll gap by the metal.

When the metal completely fills the roll gap, the rolling process can take place only subject to the condition

oc < 2β. (116)


The last inequality is obtained from the following simplifying assumptions:

(a) the pressure is homogeneously distributed along the length of the zone of deformation, and therefore the resultant of the pressure is applied in the middle of the length of the arc of contact;

(b) spread is non-existent. Recently doubts have been expressed concerning the validity of

inequality (116). Thus for instance, in the experiments of A. A. Pres-nyakov37 for the rolling of nickel ingots it was found that the ratio of the maximum bite angle during the unestablished rolling process to the same angle during the established rolling process was 1-16, and for rolling copper at a temperature of 950°C the same ratio was 1-22.

On these grounds various research workers express different hypotheses ; in particular it is suggested that the external friction coefficient during an established rolling process is substantially different from such at the initial moment of the bite, although neither experimental, nor theoretical, data of a convincing nature are presented in support of such a hypothesis.

In modern rolling theory it is also stated that the maximum possible bite angle is dependent only on the external friction coefficient, and is not dependent on spread or the thickness of the stock. At the same time experience in production confirms that the bite conditions are dependent on the thickness and the width of the stock being rolled.

B. P. Bakhtinov and M. M. Shternov3 note that the ability of the work rolls to bite the stock is dependent on the width of the stock being rolled. According to our data the maximum possible bite angle is dependent on the thickness of the stock; in this respect no confusion should arise between the difference in the thickness of the stock before and after a pass (the draft), which is geometrically connected with the bite angle, and the thickness of the stock, since at one and the same difference in the thickness the initial thickness of the stock can be different.

In what consists the radical failing in the treatment of the bite on the stock by the work rolls in modern rolling theory? In our opinion those simplifying assumptions to which reference is normally made are inapplicable here. Likewise there arf no convincing grounds for the surmise that there is a considerable change in the external friction coefficient during the transition from the initial bite on the stock by the work rolls to the established rolling process.


The investigations set out above into the inhomogeneity of the deformation during the established rolling process prompted us to think of the necessity of carrying out a similar investigation applicable to the bite on the stock by the work rolls and the filling of the roll gap by the metal. The basic aim of such an investigation is the study of the limiting conditions, or of the conditions on the contact surface. In actual fact, for the determination of the conditions of the bite on the stock by the work rolls it is normally assumed that the force of friction is equal to the product of the normal pressure and the external friction coefficient, while no limitations are made on these grounds. Such a determination of the forces of contact friction between the stock and the work rolls, as has been shown above, is valid only in the presence of slip between the contact surfaces over the whole of the contact surface. On the other hand, in the presence of a zone of adhesion during the process of filling the roll gap by the metal the limiting conditions are determined in a different manner.

From the point of view of the deformations this question is formulated in the following manner: is the deformation during the filling of the roll gap by the metal homogeneous or inhomogeneous? Even without any further investigations, and only in view of the existence of a front end of the stock during rolling between plain barrels or in box passes in billet and section mills, it may confidently be said that the deformation of the metal during the filling of the roll gap is inhomogeneous.

Taking into account the practical and theoretical importance of the problem under consideration, for the purposes of a fuller study of the interaction between the stock and the work rolls at the initial moment of the bite and during the transition to an established rolling process, we undertook the research set out below into the bite on the stock by the work rolls.

41. DEFORMATION OF THE METAL DURING THE BITE ON THE STOCK BY THE WORK ROLLS AND THE FILLING OF THE ROLL GAP BY THE METAL,

AT Low VALUES OF THE "SHAPE FACTOR"

The experiments were carried out on a two-high 210 mm mill with steel work rolls. In this work Yu. I. Odinokov also took part. Lead specimens were prepared by forging and machining of the side faces and the front end of the stock on a planing machine, so that the surface of the butt end should be at right angles to the side faces. On the upper


and lower faces of the specimen, coming into contact with the work rolls during rolling, transverse and longitudinal grid lines were applied with an indexing head. The distances between the grid lines were measured by means of an instrument microscope with an accuracy of up to 001 mm.

After setting the necessary gap between the work rolls, the stock was delivered into them while the rolling speed was very low. By switching off the mill drive motor it was possible to discontinue rolling at any given moment.

The process of filling the roll gap by the metal is not a steady-state one, as distinct from a steady-state, established rolling process. This implies that the states of deformations and stresses in any given area of the zone of deformation in the course of filling the roll gap by the metal change continuously.15 5 9 ' 4 4 This considerably complicates the investigation of the filling of the gap between the work rolls by the metal.

For purposes of revealing the changes taking place in the zone of deformation as the roll gap is filled by the metal, a number of identical specimens were rolled at a fixed setting of the work rolls, but at different degrees of filling of the roll gap by the metal. As distinct from the length of the zone of deformation during an established rolling process, /, the length of the partial filling of the roll gap we shall designate by lf (Fig. 134).

The first series of experiments were carried out with lead specimens with a transverse cross-section of 24 x 24 mm, using a roll gap of about 20 mm. Consequently, if after the bite transition to an established rolling process took place, then the draft would equal 4 mm. Since the diameter of the work rolls was 210 mm, then on complete filling of the roll gap by the metal the length of the zone of deformation would be 20-5 mm.

In practice the filling of the roll gap by the metal varied. The projection of the length of the arc of partial filling was determined as the distance between the front end face of the stock and the plane of entry of the stock into the work rolls.

From the measurements of the longitudinal distances between the transverse grid lines before and after rolling we determined the value of the logarithm of the elongation factor on an incompletely rolled length of stock for each cell. For constructing the diagrams of the increasing elongation coefficients, along the abscissa was shown the distance between the centre of the given cell and the plane of entry


into the work rolls. A diagram of an incompletely rolled specimen for a certain value of the arc of partial filling is shown in Fig. 135.

In the preceding chapters it has been shown that for an established rolling process the horizontal section of the curve of the increasing longitudinal deformation represents the section of the length of the

FIG . 135. Incompletely rolled specimen with co-ordinate grid.

zone of deformation, on which there is an absence of relative movement between the stock and the work rolls on the contact surface. Such an interpretation is possible only for a steady-state process, since only in this instance do the states of stresses and deformations of any given element of the stock, which happens at any given moment to be in the area of the zone of deformation under consideration, remain unchanged. In actual fact, a certain area of the length of the zone of deformation investigated by us remains motionless during the whole period of an established rolling process, and through it continually passes the metal of the stock being deformed, so that for any given element of the metal which at a given moment happens to be in the area of the zone of deformation under consideration, the states of stresses and deformations do remain unchanged. Therefore, during an established rolling process every element of the length of the stock being rolled passes in succession through a series of stages of the states of stresses and deformations, just in the same way as they were traversed by each of the preceding, and will be traversed by each of the following, elements of the length of the stock. Therefore, during an established rolling process there is no need for an investigation of the deformation of different elements of the length of the stock, if the mechanical properties of the metal are identical along the length of the stock, and it is possible to limit the investigation of the changes


in the states of stresses and deformations at various stages to the deformation of a single element of the length of the stock, or, which is the same thing, to limit the investigation of the states of stresses and deformations in various areas of the zone of deformation to any moment of an established rolling process.

Quite another state of affairs applies to the bite on the stock by the work rolls and to the transition to an established rolling process. The non-steady-state nature of the process of deformation during this period of rolling makes it impossible to consider that at any given degree of filling of the roll gap by the metal in the area of the zone of deformation under consideration the states of stresses and deformations will be identical. Suppose that at a certain moment of the filling of the roll gap by the metal the front end of the stock acquires the position I-I (Fig. 136), and at another moment of time the position II—II. Reviewing the states of stresses and deformations of the metal

FIG. 136. Diagram of the change in the states of deformations and stresses at various values of lf.

in a certain area of the zone of deformation AßCD, it may be said that at the first moment they were different from those at the second. Since the filling of the roll gap by the metal takes place continuously, in the absence of skidding, then the states of deformations and stresses in any given area of the zone of deformation also change continuously.

Hence it follows that during the filling of the roll gap by the metal, each element of the length of the stock entering the work rolls passes


through different stages of the states of stresses and deformations to those passed through by the preceding and following elements.

Therefore, during an established rolling process the states of stresses and deformations at any given point in the volume of the zone of deformation are not dependent on time, and during an unestablished, initial period of rolling, and also indeed in the final period, the states of stresses and deformations at one and the same point in the volume of the zone of deformation change continuously in time. In our investigation, instead of using various moments of time to indicate the degree of filling of the roll gap by the metal, as an independent

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FIG. 137. Curves of the increasing elongation factors as the roll gap is progressively filled by the metal (//= 5, 7-5, 8 and 8-5 mm).

variable we took a degree of filling of the gap with metal, lf. The differences set out here between the steady-state and non-steady-state periods of rolling are confirmed by the experimental data presented below.

In the first series of experiments, the conditions of conduct of which are described above, the minimum value for the filling of a length of the zone of deformation equalled 5 mm or /,·//= 0-25; during the rolling of each subsequent specimen this value was increased.

In Figs. 137-9 are presented the experimental curves of the increasing elongation factors, in which the length of the zone of deformation is given on the abscissa. The relevant curves of the increasing elongation factors during homogeneous deformation are shown by dashed lines. To each curve there corresponds a definite degree of filling of the length of the zone of deformation. As is evident from the figures, the curves for a given degree of filling have a sharply defined arrest in the vicinity of the plane of entry of the metal into the work rolls. We observe that the curves for the increasing longitudinal deformation during an established rolling process also have such arrests in

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the zone of transition from slip to adhesion and subsequently during the transition from adhesion to slip (taken in rolling sequence) (Fig. 47).

The value of the increasing longitudinal deformation up to the arrest on the curve we shall call the initial, longitudinal deformation and designate it by logA0. From Fig. 137 it is evident that with the increase in the degree of filling of the roll gap by the metal, the value of the initial, longitudinal deformation is somewhat reduced. But, when the relative degree of filling /,// > 3/4 (Fig. 139), the reduction in the value of the initial deformation is halted and at the same time on the curve there is a horizontal section which gradually increases, as the value of the degree of filling of the roll gap by the metal increases.

Let us explain the cause of the reduction in the initial, longitudinal deformation log λ0 with the increase in the degree of filling of the roll gap by the metal. In the preceding chapters it has been shown that the inhomogeneity of the deformation is all the greater, the less is the value of the shape factor. During an unstabilised rolling process the real length of the zone of deformation is continuously increased from zero to a certain final value, while the mean thickness of the stock continuously decreases from the initial thickness of the stock to the mean thickness during the established rolling process. Consequently with the increase in the degree of filling of the roll gap by the metal the shape factor sharply increases from zero to a certain value which remains constant after complete filling of the roll gap by the metal. Hence it follows that at the initial moment of the bite on the stock by the work rolls, when lfIHmean is very low, the surface layers of the metal are subject to strong tensile stresses up to the plane of entry into the work rolls, and also directly behind this plane as far as the arrest on the curve; after this there is no mutual movement on the contact surfaces, regardless of the absence of a horizontal section on the curve of log/l* (Figs. 137 and 138). The absence of this section will be explained below. With the increase in the shape factor the extension of the surface layers of the stock in the zone of entry into the work rolls is decreased, and in consequence the initial longitudinal deformation log λ0 is also diminished (Figs. 137 and 138). The presence of a zone of adhesion implies an absence of deformation of the surface layers of the stock, and the overall decrease in the thickness of the stock thereby takes place as a result of intensive deformation of the central layers. But at very low values of the shape factor the initial


longitudinal deformation log λ0 of the surface layers is so large that, regardless of the intensive deformation of the central layers of the stock in that section of the zone of deformation in which adhesion occurs on the contact surface, the front end face of the stock nevertheless becomes concave at the initial moment after the filling of the roll gap by the metal. Subsequently as the degree of filling increases, and in consequence the shape factor ///i/mean increases, the deformation of the central layers of the stock also continues to develop intensively, in consequence of which the concavity of the front end is decreased. In relation to the parameters of the zone of deformation during an established rolling process, primarily in relation to the shape factor, during the last moments of the filling of the roll gap by the metal not only may the concavity of the front end of the stock disappear, but even convexity may appear (Fig. 140). The last is all the more prob-

FIG. 140. Shape of the front end of the stock at various degrees of filling of the roll gap by the metal, applicable to section and sheet

rolling.

able, the greater the shape factor; on the contrary, at low values of this factor the concavity of the front end of the stock during the initial moments of the filling of the roll gap by the metal is so great that it does not disappear during the transition to the established process. As is known, such a situation normally arises during the rolling of heavy ingots in a blooming mill. In connection with the edging of the ingot concavity is produced not on one, but on two planes, i.e. 1>MR 20


on the end faces so-called funnels are formed (Fig. 141). During the rolling of ingots of killed steel this phenomenon is intensified in the top part of the ingot where the shrinkage cavity is located.

Therefore the basic cause of the continuous decrease in the value of the initial, longitudinal deformation logA0 is the increase in the ratio //// and the concomitant, continuous change in the states of stresses

FIG. 141. Formation of funnels on the ends of a heavy ingot during rolling in a blooming mill.

and deformations of the metal in the areas of the zone of deformation under consideration. But the experiments show that at a certain degree of filling of the roll gap by the metal further reduction in the initial, longitudinal deformation ceases, and instead beyond the arrest on the curve of the increasing longitudinal deformation there is a horizontal section, which continuously extends with the further increase in the degree of filling (Figs. 138 and 139). In the experiments described the appearance of the horizontal section took place when the relative degree of filling of the roll gap by the metal exceeded 0-65, at lf\l ^ 0*75 the horizontal section of the curve is expressed entirely definitely, and during the further increase in the degree of filling this section also increases continuously in length (Figs. 138 and 139).

Analysis of the experimental data makes it possible to explain the picture of the transition from the initial bite to the established rolling process. In actual fact the transition from the initial bite to the established rolling process does not take place abruptly, but gradually. In various sections of the length of the zone of deformation the transition to states of stresses and deformations which are typical for the


same sections during an established rolling process takes place at different times. The termination of the change in the initial, longitudinal deformation points to the fact that in the zone of entry of the metal into the work rolls the deformation of the metal has become the same as during the established rolling process; the extent of the horizontal sections of the curves shows that section of the length of the zone of deformation over which the deformation of metal is the same as during the established rolling process.

The last two curves in Fig. 139 relate to the moments when the distance between the front end face of the stock and the plane of entry of the metal into the work rolls was equal to 22 and 26 mm. Since the length of the zone of deformation during the established rollingprocess, or, which is the same thing, the distance between the planes of entry and exit of the metal to and from the work rolls, was equal to 20-5 mm in the experiments under review, then in consequence the front end of the stock had already passed beyond the line of the centres of the work rolls by 1-5 and 5-5 mm. But the steady-state rolling process had not yet been established over the whole length of the zone of deformation. The length between the front end of the stock and the plane of entry of the metal into the work rolls we shall still designate by // , regardless of the fact that the front end of the stock has already passed out of the work rolls and consequently // > /.

Even at lf = 26 mm an established rolling process was still not in existence over the whole length of the zone of deformation. Taking into account the shape and nature of the curves of the increasing elongation coefficients during an established rolling process for about the same specimens with the same drafts (Fig. 47), it may be said that the steady-state rolling process is established along the whole length of the zone of deformation, when // ^ 30 mm, or lf\l ^ 1-5.

Following I. M. Pavlov, G. E. Arkulis investigated the minimum length of the front rigid end, or the minimum length of the section of the stock which has emerged from the work rolls, at which a steady-state rolling process is established along the whole length of the zone of deformation in relation to the thickness of the stock and the value of the draft.2 In this investigation experimental data were obtained on the minimum lengths of the front and rear rigid ends in relation to established spread during rolling.

Above it has been remarked that the rise in the curves of the increasing elongation coefficients beyond the arrest in the vicinity of the entry plane of the metal into the work rolls during a non-steady-20*


state rolling process cannot be interpreted as absence of a zone of adhesion, as may suitably be done for a steady-state rolling process (Chapters III-V). Let us consider certain special features of the non-steady-state rolling process. Great importance attaches to the fact established experimentally that there is a gradual reduction in the initial longitudinal deformation log20, as the filling of the roll gap increases up to a certain value. Taking this into account let us consider in greater detail the problem of the mutual movements on the contact surfaces, as the roll gap is filled by the metal.

At a certain moment of filling of the roll gap by the metal let the front end of the stock be at a distance lfl from the plane of entry of the stock into the work rolls, and the value of the initial longitudinal deformation at this stage be logk01 (Fig. 142«). Let the element of

FIG. 142. Diagram of the change in the initial deformations and longitudinal deformations of the surface layer of an element A during a small increase in the degree of filling of the roll gap by the

metal.

the stock under consideration by us, A, be at a distance m from the front end of the stock, and the increasing elongation of the element equal to l o g V After a certain interval of time the degree of filling of the roll gap by the metal becomes lf2 (Fig. 142 ft). Then, as follows from the experimental data considered in this section, the value of the initial, longitudinal deformation logA0 is decreased, and the element of the length of the stock, A, will move a certain distance forward; its increasing longitudinal deformation in the general instance becomes


equal to log^2- It is apparent that the basic criterion for the assessment of the movement of the contact surface of element A relative to the surface of the work rolls during the movement of this element from the first to the second position is the change in the value of the increasing, longitudinal deformation log λχ of the contact layer. If for the surface layer of element A during its movement we obtain log/ ! = log λ2, i.e. if the value of the longitudinal deformation of the surface layer of element A remains unchanged, then on the section of movement under review the surface layer adhered to the work rolls. On the other hand in the presence of slip between the stock and the work rolls the increasing, longitudinal deformation of the surface layer of element A should change, or \ogX2 > l o g ^ .

Further let us show that during continuous entry of the metal into the work rolls and continuous increase in the degree of filling of the roll gap, the presence of a zone of adhesion is characterised on the experimental curves of the increasing elongation factors not by a horizontal, but by a rising, section. For this we shall consider the following diagram. At a small degree of filling of the roll gap by the metal let the curve of the increasing elongation factor consist of two sections, a rapidly rising and a horizontal section, At (Fig. 143). This curve represents a length of the surface layer of a certain element of the length of the stock and its initial longitudinal deformation log λΑ. After a small time interval the degree of filling of the roll gap is increased to lf 2, at which, if the surface layer of element A adheres to the work rolls, then it adopts the position A2, and its longitudinal deformation log λΑ is unchanged. At the same time the next element of the length of the stock passes into the work rolls, and a horizontal section 2?! is formed for element B, adjacent to element A. Since lfl < lf2, then in accordance with the experimental data described above (Fig. 137) the value of the initial longitudinal deformation of the element B is less than the value of the same deformation of element A (Fig. 143). In the following small time interval the degree of filling of the roll gap by the metal rises to / / 3 , the element A adopts the position A3, the element B adopts the position B2 and a new element C is formed in the zone of adhesion adopting the position Cx, so that the value of the initial longitudinal deformation of element C will be less than the value of the initial deformation of element B. Continuing such arguments, we shall come to the conclusion that during an unestablished rolling process, under conditions of continuous reduction of the initial longitudinal deformation, the presence of a


FIG. 143. Diagram of the successive entry of elements of the length of the stock into the zone of deformation.

zone of adhesion on the curve of the increasing elongation factor is characterised not by a horizontal, but by a rising section of the curve. In Fig. 144 is presented a general diagram of the production of the rising section of the curve, characterising the adhesion of the stock to the work rolls. It is understandable that as the time intervals are decreased, the elements A, B, C, etc., will be shortened, and the distance between them in height will also be decreased. Finally, at the limit for an infinitely large number of infinitely small elements we shall obtain a rising curve.

But the rising section of the curve of the increasing elongation factor, log λχ9 also occurs in the presence of slip. For the solution of the problem whether a given section of the rise in the curve of log λχ expresses slip or adhesion, a quantitative analysis of this section of the curve should be carried out. Above it has been shown that if for a given element of the stock the value of the longitudinal deformation of its surface layer does not increase, then on this section adhesion of the stock to the work rolls takes place.


It should be noted that during an unestablished rolling process we must not confuse the position of a certain section of the length of the zone of deformation, or the position of a certain section of the roll

Zone of preliminary deformation

0 Rolling direction

FIG. 144. General diagram of the production of a rising section of the curve, characterising adhesion of the stock to the work rolls.

gap, remaining motionless at any given moment of rolling, on the one hand, with an element of the length of stock moving continuously forwards along the line of rolling in accordance with the increase in the degree of filling of the roll gap by the metal.

Taking into account the special features of the non-steady-state rolling process set out above and their influence on the nature of the experimental curves of the increasing elongation factors, we shall give here a brief analysis of the experimental curves, on the basis of which the diagrams shown in Figs. 137-9 were constructed. From these diagrams it is evident that with the increase in the degree of filling of the roll gap by the metal, there is a decrease in the ordinate at which the arrest on the curve occurs, i.e. there is a decrease in the value of the initial longitudinal deformation. This reduction takes place only up to lf = 13 or lf\l = 0-65; with a further increase in the filling of the roll gap by the metal the value of the initial, longitudinal


deformation remains constant and equal to its value during the established rolling process.

For confirmation of the fact that the rising sections of the curves in Figs. 137-9 express the presence of a zone of adhesion on the contact surfaces between the stock and work rolls, let us consider the longitudinal deformation of the surface layer of the element of the length of stock 2?, located at a distance of 3 mm from the front end of the stock. From Figs. 137-9 it is evident that on increasing the degree of filling of the roll gap from 5 to 13 mm, and consequently during the movement of the element B forwards along the line of rolling by approximately 13 — 5 = 8 mm, the increasing longitudinal deformation of the surface layer of element B remained unchanged and equal to 0-045. This confirms that on increasing the degree of filling of the roll gap by the metal to 13 mm the surface layer of element B, which is at a distance of 3 mm from the front end of the stock being rolled, adhered to the work rolls. To such an analysis we may subject the deformation of the other elements of the length of the stock.

During the further increase in the degree of filling of the roll gap the surface layers of element B slip over the work rolls, since the increasing longitudinal deformation of these layers increases. Thus, at lf = 13 mm the experiments showed log λχ = 0Ό45, and at lf = 22mm, i.e. when the front end of the stock is located in front of the plane of exit of the stock from the work rolls, the experiments gave logAx = 0-055. Consequently, during the filling of the last third of the length of the roll gap by the metal the surface layers of element B are displaced relative to the work rolls.

From Fig. 138 it is also evident that, in addition to the development of slip on the section of the length of stock located close to the front end, in the rear section of the length of the zone of deformation a horizontal section forms on the curve of the increasing elongation factor, the length of which is continuously increased, as the degree of filling of the roll gap by the metal rises. Hence it follows that in these sections the deformation of the surface layers became the same as during an established rolling process. Gradually this section is lengthened, and when the front end of the stock emerges beyond the line of the centres of the work rolls to such a distance that " a rigid end of minimum length" is formed, the rolling process becomes a steady-state process. Such is the picture of the gradual transition from an unstabilised to an established rolling process.


On the basis of the experimental data set out above and their analysis, it may be asserted that the conclusions normally employed in rolling theory concerning the conditions of the bite on the stock by the work rolls and concerning the transition from the bite to an established rolling process do not accord with the real conditions of the process. In reality, in these conclusions it is normally considered that the specific forces of friction on the contact surfaces are determined by eqn. (111). But such an expression for the specific forces of friction is valid only in the presence of slip on the contact surfaces between the stock and the work rolls. As has been shown above, on the sections of the contact surfaces on which adhesion occurs, the equation cannot be considered valid. Naturally, on these sections certain shear stresses operate, but their connection with the normal stresses and the external friction coefficient is not in accordance with Coulomb's Law. At the present time it can only be said that for the bite on the stock by the work rolls and the transition to an established rolling process, it is essential that the shear forces on the contact surfaces shall make possible intensive deformation of the internal layers of the stock, for in the contrary event skidding will take place and deformation will cease.

Recently in literature it has been indicated that the external friction coefficient during the bite on the stock by the work rolls is greater than during an established rolling process.37 Such a conclusion could perhaps be drawn on the basis of the experimental data. But it should be taken into account that the change in the contact frictional forces takes place not only by reason of the coefficient of sliding friction, but also in connection with the change in the parameters of the zone of deformation, especially lfIHmean, and consequently with the change in the states of stresses and deformations.

The conclusions drawn here are valid for clearly defined rolling conditions and primarily at a particular value of the shape factor.

These conclusions are drawn on the basis of an analysis of experimental data obtained for the following rolling conditions: HQ = 24 mm; H1 = 20 mm; ΔΗ = 4 mm; D = 210 mm; / = 20-5 mm; //#mean = 20-5/220 = 0-933. The initial width of the specimens was 25 mm, and their material was lead. The work rolls were dry with a polished surface. At such a value of the shape factor and under the indicated conditions of external friction for an established rolling process in accordance with the experimental curves for the increasing longitudinal deformation we have /adh// = 0-7-0-75, i.e. the relative


extent of the zone of adhesion equals 70-75 per cent of the whole length of the geometrical zone of deformation. Therefore the data and the conclusions, set out in the present section, are valid only for such parameters of the zone of deformation, at which a sufficiently developed zone of adhesion exists during the established rolling process.

42. DEFORMATION OF THE METAL DURING THE BITE ON THE STOCK BY THE WORK ROLLS AND THE FILLING OF THE ROLL GAP, AT HIGH VALUES

OF THE "SHAPE FACTOR"

Rolling of the specimens in the second series of experiments was carried out between the same work rolls as in the first series of experiments. The dimensions of the specimens in round figures were: H0 = 6 mm; H1 = 2 mm; AH = 4 mm; / = 20-5 mm; Hmean = 4 mm; 1/Hmm = 20-5/4 = 5-12.

The initial thickness of the specimens was 24 mm. The method of conducting the experiments was the same as in the preceding series of experiments.

As is evident from the experimental data presented in Chapter III, when //i/mean = 5-12, during an established rolling process there is no zone of adhesion. Therefore the experimental data set out below, their analysis, and the conclusions from them are applicable only for those instances of rolling when no zone of adhesion exists during the established rolling process.

In Fig. 145 are presented the curves of the increasing longitudinal deformation (increasing elongation factors) at various degrees of filling of the length of the roll gap by the metal lf. As is evident from the figure, at high values of the shape factor (rolling of thin stock) the curves of the increasing longitudinal deformation of the metal during the filling of the roll gap by the metal differ sharply from the curves for low values of the shape factor (Figs. 137-9). Characteristic of all the curves is the presence of maxima, which gradually move further away from the plane of entry of the metal into the work rolls with the increase in the degree of filling of the roll gap, //. The cause of the formation of the maxima is as follows. At the initial moment of the bite on the stock by the work rolls, when the degree of filling is very small, and consequently the shape factor is also very low, on the contact surfaces adhesion occurs. But as distinct from the preceding series of experiments adhesion develops on a certain limited section of the


length of the zone of deformation. Applying to the data of this series of experiments the same method of revealing the conditions on the contact surfaces for any selected elements of the length of the stock, it is possible to establish that the falling sections AB of the curves in Fig. 145 consist both of sections of slip, and also of sections of adhesion. For confirmation of this, let us compare the sections AB of curves 1 and 2 in Fig. 145.

FIG. 145. Curves of the increasing elongation coefficients as the roll gap is filled by the metal:

1 — / / = 8 m m ; 2—//= 10 mm; 3—//= 16 mm; 4—lf= 26-5 mm; 5—lf= 30 mm.

Let us take a section of the stock of length 5 mm, reckoning from the front end. At lf = 8 mm, at point B we have logAß = 0Ό72, while at point A log/x = 0Ό90. On increasing lf to 10 mm, we have logAB = 0-068 and \ο%λΑ = 0-160, so that the projection of


the section AB onto the horizontal axis also equals about 5 mm. Hence it follows that on passing from lf = 8 mm to If = 10 mm, the front section of the length of stock of length 5 mm has not yet adhered.

A similar nature is shown by the change in the section AB on further increasing lf to 16 mm. On further increasing lf formation of the front rigid end occurs, and the section AB gradually passes beyond the limits of the zone of deformation.

In Fig. 146 are shown specimens with deformed co-ordinate grids at various degrees of filling of the roll gap by the metal, from the grids of which is clearly evident the sharp difference in the conditions on the contact surface of the various elements of the length of the stock during the process of filling the roll gap by the metal.

From Fig. 146Z? it is evident that the contact layer of a section of the length of stock adjacent to the front end of the stock, suffered less extension than the contact layer of the following section of the length of stock. This is especially clearly shown by examination of the deformed co-ordinate grid in Fig. 146 c. Consequently in the zone of entry of the stock into the work rolls the boundary conditions (or conditions on the contact surface) change sharply as the degree of filling of the roll gap by the metal proceeds. Finally the fact that at the front end of the stock extensive concavity occurred, such as normally takes place during the inhomogeneous compression of a parallelepiped, is evidence of pronounced inhomogeneity of the deformation at low degrees of filling of the roll gap by the metal and at the same time of the presence of a zone of adhesion. As // increases, the boundary conditions (or conditions on the contact surface) close to the entry of the stock into the work rolls change considerably, the zone of adhesion disappears, and the transition is completed to the established rolling processs during which there is no zone of adhesion, since in the instance under review ///fmeen = 5-12.

The two series of experiments described above to investigate the conditions on the contact surface during the initial bite by the work rolls on the metal and the gradual transition to the established rolling process are once again evidence that the conditions of deformation of the metal not only during the established, but also during the non-steady-state, process change sharply in relation to the shape of a longitudinal-vertical cross-section of the zone of deformation. It is completely evident that this factor has a decisive influence on the states of stresses and deformations during rolling.

t f f i % m§ I ; t | ,

FIG. 146. Photographs of specimens with a deformed co-ordinate grid:

a-lfll= 0-55; b-l/jl = 0-8; c - / / / / = l - 5 . 305


43. THE RELATIONSHIP BETWEEN THE MAXIMUM DRAFTS DURING THE BITE AND THE ESTABLISHED ROLLING PROCESS AND THE PARAMETERS OF

THE ZONE OF DEFORMATION

It is known that the maximum possible draft in any given pass is dependent on a series of factors, but here consideration is given to the question of the maximum possible drafts only in relation to the capacity of the work rolls to bite the stock, i.e. to the possible bite angles.

The analysis of the experimental data set out above shows that at the initial moment of the bite adhesion of the metal of the stock to the work rolls always occurs (in the absence of a lubricant, naturally) but during the rolling of thin stock, when the shape factor is high for the established rolling process, the zone of adhesion rapidly disappears as the roll gap is filled by the metal. But during the rolling of thick stock, for which the shape factor is small during the established rolling process, the zone of adhesion is extensively developed not only at the initial moment of the bite on the stock by the work rolls, but also during the course of the whole period during which the roll gap is filled by the metal, and also during the established rolling process.

These data run counter to the idea that the bite conditions on the stock by the work rolls are different at various parameters of the zone of deformation, even where the coefficient of sliding friction remains unchanged. It is known that from the maximum possible bite angle at a given work roll diameter, ocmax, hite9 it is possible to calculate the maximum possible draft, AHmaXmbite9 and vice versa from dHmax.hite it is possible to find<%max.bite. Taking into account this connection, in future we shall speak either of the maximum possible reduction Ά max. bite or of the maximum possible bite angle, #max. b i t e .

Taking into account that at a given coefficient of sliding friction the basic parameter of the zone of deformation which determines the inhomogeneity of the deformation of the metal and the development of a zone of adhesion is the characteristic of the shape of the vertical-longitudinal cross-section of the zone of deformation or the ratio ////mean, and taking into account that

l ]/ 2 D #mean HQ ]_ ΔΗ

D 2D


or / .(AH H0

it may be said that at a given work roll diameter the degree of inhomo-geneity of the deformation increases with the increase in the thickness of the stock before the pass (or after the pass) for a given draft, and decreases with the increase in the draft at a given stock thickness. In addition it should be kept in view that the presence of a zone of adhesion in certain sections of the length of the zone of deformation is a consequence of the fact that under the given conditions of the process the work done in rolling is minimal with intense deformation of the internal layers and an absence of deformation of the surface layers. It is understandable that the thicker the stock, the more favourable are the conditions for the development of inhomogeneous deformation, and it should therefore be assumed that at a given work roll diameter and coefficient of sliding friction the maximum possible bite angle, and correspondingly the maximum possible draft, will be greater, the greater is the thickness of the stock. Starting from these hypotheses, experiments were carried out to reveal the relationship between the maximum possible bite angle, or the maximum possible draft at the in itial moment of the bite, and the thickness of the stock under otherwise equal conditions. Yu. I. Odinokov took part in the conduct of these experiments.

In the first series of experiments lead specimens were rolled with a rectangular transverse cross-section, identical along the whole length of the stock, between dry steel work rolls of 200 mm dia. with finish machined surfaces. The front end of each specimen was machined so that the end face should be perpendicular to the side faces. The specimen was moved towards the rotating work rolls between which there was a small gap, and after increasing the gap between the rolls, which is equivalent to a reduction in the draft and the bite angle, when the moment of the bite occurred. In this way determinations were made of the maximum possible draft and the maximum possible bite angle at various initial thicknesses of the stock and under otherwise equal conditions. The dimensions of the specimens before and after rolling, and the maximum drafts and bite angles are shown in Table 26. As is evident from this table, specimens of three different thicknesses were used, and to obtain more reliable results several specimens of roughly the same thickness were rolled. In the table are given the


mean values of the dimensions of the specimens, the drafts and the bite angles for each group of specimens. From the data of this table a graph was constructed of the relationship between the maximum possible draft and the initial stock thickness (Fig. 147). From the table and the diagram it is evident that at H0 = 10-3 mm AHmax. bite was 4-6 mm (first group of specimens), while at H0 = 34-5 mm ^ " m a x .

bite was 7-5 mm (third group of specimens). TABLE 26. MAXIMUM DRAFTS AND BITE ANGLES DURING THE ROLLING OF RECTANGULAR LEAD SPECIMENS BETWEEN FINISH MACHINED WORK ROLLS OF 200 mm dia.

Ho mm

10-4 10-1 10-5

10-3

23-5 23-8 230

23-4

34-5 34-5

34-5

Hi mm

600 5-86 5-20

5-62

17-50 16-70 16-60

16-90

27-00 27-00

27-00

ττ ■"rman mm

8-20 7-98 7-85

8-01

20-50 20-25 19-80

20-15

30-75 30-76

30-76

Bo mm

30-5 30-5 30-5

30-5

300 30-0 300

300

30-0 30-0

30-0

/ mm

21-40 21-40 23-80

22-20

25-00 27-20 25-95

26-75

28-00 28-00

28-00

/ ^raenn

2-62 2-64 300

2-76

1-22 1-34 1-31

1-29

0-91 0-91

0-91

^ " i i n x . bite mm

4-40 4-24 5-30

4-63

500 7-10 6-40

6-52

7-50 7-50

7-50

Ämax. bite

degrees

12°06' 11°54' 13°00'

12°26'

14°05' 15°18' 14°30'

14°49'

15°48' 15°48'

15°48'

Since the maximum possible draft for the bite, AHmax. bite, is dependent on the inhomogeneity of the deformation and the development of a zone of adhesion, which at a given coefficient of sliding friction is dependent on the shape factor, then from the data of Table 26 a graph was also constructed for AHmax. bite in relation to the shape factor (Fig. 148). On this diagram it is evident that the maximum possible draft decreases with the increase in the shape factor, or with the decrease in the inhomogeneity of the deformation, which confirms the hypotheses set out above concerning the relationship between the maximum possible draft, AHmax. bite5 and the maximum possible bite angle, <xmax> bite, on the one hand, and not only the


coefficient of sliding friction, but also the state of stresses and deformations as determined by the parameters of the zone of deformation, on the other.

·

10 75 20 25 30 35 H0 FIG. 147. Relationship between the maximum possible draft and the initial thickness of the stock during the bite. Finish machined

work rolls of 200 mm dia.

In the second series of experiments lead specimens with a rectangular transverse cross-section, identical over the whole length of the stock were rolled between steel work rolls. The roll diameter was almost the same (210 m) as in the first series of experiments, but the surface of the work rolls was fairly rough and coarse (rough machined rolls). In this way the coefficient of sliding friction in the second series of experiments was considerably higher than in the first. In order to discover the relationship between the maximum possible draft, AHmaXt b i t e , and the maximum bite angle, <xmax. b i t e, on the one hand and the thickness of the stock on the other, lengths of stock were rolled with various initial thicknesses, H0, but with the same initial width. In Table 27 are shown the dimensions of the specimens before and after rolling, and also the maximum drafts and bite angles. From the data of this table was constructed the diagram of the relationship between the maximum possible draft and the thickness of the stock (Fig. 149). As is evident from the table and the diagram, with the increase in the thickness of the stock there is an increase in the maximum possible draft, AHmax. b i t e , and in the maximum possible bite angle,

ß-max. bite· Ifl Table 27 are also given the values of the shape factor. From a comparison of the relevant data of the table it is evident that the maximum possible draft increases with the reduction in the shape factor, i.e. as the inhomogeneity of the deformation increases and the zone of adhesion develops. DMR 21


In the third series of experiments we investigated the relationship between the maximum possible draft, A i/max< bite, and consequently the maximum possible bite angle on the one hand, and the thickness

Ahfmax. bite 8

7

6

5

4

j 0 1 2 3 * rr—

"mean FIG. 148. Relationship between the maximum possible draft during the bite and the shape factor during the established process. Finish

machined work rolls of 200 mm dia.

of the stock and the shape of the longitudinal-vertical cross-section of the zone of deformation during the established rolling process. Wedge-shaped lead specimens were rolled, having an identical width throughout their length, between the same work rolls as in the second series of experiments. At a certain moment of rolling skidding com-

TABLE 27. MAXIMUM DRAFTS AND BITE ANGLES FOR THE ROLLING OF RECTANGULAR SPECIMENS BETWEEN ROUGH MACHINED WORK ROLLS OF 210 mm dia.

Ho mm

13-5 24-8 24-5 36-0 38-0 38-5 35-5

Bo mm

30-0 300 30-0 30-0 30-0 300 30-0

Hi

mm

0-3 8-5 8-6

17-0 19-0 19-0 17-0

■"mean

mm

6-90 16-65 16-80 26-50 28-50 28-75 26-25

/ mm

38Ό 41-3 42-0 45-0 45-0 45-3 44-1

/ TLT -1-1 mean

5-50 2-48 2-50 1-75 1-57 1-57 1-74

^ -"max. bite

mm

13-2 16-3 15-9 19-0 19-0 19-5 18-5

am a x . bite

degrees

20°44' 22°50' 22°50' 24°31' 25°06' 25°00' 24°48'


menced. Then the mill was stopped, the stock was withdrawn from the work rolls, and from the thickness of the wedge in the plane of entry into the work rolls the maximum draft was determined for the established rolling process, ^//max. bite> and correspondingly the maximum bite angle, #max. bite.

For the sake of accuracy it should be mentioned that during the rolling of wedges there is no established rolling process,47 since the parameters of the zone of deformation are continuously changing, but the deviations from an established rolling process do not have great importance for the question under consideration, and the most important factor here is the filling of the whole of the roll gap by the metal and the presence of rigid ends.

&H max. bit* 23t r 1 1 1 1 . 1

2l\

19\— \ y ^ \ —

n\— — ^ s \

1S\ L·^

f 3 \ \ 15 20 25 JO 35 40 H0

FIG. 149. Relationship between the maximum possible draft and the initial thickness of the stock during the bite. Rough machined work

rolls of 210 mm dia.

In Table 28 are given the dimensions of the wedge-shaped specimens before rolling, where H0k and Hlk denote the greatest and least thicknesses of the wedge before rolling, and H0 the thickness of the wedge in the plane of entry of the stock into the work rolls at the moment when skidding started (Fig. 150).

Since in this series of experiments large bite angles occurred, the mean thickness of the stock in the zone of deformation was determined not as half the sum of the thicknesses of the stock before and after DMR 21a


FIG. 150. Diagram of the rolling of a wedge until the start of skidding in the work rolls.

rolling, but as the height of a rectangle, the area of which equals the area of the figure AB CD and the length of which equals the length of the arc of the bite. The area of the figure AB CD was determined as the difference in the areas of the trapezium OAB01 and the two sectors OAD, so that

H0 + Hx + 2R JTT ' " m e a n I- R2d,

where R is the radius of the work rolls; and (x is the bite angle in radians.

Taking into account that / = R sin oc9 we obtain Ho + Hi

H„ R sin(%

(117)

From this equation it is evident that at small bite angles, when it may be assumed without any substantial error that oc = sin^x, we obtain the normally accepted value for Hmean in the form of half the sum of the thicknesses of the stock before and after rolling. At large bite angles the mean thickness of the stock in the zone of deformation,


calculated from the more accurately defined equation, differs considerably from half the sum of the thicknesses of the stock before and after rolling. Thus for instance, from Table 28 it is clear that at oc = 36°20' from the accurate eqn. (117) a value Hmean = 15-2 mm was obtained, while half the sum of the thicknesses before and after rolling equals 24 mm. The relative difference in the values obtained through the use of the accurate equation on the one hand, and of the simplified equation on the other, is dependent not only on the bite angle, but also on the ratio of the thickness of the stock to the work roll diameter.

TABLE 28. MAXIMUM DRAFTS AND BITE ANGLES DURING AN ESTABLISHED PROCESS OF ROLLING OF WEDGE-SHAPED SPECIMENS BETWEEN ROUGH MACHINED WORK

ROLLS OF 210 mm dia.

# 0 k mm

37-5 37-5 530 530 560 600 600 650 650 710

# l k mm

30 30 30 30 60

100 100 150 150 310

Ho mm

35-2 36-0 42-5 44-5 49-9 52-6 53-8 560 59-8 660

B0 mm

37-5 37-5 450 450 450 450 45-0 45-0 45-0 450

Hi mm

1-9 2-2 30 30 3-7 50 6-5 6-9

10-5 110

-"mean mm

111 11-7 14-3 15-2 16-7 18-5 201 20-9 24-6 270

/ mm

60-5 61-0 660 67-0 70-5 72-1 72-2 73-5 74-0 78-0

/ n mean

5-45 5-21 4-62 4-40 4-23 3-90 3-60 3-52 301 2-90

^ -"max. bite mm

33-3 33-8 39-8 410 45-2 47-0 47-3 49-1 49-3 550

am a x . bite

degrees

32°40' 33°00' . 35°50' 36°20' 38°20' 39° 10' 39°20' 40°00' 40°10' 42°20'

As the thickness of the stock increases, it is evident from Table 28 that there is an increase in the maximum possible draft, zl//max- bite» and correspondingly in the maximum possible bite angle, ocmax b i t e , for the established rolling process. Thus attention is drawn to the fact that for very thick stock (H0 = 66 mm at D = 210 mm) a heavy draft was obtained, AHmaXt bi te = 55 mm and a bite angle, <%max. bi te = 42°20'. From the data of Table 28 were constructed the graph for the maximum possible draft for the established rolling process as a function of the initial stock thickness (Fig. 151) and as a function of the shape factor (Fig. 152).

The three series of experiments described in this section and their analysis show convincingly that the conditions of the bite on the stock by the work rolls, and also the conditions under which the established 21a*


Δ H max. bite 60

55

50

k5

W

35

30

I I I I I I 9

kO *5 50 55 60 65 70H,

FIG. 151. Relationship between the maximum possible draft and the initial stock thickness for the established rolling process. Rough

machined work rolls of 210 mm dia.

aHnnoxbitt 60{

55

50

tt

40

35

30

T rV

1 N L O

1 I Ps? I

o \ o

9W fimean

FIG. 152. Relationship between the maximum possible draft and the shape factor during an established rolling process. Rough machined

work rolls of 210 mm dia.


rolling process is initiated, are dependent not only on the coefficient of sliding friction, but also on the state of stresses and deformations which are expressed in the inhomogeneity of the deformation and the presence of a zone of adhesion, which in their turn can be characterised by certain parameters of the zone of deformation.

Under practical conditions of rolling production large drafts are necessary for rolling very thick stock, especially for rolling ingots in a blooming mill, since in order to attain the intended elongation factors for the passes the draft should be correspondingly greater, the greater is the thickness of the ingot or rolled stock. The experimental data show that precisely under these conditions it is possible to obtain heavy drafts. During the rolling of thin stock high elongations can on the other hand be achieved with relatively small drafts.

In this way the development of the theory of the inhomogeneity of the deformation will be able to promote an intensification of rolling schedules in blooming and billet mills in general.

Conclusions

1. The theory of rigid ends in rolling theory is analogous to the hypothesis of flat cross-sections in the applied theory of elasticity. In all instances of rolling between plain work rolls the basic hypothesis of the theory of rigid ends concerning the interaction between the metal of the zone of deformation and the external ends, which are not undergoing plastic deformation at a given moment, should be considered in the form of the boundary conditions at the start and end of the zone of deformation. The other hypotheses of the theory of rigid ends remain valid for sheet and strip rolling, where the ratio of the length of the zone of deformation to its mean thickness is relatively high (//#mean > 2). At small values of this ratio, typical of section rolling, and especially for the rolling of heavy ingots in blooming and slabbing mills, pronounced inhomogeneity of the deformation of each element of the length of the stock occurs during its passage through the zone of deformation. This is confirmed by the numerous experimental data presented in this book.

2. At low ratios of the length of the zone of deformation to its mean thickness, at the start of the zone of deformation the stock undergoes heavy vertical deformation, while the layer of metal at mid-height is only slightly deformed, and in certain instances it is not merely not reduced, but even increased in thickness. In the central part of the zone of deformation the contact layer is scarcely deformed at all, while the central layer of the thickness of the stock undergoes heavy deformation. Finally in the area of the emergence of the stock from the work rolls the distribution of the deformations over the height of the stock is qualitatively analogous to that in the area of the entry into the rolls, but it is known that the local deformations in the exit zone are many times smaller than in the entry zone.

3. The presence and relative extent of the zone of adhesion are dependent on the ratio of the length of the zone of deformation to its mean height, 1/Hmean, and the coefficient of external friction, μ. At high values of //i/mean the zone of adhesion is almost non-existent

316

CONCLUSIONS 317

even during hot rolling, when the value of μ is sufficiently high. The relative extent of the zone of adhesion increases with the decrease in l/Hmean and with the increase in μ.

4. The work rolls fulfil a dual function: they move the stock in the direction of rolling as a solid undeformable body, and at the same time deform it. Therefore the longitudinal velocity of the movement of the particles of metal is the sum of the velocity of the rigid movement on the one hand, and of the velocity of the plastic movement on the other. The absolute longitudinal velocities at mid-height and on the contact surface in any given vertical cross-section of the zone of deformation differ inconsiderably one from another. The transport velocity, or velocity of the rigid movement, has no direct relationship to the plastic deformation of the metal and the inhomogeneity of this deformation. The inhomogeneity of the deformation is most fully characterised by the inhomogeneity of the longitudinal velocities of the plastic movements in any given vertical cross-section of the zone of deformation. Experiments show that the difference in the longitudinal velocities of the plastic movements on the contact surface, and at mid-height, of the stock is great at low values of ljHmean and high values of the external friction coefficient.

5. During the rolling of very thick stock, or more precisely during rolling at low ratios of //i/mean ·> a series of important special features occur, above all the presence of tensile stresses in the layers of the rolled stock or ingot at mid-height.

6. The experimental research into the deformation of the metal and the conditions on the contact surfaces during the initial bite on the stock, the filling of the roll gap by the metal and the transition to the established rolling process has shown that for hot rolling, i.e. at a considerable value of the external friction coefficient, at the initial moment of the bite on the stock by the work rolls adhesion of the metal of the stock to the work rolls takes place. The maximum possible draft subject to the condition that the work rolls will bite the stock at a given roll diameter is dependent not only on the external friction coefficient, but also on the dimensions of the stock and especially its thickness.

7. In this book an attempt has been made to employ an origin (Lagrange origin) to examine possible changes in the deformed state for the theoretical study of the deformed state and of the force conditions during the mechanical working of metals. By using varia-tional methods, analytical solutions have been produced for the

318 CONCLUSIONS

determination of the deformed state during the compression of cylinders, equations for the determination of the location of the neutral cross-section during rolling, and also for the determination of the forces for rolling thick stock.

8. There exist ample grounds for the assertion that by means of variational methods it is possible to produce theoretical formulae for calculations of the deformations and stresses both during rolling, and also for other forms of mechanical working of metals.

References

1. P. A. ALEKSANDROV, Contradictions in the present trend of construction of blooming mills and ways of resolving them, Obrabotka metallov davleniyem (symposium of articles), No. 2, Metallurgizdat, 1953.

2. G. E. ARKULIS, The inhomogeneity of deformation along the length of the stock in relation to the minimum length of the rigid ends, Obrabotka metallov davleniyem (symposium of articles), No. 1, Metallurgizdat, 1952.

3. B. P. BAKHTINOV and M. M. SHTERNOV, Pass designing of rolling mill rolls (Kalibrovka prokatnykh valkov), Metallurgizdat, 1953.

4. B. P. BAKHTINOV and M. M. SHTERNOV, The theory of rigid ends, StaV, No. 1, 1953.

5. N. I. BEZUKHOV, The theory of elasticity and plasticity (Teoriya uprugosti i plastichnosti), Gostekhizdat, 1953.

6. M. I. BOYARSHINOV and V. V. MEL'TSER, Accurate analysis of the equilibrium of forces during rolling, taking into account spread, Sbornik nauchnykh trudov Magnitogorskogo gorno-metallurgicheskogo institute, No. 7, Magnitogorsk, 1954.

7. A. P. VINOGRADOV, Fundamentals of pass designing of rolling mill rolls (Osnovy kalibrovki prokatnykh valkov), Gosizdat Ukrainy, 1925.

8. A. F. GOLOVIN, Prokatka, Metallurgizdat, pt. I, 1933; pt. II, 1934; pt. Ill, 1936. 9. A. F. GOLOVIN, Forward slip, the maximum bite angle and the coefficient of

friction, StaV, No. 4, 1947. 10. T. M. GOLUBEV and M. A. ZAIKOV, Forward slip and the coefficient of friction

during rolling, Trudy Sibirskogo metallurgicheskogo instituta im. S. Ordzhoni-kidze, Issledovanye protsessov obrabotki metallov davleniyem, No. I, Metallurgizdat, 1954.

11. T. M. GOLUBEV, Distribution of horizontal velocities in the zone of deformation during rolling, Ibid, Metallurgizdat, 1954.

12. T. M. GOLUBEV, Determination of movements in metal being rolled, StaV, No. 2, 1952.

13. V. Y. GRUM-GRZHIMAILO, Elementary theory of rolling and pass designing of mill rolls, Metallurgiya stali, pt. II, St. Petersburg, 1909.

14. S. I. GUBKIN, Theory of mechanical working of metals (Teoriya obrabotki metallov davleniyem), Metallurgizdat, 1947.

15. S. I. GUBKIN, I. M. Pavlov's theory of rigid ends, StaV, No. 12, 1951. 16. B. V. DERYAGIN, What is Friction? (Chto takoye treniye?), Acad. Sei. U.S.S.R.

1952. 17. M. L. ZAROSHCHINSKII, Movement of the metal in the zone of deformation

during rolling, Obrabotka metallov davleniyem (symposium of articles), No. 1, Metallurgizdat, 1952.

319

320 REFERENCES

18. M. L. ZAROSHCHINSKII, Movement of the metal in the zone of deformation during rolling, StaV, No. 8, 1950.

19. M. L. ZAROSHCHINSKII, The bite of the stock by the work rolls during rolling, Obrabotka metallov davleniyem (symposium of articles), No. 2, Metallurgizdat, 1953.

20. M. L. ZAROSHCHINSKII, Deformation of the metal during rolling in a blooming mill, Obrabotka metallov davleniyem (symposium of articles), No. 2, Metallurgizdat, 1953.

21. V. I. ZALESSKII and A. V. PUZANCHIKOV, The method of compression with conical tools, Proizvodstvo i obrabotka stali, Sb. XXIX, Metallurgizdat, 1950.

22. E. SIEBEL and A. POMP, Die Ermittlung der Formänderung und Festigkeit von Metallen durch Stauchversuch, Mitt. K.-W. Inst, für Eisenforsch., Bd. 9 and Bd. 10, 1928.

23. L. M. KACHANOV, Mechanics of plastic media (Mekhanika plasticheskikh sred), OGIZ, Gostekhizdat, 1948.

24. A. I. KOLPASHNIKOV, Flow of the metal during the rolling of aluminium alloys, Tsvetnyye Me tally, No. 4, 1954.

25. I. O. KONSTANTINOV, Forward slip in rolling, StaV, No. 9, 1940. 26. A. A. KOROLEV, New investigations of the deformation of metal during rolling

(Novyye issledovaniya deformatsii metalla pri prokatke), Mashgiz, 1953. 27. F. KASZ and P. S. VARLEY, / . Inst. Metals, LXXVI, No. 1, 1950. 28. M. A. LAVRENT'EV and L. A. LYUSTERNIK, A course in variational computation

(Kurs variatsionnogo ischisleniya), Gostekhteoretizdat, 1950. 29. N. METS, Hot rolling and roll pass designing (Goryachya prokatka i kalibrovka

valkov), ONTI, 1937. 30. O. G. MUZALEVSKII, Investigations of the velocities of the flow of metal in the

zone of deformation during the hot rolling of aluminium alloys, Trudy MAT/, No. 21, 1951.

31. E. OROWAN, The calculation of roll pressure in hot and cold flat rolling, Froc. Inst. Mech. Engr., CL, No. 4, 1943.

32. I. M. PAVLOV, Rolling, wire drawing and pressing of metals in relation to the theory of crack formation, ZhRMO, No. 1, 1927.

33. I. M. PAVLOV, Theory of rolling and fundamentals of plastic deformation of metals (Teoriya prokatki i osnovy plasticheskoi deformatsii metallov), GONTI, 1938.

34. I. M. PAVLOV, Rolling theory (Teoriya prokatki), Metallurgizdat, 1950. 35. YE. V. PAL'MOV, The action of frictional forces in the deformation process

during rolling, Trudy UraVskogo industrial'nogo instituta im. S. M. Kirova, No. 19, GONTI, 1944.

36. A. A. PRESNYAKOV, The problem of the conditions of the bite on the metal by the work rolls during rolling and the coefficient of external friction, Obrabotka metallov davleniyem (symposium of articles), No. 1, Metallurgizdat, 1952.

37. A. A. PRESNYAKOV, Maximum bite angles during hot rolling of non-ferrous metals, Obrabotka tsvetnykh metallov i splavov (symposium of articles), Metallurgizdat, 1953.

38. A. A. PRESNYAKOV, The relationship between the bite angles during the entry of the metal into the work rolls and during an established rolling process, StaV, No. 3, 1952.

REFERENCES 321

39.1. L. PERLIN and K. K. GODERZIAN, Graphical-analytical investigation of the nature of the distribution of the pressure along the arc of contact during rolling between flat rolls, Obrabotka metallov davleniyem (symposium of articles), No. 3, Metallurgizdat, 1954.

40. P. I. POLUKHIN, Analysis of spread during rolling, StaV, No. 7, 1947. 41. A. F. RODZEVICH-BELEVICH, Theory of forward slip during rolling, ZhRMO,

1912. 42. N. A. SOBOLEVSKII, Basic phenomena of the rolling process, Sovetskaya

metallurgiya, No. 8-9, 1933. 43. V. S. SMIRNOV, Cross rolling (Poperechnaya prokatka), Mashgiz, 1948. 44. I. YA. TARNOVSKII, The change in shape during the plastic working of metals

(Formoizmeneniye pri plasticheskoi obrabotke metallov), Metallurgizdat, 1954.

45.1. YA. TARNOVSKII, I. M. Pavlov's theory of rigid ends, StaV, No. 11, 1952. 46. I. YA. TARNOVSKII, A. A. POZDEYEV and N. N. KRASOVSKII, The problem of

the determination of the forces during the mechanical working of metals, Obrabotka metallov davleniyem (symposium of articles), No. 3, Metallurgizdat, 1954.

47.1. YA. TARNOVSKII and V. B. LYASHKOV, The analysis of experimental data after rolling, Ibid, 1954.

48. W. TRINKS, Blast Furnace and Steel Plant, 1951. 49. L. R. UNDERWOOD, The Rolling of Metals, Sheet Metal Industries, 1945,

p. 218. 50. YE. P. UNKSOV, Plastic deformation during forging and stamping (Plasticheskaya

deformatsiya pri kovke i shtampovke), Mashgiz, 1939. 51. YE. P. UNKSOV, New developments in hot stamping technology (Novoye v

tekhnologii goryachei shtampovki), Mashgiz, 1948. 52. A. YA. KHEIN, The process of strip and thin sheet rolling, GONTI, 1941. 53. A. HOLLENBERG, Comments on the processes occurring during the rolling of

iron, Stahl und Eisen, No. 2, 1883. 54. A. I. TSELIKOV, Distribution of pressure along the arc of contact, StaV, No. 5-6,

1944. 55. A. I. TSELIKOV, Rolling Mills (Prokatnyye stany), Metallurgizdat, 1946. 56. A. I. TSELIKOV, Forward slip of the rolled metal in the zone of deformation,

Obrabotka tsvetnykh metallov i splavov (symposium of articles), Metallurgizdat, 1953.

57. A. I. TSELIKOV and V. V. SMIRNOV, The effect of external zones on the resistance to deformation during rolling, StaV, No. 7, 1952.

58. Yu. M. CHIZHIKOV, The problem of the determination of the coefficient of friction during rolling, Obrabotka metallov davleniyem (symposium of articles), No. 2, Metallurgizdat, 1953.

59. A. P. CHEKMAREV, Spread during rolling, Nauchnyye trudy Dnepropetrovskogo metallurgicheskogo instituta im. I. V. Stalina, No. 12, Gostekhizdat Ukrainy, 1948.

Index

Adhesion 8, 302 investigation of 29-35

Adhesion zone 3, 7, 12, 24, 26, 23, 30-35, 48, 63, 65, 100, 101, 153, 179, 194, 217, 232, 240, 252, 257, 263, 265-266, 270, 272, 279 292, 297, 298, 300, 302, 304, 306, 309, 315, 316

longitudinal 87-95 ALEKSANDROV, P. A. 319 Angle of neutral cross-section 12-14,

264-265, 279 Anisotropy of friction 261 ARKULIS, G. E. 295, 319

BAKHTINOV, B. P. 248, 285, 319 Barrel

double 11, 152, 155, 178, 240 single 240

Barrelling 26 double 48-49

BEZUKHOV, N. I. 319 Bite 263,317

by work rolls 10 deformation of metal during 286-

306 literature data on conditions of

284-286 Bite angle 245, 252, 263, 279, 285,

306-315 Blooming mill

flow of metal in, longitudinal velocities of 196-199

steel ingot in, works experiment 190-193

Boundary conditions, principle of modification of 35

BOYARSHINOV, M. I. 319

Bulk deformation during rolling, in-homogeneity of increasing 116— 118

Bulk strain, heavy ingots in blooming mill 170-190

Castillian origin 38 CHEKMAREV, A. P. 321 CHIZHIKOV, YU. M. 276, 321 Closed contours, rule of 17 Compression

"fan"during 161 forces during, plane problem 202-

203 no contact slip 209-216

inhomogeneity of deformation during 149-199

of cylinders 92-94, 203-206, 261 as ductility test 149-151 forces during 203-206

of high bodies with rigid ends 158-166

state of stresses 164-166 of high cylinders 151-157

deformations during 154-157 of long strip 52-56 of parallelepipeds 157-189 of stock 98 of wedge-shaped lead specimens 250 of wedge-shaped specimens 218-225 pressure distribution during 215 vertical deformations during, distri

bution of 161-164 Co-ordinate grid 170-172, 304 Co-ordinate grid experiment 116-118 Coulomb's Law 260, 266, 301 Cross-section, neutral. See Neutral

cross-section 323

324 INDEX

Cumulative deformation 18-20 Cylinders

compression of 92-94, 261 as ductility test 149-151

forces during compression of 203-206

high, compression of 151-157 deformations during compression

of 154-157

Defects, prevention of 15 Deformation

determination of 37-48 distribution of 8, 37-48 during bite on stock 286-306 during compression, inhomogeneity

of 149-199 of high cylinder 154-157

inhomogeneity of 9-15, 279, 292, 307, 315, 317

unevenness of, causes of 23 velocities in zone of, diagrams of

105-110 work done in 222-223, 237

use of integral equations 200-202 See also under specific types of defor

mation Deformation zone 24, 27, 32, 35,

288-299, 302, 316 long 143-148 parameters of 306, 309-311 trajectories of particles in 99-100 velocities in 129

diagrams of 105-110 velocities of particles in 97-115

Deformed co-ordinate networks 69 Deformed state calculation, variational

methods 48-59 DERYAGIN, B. V. 266,319 Double barrel 11, 152, 155, 178, 240 Double barelling 48-49 Draft 187

maximum possible 306-315, 317 Ductility 145-151, 166 Ductility test, compression of cylinders

as 149-151

Elongation coefficients 17, 19-20, 287 curves of 295

Elongation factor 119, 126, 297 curves of 125, 128-129, 137-140,

186-187, 302-305 as roll gap is filled by metal

290-292, 299-300 long zone of deformation 144-

148 using lubricant 141

for elementary layers, change in 132-137

EULER 44 Expansion coefficient, linear 271 Experimental works, review of 29-35 External forces, work done in over

coming 8 External friction. See Friction External zones. See Rigid ends

"Fan" during compression 161 Filling of roll gap 286-306, 311 Flat cross-sections, hypothesis of 1,

15, 34, 37, 89,148, 202, 209, 216, 253, 316

application of 4-9 Flat deformation 5 Flow of metal 60-62

during rolling 1 in blooming mill, longitudinal veloc

ities of 196-199 Forces during compression

of cylinder 203-206 plane problem 202-203

no contact slip 209-216 Forces during rolling

of thick stock, determination of 253-257

plane problem 206-209 Friction

angle of 10 anisotropy of 261 at rest 266 external, work done in overcoming

7 external sliding 242 slip 10 static 10, 266

INDEX 325

Friction work done in overcoming internal

resistance of 8 Friction coefficient 32, 94, 156

external 13-14, 91, 93, 251, 285, 301, 316

curves of 272-278, 282-283 determination 260-264 determination from experimental

forward slip 264-284 sliding 308

Frictional forces, distribution of 27-28

Funnels 294

GALERKIN, B. G. 46 GODERZIAN, K. K. 263, 321 GOLOVIN,A. F. 2-3, 9, 22, 30, 31,

33, 34, 43, 94, 200, 276, 280, 319 GOLUBEV, T. M. xii, 30, 35, 64, 65,

280, 282, 319 GRIM-GRZHIMAILO, V. YE. 1,34,319 GUBKIN, S. I. 69, 85, 95, 150, 261,

319

High bodies 36-37 rolling of, stresses during 194-195 with rigid ends, compression of

158-166 state of stresses 164-166

High cylinders, compression of 151-157 deformations during compression of

154-157 upsetting 48-52

HOLLENBERG, A. 60, 61, 321 Homogeneous deformation 9 Horizontal deformations 66-67 Horizontal displacement 53 Hot rolling 91 Hypothesis of flat cross-sections 1,15,

34, 37, 89, 148, 202, 209, 216, 253, 316

application of 4-9

Inhomogeneity of bulk strain 120-148 of deformation 9-15, 279, 292, 307,

315,317

Inhomogeneity, of deformation during compression 149-199 during rolling 9-15 during rolling, manifestations of

16-28 during rolling, methods of investi

gation of 60-70 external 23-24 internal 23-24 internal, present state of study of

29-35 of increasing bulk deformation dur

ing rolling 116-148 of velocities of particles 111-115

curves of 114-115 of vertical deformation 95-97

during rolling, experimental data on 70-86

Inhomogeneous deformation, theory of 16-59

Integral equations for work done in deformation 200-202

Internal stresses, work done in overcoming 7-8

KACHANOV, L. M. 320 KASHCHENKO, G. A. 271 KASZ, F. 32, 320 Κ Η Ε Ι Ν , Α . Υ Α . 321 KOLEMAGIN, Y U . I. 92, 170 KOLPASHNIKOV, A. I. 85, 86, 92, 320 KONSTANTINOV, I. O. 320 KOROLEV, A. A. 28, 32, 33, 94, 320 KOSTYCHEV, P. S. 261 KRASOVSKH, N. N. 321

LAGRANGE 46 Lagrange origin 38, 317 LAVRENT'EV, M. A. 320 Local deformation 18-20 Local vertical deformations, heavy in

gots in blooming mill 166-169 Longitudinal deformations, curves of

171-177, 182 Longitudinal velocities of flow of metal

in blooming mill 196-199 Lubricant, use of 140-143

326 INDEX

LYASHKOV, V. B. xii, 321 LYUSTERNIK, L. A. 320

Mechanical working of metals 261 theory of, application of Saint

Venant principle 35-37 variational methods 37-48

MEL'TSER, V. V. 319 Method of vertical grids 248 METS, N. 61, 62, 64, 320 Migration 53 Moderately high bodies 36-37 MUZALEVSKII, O. G. 29, 32, 35, 64,

85, 86, 92, 320

Neutral cross-section 12 angle of 12-14, 264-265, 279

equation for 267-270 determination of location 216-253

experimental 246-253 for medium in state of flow 233-

239 for simultaneous occurrence of zones of adhesion and slip 239-246

during rolling of thick bodies 231-232

relationship of location and bite angle and shape factor 245

relationship of location to thickness of stock and reduction 244

thick lengths and moderate height stock 216-218

wedge-shaped specimens 218-233 Non-steady-state rolling 296-299

ODINOKOV, Y U . I. 286, 307 OROWAN, E. 30, 32, 33, 85, 86, 92, 94,

320

PAL'MOV,YE. V. 3, 31,33,320 Parallelepipeds, compression of 157-189 PAVLOV, I. M. xi, 1, 2, 17, 23, 159,

195, 245, 261, 262, 263, 295, 320 PERLIN, I. L. 263, 321 Plane strain, state of 52

Plastic deformation 7, 80, 85-86, 171 deformation zone 173, 194 movement 102

velocity of 98, 100, 317 working of metals 260

POLUKHIN, P. I. 321 POMP, A. 261, 320 POZDEYEV, A. A. xii, 321 PRESNYAKOV, A. A. 10, 263, 285, 320 Pressure

distribution diagram 256 distribution during compression 215 during rolling 27

Principle of modification of boundary conditions 35

PUZANCHIKOV, A. V. 21, 261, 320

Reduction coefficient 17, 19-20 Reduction factor 119

curves of 130, 137, 138, 140 long zone of deformation 147 using lubricant 142-143

Restricted deformation zone 26, 87, 92, 127,

137, 156, 181, 194 working zones 91-92

Rigid ends 129, 142, 199, 295, 311 high bodies with, compression of

158-166 state of stresses 164-166

theory of 1, 4, 8, 316 velocity of 102-104

Rigid movement, velocity of 98, 102 RODZEVICH-BELEVICH, A. F. 1, 34,321 Roll-gap, filling of 286-306, 311 Rolling

between smooth rolls 6, 7 flow of metal during 1 forces during, plane problem 206-209 non-steady state 296 pressure during 27 steady-state 288, 300 variational principles 200-259

Rule of closed contours 17

Saint Venant principle 39, 48, 152, 157

INDEX 327

Saint Venant principle application of 35-37

Shape factor 30, 88, 94, 245, 270, 272-279, 281-284, 292, 293, 301, 302, 306, 308, 309, 313

high values of 302-304 low values of 286

Shear deformation velocities 5 stresses 6

SHTERNOV, M. M. 248, 285, 319 SIEBEL, E. 261, 320 Single barrel 240 Skidding 301,310 Slip 2-3, 8, 9, 30-31, 53, 89, 102, 126,

220, 297, 298, 303 backward 11, 100, 102, 104-105,

112, 206-208, 218, 229, 235-236, 254-257

forward 11,12,13-14,102,104-105, 112, 208-209, 218, 229, 235-236, 253-257, 262

calculation of 257-259 experimental 264-271 experiments in determination of

270-271 velocity of 280-284

Slip zone 240,292 SMTRNOV, V. S. 321 SMIRNOV, V. V. 159, 321 SOBOLEVSKII, N. A. 3, 31, 321 Spread 11-12, 18, 56, 121, 127, 134,

176, 192 coefficient of 19-20 using lubricant 142

Spread factor 119 curves of 130, 137, 139

long zone of deformation 144-148 using lubricant 141

State of plane strain 52 Steady-state rolling 288, 300 Steel ingot in blooming mill, works ex

periment 190-193 Strain

bulk, heavy ingots in blooming mill 170-190

inhomogeneity of 120-148 distribution of 8

Stress determination of 37-48 distribution of 8, 37-48, 95-97 during compression of high bodies

with external zones 164-166 during rolling of high bodies

194-195 tensile 292, 317

TARNOVSKII, I. YA. xii, 9, 21, 92, 159, 250, 321

Tensile stresses 292, 317 Thick

bodies 36-37 stock

forces during rolling of, determination of 253-257

rolling of 259 Thin bodies 36-37 Tongs mechanism of I. M. Pavlov 262 Trajectories of particles in deformation

zone 99-100 Transverse deformation 180

curves of 171-177, 183-190 mechanism of 11

Tri-axial deformation 18 TRINKS,W. 62,321 TRUBIN, V. N. xii, 51, 58, 92 TSELIKOV, A. I. 3, 27, 28, 31, 33, 35,

85, 94, 111, 159, 207, 208, 209, 227, 246, 266, 321

UNDERWOOD, L. R. 32, 63, 321 Unevenness of deformation, causes of

23 UNKEL, G. 62, 63, 67 UNKSOV,YE. P. 202, 203, 205, 209,

212, 321 Upsetting 21-22, 26, 40

between plane parallel plates 7 flat deformed state during 5 of cylinder of moderate height

56-59 of high cylinder 48-52

Urals Institute of Ferrous Metallurgy 65

328 INDEX

Variational principles for rolling theory 200-259

VARLEY, P. S. 32, 320 Velocities

in deformation zone 129 diagrams of 105-110

longitudinal, of flow of metal in blooming mill 196-199

of movement of particles 317 of metal 24-26, 33

of particles in deformation zone 97-115

inhomogeneity of 111-115 of particles of metal 29 of plastic movement 98, 100, 317 of rigid ends 102-104 of rigid movement 98, 102, 317 of slip 280-284

Vertical deformation 22, 24-26, 66-67

curves of 171-177, 178-180, 183-190

distribution of 95-97 during compression, distribution of

161-164 during rolling, inhomogeneity of,

experimental data on 70-86 inhomogeneity of 95-97

Vertical, deformation local, heavy ingots in blooming

mills 166-169 grids, method of 248

VINOGRADOV, A. P. 1, 34, 319

Wedge cross-section 233 Wedge-shaped

lead specimens, compression of 250 specimens 225, 311

neutral cross-section during compression of 218-225

Work done during deformation on 222-223,

237 use of integral equations 200-202

in overcoming external forces 8 in overcoming external friction 7 in overcoming internal resistance of

friction 8 in overcoming internal stresses 7-8

ZAIKOV, M. A. xii, 280, 282, 319 ZALESSKII, V. I. 21, 261, 320 ZAROSHCHINSKII, M. L. 30, 32, 63,

319, 320

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Deformation of Metals During Rolling