Theory of Thermomechanical Processes in Welding

173 Pages·2005·1.48 MB·English

by  by Andrzej Sluzalec

THEORY OF THERMOMECHANICAL PROCESSES IN WELDING
Theory of Thermomechanical
Processes in Welding
by
Andrzej Służalec
Technical University of Czestochowa,
Poland
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-2990-X (HB)
ISBN 1-4020-2991-8 (e-book)
Published by Springer,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
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©2005 Springer
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Printed in the Netherlands.
Contents
Preface.......................................................................................................ix
Part I FUNDAMENTALS OF WELDING THERMOMECHANICS
Chapter 1. Description of Welding Deformations.................................3
1.1 Introduction....................................................................................3
1.2 The Referential and Spatial Description........................................3
1.3 The Deformation Gradient.............................................................4
1.4 Strain Tensors................................................................................7
1.5 Particulate and Material Derivatives............................................12
1.6 Mass Conservation.......................................................................14
Chapter 2. Stress Tensor.......................................................................17
2.1 Momentum...................................................................................17
2.2 The Stress Vector.........................................................................18
2.3 Momentum Balance.....................................................................21
2.4 Properties of the Stress Tensor....................................................24
2.5 The Virtual Work Rate.................................................................26
2.6 The Piola-Kirchhoff Stress Tensor..............................................28
2.7 The Kinetic Energy Theorem.......................................................29
Chapter 3. Thermodynamical Background of Welding Processes....31
3.1 Introduction..................................................................................31
3.2 The First Law of Thermodynamics..............................................32
3.3 The Energy Equation...................................................................33
3.4 The Second Law of Thermodynamics.........................................35
vi Theory of Thermomechanical Processes in Welding
3.5 Dissipations..................................................................................38
3.6 Equations of State for an Elementary System..............................40
3.7 The Heat Conduction Law...........................................................42
3.8 Thermal Behaviour......................................................................44
3.9 Heat Transfer in Cartesian Coordinates.......................................45
3.10 Heat Convection........................................................................46
3.11 Heat Radiation...........................................................................47
3.12 Initial and Boundary Conditions................................................48
Chapter 4. Motion of Fluids..................................................................51
4.1 Viscous Fluids..............................................................................51
4.2 Navier-Stokes Equation...............................................................54
Chapter 5. Thermomechanical Behaviour..........................................57
5.1 Thermoelasticity..........................................................................57
5.2 Plastic Strain................................................................................58
5.3 State Equations............................................................................59
5.4 Plasticity Criterion.......................................................................62
5.5 The Plastic Flow Rule..................................................................66
5.6 Thermal Hardening......................................................................71
5.7 The Hypothesis of Maximal Plastic Work...................................73
5.8 The Associated Flow Rule...........................................................74
5.9 Incremental Formulation for Thermal Hardening........................75
5.10 Models of Plasticity...................................................................76
Part II NUMERICAL ANALYSIS OF WELDING PROBLEMS
Chapter 6. Numerical Methods in Thermomechanics........................83
6.1 Introduction..................................................................................83
6.2 Finite-Element Solution of Heat Flow Equations........................83
6.2.1 Weighted Residual Method.................................................83
6.2.2 Variational Formulation......................................................87
6.3 Finite-Element Solution of Navier-Stokes Equations..................89
6.4 Time Discretization......................................................................92
6.5 Time Integration Schemes for Nonlinear Heat Conduction.........92
6.6 Solution Procedure for Navier-Stokes Equation..........................96
6.7 Modeling of the Phase Change Process.......................................97
6.8 The Theorem of Virtual Work in Finite Increments....................99
6.9 The Thermo-Elasto-Plastic Finite Element Model....................100
6.10 Solution Procedure for Thermo-Elasto-Plastic Problems........101
Contents vii
Part III HEAT FLOW IN WELDING
Chapter 7. Analytical Solutions of Thermal Problems in Welding.105
7.1 Introduction................................................................................105
7.2 Heat Flow in Spot Welding........................................................106
7.3 Temperature Distribution in a Thin Plate..................................109
7.4 Temperature Distributions in an Infinitely Thick Plate.............116
7.5 Heat Flow in Friction Welding..................................................118
7.5.1 Thermal Effects in Friction Welding................................118
7.5.2 Analytical Solutions..........................................................120
Chapter 8. Numerical Solutions of Thermal Problems in Welding129
8.1 Introduction................................................................................129
8.2 Temperature Distributions in Laser Microwelding....................129
8.3 Temperature Distribution in Electroslag Welding.....................141
Part IV WELDING STRESSES AND DEFORMATIONS
Chapter 9 Thermal Stresses in Welding............................................147
9.1 Changes of Stresses During Welding........................................147
9.2 Residual Stresses........................................................................149
9.2.1 Two-Dimensional Case.....................................................149
9.2.2 Plug Weld.........................................................................151
9.2.3 Circular Patch Weld..........................................................152
9.2.4 Welded Structures.............................................................153
Chapter 10 Welding Deformations.....................................................157
10.1 Distortion.................................................................................157
10.2 Deformations in Friction Welding ..........................................159
References and Further Reading..........................................................165
Subject Index..........................................................................................171
Preface
The main purpose of this book is to provide a unified and systematic
continuum approach to engineers and applied physicists working on models
of deformable welding material. The key concept is to consider the welding
material as an thermodynamic system.
Significant achievements include thermodynamics, plasticity, fluid flow
and numerical methods.
Having chosen point of view, this work does not intend to reunite all the
information on the welding thermomechanics. The attention is focused on
the deformation of welding material and its coupling with thermal effects.
Welding is the process where the interrelation of temperature and
deformation appears throughout the influence of thermal field on material
properties and modification of the extent of plastic zones. Thermal effects
can be studied with coupled or uncoupled theories of thermomechanical
response. A majority of welding problems can be satisfactorily studied
within an uncoupled theory. In such an approach the temperature enters the
stress-strain relation through the thermal dilatation and influences the
material constants. The heat conduction equation and the relations governing
the stress field are considered separately.
In welding a material is either in solid or in solid and liquid states. The
flow of metal and solidification phenomena make the welding process very
complex. The automobile, aircraft, nuclear and ship industries are
experiencing a rapidly-growing need for tools to handle welding problems.
The effective solutions of complex problems in welding became possible
in the last two decades, because of the vigorous development of numerical
methods for thermal and mechanical analysis.
The book has been divided into four parts. Part I Fundamentals of
Welding Thermomechanics is devoted to the description of the deformation
of the welding material. Both the Lagrangian and the Eulerian standpoints
x Theory of Thermomechanical Processes in Welding
are considered. The concept of stress is introduced. The derivations of the
theorem of virtual work rate and the kinetic energy theorem are provided. A
general thermodynamic framework for the formulation of all constitutive
equations is given. In this framework the laws governing the heat flow are
formulated. The background of plasticity in welding is introduced.
Hardening and softening phenomena are refined in the global context of
thermodynamics and the modeling of thermal hardening in welding is
discussed. In Part II Numerical Solutions of Welding Problems all the
constitutive models of the previous chapters are discussed extending well-
established standard procedures for solid and liquid media. The numerical
approach to thermo-elastic-plastic material model is discussed. The phase-
change problems are analyzed. The numerical solution of fluid flow in
welding is presented. Part III Heat Flow in Welding presents in details the
solution of heat transfer equations under conditions of interest in welding.
The restrictive assumptions will however limit the practical utility of the
results. Nevertheless the results are useful in that they emphasize the
variables involved and approximate the way in which they are related. In
addition, such solutions provide the background for understanding more
complicated solutions obtained numerically and provide guidance in making
judgements. The numerical solutions of welding problems are focused on
laser and electroslag welding. In Part IV Welding Stresses and
Deformations coupled thermomechanical phenomena are discussed. First of
all thermal stresses being the result of complex temperature changes and
plastic strains in regions near the weld are analyzed. Deformations and
residual stresses in welding structures are considered for chosen welding
processes and geometry of elements.
Cz(cid:266)stochowa Andrzej S(cid:225)u(cid:298)alec
May 2004
Part I
FUNDAMENTALS OF WELDING
THERMOMECHANICS