rees - basic engineering plasticity - introduction with applications (elsevier, 2006)

BASIC ENGINEERING
PLASTICITY
BASIC ENGINEERING
PLASTICITY
This page intentionally left blank
BASIC ENGINEERING
PLASTICITY
An Introduction with Engineering
and Manufacturing Applications
D. W. A. Rees
School of Engineering and Design,
Brunel University, UK
BASIC ENGINEERING
PLASTICITY
An Introduction with Engineering
and Manufacturing Applications
D.
W. A. Rees
School of Engineering and Design,
Brunei University, UK
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS
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ELSEVIER Butterworth-Heineman n is an imprint of Elsevier
Butterworth-Heinemann is an imprint of Elsevier
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First edition 2006
Copyright © 2006, D. W. A. Rees. Published by Elsevier Ltd. All rights
reserved
The right of D. W. A. Rees to be identified as the author of this work has been

asserted in accordance with the Copyright, Designs and Patents Act 1988
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
form or by any means electronic, mechanical, photocopying, recording or otherwise without the
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Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons or
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of any methods, products, instructions or ideas contained in the material herein. Because of rapid
advances in the medical sciences, in particular, independent verification of diagnoses and drug
dosages should be made
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
ISBN-13: 978-0-7506-8025-7
ISBN-10: 0-7506-8025-3
For information on all Butterworth-Heinemann publications
visit our web site at
Printed and bound in the UK
0607080910 10987654321
Butterworth-Heinemann is an imprint of Elsevier
Linacre House, Jordan Hill, Oxford 0X2 8DP
30 Corporate Drive, Suite 400, Burlington, MA 01803
First edition 2006
Copyright © 2006, D. W. A. Rees. Published by Elsevier Ltd. All rights

asserted
The right of D. W. A. Rees to be identified as the author of this work has been
asserted in accordance with the Copyright, Designs and Patents Act 1988
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
form or by any means electronic, mechanical, photocopying, recording or otherwise without the
prior written permission of the publisher
Permissions may be sought directly from Elsevier's Science & Technology Rights
Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333;
email: Alternatively you can submit your request online by visiting
the Elsevier web site at , and selecting Obtaining
permission to use Elsevier material
Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons or
property as a matter of products liability, negligence or otherwise, or from any use or operation
of any methods, products, instructions or ideas contained in the material herein. Because of rapid
advances in the medical sciences, in particular, independent verification of diagnoses and drug
dosages should be made
British Library Cataloging in Publication Data
A catalog record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
ISBN-13:
978-0-7506-8025-7
ISBN-10: 0-7506-8025-3
For information on all Butterworth-Heinemann publications
visit our web site at
Printed and bound in the UK
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ELSEVIER ?
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t
£S Sabre Foundation
CONTENTS
Preface xi
Acknowledgements xii
List of Symbols xiii
CHAPTER 1
STRESS ANALYSIS
1.1 Introduction 1
.,2 Cauchy Definition of Stress 4
1.3 Three Dimensional Stress Analysis 7
1.4 Principal Stresses and Invariants 15
1.5 Principal Stresses as Co-ordinates 21
1.6 Alternative Stress Definitions 27
Bibliography 31
Exercises 31
CHAPTER 2
STRAIN ANALYSIS
2.1 Introduction 33
2.2 Infinitesimal Strain Tensor 33
2.3 Large Strain Definitions 40
2.4 Finite Strain Tensors 47
2.5 Polar Decomposition 58
2.6 Strain Definitions 62
References 62
Exercises 63

vi
CONTENTS
CHAPTER
3
YIEL D CRITERI A
3.1 Introductio n 6 5
3.2 Yieldin g of Ductil e Isotropi e Material s 6 5
3.3 Experimenta l Verificatio n 7 1
3.4 Anisotropi c Yieldin g in Polyerystal s 8 3
3.5 Choic e of Yield Functio n 9 0
Reference s 9 1
Exercise s 9 3
CHAPTE R 4
NON-HARDENIN G PLASTICIT Y
4.1 Introductio n 9 5
4.2 Classica l Theorie s of Plasticit y 9 5
4.3 Applicatio n of Classica l Theor y to Unifor m Stres s States 9 8
4.4 Applicatio n of Classica l Theor y to Non-Unifor m Stres s Slates 11 1
4.5 Henck y versu s Prandtl-Reus s 12 3
Reference s 12 4
Exercise s 12 4
CHAPTE R 5
ELASTIC-PERFEC T PLASTICIT Y
5.1 Introductio n 12 7
5.2 Elastic-Plasti c Bendin g of Beam s 12 7
5.3 Elastic-Plasti c Torsio n 13 7
5.4 Thick-Walled , Pressurise d Cylinde r with Closed-End s 14 4
5.5 Open-Ende d Cylinde r and Thin Disc Unde r Pressur e 14 9
5.6 Rotatin g Disc 15 4
Reference s 15 9

Exercise s 15 9
CONTENTS
CHAPTER
6
SLIP LINK FIELD S
6.1 Introductio n 16 1
6.2 Slip Line Field Theor y 16 1
6.3 Frictionles s Extrusio n Throug h Paralle l Dies 18 0
6.4 Frictionles s Extrusio n Throug h Incline d Dies 19 1
6.5 Extrusio n With Frictio n Throug h Paralle l Dies 19 5
6.6 Notche d Bar in Tensio n 19 7
6.7 Die Indentatio n 19 9
6.8 Roug h Die Indentatio n 20 4
6.9 Lubricate d Die Indentatio n 20 7
Reference s 21 0
Exercise s 21 1
CHAPTE R 7
LIMI T ANALYSI S
7.1 Introductio n 21 3
7.2 Collaps e of Beam s 21 3
7.3 Collaps e of Structure s 21 5
7.4 Die Indentatio n 22 1
7.5 Extrusio n 22 5
7.6 Strip Rollin g 23 0
7.7 Transvers e Loadin g of Circula r Plates 23 4
7.8 Concludin g Remark s 23 8
Reference s 23 9
Exercise s 23 9
vili CONTENTS
CHAPTER

8
CRYSTA L PLASTICIT Y
8.1
Introductio n
24 1
8.2
Resolve d Shear Stress
and
Strai n
24 2
8.3
Lattic e Slip System s
24 6
8.4
Hardenin g
24 8
8.5
Yield Surfac e
25 0
8.6
Flow Rule
25 5
8.7
Micro -
to
Macro-Plasticit y
25 7
8.8
Subsequen t Yield Surfac e
26 2

8.9
Summar y
26 6
Reference s
26 7
Exercise s
26 8
CHAPTE R
9
THE FLOW CURV E
9.1
Introductio n
26 9
9.2 Equivalenc e
in
Plasticit y
26 9
9.3
Uniaxia l Tests
27 4
9.4 Torsio n Tests
28 0
9.5
Uniaxia l
and
Torsiona l Equivalenc e
28 3
9.6 Modifie d Compressio n Tests
28 6
9.7 Bulg e Test

29 0
9.8 Equation s
to the
Flow Curve
29 4
9.9 Strai n
and
Wor k Hardenin g Hypothese s
29 8
9.10 Concludin g Remark s
30 4
Reference s
30 4
Exercise s
30 5
CHAPTE R
10
PLASTICIT Y WIT H HARDENIN G
10.1 Introductio n
30 9
10.2 Condition s Associate d with
the
Yield Surfac e
309
10.3 Isotropi c Hardenin g
31 3
10.4 Validatio n
of
Levy Mises
and

Drucke r Flow Rule s
318
10.5 Non-Associate d Flow Rules
32 5
10.6 Prandtl-Reus s Flow Theor y
32 6
10.7 Kinemati c Hardenin g
33 1
10.8 Concludin g Remark s
33 6
Reference s
33 6
Exercise s
33 7
CONTENTS
ta
CHAPTER
11
ORTHOTROPI C PLASTICIT Y
11.1 Introductio n 33 9
11.2 Ortnotropi e Flow Potentia l 33 9
11.3 Qrtholropi c How Curves 34 3
11.4 Plana r Isotrop y 34 8
11.5 Rolled Sheet Metal s 35 1
11.6 Extrude d Tubes 35 7
11.7 Non-Linea r Strain Paths 36 2
11.8 Alternativ e Yield Criteri a 36 5
11.9 Concludin g Remark s 36 6
Reference s 36 7
Exercise s 36 8

CHAPTE R 12
PLASTI C INSTABILIT Y
12.1 Introductio n 37 1
12.2 Inelasti c Bucklin g of Struts 37 1
12.3 Bucklin g of Plates 37 8
12.4 Tensil e Instabilit y 38 8
12.5 Circula r Bulge Instabilit y 39 3
12.6 Ellipsoida l Bulgin g of Orthotropi c Sheet 39 5
12.7 Plate Stretchin g 39 9
12.8 Concludin g Remark s 40 8
Reference s 40 9
Exercise s 40 9
CHAPTE R 13
STRES S WAVE S IN BARS
13.1 Introductio n 41 1
13.2 The Wave Equatio n 41 1
13.3 Particl e Velocit y 41 2
13.4 Longitudina l Impact of Bars 41 5
13.5 Plastic Waves 42 1
13.6 Plastic Stress Levels 43 2
13.7 Concludin g Remark s 43 6
Reference s 43 6
Exercise s 43 6
CONTENTS
CHAPTER
14
PRODUCTION PROCESSES
14.1 Introduction 439
14.2 Hot Forging 439
14.3 Cold Forging 442

14.4 Extrusion 444
14.5 Hot Rolling 448
14.6 Cold Rolling 454
14.7 Wire and Strip Drawing 457
14.8 Orthogonal Machining 461
14.9 Concluding Remarks 475
References 475
Exercises 475
CHAPTER 15
APPLICATIONS OF FINITE ELEMENTS
15.1 Introduction 479
15.2 Elastic Stiffiiess Matrix 479
15.3 Energy Methods 482
15.4 Plane Triangular Element 484
15.5 Elastic-Plastic Stiffiiess Matrix 490
15.6 FE Simulations 496
15.7 Concluding Remarks 502
References 503
Exercises 503
Index 505
PREFACE
This book brings together the elements of the mechanics of plasticity most pertinent to
engineers. The presentation of the introductory material, the theoretical developments and
the use of appropriate experimental data appear within a text of 15 chapters. A textbook
style has been adopted in which worked examples and exercises illustrate the application of
the theoretical material. The latter is provided with appropriate references to journals and
other published sources. The book thereby combines the reference material required of a
researcher together with the detail in theory and application expected from a student. The
topics chosen are primarily of interest to engineers as undergraduates, postgraduates and
practitioners but they should also serve to capture a readership from among applied

mathematicians, physicists and materials scientists. There is not a comparable text with a
similar breath in the subject range. Within this, much new work has been drawn from the
research literature. The package of topics presented is intended to complement, at a basic
level, more advanced monographs on the theory of plasticity. The unique blend of topics
given should serve to support syllabuses across a diversity of undergraduate courses
including manufacturing, engineering and materials.
The first two chapters are concerned with the stress and strain analyses that would
normally accompany a plasticity theory. Both the matrix and tensor notations are employed
to emphasise their equivalence when describing constitutive relations, co-ordinate
transformations, strain gradients and decompositions for both large and small deformations.
Chapter 3 outlines the formulation of yield criteria and their experimental confirmation for
different initial conditions of material, e.g. annealed, rolled, extruded etc. Here the identity
between the yield function and a plastic potential is made to provide flow rules for the ideal
plastic solids examined in Chapters 4 and 5. Chapter 4 compares the predictions from the
total and incremental theories of classical plasticity with experimental data. Differences
between them have been attributed to a strain history dependence lying within non-radial
loading paths. Chapter 5 compiles solutions to a number of elastic-perfect plastic structures.
Ultimate loads, collapse mechanisms and residual stress are among the issues considered
from a loading beyond the yield point.
In Chapter 6 it is shown how large scale plasticity in a number of forming processes can
be described with slip line fields. For this an ideal, rigid-plastic, material is assumed. The
theory identifies the stress states and velocities within a critical deformation zone. The
rolling Mohr's circle and hodograph constructions are particulary useful where a full field
description of the deformation zone is required. Alternative upper and lower bound analyses
of the forming loads for metal forming are given in Chapter 7. Bounding methods provide
useful approximations and are more rapid in their application.
Chapters 8-10 allow for material hardening behaviour and its influence upon practical
plasticity problems. Firstly, in Chapter 8, a description of hardening on a micro-scale is
given. It is shown from the operating slip processes and their directions upon closely packed
atomic planes, that there must exist a yield criterion and a flow rule. There follows from this

the concept of an initial and a subsequent yield surface, these being developed further in
later chapters. The measurement and description of the flow curve (Chapter 9) becomes an
essential requirement when the modelling the observed, macro-plasticity behaviour. The
xli PREFACE
simplest isotropic hardening model is outlined in chapter 10. Also discussed here is the
model of kinematic hardening for when a description of the Bauschinger effect is required.
In Chapter 11 the theory of orthotropic plasticity for rolled sheet metals and extruded
tubes is given. These two models of hardening behaviour are extended in Chapter 12 to
provide predictions to plastic instability in structures and necking in sheet metal forming.
A graphical analysis of the plasticity induced by longitudinal impact of bars is given in
Chapter 13. The plasticity arising from high impact stresses is shown to be carried by a
stress wave which interacts with an elastic wave to disfribute residual stress in the bar.
Chapter 14 considers the control of plasticity arising in conventional produetion processes
including: forging, extrusion, rolling and machining. Here, the detailed analyses of ram
forces, roll torques and strain rates employ the principles of force equilibrium and strain
compatibility. This approach recognises that there are alternatives to slip lines and bounding
methods, all of which are complementary when describing plasticity in practice.
Thanks are due to the author's past teachers, students and conference organisers who
have kept him active in this area. The subject of plasticity continues to develop with many
solutions provided these days by various numerical techniques. In this regard, the material
presented here will serve to provide the essential mechanics required for any numerical
implementation of a plasticity theory. Examples of this are illustrated within the final
Chapter 15, where my collaborations with the University of Liege (Belgium) and the
Warwick Manufacturing Centre (UK) are gratefully acknowledged.
ACKNOWLEDGEMENTS
The figures listed below have been reproduced, courtesy of the publishers of this author's
earlier articles, from the following journals:
Acta Mechanica, Springer-Verlag (Figs 3.11, 3.13,11,6)
Experimental Mechanics, Society for Experimental Mechanics (Figs 10.5,11.10,11.11)
Journal of Materials Processing Technology, Elsevier (Fig. 12.19,12.23)

Journal Physics IV, France, EDP Sciences (Fig. 11.15)
Research Meccanica, Elsevier (Figs 8.13,10.9,10.13)
Proceedings of the Institution of Mechanical Engineers, Council I. Mech. E. (Fig. 10.15)
Proceedings ofthe Royal Society,
RoyaL
Society (3.7,3.14,4.1,4.5,9.20,10.7,10.8,10.12)
ZeitschriftfurAngewandteMathematikundMechamk, Wiley VCH (Figs 5.15, 5.16)
and from the following conference proceedings:
Applied SolidMechanics 2 (eds A. S. Tooth andJ, Spence) Elsevier Applied Science, 1988,
Chapter 17 (Figs 4,8,4.9,4.10).
sili
LIST OF SYMBOLS
The intentio n within
the
variou s theoretica l development s given
in
this book
has
been
to
define each
new
symbo l where
it
first appear s
in the
text.
In
this regard each chapte r should
be treate d

as
self-containe d
in its
symbo l content . There
are,
however , certai n symbol s that
re-appear consistentl y throughou t
the
text, such
as
those representin g force, stress
and
strain .
These symbol s
are
given
in the
followin g list along with others most commonl y employe d
in
plasticit y theory .
O.P
aJ, ft?
P>*
3
e, Y
e
a, T
Of % °3
a
m

*
i
&
M
p.,
v
4>,Ji
0
d,m
f
V
P
A
I,H,F
curvilinea r co-ordinate s (slip lines)
kinemati c hardenin g translation s
Schmidt' s orientatio n factor s
friction and shear angles
rolling draft
norma l
and
shear strains
norma l
and
shear rates
of
strain
micro-plasti c strain tensor
equivalen t plastic strain
direct

and
shear stress
principa l stresse s
mean
or
hydrostati c stress
micro-stres s tensor
transforme d stress
tensile
and
compressiv e strength s
equivalen t stress
friction coefficien t
Lode's parameter s
scalar multiplier s
angula r twist
die angles
hardenin g measur e
Poisson' s ratio
density
extensio n (stretch ) ratio
hardenin g function s
a, h, I, z
length s
A sectio n
or
surfac e area
b,
t
breadt h

and
thicknes s
c propagatio n velocit y
C, T
torqu e
«?!,
«j,
e
%
principa l engineerin g strain s
e
ti
distortion s
ep subscript s denotin g elastic-plasti c
E superscrip t denotin g elastic
MV
E,G,K
f
F,G,H,L
F,P
Hfj, C
m
H
m
, Hy
Uma
I,J
?uh>h
/„
J

2
,
J%
K
l,m, n
m,
n
M
n
P
P
Q
Q,S
rB,z
r
n
r
t
R
R
U
R
2
u,v,w
U
v, a
y
W
x,y,z
x

.
X
(
x,z
Y(=a
o
),k
z
LIST OF SYMBOLS
elastic constants
yield function (plastic potential)
anisotropy parameters
force
orthotropic tensors
orthotropic tensors continued
second moments of area
strain invariants
stress invariants
stress deviator invariants
buckling coefficient
direction cosines
P
r
.
anisotropy parameters continued
half-waves in buckling
bending moment
hardening exponent
pressure
superscript denoting plastic

stress ratio
shape and safety factors
polar co-ordinates
incremental strain ratios (r values)
extrusion ratio
radii of curvature
back tensions
displacements
strain energy
linear and angular velocities
volume
work done
Cartesian co-ordinates
spacial co-ordinates
material co-ordinates
equivalence coefficients
tensile and shear yield stresses
Considere's subtangent
Q (= 6^) rotation tensor/matrix
B,
C, G, L deformation tensors
E(=£
g
) infinitesimal strain tensor/matrix
F,
H deformation gradients
m, n, u unit vectors
M (=
IQ)
rotation matrix

S nominal stress tensor
T(=er
9
) stress tensor/matrix
T (=00 deviatoric stress tensor/matrix
U, V stretch tensors
CHAPTER 1
STRESS ANALYSIS
1,1 Introduction
Before we can proceed to the study of flow in a deforming solid it is necessary to understand
what is meant by the term stress. Various definitions of stress have been used so it is
pertinent to begin with explanations as to how it arises and is quantified. Firstly, it is
essential that the tensorial nature of stress is appreciated. It will be shown that stress is a
symmetrical second order Cartesian tensor. Where deformation is small (infinitesimal) we
can represent stress in both the tensor component and matrix notations. Stress is first
introduced for simple uniaxial and shear loadings. A combination of these loadings gives
both normal and shear stress, these eomprising two of the six independent components that
are possible within a stress tensor. The transformation properties of stress are to be
examined following a rotation in the orthogonal co-ordinates chosen to define the stress state
at a point. Alternative stress definitions are given when it becomes necessary to distinguish
between the initial and current areas for large (finite) deformations. Finite deformation will
affect the definition of stress because the initial and current areas can differ appreciably. The
chosen definition of stress becomes important when connecting the stress and strain tensors
within a constitutive relationship for elastic and plastic deforming solids.
The following analyses will alternate between the engineering and mathematical co-
ordinate notations listed in Table 1.1. This will enable the reader to interchange between
notations in recognition of the equivalence between them.
Table 1.1 Symbol Equivalence fa Engineering and Mathematical Notations
Quantity
Material co-ordinates

Spacial co-ordinates
Material displacements
Spacial displacements
Unit co-ordinate vectors
Direction cosines
Unit normal equation
Unit normal column matrix
Normal stress
Shear stress
Normal strain (see Ch. 2)
Shear strain (see Ch. 2)
Stresses on oblique plane
Engineering Notation
x,y,z
X,T,Z
u,v,w
U,V,W
/, m, n
u
B
= lu» + wu^+«u
a
ff
x
, a,, ff
t
e
x
,
e

P
e
x
'
a,
r
Mathematical Notation
*H-^2!^"3
«!»«a> «3
u
u
u
2
,
u
3
tti, U
25
U
3
n = /
t
Uj + l
2
u
2
+
l
3
Uj

B={/ ,
/
2
/,}
T
^11» ^22' ^33
*U '
^13' ^23
a
u
',a
n
',a
M
'
BASIC ENGINEERING PLASTICITY
Note that a rotation matrix M employs the direction cosines in the above table for a co-ordinate
transformation between Cartesian axes 1, 2 and 3, in each notation as follows:
M =
hi
*31
hi
hi
<B
hi
*33
s
h
h
m

l
"h
«i
"a
«3
1.1.1 Direct Stress
Direct stress a measures the intensity of a reaction to externally applied loading. In fact, a
refers to the internal force acting perpendicular to a unit of area within a material. For
example, when a uniaxial external force is either tensile(+) or compressive(-), cis simply
a=±W/A
(1.1)
where W is the magnitude of the externally applied force and A is the original normal area
(see Fig.
1,1a).
The elastic reduction in a section area under stress is negligibly small and
hence it is unnecessary to distinguish between initial and current areas within eq(l.l).
Elasticity is clearly evident from the initial linear plot of stress versus strain in Fig. Lib.
W
W
Ultimate tensile strength
s*—^
"Tensile yield stress
l + x
Engineering strain, e-x/l
(a)
(b)
Figure 1.1 Direct tensile stress showing elastic and plastic strain responses
Note, from' Fig, 1.1a, that the corresponding direct strain e is the amount by which the
material extends per unit of its length as shown. For displacements under tension or
compression,i.e. ± x, occurring over a length I, the corresponding strains are:

e = ±x/l (1.2)
This engineering definition of strain applies to small, elastic displacements. With larger
deformations in the plastic range a true stress is calculated from the current area and plastic
strains are calculated from referring the displacement to the current length. The true stress
and true strain are developed further in this and the following chapters.
STRESS ANALYSIS
1.1.2 Shear Stress
Let
an
applie d shea r forc e
F act
tangentiall y
to the top
area
A, as
show n
in
Fig .
L2a.
Ultimate shear strength
*. F (acting on area A) 5
^
a
j Engineerin g shear strain, y = tan
4>
(b)
Figure 1.2 Shear distortion showing elastic and plastic strain responses
The shea r sfres s intensit y
t,
sustaine d

by the
materia l
as it
maintain s equilibriu m with this
force ,
is
give n
by
T=F/A
(1.3)
The absciss a
in the
shea r stres s versu s shea r strai n plo t
Fig. 1.2b
refer s
to the
angula r
distortio n that
a
materia l suffer s
in
shear .
The
shea r strai n
is a
dimensionles s measur e
of
distortio n
and is
define d

in
Fig.
1.2a as
y=tan(fr=x/1
(1.4)
In eq(
1,4)
<f>
is the
angula r chang e
in the
right angl e measure d
in
radians . Withi n
the
elasti c
regio n
the
shea r displacemen t
x is
smal l whe n
it
follow s from eq{1.4 ) that , wit h
a
correspondingl y smal l
tf),
the
shea r strai n
may be
approximate d

as y ~ $
(rad) .
The origina l area
.4 in
eq(1.3 ) will depen d upo n
the
mod e
of
shear .
For
example ,
conside r
the two
plates ,
in
Fig .
1.3a
joine d with
a
singl e rivet, subjecte d
to
tensil e forc e
F.
Since
the
rivet
is
place d
in
singl e shear ,

A
refer s
to its
cross-sectiona l area
and F to the
transvers e shea r force .
In a
doubl e shea r
lap
join t
in
Fig .
1.3b the
effectiv e area resistin g
F
is double d
and so ris
halved .
(a)
X \ SI—» r
77
\ \ X XX X
jv
(b)
Figure U Riveted joints in single and double shear
4 BASIC ENGINEERING PLASTICITY
1.2 Cauchy Definition of Stress
Consider an elemental area da, on a plane B, mat cuts through a loaded body in its deformed
configuration (see Fig. 1.4).
Figure 1.4 Force <5F transmitted ttou$i area da

Let a unit vector n, lying normal to & at P, be directed outward from the positive side of B
as shown. Due to the applied loading, an elemental resultant force vector 3F, acting in any
direction on the positive side of da, must also be transmitted to the negative side of B if the
continuum is to remain in equilibrium. The traction acting across da may be found from
considering the lower half as a free body.
1.2.1 Stress Intensity
Let an average stress intensity, or traction vector t
m
, be the average force per unit area of
da, so that
<5F = r
w
«5a or dF
i
= r* da (1.5a,b)
The alternative expression (1.5b) has employed the componente r^"
5
of r
m
in co-ordinates,
x,
(where r= 1, 2 and 3). Equation (1.5a) shows that dF will depend upon the size and
orientation of da. The vector r
s
* emphasises this dependence upon the chosen area da at
P.
For a given P, r'
m)
is uniquely defined at the finite limit when & tends to zero. This limit
will fiirfher eliminate any momente of $F acting on 3a. Thus, fromeq(l .5a), the traction forany

given normal direction n, through P, becomes
jtf?
J i? dF
r
W _ jj
m
_ —
or
j,M _ —i (l.5c,d)
,5o-,o da da da
Equation (l.Sd) reduces to the simple forms given in eqs(l.l) and (1.3) when a single force
acte normal or parallel to a given surface. Where oblique forces act, the total stress vector
r
(l
* may be resolved into chosen co-ordinate directions, x
f
. To define a general stress state
STRESS ANALYSIS S
completely ,
it is
sufficien t
to
resolv e
r
(n>
into
one
norma l
and two
shea r stres s component s

for
the
positiv e side s
of
orthogona l co-ordinat e plane s passin g throug h poin t
P.
Suc h
resolutio n reveal s
the
tensoria l natur e
of
stres s sinc e
it
follow s
mat
mere will
be
nin e tractio n
component s when thre e orthogona l plane s
are
considered .
To
show this,
let n, (/ = 1,2,3)
be unit vector s
in the
directio n
of the
co-ordinate s
x

t
so
tha t
r °' , r *', r ""
becom e
the
tractio n vector s
on the
thre e face s show n
in
Fig.
1.5.
Figure IS Tractions across the three faces of a Cartesian element
B/
(i
The thre e tractio n vector s
r
B/
(in
whic h
n^ are
also unit
planes )
may b e
writte n
in
term s
of
the scala r intercept s
r

t
'
t normal s
to the
thre e orthogona l
that each vecto r mate s with
x,
as follows ;
r
= r
T
n
x
+ r
2
n
2
+ r
3
n
3
= r
s
n
;
r
z
= rj iij + r
2
n

2
+ i
r
' -r, r
2
n
3
n
3
= r, n,
n, = r, ' n,
which ,
by the
summatio n convention ,
may be
contracte d into
a
singl e equation ;
wher e
i,j =1,2 and 3. The
nine scala r component s
r
t
'
for m
the
component s
of a
orde r Cartesia n stres s tenso r
a

v
= r^ .
Thus ,
the
syste m
of
eqs( 1.6a) becomes :
(1.6a )
i secon d
Equation(1.6b ) satisfie s force equilibriu m paralle l
to
each co-ordinat e directions . Thi s
equilibriu m conditio n will appea r late r with
the
alternativ e engineerin g stres s notatio n
(see
BASIC ENGMffiRING PLASTICITY
eqs(l.lla,b,c)). The Cauehy stress tensor, T, with components
&y
(where i,j = 1,2, 3), is
defined in from eq(i.6b) when the co-ordinates x
t
are referred to the deformed configuration.
1.2.2 General Stress State
WitMn a general three-dimensional stress state both normal and shear stresses components
comprise the tensor components a
v
within eq(1.6b). Two conventions are employed to
distinguish between these components and to identify the directions in which they act. In
the engineering notation,

IT
denotes normal stress and r denotes shear stress. Let these
appear with Cartesian co-ordinates x, y and z, as shown in Fig. 1.6a.
i
V -
1
r
-
/
/
Y
/
/
—1»
ff
,
\ ,—+>
y
/ '
4
0
I
h
/
f
(v
°
n
/
On

(a)
(b)
Figure 1.6 General stress sates in (a) engineering and (b) mathematical notations
A single subscript on a identifies the direction of the three normal stress components. The
double subscript on r distinguishes between the six shear components. The first subscript
denotes the direction of the stress and Ihe second the direction of the normal to the plane on
which that stress acts, e.g. T
V
is a shear stress aligned with the jc-direction on the plane whose
normal is aligned with the y-direetion (Note: some texts interchange these subscripts by
writing me normal direction first). Only three shear stresses components are independent.
The complementary nature of the shear stresses: r
v
= t
yM
, r
s
= t
m
and
f
yz
=
T
V
, ensures that
moments produced by the force resultants about any point are in equilibrium. To show this,
take moments on four faces in the x-y plane about a point along the z-axis in Fig. 1.6a:
which leads to T
V

= r^. In Fig. 1.6b, an alternative Cartesian frame x
t
(x
lt
x
2
and x$) is
employed to identify the stress components according to the mathematical tensor notation.
Here, the single symbol
er
is used for both normal and shear stress components. They are
distinguished with double subscripts referring to directions and planes as before. Thus, a
lx
is a normal stress aligned with the
1-direction
and the normal to its plane is also in the x
r
direction. Normal stresses will always appear with two similar subscripts in this notation.
STRESS ANALYSE 7
Different subscripts denote shear stresses, e.g. a
l2
is aligned with the ^-direction but acts on the
plane whose normal is aligned with the x,-direction. Great care must be token not to confuse
generalised co-ordinates:
JC, ,
X
2
and x, with the system of co-ordinates 1, 2 and 3 used in the
following section to identify principal stresses. Since shear stress is absent along principal
directions we may employ a single subscript 1, 2 or 3 with a to identify principal stresses

unambiguously.
1.2.3 Stress Tensor
It is seen that six independent scalar components of stress are required to define the general
state of stress at a point. This identifies sttess as a Cartesian tensor of second order. The
components appear in the tensor notation as a
l}
=
a
M
{where i =j =1,2 and 3). Note that a
vector is a tensor of the first order since it is defined from the three scalar intercepts the
vector makes with its co-ordinate axes. The following section shows that the scalar
components of the stress tensor may be toinsfcrmed for any given rotation in the co-ordinate
axes.
These tensor components are often expressed in the form of a symmetrical 3x3
matrix T. The following matrices of stress tensor components are thus equivalent and we
shall alternate between them throughout this and other chapters.
a
y
a
n
°22 °23
(1.7a,b)
The matrices (1.7a and b) are symmetrical about a leading diagonal composed of the three
independent direct stress components.
1.3 Three-Dimensional Stress Analysis
Let an oblique triangular plane ABC in Fig. 1.7a cut through the stressed Cartesian element
in Fig. 1.6a to produce a tetrahedron OABC. The six known independent stress components:
a
x

, a
y
,
O J
, t
v
= r
yx
,
T
ia
=
T^ and t
n
= t
v
now act on the back three triangular faces OAB,
OBC and OAC in the negative co-ordinate directions.
n(Z, m, n)
(b)
Figure 1.7 General stress state far a tetrahedron showing direction cosines to oblique plane ABC
BASIC
ENGINEERING PLASTICITY
Since the element must remain in equilibrium, the force resultants produced by the action
of these stresses are equilibrated by a normal stress a and a shear stress ron the oblique
plane ABC in Fig. 1.7a. The objective is to find this stress state (0, tf in both magnitude and
direction, by the methods offeree resolution and tensor transformation.
1.3.1
Direction Cosines
It is first necessary to find the areas of each back face. Let the area ABC in Fig. 1.7b be

unity. Construct a perpendicular CD to AB and join OD. A normal vector n to plane ABC
is defined by direction cosines I, m and n, measured relative to x, y and z respectively as
follows;
I = eosa; m = cos/? and « = cosy (1.8a,b,c)
Then, as Area ABC = %AB x CD and Area OAB = &AB x OD:
(Area OAB) / (Area ABC) = OD / CD = cos y= n
Hence: Area OAB = n. Similarly: Area OBC = / and Area OAC = m. The direction cosines
are not independent. Their relationship follows from the equation of vector n:
n =
(1.9a)
where u*, Uy and u
z
are unit vectors and n
x
n ^and n are scalar intercepts with the co-
ordinates x, y and z, as shown in Fig. 1.8a.
(a)
Figure
1.8 Scalar intercepts for (a) normal vector n and (b) unit normal u
n
The unit vector n
n
, for the normal direction (see Fig.
1.8b),
is found from dividing eq(1.9a)
by the magnitude |n|:
i^, = («
I
/ln|)«»+(VM)n, + (VW)n« Cl-9b)
Substituting from eqs(1.8a,b,c): I = cosa= n

x
/\n\, m = cos/?=
n
T
/\n\
and n = CQSJ^ nj\n\,
eq(1.9b) becomes
STRESS ANALYSIS
It follows that I, m and « are also the intercepts that the unit normal vector u, makes with x,
y and z (shown in Fig.
1
Jb). Furthermore, since
the direction cosines obey the relationship:
1.3.2 Force Resolution
(1.10)
(a) Magnitudes of a and t
Let a and r be the normal and shear stress components of the resultant force or traction
vector r, acting upon plane ABC in Fig. 1.9a.
F^ure 1^ Stress state for the oblique plane ABC
The components of vector r are r
x
, r
y
and r
z
as shown. Since r must equilibrate the forces
due to stress components applied to the back faces (see Fig.
1.7a),
it follows that
(1.1 la)

(1.11b)
(1.11c)
r, =
It-
Writing eqsfl.l la,b.c) in the contracted form: r
i
= Oytij, it is seen that these become a re-
statement of eq(1.6) in which tfy = a
fl
. Using the engineering notation, the corresponding
matrix equation, r = Tn, gives
Now as the area of ABC is unity, eris the sum of the r
s
, r
y
and r
z
force components resolved
10
BASIC ENGINEERING PLASTICITY
into the norma l direction . This gives
c= r^cos*? + r
y
ca%fl + r
t
cosy= rj+ r
f
m + r
z
n (1.12a )

where , from eqs(l. l la,b,c )
a= aj
i
+ a
y
m
2
+ 0
z
n
%
+
2{lmT
v
+
mnT
n
+ lnT
a
} (1.12b )
The magnitud e of the resultan t force on ABC is expresse d in two ways :
r
2
= r
x
+ r* + r? = a
2
+
t
2

= r
1
- o
2
= r? + r/ + r/ - a
2
(1.12c )
and substitutin g eqs(l.lla-c ) into (1.12c) , t can be found .
(b) Directions of a and r
Since ff lies paralle l to n» th e directio n of a is also define d by I, m and n for the plane ABC .
The directio n of r in the plane ABC is define d by the directions : l
s
= eosar, , m
s
= cos/ | and
n
s
= cos f
5
(see Fig.
1.9b).
Becaus e r
x
, r
y
and r
2
are the resultant force s for the x, y and z
component s of can d r, this give s
r

x
= a cosa
+
tcosa
s
= la+ l
s
r
r
f
= acosfl+ rcosfl, = ma+ m,r
r
t
=
CTCQS
j^+
rcosf
s
= na+ n
g
r
Re-arrangin g give s
l
s
=
(r
x
-la)fr
(1.13a )
(1.13b )

Exampl e 1.1 A stress resultan t of 140 MP a make s respectiv e angle s of 43°, 75° and 5O°53 '
with the x, y and z-axes . Determin e the norma l and shea r stresses , in magnitud e and
direction , on an obliqu e plane whos e norma l make s respectiv e angle s of 67°13' , 30° and
71°34 ' with these axes .
Referrin g to Fig. 1.9a, first resolv e r = 140 MPa in the x, y and z direction s to give its
component s as
r
x
= 140 cos 43° = 102.3 9 MPa
r, = 140 cos 75° = 36.24 MPa
r
t
= 140 cos 50°53 " = 88.3 3 MPa
The norma l stres s is found from eq(1.12a) , in whic h I, m and n are the directio n cosine s for
the normal :
ff= r
x
l + r
y
m + r
z
n = r,cosf f + r
y
eos/? + r
8
cosy
= 102.3 9 cos 67°13 ' + 36.2 4 cos 30° + 88.3 3 cos 71°34 ' = 98.9 6 MPa
Equatio n (1.12c ) supplie s the shea r stres s on this plane as
r= y/ (r
2

- a
2
) = v' (140
2
- 98.96
2
) = 99.0 3 MPa