This document provides information about the book "Metal Forming and the Finite-Element Method" published in 1989. It includes the authors, publisher, and copyright information. The preface describes the main purpose of the book, which is to present the fundamentals and applications of the finite-element method in analyzing and simulating metal forming processes. It focuses on flow formulation, which assumes negligible elastic response, but also discusses recent advances in solid formulation.

1. SHIRO KOBAYASHI
SOO-IK OH
TAYLAN ALTAN
New York Oxford
OXFORD UNIVERSITY PRESS
1989
METAL FORMING AND THE
FINITE-ELEMENT METHOD

2. Oxford University Press
Oxford New York Toronto
Delhi Bombay Calcutta Madras Karachi
PetalingJaya Singapore Hong Kong Tokyo
Nairobi Dar es Salaam Cape Town
Melbourne Auckland
and associated companies in
Berlin Ibadan
Copyright © 1989 by Oxford University Press, Inc.
Published by Oxford University Press, inc.,
200 Madison Avenue, New York, New York 10016
Oxford in a registered trademark of Oxford University Press
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, mechanical, photocopying, recording, or otherwise,
without the prior permission of Oxford University Press.
Library of Congress Cataloging-in-Publication Data
Kobayashi, Shiro.
Metal forming and the finite-element method /
Shiro Kobayashi, Soo-lk Oh, Taylan Altan.
p. cm. -- (Oxford series on advanced manufacturing;
4) Bibliography: p. Includes index.
ISBN 0-19-504402-9
1. Metal-work--Mathematical models. 2. Finite-element method.
I. Oh, Soo-lk. II. Altan, Taylan. III. Title. IV. Series.
TS213.K56 1989
671'.072'4-Mc19 88-11995
CIP
135798642
Printed in the United States of America
on acid-free paper

3. PREFACE
The application of computer-aided engineering, design, and manufactur-
ing, CAE/CAD/CAM, is essential in modern metal-forming technology.
Thus, process modeling for the investigation and understanding of
deformation mechanics has become a major concern in research, and the
finite-element method (FEM) has assumed increased importance, particu-
larly in the modeling of forming processes.
There are many excellent textbooks on the principles and fundamentals
of metal forming, but only a few describe the application of FEM to the
analysis and simulation of metal-forming processes.
The main purpose of this book is to present the fundamentals and
applications of FEM in metal-forming analysis and technology. The book
is primarily written for graduate students and researchers. However, it
should also be useful to practicing engineers who have a good background
in FEM and who are interested in applying this technique to the analysis of
metal-deformation processes.
In the application of FEM to metal forming, there are two formulations,
namely, flow formulation and solid formulation. Flow formulation assumes
that the deforming material has a negligible elastic response, while solid
formulation includes elasticity. Despite recent advances, the application of
solid formulation to the analysis of metal-forming problems remains
limited. On the other hand, flow formulation has found applications in a
wide variety of important forming problems. This book, therefore, is
mainly devoted to the applications that are based on flow formulation
(purely plastic and viscoplastic). However, recent advances achieved in
solid formulation have made it applicable to the analysis of some forming
problems. In order not to neglect these investigations, comparisons of
solutions based on both formulations, solid and flow, are presented in
Chapter 16.
The book begins with a general background on the subject in Chapter 1,
The description of metal-forming processes is given in Chapter 2, and
Chapter 3 details important technological aspects of these processes.
Chapters 4 and 5 present the theory of plasticity and methods of analysis as
applied to metal forming. The FEM formulations are described in
Chapters 6 and 7, and the applications of the method to the analyses of
various forming processes are presented in Chapters 8 through 11. Chapter

4. vi Preface
12 presents a thermo-viscoplastic analysis and Chapters 13, 14, and 15
include developments in the areas of deformation of porous materials,
three-dimensional problems, and preform design. The book concludes with
Chapter 16, in which further developments are discussed, along with the
outline of solid formulation and comparison of the results by both solid
and flow formulations.
Although this book primarily deals with metals, some of the principles
and solution techniques should be applicable to deformation analyses of
other materials, such as polymers and composites.
Sincere thanks are due to a number of individuals. First of all, we wish
to express our appreciation to Professor E. G. Thomsen, Professor
Emeritus, University of California at Berkeley, who helped us to devote
our careers to research in metal forming. We also thank Professor M. C.
Shaw, Arizona State University, for his encouragement and support in
writing this book, and Professor W. Johnson, Emeritus Professor, Univer-
sity of Cambridge, for his critical comments during the preparation of the
manuscript.
The senior author wishes to thank his former graduate students in the
Department of Mechanical Engineering, University of California at
Berkeley, who have contributed to the advances in the application of FEM
to metal forming.
The contents of this book are largely the results of research supported
by the Air Force Wright Aeronautical Laboratory, the National Science
Foundation, and the Army Research Office, and their support is
acknowledged.
We also thank Mr. and Mrs. Joe Bavonese for typing the manuscript.
Berkeley S. K.
Columbus S. O.
May 1988 T. A.

5. SYMBOLS
A
A
Ao
AjN
B
B
Bo
B
C
C
C
C
D
Do
D
E
e(~,j)
F
F(o,;)
F,
G
G
H
Cross-sectional area
Function of relative
density for porous
materials
Initial cross-sectional
area
area contribution of the
jth element to node N
Function of relative
density for porous
materials
Breadth
Initial breadth
Strain-rate matrix
Constant
Class of functions with
continuous derivatives of
all orders up to and
including r
Volumetric strain-rate
vector
Heat capacity matrix
Diameter
Initial diameter
Effective strain-rate
coefficient matrix
Young's modulus
Work function
Energy-rate
Lagrangian strain
Coefficient of anisotropy
Function of stresses
Traction
Shear modulus
Coefficient of anisotropy
Coefficient of anisotropy
H
Ho
H(~)
/t
AH
t,
&
J,
J~
J~
J
K
K
L
L,, L2, L3
Lqk!
M
M
N
N
P
P
Height
Initial height
Final height
Work-hardening function
Time derivative of height
Increment of height
Linear invariant of stress
tensor
Quadratic invariant of
stress tensor
Cubic invariant of stress
tensor
Linear invariant of
deviatoric stress tensor
Quadratic invariant of
deviatoric stress tensor
Cubic invariant of
deviatoric stress tensor
Jacobian of coordinate
transformation
Penalty constant
Stiffness matrix
Heat conduction matrix
Coefficient of anistropy
Area coordinate
Small-strain moduli
Constitutive moduli
Coefficient of anisotropy
Gradient matrix of shape
function vector N
Coefficient of anisotropy
Shape function matrix
Load
Effective strain-rate
matrix

6. xiv
P.
Q
R
R
Ro
Ro
Ro
Ri
Re
R.
RD
Rp
S
S
s~
s~,
s,
s.
s~
T
T
T~,
Tb
T,
T~
TR
T~
Element of strain-rate
matrix B
Heat flux vector
Roll radius
Relative density of
porous materials
Average relative density
of porous materials
Initial relative density of
porous materials
Initial radius
Internal radius of tings
and tubes
Radius of extruded or
drawn bars
Radius of neutral point in
ring compression
Die comer radius
Punch radius
Microstmcture
Surface
Surface of tool-
workpiece contact
Surface of discontinuity
Surface where traction is
prescribed
Internal surface
Surface where velocity is
prescribed
Surface where heat flux is
prescribed
Thickness
Temperature
Nodal-point temperature
Temperature of base
metal in porous materials
Die temperature
Environmental
temperature
Apparent temperature of
porous materials
Surface temperature
Symbols
Tw
T
v~
Uo
u~
up
V
Vo
lib
V~
AV
W
Wo
w.
wp
x~
Y
Y0
Y~
YR
Y.
z~
a
c
Workpiece temperature
Time derivative of
temperature
Coordinate
transformation matrix
Die or roll velocity
Entrance velocity in
rolling
Exit velocity in rolling
Punch velocity
Volume
Initial volume
Volume of base metal in
porous materials
Volume of void in porous
materials
Volume change
Width
Initial width
Average width
Time derivative of width
Total plastic work per
unit volume
Plastic work-rate per unit
volume
Work-rate per unit
volume in reference state
Element of strain-rate
matrix B
Yield stress in uniaxial
tension
Initial yield stress
Yield stress of base metal
in porous materials
Apparent yield stress of
porous materials
Element of strain-rate
matrix B
Element of strain-rate
matrix B
Height-to-diameter ratio
Specific heat

7. xvi
vo
Av
wj
w/
x=, y~, z=
tg
t~
P
y, y', y"
6b
6o
6~p
E
g
gb
deq
~o
TI
Symbols
Initial velocity vector at r/
nodal point r/~
Velocity corrections of
nodal values 0
Virtual velocity x
Weight factors
x, y, z-Coordinates of 3,
trth node
Die semi-angle
Deceleration coefficient d3,
Coupling coefficient in
temperature calculation
Viscosity coefficients #
Radial displacement in v
bore expanding
Radial displacement in ~
flange drawing
Kronecker delta ~r
Emissivity
Effective strain Jr
Effective strain of base 6~
metal in porous materials 6~(,
Effective strain value at &to
node N
Strain-rate 6~tp
Infinitesimal strain 6~rs~
Volumetric strain-rate
Plastic strain-rate 6~
Elastic strain-rate
Effective strain-rate
6~rs,
Effective strain-rate of
base metal in porous
materials P
Apparent effective strain- Po
rate of porous materials Pb
Limiting strain-rate
Pd
Natural coordinate
PR
t-Coordinate of trth
node p~
Function of relative
density in porous o
materials
Natural coordinate
r/-Coordinate of orth
node
Angle
Heat generation
efficiency factor
Lagrangian multiplier
Proportionality factor
(rate) in flow rules
Proportionality factor
(infinitesimal) in flow
rules
Coefficient of friction
Poisson's ratio
Natural coordinate
~-Coordinate of o:th
node
Plane of zero mean stress
in stress space
Functional
Variation of functional
6at-value at jth element
Term due to deformation
energy-rate in 6~r
Penalty term in 6~r
Term due to traction in
&r
Term that includes
Lagrangian multiplier in
&r
Term due to friction in
&r
Density
Initial density
Density of base metal in
porous materials
Density of die material
Apparent density of
porous materials
Density of void in porous
materials
Stephan-Boltzman
constant

8. Cd
Cb
c~
CR
d
e
f
f
f(o,~)
g
g(a,j)
h
h~
hlub
h
h(oo)
k
k,
kl
k~
k~
!
l
lo
Symbols
Specific heat of die t~
material m
Specific heat of base m
metal in porous materials
m
Specific heat of void in
n
porous materials
Apparent specific heat of
n
porous materials
n
Punch depth in sheet-
metal forming
Engineering strain P
Engineering strain-rate P°
Coefficient of anisotropy Po
Frictional stress P#
Nodal-point force vector
ql
Yield function
Coefficient of anisotropy q.
Scalar function of stress
invariants q~
Heat transfer coefficient r
Heat transfer coefficient rx, r4s, ry
at tool-workpiece
contact surface
Heat transfer coefficient i
of lubricant
Sq
Coefficient of anisotropy
Scalar function of stress t
invariants
At
Shear yield stress ui
Apparent shear yield u~=)
stress of porous materials
Thermal conductivity
Uo
Apparent thermal
conductivity of porous u,
meterials u.
Thermal conductivity of
base metal in porous ut
materials
Gage length in tensile Au
test v,
Coefficient of anisotropy
Initial gage length in v
tensile test
XV
Unit tangent vector
Friction factor
Strain-rate exponent
Coefficient of anisotropy
Strain-hardening
exponent
Coefficient of anisotropy
Unit normal to the
surface
Pressure
Average pressure
Die pressure in drawing
First Piola-Kirchhoff
stress
Heat generated through
friction
Heat flux across surface
s~
Shape functions
r-Value in sheet forming
r-Values in the rolling,
45°, and transverse
directions, respectively
Heat generation-rate
Second Piola-Kirchhoff
stress
Time
Time-increment
Velocity component
Velocity component at
the ¢~thnode
Initial velocity
Relative sliding velocity
Velodty component
normal to a surface
Velocity component
tangent to a surface
Velocity discontinuity
Relative sliding velocity
at nodal point
Velocity vector at nodal
point

9. Symbols xvii
0
Gm
Cauchy stress
Deviatoric stress
Effective stress, flow
stress
Mean stress
Kirchhoff stress
Shear traction in Hill's
method
,/,(z)
~(F)
tbo
Bulgefunction in simple
compression
Strain-rate sensitivity
function
Rate of rotation