Advances in Engineering Plasticity and its Applications

1111 Pages·1993·27.705 MB·English

by  W.B. Lee (Eds.)

ADVANCES IN
ENGINEERING
PLASTICITY
AND ITS
APPLICATIONS
Proceedings of the Asia-Pacific Symposium on
Advances in Engineering Plasticity and its Applications - AEPA '92
Hong Kong, 15-17 December, 1992
Edited by
W.B. LEE
Divisions of Engineering and Construction & Land Use,
Hong Kong Polytechnic
Hung Horn, Hong Kong
1993
ELSEVIER
AMSTERDAM - LONDON - NEW YORK - TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress CatalogIng-in-Pub1Ication Data
Asia-Pacific Symposium on Advances in Engineering Plasticity and Its
Applications (1992 : Hong Kong)
Advances in engineering plasticity and its applications :
proceedings of the Asia-Pacific Symposium on Advances in Engineering
Plasticity and Its Applications, AEPA '92, Hong Kong, 15-17
December, 1992 / edited by W.B. Lee.
p. cm.
Includes bibliographical references and index.
ISBN 0-444-89991-X (acid-free paper)
1. Plastic analysis (Engineering)—Congresses. I. Lee, W. B.
II. Title.
TA652.A83 1992
620. T 1233—dc20 93-10542
CIP
ISBN: 0 444 89991 X
© 1993 Elsevier Science Publishers B.V. All rights reserved.
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V
Preface
The papers in this bound volume were presented at the first Asia-Pacific Symposium on
Advances in Engineering Plasticity and its Applications held from 15-17 December, 1992 at
the Hong Kong Polytechnic, Hong Kong. The objective of the Symposium was intended to
review the latest developments in both macroplasticity and microplasticity theories, their
interactions and applications in various engineering disciplines such as solid mechanics,
structural analysis and geo-mechanics, materials science and technology, and metal forming
and machining. The Symposium was a part of a series of academic activities held in
celebration of the 20th Anniversary of the Hong Kong Polytechnic.
Classical plasticity is a fairly well founded domain of mechanics and engineering. It serves
as the basis of many engineering structural design, manufacturing processes and natural
phenomena. Simulated by the rapid progress of computer science and experimental
techniques, and by the urgent research need of engineering and materials science, modern
plasticity has developed vigorously in the last two decades. Among the important
characteristics is the cooperated approaches of micro-, meso- and macro-mechanics, and of
analysis, experiments and computation. The size scale ranges from dislocation to geological
distances, from micrometers to kilometres. The cooperation of mechanics and materials
scientists is introducing tremendous changes in the two disciplines, and is not too remote in
future that materials can be processed on the microscale to achieve the desired macroscopic
properties.
Thanks are due to the following Organizations and Institutes for their support of the
Symposium: Tsinghua University of China, the Chinese Society of Theoretical and Applied
Mechanics (Plasticity Division), University of Hong Kong (Department of Mechanical
Engineering), the Open Learning Institute of Hong Kong, the Hong Kong Institute of
Engineers (Materials, Civil and Structural Divisions), the Beijing-Hong Kong Academic
Exchange Centre and the Croucher Foundation. I would like to thank the members of the
Organization Committee and members of the International Scientific Committee, keynote
speakers and session chairmen for their various contribution and active participation in the
programme. Special thanks are given to Professor Yang Wei for the running of the Pre-
Conference Workshop on macroplasticity and meso-damage theory. Last but not least, I
wish to express my sincere appreciation and gratitude to To Suet, Janice Chung, Kwok
Siu-keung, Zhou Ming, Cai Mingjie, Lu Hongyuan, Ma Zhirong, Li Hailong, Xu Yu and
Tang Chak-yin who made significant contribution to various aspects of the conduct of the
Symposium.
W.B. Lee
Chairman, AEPA'92,
Department of Manufacturing Engineering,
Faculty of Engineering,
Hong Kong Polytechnic
January 1993
vi
Organizing Committee
W.B. Lee (Chairman) T.P. Leung
K.C. Chan (Secretary) W.S. Lau
B.Y. Xu M. Anson
C.N. Reid S.L. Chan
B.J. Duggan L.H. Yam
L.M. Yu Y.M. Cheng
D. Poon J. Song
S.Y. Lee K.C. Wong
International Scientific Committee
R.J. Asamx University of California, San Diego, U.S.A.
H.J. Bunge Technical University of Clausthal, Germany.
W.F. Chen Purdue University, U.S.A.
J.L. Chenot Ecole Nationale Superieure, France.
E.J. Hearn Hong Kong Polytechnic, Hong Kong.
K.C. Hwang Tsinghua University, China.
J.J. Jonas McGill University, Canada.
S. Kobayashi University of California, Berkeley, U.S.A.
T. Leffers RisNational Laboratory, Denmark.
J. Lemaitre University Paris, France.
Y.W. Mai University of Sydney, Australia.
K.W. Neale Shebrooke University, Canada.
A. Needleman Brown University, U.S.A.
D.A. Nethercot Nottingham University, U.K.
S.R. Reid UMIST, U.K.
S. Storen Norwegian Institute of Technology, Norway.
M. Tokuda Mie University, Japan.
Y. Tomita Kobe University, Japan
P. Van Houtte Katholieke Universiteit, Belgium.
R.H. Wagoner Ohio University, U.S.A.
R. Wang Chinese Society of Theoretical & Applied Mechanics, China.
S.L. Wong Hong Kong Polytechnic, Hong Kong.
T.X. Yu Peking University, China & UMIST, U.K.
Advances in Engineering Plasticity and its Applications
W.B. Lee (Editor)
1993 Elsevier Science Publishers B.V. 3
MODELLING THE DEFORMATION OF POLYCRYSTALS:
EXPLAINING THE LENGTH CHANGES THAT TAKE PLACE
DURING TORSION TESTING
J.J. Jonas
Department of Metallurgical Engineering, McGill University,
3450 University Street, Montreal, Canada H3A 2A7
ABSTRACT
Since the early experiments of Swift (1947), it has been recognized that metal
polycrystals lengthen when twisted at room temperature under free-end testing
conditions and shorten when similarly strained at elevated temperatures. Glide
modelling using the conventional methods of crystal plasticity has provided a
detailed explanation of the lengthening behaviour in terms of texture effects. This
arises because the lattice rotations caused by shear move more grains into
'lengthening* than into 'shortening* orientations. The explanation for the
shortening behaviour has proved to be much more elusive and cannot be provided
by glide simulations alone. It is shown that shortening is caused by the occurrence
of dynamic recrystallization during deformation at elevated temperatures.
Methods of modelling the grain rotations produced by recrystallization are
described. Account must be taken of both oriented nucleation and selective growth.
When the grain rotation effects of recrystallization are incorporated into a suitable
crystal plasticity model, the shortening behaviour is readily reproduced.
1. INTRODUCTION
The length changes that take place when polycrystalline metal rods are twisted
under conditions of free axial movement were first described in detail by Swift in
1947 [1], He showed that a selection of 5 fee and 2 bec metals all lengthened when
deformed at room temperature to shear strains of about 6. (For the alloys selected,
this corresponded to homologous temperatures T p/T of about 0.2 or less.) By
ex mp
contrast, when experiments were carried out to the same strains on lead, for which
room temperature corresponds to a homologous temperature of 0.5, shortening was
observed instead. Even in this case, however, shortening was preceded by an
initial period of lengthening, which persisted for shear strains of about 0.5 to 1.0.
Since the classical experiments of Swift, numerous other researchers have
confirmed these general trends [2-7]. There is lengthening at the initiation of
straining, even at elevated temperatures, followed in these cases by shortening at
larger strains. More recently, it has been shown that, even at room temperature,
copper wires containing an initial <111> texture also exhibit shortening
behaviour [8].
4
The relation between the presence of particular texture components and the
tendency for twisted samples to lengthen or shorten was first pointed out by
Montheillet and co-workers [9, 10] in fixed end torsion tests. Making use of the
techniques of crystal plasticity, several groups of researchers have since succeeded
in simulating the length increases and axial compressive stresses that develop
during free end and fixed end twisting, respectively [11-15]. The explanation for
the shortening behaviour has been much more elusive and has only recently begun
to be clarified [16-18], It is the aim of this brief review to summarize the salient
features of the lengthening simulations and to describe the additional mechanisms
and procedures which have had to be introduced into the torsion model so that the
shortening behaviour could be reproduced.
2. THE SIMULATION OF LENGTHENING BY MEANS OF
CRYSTAL PLASTICITY
The earliest simulations of torsion testing were carried out using the methods of
rate independent crystal plasticity [11]. However, in part because of the ease of use
of the rate dependent models, the more recent calculations were performed using
the latter methods [12-15]. One of the features of these calculations is that they
provide unambiguous values of the glide rotation rate, W, for individual crystals,
which depends on their orientations, as expressed in terms of the Euler angles <}>i,
$2» for example. Once the glide rotation rates are known, the lattice rotation
rates Q, of individual grains can be readily obtained from the rigid body rotation
rate W, which does not depend on orientation:
a = W-W (1)
2 2 2 1/2
Here |Q(<|>1, <|>2)| = (Q + & + Q ) > where the reference axes 1 and 2
23 31 12
represent the shear direction and shear plane normal, respectively. The method of
deducing the glide rotation rate from the strain rate tensor and the
crystallographic slips is described in detail in refs. [12-15].
The Rotation Field In the present context, it is useful to represent the orientation
flow produced by the lattice rotation rate directly in Euler space for comparison
with experimental ODF's (orientation distribution functions) of the texture. Thus
the orientation change Ag = (A<J>i, A$, A<J>2) at selected points g = g(<J>i, $2) can
be obtained from [14]:
cos<J>
2(
31
(2)
<j> = -^coscj) 4- Q
2
5
ltv«li . 0.4 01 1.2 II 2.0 2.4 2.1 S.2
Fig. 1. The rotation field in the case of free end torsion for a shear increment of
0.05 and m = 0.125. The scale of the rotation rate vectors is expanded by a factor
of three (over that for the <j>'s) in order to make the arrows more visible [14].
An example of a rotation field pertaining to free end torsion and obtained in this
way is presented in Fig. 1. Here a shear increment of 0.05 was employed in
conjunction with a rate sensitivity m = 0.125. The left and right hand diagrams
correspond to <$>i = constant and <j>2 = constant sections, respectively. It should be
pointed out that the rigid body rotation rate for positive shear corresponds to
motion in the decreasing $1 direction, with ($> and $2 remaining constant. This
means that the orientation changes take place solely around the r or 3 axis of the
specimen. It is evident from the right hand side of the figure that the rotation rate
vectors are oriented mainly in the direction of rigid body spin.
6
The Orientation Stability Map Regions in which the lattice spin is much less
than the rigid body spin have been identified in the diagram by means of contour
lines of equal spin. These are centered about 'tubes' or 'fibres' that correspond to
orientations possessing very low rates of lattice rotation. At these locations in
Euler space, the glide spin'W is opposite and nearly equal in magnitude to the
rigid body spin W, leading to low values of Q from eq. (1). The low rotation rate
regions can also be identified by the corresponding high values of the orientation
stability parameter S [19] that apply to them, where this parameter is defined as:
S(4>,,4>,4>o) = In ; -=? (3)
Here D is the strain rate tensor and D is the 'effective' strain rate given by D =
2Di2/V3.
An example of an orientation stability map in Euler space pertaining to free end
torsion is presented in Fig. 2. This was calculated using a rate sensitive model with
m = 0.125 [14]. Experimental textures are expected to show high intensities
within the contours, as these fibres represent the regions in Euler space where the
orientation flow caused by the rigid body spin is largely offset by the reverse
rotations attributable to the glide spin pertaining to these geometrically favoured
locations.
The favoured locations^are known as ideal orientations and are identified in Fig. 2
by the letters A/K, B/B, C, A* A* A* and A* . The first three textures are the
p s L
preferred orientations known as {111}<110>, {112}<110> and {100}<011>,
where the first index identifies the crystallographic plane that is parallel to the
shear plane, and the second the crystallographic direction that is parallel to the
shear direction. The orientations A* and A* belong to the set {111}<211> and
x 2
are observed under fixed end testing conditions, while A* and A* are displaced
s L
slightly from the {111}<211> ideal locations and are observed under free end
testing conditions. As will be seen below, grains located near A* cause shortening,
s
while those whose orientations lie close to A* cause lengthening [14]. Grains
L
whose initial orientations are located outside the fibres displayed in Figs. 1 and 2
have high lattice spins (nearly equal to the rigid body spin) and are thus rotated
rapidly into the fibres.
The Axial Strain Rate Map Some experimental textures will be presented below,
from which it will be seen that shear deformation does indeed lead to the
concentration of grain orientations within the tubes or fibres presented in Figs. 1
and 2. It is now of interest to examine how the axial effects (length changes in free-
end testing and axial stresses in fixed-end testing) contributed by individual grains
are related to their locations in Euler space, and therefore to their positions with
respect to the orientation tubes of Figs. 1 and 2.
An example of an axial strain rate map is presented in Fig. 3 [14]. It was calculated
using a rate sensitivity m = 0.125 and the values of strain rate displayed here have
been normalized using the applied shear rate. The contour lines in the unshaded
(lengthening) regions represent normalized extension rates of 0.08, 0.16 and 0.24,
whereas those in the shaded (shortening) regions represent normalized contraction
7
Ltvcls • 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 <J>,
Fig. 2. Orientation stability map in Euler space for free end torsion (m = 0.125)
A*x and A*2 mark the locations of fixed end ideal orientations; the rotated
versions in the free end case are the A* and A* components, respectively. The
s L
commonly observed A/A, B/B and C orientations are also identified [14].
rates of - 0.08, - 0.16 and - 0.24. It is evident that the left hand (A ) fibre, which
sf
runs from B to A* to B and is centred on A* , is entirely within the shaded
s s
(shortening) regions of the map, while conversely the right hand (A ) fibre, which
Lf
runs from B to A* to B and is centred on A* , is located entirely within the
L L
unshaded (lengthening) regions of the map. The centre fibre, which runs from B to
C to B and is centred on C, runs along the frontier between the shaded and
unshaded regions.
When a polycrystalline sample containing randomly oriented grains is initially
strained in shear (torsion), the axial strain rate is zero because the lengthening and
shortening regions in Fig. 3 are equally densely populated. (The overall behaviour
8
levels - -0.24 -0.16 -0.08 0 4-0.08 +0.16 4-0.24
Aft Bf At
Fig. 3. Map of the axial strain rate/shear rate ratio in Euler space
under free end testing conditions. Shaded and white areas
correspond to shortening and lengthening, respectively [14].