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CHRISTIAN: The Theory of Transformations in Metals, 2nd Ed., Part 1
EASTERLING: Mechanisms of Deformation and Fracture
HAASEN et al: Strength of Metals and Alloys (ICSMA 5), 3 Vols.
HEARN: Mechanics of Materials, 2 Vols.
HOPKINS & SEWELL: Mechanics of Solids
KRAGELSKY: Friction and Wear: Calculation Methods
MARSHALL & MARINGER: Dimensional Instability
MASUBUCHI: Analysis of Welded Structures
MILLER & SMITH: Mechanics of Materials (ICM 3), 3 Vols.
REID: Deformation Geometry for Materials Scientists
SMITH: Fracture Mechanics: Current Status, Future Prospects
SMITH: Internal Friction and Ultrasonic Attenuation in Solids
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ACTA METALLURGICA
ENGINEERING FRACTURE MECHANICS
FATIGUE OF ENGINEERING MATERIALS AND STRUCTURES
INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES
JOURNAL OF PHYSICS AND CHEMISTRY OF SOLIDS
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS
MATERIALS RESEARCH BULLETIN
THE PHYSICS OF METALS AND METALLOGRAPHY
PROGRESS IN MATERIALS SCIENCE
SCRIPTA METALLURGICA
SOLID STATE COMMUNICATIONS
Plane-Strain
Slip-Line Fields
For
Metal-Deformation Processes
A SOURCE BOOK AND BIBLIOGRAPHY
By
W. JOHNSON
University Engineering Department
Cambridge, England
R. SOWERBY
Department of Mechanical Engineering
McMaster University, Canada
and
R. D. VENTER
Department of Mechanical Engineering
University of Toronto, Canada
PERGAMON PRESS
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FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus,
OF GERMANY Hammerweg 6, Federal Republic of Germany
Copyright © 1982 W. Johnson, R. Sowerby, R. Venter
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in
any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers.
First edition 1982
British Library Cataloguing in Publication Data
Johnson, W.
Plane strain slip line fields for metal
deformation processes.
1. Dislocations in metals
2. Deformations (Mechanics)
I. Title II. Sowerby, R.
III. Venter, R.
620.Π233 TA418.4
ISBN 0-08-025452-7
Library of Congress Catalog Card no: 81-81220
In order to make this volume available as economically
and as rapidly as possible the typescript has been
reproduced in its original form. This method unfor
tunately has its typographical limitations but it is hoped
that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
Preface
The last thirty years has seen the emergence, establishment and con
solidation of a body of well-defined knowledge, the theory of metal
plasticity. It has tended to be composed of two principal parts, one
concerned with structural elements, the other with metal processing.
Within the scope of the latter there has appeared a fairly well-defined
set of techniques for analysing and predicting loads, pressures and
deformations using what is known as Slip Line Field Theory (s.l.f.t.).
The situations for which this field theory may be applied are ones in
which there is either plane strain, plane stress or axial symmetry.
It is applied to isotropie materials, anisotropie materials and to
materials such as soils to which a Coulomb-type criterion belongs.
However, for metals, the plane-strain category is the one which has
been very intensively exploited and very successfully cultivated. It
is in the subject of plane-strain metal deformation that s.l.f.t. has
been extremely valuable, especially as a tool for students of engineer
ing and solid mechanics in helping them to appreciate the mechanics
of metal processing. Axial symmetry, though of enormous industrial
importance and despite the fact that the computer can facilitate iter
ative numerical procedures for resolving problems,seems still to pre
sent theoretical difficulties which render it hard to develop solutions
that can be of great practical use. The number of practical problems
for which useful solutions have been given is small. This is not so
for plane-strain, where there now exist solutions for very complex
situations. It is, however, still a subject in which there is much
scope for theoretical development, witness the matrix method described
in Chapter 6 and its attendant references, and practical application,
e.g. to explain various kinds of defects.
The authors, as research engineers, have for several years worked in
this area, and as university teachers have been preoccupied with the
understanding of the mechanics of industrial metal-forming operations
and related design situations, with the aim of imparting an apprecia
tion of the principles involved to their students. The number of
papers in this area is now very large, and as many teachers of the
mechanics of deformable solids are now lecturing this particular topic
at advanced first degree and post-graduate-degree level it seemed very
woxthwhile to update the catalogue of contributions to the field which
we first made eleven years ago. The present monograph comprises our
v
PLANE-STRAIN SLIP-LINE FIELDS
previous one,t describes most of the advances in the field developed
during the last decade and includes references to many new papers
which give results for specific problems. A co-author in the previous
monograph was Professor J. B. Haddow of the Mechanical Engineering
Department at the University of Edmonton, Canada. On this occasion
he felt unable to be associated with the book but he has, none-the-less,
kindly raised no objection to the incorporation of any work used in
the earlier publication. As before, our prime objective has been to
provide teachers and researchers with basic material and a bibliography
of papers on the theory and application of plane-strain slip fields
to metal-working problems.
It remains, again, to repeat an apology and make an explanation. The
references are predominantly to work in the English language. There
is little reference to work in Russian, Polish, Japanese and Chinese,
and for this we apologise. We are certainly conscious of the great
amount of work now appearing in these languages, but as before it has
proved to be beyond the resources of the authors to explore in depth
what has been written in foreign-language journals on Plasticity. We
can but hope that this survey of the work published in the English
language will provoke similar summaries in the languages referred to.
In conclusion, we have endeavoured to be as comprehensive as possible
in compiling English-language references, but it is too much to hope
that we have found a very great fraction of all those which are in
print. Where we have omitted references this is indeed due to over
sight, and if readers will therefore bring notice of oversights (or
of their own published work) to the authors1 attention we shall be
very grateful.
ACKNOWLEDGMENT
We would like to thank Mrs S. PurIan for typing this text for press
and Mr R. PurIan for attending to the setting of the figures.
■\Plane Strain Slip Line Fields: Theory and Bibliography published by
3
Edward Arnold, 1970.
vi
CHAPTER 1
Introduction
THE METAL-FORMING PROCESS IN HISTORY
The working of metals is at least as old as recorded history, although
some of the processes for doing it are not. Coining, forging and
hammering, drifting or making holes, cutting or parting or indenting
are clearly among the oldest processes. In the Bible, Exodus* xxxix,
3, we read: "... and they did beat the gold into thin plates and
cut it into wires . . .". Rolling, swaging and drawing were known
and well developed by A.D. 1500. Extrusion, metal machining, drawing
sheet and tube, and section rolling depend predominantly on applying
more energy than can be supplied by one person, and these processes
were the results of nineteenth-century innovation. The twentieth
century has seen the working of large or strong parts and the conse
quent need for large sources of energy. Machines have been built
which simultaneously employ a variety of the more primitive processes,
e.g. planetary and pendulum mills and the use of explosive energy.
References 1 to 10 are useful sources of information on technology
and machine and press innovations in mechanical engineering.
PLANE-STRAIN SLIP-LINE FIELDS: HISTORICAL NOTE
Slip lines appear to have first been studied when they became obvious
in catastrophies associated with soil and the foundations of struct
ures. The history of the subject need not be developed here but the
earliest original theory in this field appears to have stemmed from
work by Coulomb (1773), followed by Rankine (1857)12 and later
Levy (1873).13 Mohr,14 especially, gave a slip theory of strength
and pointed to the importance of slip surfaces in about 1914.
Tresca15 also had studied slip directions in his work on extrusion,
reported about 1864.
Luders lines in strained sheets and curved spiral lines emanating
from the bore of an internally pressurized cylinder or shot lines
from plate penetrated by shells have been noted for more than a
century.
Descriptions of the historical development of some of this work will
1
PLANE-STRAIN SLIP-LINE FIELDS
be found by referring to an article by Sobotka1 6 and to the article
in the Encyclopaedic Dictionary of Physics1 7 by Freudenthal and
Geiringer, both of which recount the development in detail. We refer
below only to contributions which seem to have been the most influen
tial for contemporary work.
Prandtl's18 consideration of the indentation of a semi-infinite block
using a slip-line field was perhaps the first practical solution to
be given in 1921. Nadai19 adapted this to considering the crushing
of blunt wedges in 19 21 and Hencky2° in 192 3 stated the well-known
equations which now carry his name, though Kotter21 in 1903 had given
them in a more general form. Geiringer22 in 1930 seems to have first
clearly formulated the velocity equations. Many of the early attempts
at solutions to metal-forming problems considered only the stress
boundary conditions and failed to establish a kinematically accept
able velocity field. The book by van Iterson (1946),23 which carries
many potential but incorrect solutions, has been a successful starting
point for many, while Sokolovskii's24 Russian monograph of 19 46
marked a great advancement in the whole subject, despite some few
errors which it carried. Books by Tomlenov,2sGubkin 26 and Unksov27
greatly contributed to the work, and Hill, Lee and Tupper and others
appear to have brought the subject very prominently to the fore,
especially when they succeeded in solving a number of very practical
problems and thereby showing the power of the technique. The book
by Hill,28 and that by Prager and Hodge,29 first presented systematic
accounts of slip-line theory and showed the engineering worth of the
approach. Präger1s introduction of the hodograph - discernible in
Hill's unit diagram - or velocity-plane diagram in 1953 introduced
a vast and welcome simplification into the handling of slip-line
field solutions and removed, what were to many, conceptual difficulties.
(The use of the cycloid in the stress plane was a further but later
innovation of Prager.30) Many solutions and much useful discussion
of the subject was given by Lee, and a series of solutions presented
by A. P. Green in the early 19 50s were notable achievements. In the
same period certain aspects of the theory which were imperfectly
understood, or where theoretical weaknesses were known to exist -
for example, that materials might be undergoing a negative rate of
working, or concerning questions of overloading - were removed.
Since the first printing of this monograph the principal advance in
s.l.f,t. has been the development of the matrix method especially
in respect of the solution of statically indeterminate problems.
This has been achieved mainly by methods devised by I. F. Collins;
the method rests on work by Hill and Ewing with some notable contri
butions by Dewhurst. The matrix method is now so important that we
considered it warranted a separate chapter in this monograph, see pp.
160-265. It would be invidious to select further the names of men
who have made distinctive contributions in the last fifteen years
but these will be clear from references scattered through the text.
The collection of slip-line solutions now available is large and they
have been steadily extended since the first edition of this book.
Fields analysed and problems understood have become more complex
especially because of the availability of the matrix method and
modern computers. The latter has raised the level of problem solving
by the use of slip-line fields to a new level of sophistication.
The formulation of a matrix solution itself for a given specific
circumstance followed by programming for the computer have tended to
2
INTRODUCTION
make this theatre of research now more the province of the applied
mathematician than the engineer.
PHYSICAL OBSERVATIONS
The primary objective in working metal is to secure a specific shape,
and secondarily and occasionally it is to secure certain properties
in the worked metal. How the sought shape is arrived at is often of
subsidiary importance. However, a desire to know - curiosity - a
search for the origin of certain product failures and defects as a
result of certain forming processes, the wish to improve product
quality and eventually the need to be able to predict the load on
machines in order to assess probable design performances, are all
stimuli for studying the details of metal flow. It thus transpires
that at a low level of sophistication there are several categories
of physical observation which have contributed substantially to the
progress of research into metal deformation, and these deserve to be
listed. They are:
(i) Surface Coatings
When working hot steel, as say in hammering, the black oxide coating
falls away in a manner which depends on the geometry of the tools and
workpiece, and to some extent the flaking reflects the surface plastic
deformation undergone by the workpiece.
Painting a part which is to be stressed with some material which is
brittle when it is dry, such as whitewash or a varnish or Stress
Coat, can be used to reveal critically stressed regions. Cracking of
the whitewash is characteristic, and distinctive patterns will indicate
regions of severe deformation. The use of coatings is of limited value
because only (literally) superficial information is forthcoming and
even this is crude.
(ii) Surface Markings on the Metal
Of particular and outstanding value for work on slip-line fields is
the use of Lüders bands (or stretcher-strain markings). The network
of lines which appear on the surface of certain steels when the mat
erial undergoes yielding were first publicized by Lüders31 in 1860.
They were also studied in detail by Hartmann32 and independently
again by Chernov.33 The lines on the surface of the material can also
be made clear by etching with a weak solution of nitric acid.
Fry34 perfected an etching technique (see also Chapter 5) for revealing
what he termed strain markings on the surface of some steels. The
technique, however, is not limited to surface markings; a deformed
component can be cut along any appropriate section and the yielded
zones revealed by the etching reagent. The etch patterns, which
usually appear black against a grey background, are often similar in
shape and size to slip-line field patterns proposed for accounting
for the deformation mode, see Plates 1, 2 and 3. Certainly the char
acteristic Lüders bands and the etching technique of Fry were known
long before slip-line-field theory as such was invented. However,
these methods are now of great importance, both for suggesting the
3
PLANE-STRAIN SLIP-LINE FIELDS
form of theoretical slip-line-field patterns and for corroborating
those which have been proposed. Specific etching and ageing tech
niques are:
(a) Mild steel
(i) Annealed mild steel with a high nitrogen content, e.g.
0.021%, responds well to etching. After unloading, specimens
are "aged" for about 30 minutes at 250 C in an electric furn
ace and then air cooled. After sectioning, the new surface
is highly polished by grinding-down with emery papers to 00
or 000 grade. Etching is performed by immersing the specimen
in Fry1s reagent, a mixture of 45 g of cupric chloride crystals,
180 ml hydrochloric acid and 100 ml of water for a few (say 3
or 4) minutes. After etching, a specimen is swilled with
concentrated hydrochloric acid or methyl alcohol to prevent
staining and then washed thoroughly with water, swilled in
alcohol and dried.
In practice, due to the expansion of plastically deformed zones
with increases in load, it is difficult to choose the approp
riate point at which to cease increasing it in order to secure
the best result. Experience and practice are unavoidable.
(ii) A highly polished, mirror-like surface when viewed opt
ically at glancing incidence after being subjected to deform
ation will also reveal a deformation zone in form and size.
(b) Aluminium
To reveal macroscopically the plastic regions in an aluminium
specimen, before loading, one procedure is to anneal it for
4 hours at 450 C in a muffle-type electric furnace and then
furnace cool. The specimen is then loaded and if, say, this
was already in two parts, it is separated and mechanically
polished by grinding-down with emery paper to 000 grade and
polishing on rotating pads impregnated with diamond; a final
polish uses 0.5 ym diamond powder. The newly polished surface
is macro-etched by immersing the specimen in a solution of
approximately HN0-. : 30%; HC : 20%; H 0 : 50% and HF (hydro
2
fluoric acid) : 5%. The resultant plastically deformed and
etched surface is usually quite distinct. (Note that HF is
extremely dangerous and great care must be taken in handling
it. Masks should be worn.)
(iii) Grids or Nets of Lines
Grids of rectangular lines (and circles) may be scribed or printed3 s's *on
to various surfaces and examined before and after deformation to
reveal strain distribution patterns (see Plates 4-6). More directly,
the pattern of deformation of a rectangular grid may be compared with
that which would be predicted by a particular pattern of slip lines.
The use of grids on the outside stress-free surface of metal being
processed should,however,be viewed with caution because the condition
there may not be precisely that which is being assessed theoretically.
Blocks of metal about to be processed may be divided usually on a
4
d = 1/16" dia. d = 1/4" dia.
PLATE 1(c). DEFORMATION PATTERNS IN RINGS COMPRESSED BETWEEN DIES OF ï in. WIDTH.
