Theory of Metal Forming Plasticity: Classical and Advanced Topics

272 Pages·2004·4.41 MB·English

by  Professor Dr. Andrzej Służalec (auth.)

Andrzej Sluzalec
Theory of Metal Forming Plasticity
ONLINE LIBRARY
http://www.springer.de/engine/
Springer-Verlag Berlin Heidelberg GmbH
Andrzej Sluzalec
Theory of
Metal Forming Plasticity
Classical and Advanced Topics
With 95 Figures
i
Springer
Professor Dr. Andrzej Sluialec
Technical University of Czestochowa
ul. Akademicka 3
42-200 Czestochowa
Poland
Library of Congress Cataloging-in-Publieation Data
Sluzalee, Andrzej.
Theory of metal forming plastieity: classieal and advanced topies I Andrzej Sluzalee.
p.em.
Includes bibliographieal referenees and index.
ISBN 978-3-642-07370-0 ISBN 978-3-662-10449-1 (eBook)
DOI 10.1007/978-3-662-10449-1
1. Metal-work. 2. Metals--Plastie properties. I. Title.
TS205.S565 2003
671--de22 2003059107
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Originally published by Springer-Verlag Berlin Heidelberg New Yark 2004
Softcover reprint of the hardcover lst edition 2004
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Preface
The intention of this book is to reveal and discuss some aspects of the metal form-
ing plasticity theory. The modern theory describes deformation of metallic bodies
in cold and hot regimes under combined thermal and mechanical loadings. Ther-
mal and deformation fields appear in metal forming in various forms. A thermal
field influences the material properties, modifies the extent of plastic zones, etc.
and the deformation of metallic body induces changes in temperature distribution.
The thermal effects in metal forming plasticity can be studied at two levels, de-
pending on whether uncoupled or coupled theories of thermo-plastic response
have to be applied. A majority of metal forming processes can be satisfactorily
studied within an uncoupled theory. In such an approach the temperature enters
the stress-strain relation through the material constants and through the thermal
dilatation. The description of thermo-plastic deformation in metal forming is car-
ried out on the ground of thermodynamics.
The effective solutions of metal forming problems have become possible only
recently. In the last two decades one observes the vigorous development of effec-
tive nonlinear methods in computational plasticity. Manufacturing industry is ex-
periencing a rapidly-growing need for the analytical tools to handle complex prob-
lems of metal forming. Efficient analytical methods for combining geometrically,
materially and thermally nonlinear problems are needed because experimental
testing in such cases is often prohibitively expensive or physically impossible. In
the book the theory of numerical solutions for metal forming plasticity is dis-
cussed because recent advances in computational plasticity make it possible to
perform extensive calculations with great accuracy, at significantly reduced execu-
tion times and at reasonable cost.
The theory of sensitivity discussed in the book originates form purely mathe-
matical studies of the influence of coefficient variations on differential equations.
It was much later that the theory of sensitivity became the subject of studies in the
field of metal forming plasticity. In metal forming a set of equations that defines
the relationship between external loads, prescribed displacements, stress, etc. in
metallic body are considered. A sensitivity of metal forming process to variations
of its parameters is one of the most important aspects necessary for a proper un-
derstanding of the process.
Stochastic methods considered in the book have recently become an area of re-
search in metal forming plasticity. As the name suggests, these methods combine
two crucial methodologies developed to deal with problems of metal forming:
analytical or numerical analysis with the stochastic one. The stochastic analysis in
the broadest sense refers to the explicit treatment of uncertainty in any quantity
vi Preface
entering the corresponding deterministic analysis. The exact values of these quan-
tities are usually unknown because they cannot be precisely measured. Stochastic
approach to metal forming problems is important not only because of random ma-
terial parameters, but particularly because of boundary problems appearing in
these processes. Contact problems die-workpiece have exceptional random char-
acter and lead to determine the boundary forces in the contact die-workpiece con-
sidering random character of friction between them. Existing uncertain variations
in parameters may have significant effects on such fundamental final characteris-
tics as strain and stress distributions, and they must affect the final design.
Standard analytical solutions evaluating plastic deformation have been included
in the book. It describes the deformation in the simple way, which sometimes is
very helpful.
This book is divided into eleven parts. It contains: deformation of metallic body
(Part I), metal forming thermodynamics (Part II), plasticity (Part III), powder
forming plasticity (Part IV), viscoplasticity (Part V), discontinuous fields
(Part VI), numerical solution methods (Part VII), sensitivity in metal forming plas-
ticity (Part VIII), stochastic metal forming process (Part IX), contact and friction
(Part X) and simplified equations (Part XI).
Part VIII and Part IX provide new elements to the theory of metal forming plas-
ticity not met yet in literature of the subject. The remaining parts are added to de-
scribe the theory comprehensively.
I would like to make an acknowledgement to Malgorzata Kowalik and my son
Tom for their involvement in preparing the manuscript in its camera ready form.

Cz(cid:2)stochowa Andrzej S(cid:3)u(cid:4)alec
April 2003
Contents

Part I Deformation of Metallic Body
1 Description of Deformation...............................................................................3
1.1 Introduction..................................................................................................3
1.2 Description of Motion..................................................................................3
1.3 The Deformation Gradient...........................................................................3
1.4 The Polar Decomposition Theorem.............................................................4
1.5 Local Description of Deformation...............................................................4
1.6 The Green-Lagrange Strain Tensor..............................................................6
1.7 The Logarithmic Strain Tensor....................................................................7
1.8 Infinitesimal Transformation.......................................................................7
1.9 Lagrangian and Eulerian Strain Rate Tensors..............................................8
1.10 Particulate and Material Derivatives........................................................10
1.11 Mass Conservation...................................................................................12
2 Stress Tensor.....................................................................................................14
2.1 Momentum Balance...................................................................................14
2.2 The Cauchy Stress Tensor.........................................................................16
2.3 The Virtual Work Rate..............................................................................18
2.4 The Piola-Kirchhoff Stress Tensor............................................................19
2.5 The Kinetic Energy Theorem.....................................................................20
3 Components of Stress and Strain Tensors......................................................22
3.1 Components of the Green-Lagrange Strain Tensor....................................22
3.2 Components of Stress and Strain in Infinitesimal Transformation............24
3.2.1 Stress and Strain Tensors in Cartesian Coordinates...........................24
3.2.2 Stress and Strain Tensors in Cylindrical Coordinates........................26
3.2.3 Stress and Strain Tensors in Spherical Coordinates...........................28

Part II Metal Forming Thermodynamics
4 Thermodynamical Considerations..................................................................33
4.1 Introduction................................................................................................33
v i i i Contents
4.2 The Local State Postulate..........................................................................34
4.3 The First Law of Thermodynamics...........................................................35
4.4 The Energy Equation.................................................................................36
4.5 The Second Law of Thermodynamics.......................................................37
4.6 Dissipations...............................................................................................40
4.7 Equations of State for an Elementary System............................................41
4.8 The Heat Conduction Law.........................................................................43
4.9 Equations of the Thermodynamical Process..............................................45
5 Temperature Field in Material........................................................................47
5.1 Introduction...............................................................................................47
5.2 Thermal Behaviour....................................................................................47
5.3 Heat Transfer in Cartesian Coordinates.....................................................48
5.4 The Heat Conduction Equation..................................................................49
5.5 Heat Convection........................................................................................52
5.6 Heat Radiation...........................................................................................52
5.7 Initial and Boundary Conditions................................................................53
5.8 Thermomechanical Behaviour...................................................................54

Part III Plasticity
6 Plastic Behaviour of Material..........................................................................59
6.1 Introduction...............................................................................................59
6.2 Plastic Strain..............................................................................................61
6.3 State Equations..........................................................................................61
6.4 Evolution Relations...................................................................................64
6.4.1 The Plastic Work Rate........................................................................64
6.4.2 Plasticity Criterion..............................................................................65
6.4.3 The Plastic Flow Rule........................................................................69
6.4.4 Thermal Hardening............................................................................73
6.5 Plastic Behaviours.....................................................................................74
6.5.1 The Hypothesis of Maximal Plastic Work.........................................75
6.5.2 The Associated Flow Rule.................................................................76
6.5.3 Stability..............................................................................................77
6.5.4 The Drucker Stability Postulate ........................................................84
6.6 The Non-associated Flow Rule..................................................................84
6.7 Incremental Formulation...........................................................................86
6.8 Incremental Formulation for Thermal Hardening......................................92
6.9 Models of Plasticity...................................................................................93
6.9.1 The Isotropic Model...........................................................................94
6.9.2 Loading Functions..............................................................................98
6.9.3 The Flow Rule..................................................................................105
6.9.4 Hardening.........................................................................................107
Contents ix

Part IV Powder Forming Plasticity
7 Description of Powder Material....................................................................111
7.1 Introduction..............................................................................................111
7.2 Infinitesimal Transformation...................................................................111
7.3 Mass Conservation...................................................................................112
7.4 Momentum Balance.................................................................................114
7.5 Physical Laws..........................................................................................115
7.6 Plastic Porosity........................................................................................116
8 State Equations...............................................................................................119
8.1 The Poroplastic Flow Rule.......................................................................120
8.2 The Associated Flow Rule.......................................................................122
8.3 The Non-associated Flow Rule................................................................123
8.4 Plasticity Models......................................................................................125

Part V Viscoplasticity
9 Viscoplastic Behaviour...................................................................................131
9.1 Introduction..............................................................................................131
9.2 Dissipation Potential................................................................................131
9.3 State Equations........................................................................................132
9.4 Evolution Laws........................................................................................133
9.5 Viscoplastic Material...............................................................................134
9.6 Stability in Viscoplasticity.......................................................................135
9.7 Viscoplastic Models.................................................................................135

Part VI Discontinuous Fields
10 Surfaces of Discontinuity.............................................................................139
10.1 The Jump Operator................................................................................139
10.2 Discontinuity of Stress...........................................................................139
10.3 Discontinuity of Velocity.......................................................................143
10.4 The Virtual Work Theorem for Discontinuous Material.......................144
10.5 Strain in the Discontinuity Field............................................................146
10.6 Models of Plasticity with Discontinuities..............................................148
10.7 Hardening Materials..............................................................................150
x Contents

Part VII Numerical Solution Methods
11 Numerical Solutions.....................................................................................155
11.1 Introduction...........................................................................................155
11.2 Uniqueness of Solution..........................................................................155
11.3 Time Discretization...............................................................................157
11.4 The Theorem of Virtual Work...............................................................158
11.5 Variational Formulation.........................................................................159
11.6 Geometrical Interpretation.....................................................................162
11.7 Convergence..........................................................................................165
12 Numerical Models of Plasticity....................................................................166
12.1 The Thermo-Elasto-Plastic Finite Element Model................................166
12.2 The Rigid-Viscoplastic Finite Element Model......................................167
12.3 The Rigid-Poroplastic Finite Element Model........................................167
12.4 The Finite Differences Model................................................................168

Part VIII Sensitivity in Metal Forming Plasticity
13 Sensitivity......................................................................................................171
13.1 Introduction...........................................................................................171
13.2 Notations and Terminology...................................................................171
13.3 Discretized Equation of Rigid-Viscoplasticity......................................173
13.4 Continuous Formulation for Rigid-Viscoplasticity................................176
13.5 Adjoint System Method for Rigid-Viscoplasticity................................178
13.6 Shape Sensitivity...................................................................................180
13.6.1 The Control Volume Approach......................................................180
13.6.2 Design Sensitivity Analysis............................................................181
13.6.3 Variation of Variables....................................................................182
13.6.4 Sensitivities in the Virtual Work Theorem.....................................183
13.7 Sensitivity in Elasto-Plasticity...............................................................184

Part IX Stochastic Metal Forming Process
14 Stochastic Viscoplasticity.............................................................................189
14.1 Introduction...........................................................................................189
14.2 The Stochastic Virtual Energy Principle................................................190
14.3 Discretized Random Variable................................................................194
14.4 The Numerical Stochastic Rigid - Viscoplastic Model..........................196