Deformation of Metals During Rolling

332 Pages·1965·8.13 MB·English

by  I. Ya. Tarnovskii, A. A. Pozdeyev and V. B. Lyashkov (Auth.)

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 hypo­
thesis 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 diame­
trically 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 condi­
tions 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 construc­
tion 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 approx­
imation, while in a number of cases, especially during the rolling of
sufficiently thick stock, they must be considered useless. Further devel­
opment of a rolling theory which takes into account the fundamental,
real factors of the deformation of the metal, will bring about an im­
provement 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-
7 13 41
gradov, V. Ye. Grum-Grzhimailo, A. F. Rodzevich-Belevich 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
32
metal during rolling, 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
32
manner, ,. .".
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 evi­
dent 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 trans­
verse vertical section of the zone of deformation the longitudinal vel­
ocities 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 trans­
verse 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 CROS-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 longi­
8
tudinal slip and adhesion.
N. A. Sobolevskii came to the conclusion that during rolling along
the whole length of the contact surface between the stock and the
42
rolls there is no slip, 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
35
of metal in any given vertical section of the zone of deformation.
The opinions set out above and the conclusions of the various in­
vestigators 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 exist­
54
ence of an adhesion zone is possible. 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 para­
meters 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 defor­
55
mation.
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-
4 DEFORMATION OF METALS DURING ROLLING
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 defor­
mation, 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 defor­
mation 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 trans­
verse vertical sections, which weref latb efore 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 mechani­
cal 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-sec­
tion 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 con­
clusions 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 hypo­
thesis, similarly to the way in which this is done in the study of the
resistance of material and in the theory of elasticity.
FLAT CROS-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 equa­
tion will satisfy the latter identically. Under such circumstances the
shear stresses are proportional to the shear velocities, and the plast­
icity 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