Grong - Metallurgical Modelling of Welding

METALLURGICAL MODELLING

OF WELDING

ALSO PUBLISHED BY THE INSTITUTE OF MATERIALS

Mathematical Modelling of Weld PhenomenaEdited by H. Cerjak and K. E. Easterling

Book 533 ISBN 0 90171616 2. 1992

Mathematical Modelling of Weld Phenomena 2Edited by H. Cerjak and H. K. D. H. Bhadeshia

Book 594 ISBN 0 901716 63 4. 1994

Mathematical Modelling of Weld Phenomena 3Edited by H. Cerjak and H. K. D. H. Bhadeshia

Book 650 ISBN 1 86125 010 X

MetallurgicalModellingof WeldingSECOND EDITION

0YSTEIN GRONG

Norwegian University ofScience and Technology,

Department of Metallurgy,N-7034 Trondheim, Norway

MATERIALS MODELLING SERIES

Editor: H. K. D. H. BhadeshiaThe University of Cambridge

Department of Materials Scienceand Metallurgy

THE INSTITUTE OF MATERIALS

Book 677First published in 1997 byThe Institute of Materials1 Carlton House TerraceLondon SWIY 5DB

First edition (Book 557)Published in 1994

The Institute of Materials 1997All rights reserved

ISBN 1 86125 036 3

Originally typeset byPicA Publishing Services

Additional typesetting and corrections byFakenham Photosetting Ltd

Printed and bound in the UK atThe University Press, Cambridge

TO TORHILD, TORBJ0RN AND HAvARD

without your support, this book would never have been finished.

Contents

Preface to the second edition xiiiPreface to the first edition xiv

Chapter 1: Heat Flow and Temperature Distribution in Welding 11.1 Introduction................................................. 11.2 Non-Steady Heat Conduction 11.3 Thermal Properties of Some Metals and Alloys p 21.4 Instantaneous Heat Sources ~ 41.5 Local Fusion in Arc Strikes 71.6 Spot Welding 101.7 Thermit Welding.... 141.8 Friction Welding '181.9 Moving Heat Sources and Pseudo-Steady State 241.10 Arc Welding 24

1.10.1 Arc efficiency factors 261.10.2 Thick plate solutions ; 26

1.10.2.1 Transient heating period 281.10.2.2 Pseudo-steady state temperature distribution 311.10.2.3 Simplified solution for a fast-moving high power source 41

1.10.3 Thin plate solutions 451.10.3.1 Transient heating period 481.10.3.2 Pseudo-steady state temperature distribution 491.10.3.3 Simplified solution for a fast moving high power source 56

1.10.4 Medium thick plate solution 591.10.4.1 Dimensionless maps for heat flow analyses 611.10.4.2 Experimental verification of the medium thick plate solution 721.10.4.3 Practical implications " 75

1.10.5 Distributed heat sources 771.10.5.1 General solution 771.10.5.2 Simplified solution 80

1.10.6 Thermal conditions during interrupted welding 911.10.7 Thermal conditions during root pass welding 0- 951.10.8 Semi-empirical methods for assessment of bead morphology 96

1.10.8.1 Amounts of deposit and fused parent metal 961.10.8.2 Bead penetration , 99

1.10.9 Local preheating 100References 103Appendix 1.1: Nomenclature...................................................................................... 105Appendix 1.2: Refined Heat Flow Model for Spot Welding 110Appendix 1.3: The Gaussian Error Function . 111Appendix 1.4: Gaussian Heat Distribution 112

CONTENTS vii

Chapter 2: Chemical Reactions in Arc Welding 1162.1 Introduction.................................................................................................................. 1162.2 Overall Reaction Model 1162.3 Dissociation of Gases in the Arc Column 1172.4 Kinetics of Gas Absorption 120

2.4.1 Thin film model 1202.4.2 Rate of element absorption..... 121

2.5 The Concept of Pseudo-Equilibrium 1222.6 Kinetics of Gas Desorption 123

2.6.1 Rate of element desorption 1232.6.2 Sievert's law 124

2.7 Overall Kinetic Model for Mass Transfer during Cooling in the Weld Pool........ 1242.8 Absorption of Hydrogen 128

2.8.1 Sources of hydrogen 1282.8.2 Methods of hydrogen determination in steel welds 1282.8.3 Reaction model 1302.8.4 Comparison between measured and predicted hydrogen contents 131

2.8.4.1 Gas-shielded welding 1312.8.4.2 Covered electrodes 1342.8.4.3 Submerged arc welding 1382.8.4.4 Implications of Sievert's law 1402.8.4.5 Hydrogen in multi-run weldments 1402.8.4.6 Hydrogen in non-ferrous weldments 141

2.9 Absorption of Nitrogen 1412.9.1 Sources of nitrogen ~ 1422.9.2 Gas-shielded welding 1422.9.3 Covered electrodes 1432.9.4 Submerged arc welding 146

2.10 Absorption of Oxygen 1482.10.1 Gas metal arc welding 148

2.10.1.1 Sampling of metal concentrations at elevated temperatures 1492.10.1.2 Oxidation of carbon 1492.10.1.3 Oxidation of silicon 1522.10.1.4 Evaporation of manganese 1562.10.1.5 Transient concentrations of oxygen 1602.10.1.6 Classification of shielding gases 1662.10.1.7 Overall oxygen balance 1662.10.1.8 Effects of welding parameters 169

2.10.2 Submerged arc welding 1702.10.2.1 Flux basicity index 1712.10.2.2 Transient oxygen concentrations 172

2.10.3 Covered electrodes 1732.10.3.1 Reaction model 1742.10.3.2 Absorption of carbon and oxygen 1762.10.3.3 Losses of silicon and manganese 1772.10.3.4 The product [%C] [%0] 179

2.11 Weld Pool Deoxidation Reactions. 1802.11.1 Nucleation of oxide inclusions.. 182

viii CONTENTS

2.11.2 Growth and separation of oxide inclusions 1842.11.2.1 Buoyancy (Stokes flotation) 1852.11.2.2 Fluid flow pattern 1862.11.2.3 Separation model........................................................................................................ 188

2.11.3 Predictions of retained oxygen in the weld metal 1902.11.3.1 Thermodynamic model 1902.11.3.2 Implications of model 192

2.12 Non-Metallic Inclusions in Steel Weld Metals 1922.12.1 Volume fraction of inclusions 1932.12.2 Size distribution of inclusions 195

2.12.2.1 Effect of heat input 1962.12.2.2 Coarsening mechanism 1962.12.2.3 Proposed deoxidation model 201

2.12.3 Constituent elements and phases in inclusions 2022.12.3.1 Aluminium, silicon and manganese contents 2022.12.3.2 Copper and sulphur contents 2022.12.3.3 Titanium and nitrogen contents 2032.12.3.4 Constituent phases 204

2.12.4 Prediction of inclusion composition 2042.12.4.1 C-Mn steel weld metals 2042.12.4.2 Low-alloy steel weld metals 206

References 212Appendix 2.1: Nomenclature 215Appendix 2.2: Derivation of equation (2-60) 219

Chapter 3: Solidification Behaviour of Fusion Welds .................................................. 2213.1 Introduction 2213.2 Structural Zones in Castings and Welds 2213.3 Epitaxial Solidification 222

3.3.1 Energy barrier to nucleation 2253.3.2 Implications of epitaxial solidification 226

3.4 Weld Pool Shape and Columnar Grain Structures 2283.4.1 Weld pool geometry 2283.4.2 Columnar grain morphology 2293.4.3 Growth rate of columnar grains 230

3.4.3.1 Nominal crystal growth rate 2303.4.3.2 Local crystal growth rate 234

3.4.4 Reorientation of columnar grains 2393.4.4.1 Bowing of crystals 2403.4.4.2 Renucleation of crystals 242

3.5 Solidification Microstructures 2513.5.1 Substructure characteristics 2513.5.2 Stability of the solidification front 254

3.5.2.1 Interface stability criterion 2543.5.2.2 Factors affecting the interface stability 256

3.5.3 Dendrite morphology 2603.5.3.1 Dendrite tip radius 2603.5.3.2 Primary dendrite arm spacing 261

CONTENTS ix

3.5.3.3 Secondary dendrite arm spacing 2643.6 Equiaxed Dendritic Growth 268

3.6.1 Columnar to equiaxed transition 2683.6.2 Nucleation mechanisms --272

3.7 Solute Redistribution 2723.7.1 Microsegregation 2723.7.2 Macrosegregation 2783.7.3 Gas porosity 279

3.7.3.1 Nucleation of gas bubbles 2793.7.3.2 Growth and detachment of gas bubbles 2813.7.3.3 Separation of gas bubbles 284

3.7.4 Removal of microsegregations during cooling . 2863.7.4.1 Diffusion model 2863.7.4.2 Application to continuous cooling 286

3.8 Peritectic Solidification 2903.8.1 Primary precipitation of the 'Yp -phase 2903.8.2 Transformation behaviour of low-alloy steel weld metals 290

3.8.2.1 Primary precipitation of delta ferrite 2903.8.2.2 Primary precipitation of austenite ~ 2923.8.2.3 Primary precipitation of both delta ferrite and austenite 292

References 293Appendix 3.1: Nomenclature 296

Chapter 4: Precipitate Stability in Welds 3014.1 Introduction 3014.2 The Solubility Product ., 301

4.2.1 Thermodynamic background 3014.2.2 Equilibrium dissolution temperature 3034.2.3 Stable and metastable solvus boundaries 304

4.2.3.1 Equilibrium precipitates 3044.2.3.2 Metastable precipitates 308

4.3 Particle Coarsening 3144.3.1 Coarsening kinetics 3144.3.2 Application to continuous heating and cooling 314

4.3.2.1 Kinetic strength of thermal cycle 3154.3.2.2 Model limitations 315

4.4 Particle Dissolution 3164.4.1 Analytical solutions 316

4.4.1.1 The invariant size approximation ;..- 3194.4.1.2 Application to continuous heating and cooling 322

4.4.2 Numerical solution 3254.4.2.1 Two-dimensional-diffusion model 3264.4.2.2 Generic model 3284.4.2.3 Application to continuous heating and cooling 3294.4.2.4 Process-diagrams for single pass 6082- T6 butt welds 332


x CONTENTS

Chapter 5: Grain Growth in Welds 3375.1 Introduction 3375.2 Factors Affecting the Grain Boundary Mobility 337

5.2.1 Characterisation of grain structures 3375.2.2 Driving pressure for grain growth 3395.2.3 Drag from impurity elements in solid solution 3405.2.4 Drag from a random particle distribution 3415.2.5 Combined effect of impurities and particles 342

5.3 Analytical Modelling of Normal Grain Growth 3435.3.1 Limiting grain size 3435.3.2 Grain boundary mobility 3455.3.3 Grain growth mechanisms 345

5.3.3.1 Generic grain growth model 3455.3.3.2 Grain growth in the absence of pinning precipitates 3475.3.3.3 Grain growth in the presence of stable precipitates 3485.3.3.4 Grain growth in the presence of growing precipitates 3515.3.3.5 Grain growth in the presence of dissolving precipitates 356

5.4 Grain Growth Diagrams for Steel Welding 3605.4.1 Construction of diagrams 360

5.4.1.1 Heat flow models 3605.4.1.2 Grain growth model 3615.4.1.3 Calibration procedure 3615.4.1.4 Axes and features of diagrams 363

5.4.2 Case studies 3645.4.2.1 Titanium-microalloyed steels 3645.4.2.2 Niobium-microalloyed steels 3675.4.2.3 C-Mn steel weld metals 3705.4.2.4 Cr-Mo low-alloy steels 3725.4.2.5 Type 316 austenitic stainless steels 375

5.5 Computer Simulation of Grain Growth 3805.5.1 Grain growth in the presence of a temperature gradient 3805.5.2 Free surface effects 382


Chapter 6: Solid State Transformations in Welds 3876.1 Introduction 3876.2 Transformation Kinetics 387

6.2.1 Driving force for transformation reactions 3876.2.2 Heterogeneous nucleation in solids 389

6.2.2.1 Rate of heterogeneous nucleation 3896.2.2.2 Determination of ~G~et. and Qd 3906.2.2.3 Mathematical description of the C-curve 392

6.2.3 Growth of precipitates 3966.2.3.1 Interface-controlled growth 3966.2.3.2 Diffusion-controlled growth 397

6.2.4 Overall transformation kinetics 4006.2.4.1 Constant nucleation and growth rates 400

CONTENTS xi

6.2.4.2 Site saturation 4026.2.5 Non-isothermal transformations 402

6.2.5.1 The principles of additivity 4036.2.5.2 Isokinetic reactions 4046.2.5.3 Additivity in relation to the Avrami equation 4046.2.5.4 Non-additive reactions 405

6.3 High Strength Low-Alloy Steels 4066.3.1 Classification of microstructures 4066.3.2 Currently used nomenclature 4066.3.3 Grain boundary ferrite 408

6.3.3.1 Crystallography of grain boundary ferrite 4086.3.3.2 Nucleation of grain boundary ferrite 4086.3.3.3 Growth of grain boundary ferrite 422

6.3.4 Widmanstatten ferrite 4276.3.5 Acicular ferrite in steel weld deposits 428

6.3.5.1 Crystallography of acicular ferrite 4286.3.5.2 Texture components of acicular ferrite 4296.3.5.3 Nature of acicular ferrite 4306.3.5.4 Nucleation and growth of acicular ferrite 432

6.3.6 Acicular ferrite in wrought steels 4446.3.7 Bainite 444

6.3.7.1 Upper bainite 4446.3.7.2 Lower bainite 447

6.3.8 Martensite 4486.3.8.1 Lath martensite 4486.3.8.2 Plate (twinned) martensite 448

6.4 Austenitic Stainless Steels 4536.4.1 Kinetics of chromium carbide formation 4566.4.2 Area of weld decay 456

6.5 Al-Mg-Si Alloys 4586.5.1 Quench-sensitivity in relation to welding 459

6.5.1.1 Conditions for ~'(Mg2Si) precipitation during cooling 4596.5.1.2 Strength recovery during natural ageing 461

6.5.2 Subgrain evolution during continuous drive friction welding 464References 467Appendix 6.1: Nomenclature 471Appendix 6.2: Additivity in relation to the Avrami Equation 475

Chapter 7: Properties ofWeldments 4777.1 Introduction 4777.2 Low-Alloy Steel Weldments 477

7.2.1 Weld metal mechanical properties 4777.2.1.1 Weld metal strength level 4787.2.1.2 Weld metal resistance to ductile fracture 4807.2.1.3 Weld metal resistance to cleavage fracture 4857.2.1.4 The weld metal ductile to brittle transition 4867.2.1.5 Effects of reheating on weld metal toughness 491

7.2.2 HAZ mechanical properties 494

xii CONTENTS

7.2.2.1 HAZ hardness and strength level 4957.2.2.2 Tempering of the heat affected zone 5007.2.2.3 HAZ toughness 502

7.2.3 Hydrogen cracking 5097.2.3.1 Mechanisms of hydrogen cracking 5097.2.3.2 Solubility of hydrogen in steel 5137.2.3.3 Diffusivity of hydrogen in steel 5147.2.3.4 Diffusion of hydrogen in welds 5147.2.3.5 Factors affecting the HAZ cracking resistance 518

7.2.4 H2S stress corrosion cracking 5247.2.4.1 Threshold stress for cracking 5247.2.4.2 Prediction of HAZ cracking resistance 525

7.3 Stainless Steel Weldments 5277.3.1 HAZ corrosion resistance 5277.3.2 HAZ strength level 5297.3.3 HAZ toughness 5307.3.4 Solidification cracking 532

7.4 Aluminium Weldments 5367.4.1 Solidification cracking 5367.4.2 Hot cracking 540

7.4.2.1 Constitutionalliquation in binary AI-Si alloys 541.7.4.2.2 Constitutionalliquation in ternary AI-Mg-Si alloys 5427.4.2.3 Factors affecting the hot cracking susceptibility 544

7.4.3 HAZ microstructure and strength evolution during fusion welding 5477.4.3.1 Effects of reheating on weld properties 5477.4.3.2 Strengthening mechanisms in AI-Mg-Si alloys 5487.4.3.3 Constitutive equations 5487.4.3.4 Predictions of HAZ hardness and strength distribution 550

7.4.4 HAZ microstructure and strength 'evolution during friction welding 5567.4.4.1 Heat generation in friction welding 5567.4.4.2 Response of AI-Mg-Si alloys and AI-SiC MMCs to friction welding 5577.4.4.3 Constitutive equations : 5587.4.4.4 Coupling of models 5587.4.4.5 Prediction of the HAZ hardness distribution 560

References

Preface to the second edition

Besides correcting some minor linguistic and print errors, I have in the second edition in-cluded a collection of different exercise problems which have been used in the training of stu-dents at NTNU. They illustrate how the models described in the previous chapters can be usedto solve practical problems of more interdisciplinary nature. 'Each of them contains a 'prob-lem description' and some background information on materials and welding conditions. Theexercises are designed to illuminate the microstructural connections throughout theweldthermal cycle and show how the properties achieved depend on the operating conditions ap-plied. Solutions to the problems are also presented. These are not complete or exhaustive, butare just meant as an aid to the reader to develop the ideas further.

Trondheim, 28 October, 1996

(/)ystein Grong

Preface to the first edition

The purpose of this textbook is to present a broad overview on the fundamentals of weldingmetallurgy to graduate students, investigators and engineers who already have a good back-ground in physical metallurgy and materials science. However, in contrast to previous text-books covering the same field, the present book takes a more direct theoretical approach towelding metallurgy based on a synthesis of knowledge from diverse disciplines. The motiva-tion for this work has largely been provided by the need for improved physical models forprocess optimalisation and microstructure control in the light of the recent advances that havetaken lplace within the field of materials processing and alloy design.

The present textbook describes a novel approach to the modelling of dynamic processes inwelding metallurgy, not previously dealt with. In particular, attempts have been made to ra-tionalise chemical, structural and mechanical changes in weldments in terms of models basedon well established concepts from ladle refining, casting, rolling and heat treatment of steelsand aluminium alloys. The judicious construction of the constitutive equations makes full useof both dimensionless parameters and calibration techniques to eliminate poorly known ki-netic constants. Many of the models presented are thus generic in the sense that they can begeneralised to a wide range of materials and processing. To help the reader understand andapply the subjects and models treated, numerous example problems, exercise problems andcase studies have been worked out and integrated in the text. These are meant to illustrate thebasic physical principles that underline the experimental observations and to provide a way ofdeveloping the ideas further.

Over the years, I have benefited from interaction and collaboration with numerous peoplewithin the scientific community. In particular, I would like to acknowledge the contributionfrom my father Professor Tor Grong who is partly responsible for my professional upbringingand development as a metallurgist through his positive influence on and interest in my re-search work. Secondly, I am very grateful to the late Professor Nils Christensen who firstintroduced me to the fascinating field of welding metallurgy and later taught me the basicprinciples of scientific work and reasoning. I will also take this opportunity to thank all myfriends and colleagues at the Norwegian Institute of Technology (Norway), The Colorado Schoolof Mines (USA), the University of Cambridge (England), and the Universitat der BundeswehrHamburg (Germany) whom I have worked with over the past decade. Of this group of people,I would particularly like to mention two names, i.e. our department secretary Mrs. Reidun0stbye who has helped me to convert my original manuscript into a readable text and Mr.Roald Skjarve who is responsible for all line-drawings in this textbook. Their contributionsare gratefully acknowledged.

Trondheim, 1December, 1993

(lJystein Grong

1Heat Flow and Temperature Distribution in

Welding

1.1 Introduction

Welding metallurgy is concerned with the application of well-known metallurgical principlesfor assessment of chemical and physical reactions occurring during welding. On purely prac-tical grounds it is nevertheless convenient to consider welding metallurgy as a profession of itsown because of the characteristic non-isothermal nature of the process. In welding the reac-tions are forced to take place within seconds in a small volume of metal where the thermalconditions are highly different from those prevailing in production, refining and fabrication ofmetals and alloys. For example, steel welding is characterised by:

High peak temperatures, up to several thousand C.High temperature gradients, locally of the order of 103oC mm'.Rapid temperature fluctuations, locally of the order of 103oC s'.

It follows that a quantitative analysis of metallurgical reactions in welding requires detailedinformation about the weld thermal history. From a practical point of view the analyticalapproach to the solution of heat flow problems in welding is preferable, since this makes itpossible to derive relatively simple equations which provide the required background for anunderstanding of the temperature-time pattern. However, because of the complexity of theheat flow phenomena, it is always necessary to check the validity of such predictions againstmore reliable data obtained from numerical calculations and in situ thermocouple measure-ments. Although the analytical models suffer from a number of simplifying assumptions, it isobvious that these solutions in many cases are sufficiently accurate to provide at least a quali-tative description of the weld thermal programme.

An important aspect of the present treatment is the use of different dimensionless groupsfor a general outline of the temperature distribution in welding. Although this practice in-volves several problems, it is a convenient way to reduce the total number of variables to anacceptable level and hence, condense general information about the weld thermal programmeinto two-dimensional (2-D) maps or diagrams. Consequently, readers who are unfamiliar withthe concept should accept the challenge and try to overcome the barrier associated with the useof such dimensionless groups in heat flow analyses.

1.2 Non-Steady Heat Conduction

The symbols and units used throughout this chapter are defined in Appendix 1.1.

2 METALLURGICAL MODELLING OF WELDING

Since heat losses from free surfaces by radiation and convection are usually negligible inwelding, the temperature distribution can generally be obtained from the fundamental differ-ential equations for heat conduction in solids. For uniaxial heat conduction, the governingequation can be written as.'

(1-1)

where T is the temperature, t is the time, x is the heat flow direction, and a is the thermaldiffusivity. The thermal diffusivity is related to the thermal conductivity A. and the volumeheat capacity pc through the following equation:

a = A./pc (1-2)

For biaxial and triaxial heat conduction we may write by analogy: 1

aT = a[a2T + a2T]at ax2 ay2 (1-3)and

(1-4)

The above equations must clearly be satisfied by all solutions of heat conduction problems,but for a given set of initial and boundary conditions there will be one and only one solution.

1.3 Thermal Properties of Some Metals and Alloys

A pre-condition for obtaining simple analytical solutions to the differential heat flow equa-tions is that the thermal properties of the base material are constant and independent of tem-perature. For most metals and alloys this is a rather unrealistic assumption, since both A., a,and .pc may vary significantly with temperature as illustrated in Fig. 1.1. In addition, thethermal properties are also dependent upon the chemical composition and the thermal historyof the base material (see Fig. 1.2), which further complicates the situation.

By neglecting such effects in the heat flow models, we impose several limitations on theapplication of the analytical solutions. Nevertheless, experience has shown that these prob-lems to some extent can be overcome by the choice of reasonable average values for A., a andpc within a specific temperature range. Table 1.1 contains a summary of relevant thermalproperties for different metals and alloys, based on a critical review of literature data. It shouldbe noted that the thermal data in Table 1.1 do not include a correction for heat consumed inmelting of the parent materials. Although the latent heat of melting is temporarily removedduring fusion welding, experience has shown this effect can be accounted for by calibratingthe equations against a known isotherm (e.g. the fusion boundary). In practice, such correc-tions are done by adjusting the arc efficiency factor 'rI until a good correlation is achievedbetween theory and experiments.

HEAT PLOW AND TEMPERATURE DISTRIBUTION IN WELDING 3

10

I Carbon steel8

2

500 1000Temperature,OC

1500...Fig. 1.1. Enthalpy increment HT-Ho referred to an initial temperature To = 20C. Data from Refs.2-4.

Table 1.1 Physical properties for some metals and alloys. Data from Refs 2-6.

X. a pet r: H -Ht aHm 0 mMaterial (W mrrr+C:') (rnm? s ') (Jmrrr? C-t) (OC) (J mm=') (J mrrr')

Carbon 0.040 8 0.005 1520 7.50 2.0Steels

Low Alloy 0.025 5 0.005 1520 7.50 2.0Steels

High Alloy 0.020 4 0.005 1500 7.40 2.0Steels

Titanium 0.030 10 0.003 1650 4.89 1.4

Alloys

Aluminium 0.230 85 0.0027 660 1.73 0.8(> 990/0 AI)

AI-Mg-Si 0.167 62 0.0027 652 1.71 0.8Alloys

AI-Mg 0.149 55 0.0027 650 1.70 0.8Alloys

t Does not include the latent heat of melting (f:Jlm).


(a)0.08..--------r--------r-----.,..-----.,

t 0.06oo

E~ 0.043:~ _A.!!0t.s~e~ _

0.02'

200 400 600 800 1000Temperature,OC ----

t 0.03ooEEO.02~

High alloy steel(b)

0.01

o 200 400 600 800 1000Temperature, c -----

Fig. 1.2. Factors affecting the thermal conductivity ~ of steels; (a) Temperature level and chemicalcomposition, (b) Heat treatment procedure. Data from Refs. 2-4.

1.4 Instantaneous Heat Sources

The concept of instantaneous heat sources is widely used in the theory of heat conduction.' Itis seen from Fig. 1.3 that these solutions are based on the assumption that the heat is releasedinstantaneously at time t = 0 in an infinite medium of initial temperature TO' either across aplane (uniaxial conduction), along a line (biaxial conduction), or in a point (triaxial conduc-tion). The material outside the heat source is assumed to extend to x = 00 for a plane sourcein a long rod, to r = 00 for a line source in a wide plate, or to R = 00 for a point source in a heavyslab. The initial and boundary conditions can be summarised as follows:

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 5

T-To = 00 for t = 0 and x = 0 (alternatively r = 0 or R = 0)T-To = 0 for t = 0 and x:j:. 0 (alternatively r> 0 or R > 0)T-To = 0 for 0 < t < 00 when x = 00 (alternatively r = 00 or R = 00).

It is easy to verify that the following solutions satisfy both the basic differential heat flowequations (1-1), (1-3) and (1-4) and the initial and boundary conditions listed above:

(i) Plane source in a long rod (Fig. 1.3a):QIA 2

T - To = 112 exp( -x 14at)pe( 4nat)

(1-5)

where Q is the net heat input (energy) released at time t = 0, and A is the cross section of therod.

(ii) Line source in a wide plate (Fig. 1.3b):

T-To= Qld exp(-r2/4at)pe( 4nat) (1-6)

where d is the plate thickness.

(iii) Point source in a heavy slab (Fig. 1.3c):

Q 2T - To = 3/2 exp( -R /4at)pe( 4nat)

(1-7)

Equations (1-5), (1-6) and (1-7) provide the required basis for a comprehensive theoreticaltreatment of heat flow phenomena in welding. These solutions can either be applied directlyor be used in an integral or differential form. In the next sections a few examples will be givento illustrate the direct application of the instantaneous heat source concept to problems relatedto welding.

(a)

Fig. 1.3. Schematic representation of instantaneous heat source models; (a) Plane source in a long rod.

6(b)

(c)

METALLURGICAL MODELLING OF WELDING

T

T

TO

R

oz

Fig. 1.3.Schematic representation of instantaneous heat source models (continued); (b) Line source in awide plate, (c) Point source in a heavy slab.


1.5 Local Fusion in Arc Strikes

The series of fused metal spots formed on arc ignition make a good case for application ofequation (1-7).

ModelThe model considers a point source on a heavy slab as illustrated in Fig. 1.4. The heat isassumed to be released instantaneously at time t = 0 on the surface of the slab. This causes atemperature rise in the material which is exactly twice as large as that calculated from equation(1-7):

2Q 2T - To = 3/2 exp( -R 14at)

pe( 41tat)(1-8)

In order to obtain a general survey of the thermal programme, it is convenient to writeequation (1-8) in a dimensionless form. The following parameters are defined for this pur-pose:

- Dimensionless temperature:

8= (T-To)(Tc-To)

(1-9)

where T; is the chosen reference temperature.

- Dimensionless time:

(1-10)

where ti is the arc ignition time.

- Dimensionless operating parameter:

(1-11)

where qo is the net arc power (equal to Qlti), and (Hc-Ho) is the heat content per unit volume atthe reference temperature.

- Dimensionless radius vector:

(1-12)

By substituting these parameters into equation (1-8), we obtain:

(1-13)

8 METALLURGICAL MODELLING OF WELDINGHeat source

Isotherms

3--Dheat flow

Fig. 1.4. Instantaneous point source model for assessment of temperatures in arc strikes.

1.4

1.2

1.0

I 0.8'1"""'4

c.......


from which

Substituting this into equation (1-13) gives:

8p~ = (e'tim )3/2 (2e / 3)3/2 JIm)3

(1-14)

where 9p is the peak temperature, and e is the natural logarithm base number.

Example (1.1)

Consider a small weld crater formed in an arc strike on a thick plate of low alloy steel. Calcu-late the cooling time from 800 to 500C (~t8/5)' and the total width of the fully transformedregion adjacent to the fusion boundary. The operational conditions are as follows:

where 1') is the arc efficiency factor. Relevant thermal data for low alloy steel are given inTable 1.1.

SolutionIn the present case it is convenient to use the melting point of the steel as a reference tempera-ture (i.e. 9 = em = 1 when T; = Tm). The corresponding values of nl and e (at 800 and 500C,respectively) are:

n = 0.75x80x35 =356I 4x7.5x(51t)3/2(0.1)1I2 .

8800

= (800 - 20) 0.52(1520-20)

(500 - 20) = 0.32(1520-20)

Cooling time L1.tS/5Since the cooling curves in Fig. 1.5 are virtually parallel at temperatures below 800 C, ~t8/5will be independent of 0"1 and similar to that calculated for the centre-line (UI = 0). By rear-ranging equation (1-13) we get:


[( J2/3 ( J2/3] [ 213 2/3]

L1'tl = ~ - ~ = (3.56) _(3.56) =1.388500 8800 0.32 0.52

and

Total width offully transformed regionZone widths can generally be calculated from equation (1-14), as illustrated in Fig. 1.6. Tak-ing the AC3-temperature equal to 890C for this particular steel, we obtain:

[( J

1I3 (J1I3] [ 1/3 1/3]L1O'lm = 1 1/2 ~ - ~ =0.74 (3.56) _(3.56) =0.23

(2e / 3) e890 em 0.58 1and

M1m =L10'Im~4ati =0.23~4x5xO.1mm=0.32mm

Alternatively, the same information could have been read from Fig. 1.5. Although it isdifficult to check the accuracy of these predictions, the calculated values for dt8/5 and M 1m areconsidered reasonably correct. Thus, because the cooling rate is very large, in arc strikes ahard martensitic microstructure would be expected to form within the transformed parts of theHAZ, in agreement with general experience.

1.6 Spot Welding

Equation (1-6) can be used for an assessment of the temperature-time pattern in spot weldingof plates.

ModelThe model considers a line source which penetrates two overlapping plates of similar thermalproperties, as illustrated in Fig. 1.7. The heat is assumed to be released instantaneously at time

Fig. 1.6. Definition of isothermal zone width in Example (1.1).


~~.--...,...Jd~----r--'--~----Iro... -f

Fig. 1. 7. Idealised heat flow model for spot welding of plates.

t = O. If transfer of heat into the electrodes is neglected, the temperature distribution is givenby equation (1-6). '

This equation can be written in a dimensionless form by introducing the following group ofparameters:


(1-15)

where th is the heating time (i.e. the duration of the pulse).


s; / dt ( 1-16)

where d, is the total thickness of the joint.


(1-17)

By substituting these parameters into equation (1-6), we get:

( 1-18)

where e denotes the dimensionless temperature (previously defined in equation (1-9)).


1.4

1.2

1.0

I 0.8N

~(J)

0.6

0.4

0.2

0 0.4 0.6 0.8 1 2 4 6 8 10 20

Fig. 1.8. Calculated temperature-time pattern in spot welding.

Figure 1.8 shows a graphical representation of equation (1-18) for a limited range of (T2 and't2' A closer inspection of the graph reveals that the temperature-time pattern in spot weldingis similar to that observed during arc ignition (see Fig. 1.5). The locus of the peak tempera-tures in Fig. 1.8 is obtained by setting dln(Oln2)/d't2 = O.

which gives

and

(1-19)

Example (1.2)

Consider spot welding of 2 mm plates of low alloy steel under the following operational con-ditions:

1= 8kA, th = 0.3s, 'l1 = 0.5, To = 20C


Calculate the cooling time from 800 to 500C (L\tS/5) in the centre of the weld, and the coolingrate (C.R.) at the onset of the austenite to ferrite transformation. Assume in these calculationsthat the total voltage drop between the electrodes is 1.6 V. The Ms-temperature of the steel istaken equal to 475C.

SolutionIf we use the melting point of the steel as a reference temperature, the parameters n2 and e (at800 and 500C, respectively) become:

0.5x8x103 x1.6n2 = 3.40

4x7.Sx4x1txS

9 = (800- 20) = 0.52soo (1520- 20)

9500

= (500- 20) 0.32(1520-20)

Cooling time Lit8/5The parameter L\tS/5 can be calculated from equation (I-IS). For the weld centre-line (a2 = 0),we get:

~'t2 = n2(_1_-_1_J = 3.40(_1 1_) = 4.099500 9S00 0.32 0.52and

..1tS/5 = ..1't2 X th = 4.09 X 0.3s = 1.23s

Cooling rate at 475CThe cooling rate at a specific temperature is obtained by differentiation of equation (1-18) withrespect to time. When 0"2 = 0 the cooling rate at e = 0.3 (475C) becomes:

-de = ~ = ~ = (0.30)2 = 0.026d't2 ('t2) n2 3.4

and

C.R.= (Tc - To) (-d9 J = (1520- 20) x 0.026Cs-1 ~ 130Cs-1th d't2 0.3

Since the cooling curves in Fig. I.S are virtually parallel at temperatures below SOOC(i.e.for 8/n2 < 0.15), the computed values of L\tS/5 and C.R. are also valid for positions outside theweld centre-line. In the present example the centre-line solutions can be applied down to(0"2m)2 ~ 2. According to equation (1-19), this corresponds to a lower peak temperature of:

9p= n2 2 = 3.40 =0.63

e(0'2m) 2e

which is equivalent with:T = T + e (T -T ) = [20 + 0.63(1520-20)]OC ~ 960Cpop C 0


It should be emphasised that the present heat flow model represents a crude oversimplifica-tion of the spot welding process. In a real welding situation, most of the heat is generated at theinterface between the two plates because of the large contact resistance. This gives rise to thedevelopment of an elliptical weld nugget inside the joint as shown in Fig. 1.9. Moreover, sincethe model neglects transfer of heat into the electrodes, the mode of heat flow will be mixed andnot truly two-dimensional as assumed above. Consequently, equation (1-18) cannot be ap-plied for reliable predictions of isothermal contours and zone widths. Nevertheless, the modelmay provide useful information about the cooling conditions during spot welding if the effi-ciency factor 11 and the voltage drop between the electrodes can be estimated with a reasonabledegree of accuracy.

A more refined heat flow model for spot welding is presented in Appendix 1.2.

1.7 Thermit Welding

Thermit welding is a process that uses heat from exothermic chemical reactions to producecoalescence between metals and alloys. The thermit mixture consists of two components, i.e.a metal oxide and a strong reducing agent. The excess heat of formation of the reaction prod-uct provides the energy source required to form the weld.

ModelIn thermit welding the time interval between the ignition of the powder mixture and the com-pletion of the reduction process will be short because of the high reaction rates involved.Assume that a groove of width 2Ll is filled instantaneously at time t = a by liquid metal of aninitial temperature T, (see Fig. 1.10). The metal temperature outside the fusion zone is To. Ifheat losses to the surroundings are neglected, the problem can be treated as uniaxial conduc-tion where the heat source (extending from -Ll to +L1) is represented by a series of elementarysources, each with a heat content of:

(1-20)

At time tthis source produces a small rise of temperature at position x, given by equation (1-5):

dT = ( dQ / All 2 JexP[ -(x - x,)2 /4at]pe( 4nat)

(1-21)(T - T )dx' 2

= l 0112

exp[-(x - x') /4at](4nat)

The final temperature distribution is obtained by substituting u = (x-x)/( 4at) 112 (i.e. dx' =- due 4at) 112) into equation (1-21) and integrating between the limits x' = -Ll and x' = +L1 Thisgives (after some manipulation):

(1-22)


TTT1~~~~0)

~3+ .0')

~33

~~.........:=iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~J-.J.....J...

Fig. 1.9. Calculated peak temperature contours in spot welding of steel plates (numerical solution). Op-erational conditions: 1= 23kA, 64 cycles. Data from Bently et al.7

T

t= 0

..x

Fig. 1.10. Idealised heat flow model for thermit welding of rails.

where erf(u) is the Gaussian error function. The error function is defined in Appendix 1.3*.Because of the complex nature of equation (1-22), it is convenient to present the different

solutions in a dimensionless form by introducing the following groups of parameters:

*The error function is available in tables. However, in numerical calculations it is more convenient to use theFortran subroutine given in Appendix 1.3.



e'= (T-To)cr, -To) (1-23)


(1-24)

- Dimensionless x-coordinate:

(1-25)

Substituting these parameters into equation (1-22) gives:

(1-26)

Equation (1-26) has been solved numerically for different values of 0 and 't3. The resultsare presented graphically in Fig. 1.11. As would be expected, the fusion zone itself (0 :::;1)cools in a monotonic manner, while the temperature in positions outside the fusion boundary(0) 1) will pass through a maximum before cooling. The locus of the HAZ peak temperaturesin Fig. 1.11 is defined by ae'la't3 = O.Referring to Appendix 1.3, we may write:

The peak temperature distribution is obtained by solving equation (1-27) for different com-binations of Om and 't3m and inserting the roots into equation (1-26).

Example (1.3)

Consider thermit welding of steel rails (i.e. reduction of Fe203 with Al powder) under thefollowing operational conditions:

Calculate the cooling time from 800 to 500C in the centre of the weld, and the total widthof the fully transformed region adjacent to the fusion boundary. The AC3-temperature of thesteel is taken equal to 890C.

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING

0.8

Definition of parameters:

el= (T - To)(lj - To>

t = V4at3 L1

.Q=~L1

0.4

0.2

o ~~~~ __ ~ ~ -L ~ ~ ~ ~ __~o 2 4 6

Fig. 1.11. Calculated temperature-time pattern in thermit welding.

SolutionFor positions along the weld centre-line (0 = 0) equation (1-26) reduces to:

17

8

Cooling time ~t8/5From the above relation it is possible to calculate the cooling time from T; = 2200C to 800 and500C, respectively:

, ( 1 J (800 - 20) .9S00 = elf - = = 0.358,1.e. 't3 = 3.04't3 (2200 - 20)

and

, ( 1 J (500 - 20) . *8500 = elf """""* = = 0.220,1.e. 't3 = 5.05't3 (2200-20)

By rearranging equation (1-24), we obtain the following expression for ~tS/5:

LI * 2 2] (6)2 [ 2 2]~t8/5 =-[('t3) -('t3) =-- (5.05) -(3.04) s=29s4a . 4x5


The computed value for fltS/5 is also valid for positions outside the weld centre-line, sincethe cooling curves at such low temperatures are reasonably parallel within the fusion zone.

Total width of fully transformed regionThe fusion boundary is defined by:

The locus of the 890C isotherm in temperature-time space can be read from Fig. 1.11.Taking the ordinate equal to 0.40, we get:

't3m ~ 1.50

By inserting this value into equation (1-27), we obtain the corresponding coordinate of theisotherm:

The total width of the fully transformed HAZ is thus:

Llx = LI (Q~-Qm) = 6 (1.24 - l)mm = 1.44mm

Unfortunately, measurements are not available to check the accuracy of these predictions.Systematic errors would be expected, however, because of the assumption of instantaneousrelease of heat immediately after powder ignition and the neglect of heat losses to the sur-roundings. Nevertheless, the present example is a good illustration of the versatility of theconcept of instantaneous heat sources, since these solutions can easily be added in space asshown here or in time for continuous heat sources (to be discussed below).

1.8 Friction Welding

Friction welding is a solid state joining process that produces a weld under the compressiveforce contact of one rotating and one stationary workpiece. The heat is generated at the weldinterface because of the continuous rubbing of the contact surfaces, which, in tum, causes atemperature rise and subsequent softening of the material. Eventually, the material at theinterface starts to flow plastically and forms an up-set collar. When a certain amount of up-setting has occurred, the rotation is stopped and the compressive force is maintained or slightlyincreased to consolidate the weld.

Model (after Rykalin et al.8)The model considers a continuous (plane) heat source in a long rod as shown in Fig. 1.12(a).The heat is liberated at a constant rate q'o in the plane x = 0 starting at time t = O. If wesubdivide the time t during which the source operates into a series of infinitesimal elements dt'(Fig. 1.12b), each element will have a heat content of:

dQ=q~dt' (1-28)


(a)

Continuous heat source

-x 0 +x

(b) q I~

qla 1

II

dQ=q~dtl~IIIIII

VdtlII

t' til = t - tl .1II

Fig. 1.12. Idealised heat flow model for friction welding of rods; (a) Sketch of model, (b) Subdivision oftime into a series of infinitesimal elements dt~

At time t this heat will cause a small rise of temperature in the material, in correspondancewith equation (1-5):

dT q~ dt' / A [ x2

]= exppc~ 41ta(t - t') '4a(t - t')

(1-29)

If we substitute t"= t - t'into equation (1-29), the total temperature rise at time t is obtainedby integrating from t" = t (t' = 0) to t" = 0 (t' = t):

o

-q~ / A J dt" [x2]pc,) 41ta {?exp - 4a;" (1-30)

In order to evaluate this integral, we will make use of the following mathematical transfor-mation:

II =Jwdp=wp- Jpdw


where

[_X2 ] [X2] [ _X

2]w=2ex -- dw= ex -- dt"

p 4at" , 2aCt,,)2 p 4at"

anddt"p =..Jt",dp =--2..Jt"

Hence, we may write:

I[ =2[..Jt" exp[ - 4::" ]] - J 2a(;/~)3/2 exp[ ~:t~']dt"t t

The latter integral can be expressed in terms of the complementary error function" erfc(u)by substituting:

u=_x_ du= -x dt"~ 4at" , 4-ra (t,,)3/2

and integrating between the limits u = x I (4at)1/2 and u = 00.This gives (after some manipulation):

q~ {i ( [x2] ({itx J [x]JT-T = exp -- - -- erfc --o Apc-[iW 4at -J4ii -J4ii (1-31)

If the temperature of the contact section at the end of the heating period is taken equal to Th,equation (1-31) can be rewritten as:

T - To = cr, - To )~t/ th (exp[ - ::t] -( ffit }ifC[ kat ]J (1-32)where tlz denotes the duration of the heating period (t ~ tlz). Measured contact section tem-peratures for different metal/alloy combinations are given in Table 1.2.

Equation (1-32) may be presented in a dimensionless form by the use of the followinggroups of parameters:


8"= T-Tor, -r; (1-33)

- Dimensionless time:'t4 = t / tlz (1-34)

*The complementary error function is defined in Appendix 1.3.


Table 1.2 Measured contact section temperatures during friction welding of some metals and alloys.Data from Tensi et al.lO

MetaUAlloy Measuring Temperature Level PartialCombination Method [OC] Melting

Steel Thermocouples 1080-1340 No

Steel-Nickel Direct readings 1 1260-1400 NolYes

Steel- Titanium Direct readings 1 1080 No

Copper-Al Direct readings 1 548 Yes

Copper-Nickel Direct readings 1 1083 Yes

AI-Cu-2Mg Thermocouples 506 Yes

AI-4.3Cu Thermocouples 562 Yes

AI-12Si Thermocouples 575 Yes

AI-5Mg Thermocouples 582 Yes

1Based on direct readings of the voltage drop between the two work-pieces.


Q'=_X_~4ath

(1-35)

By substituting these parameters into equation (1-32), we obtain:

" [[ (Q')2] ( {ito.' J [il']Je =.[t:; exp -~ - .[t:; erfc .[t:; (1-36)Equation (1-36) describes the temperature in different positions from the weld contact sec-

tion during the heating period. However, when the rotation stops, the weld will be subjected tofree cooling, since there is no generation of heat at the interface. As shown in Fig. 1.13(a) thiscan be accounted for by introducing an imaginary heat source of power +q' 0 at time t = lizwhich acts simultaneously with an imaginary heat sink of negative power -q' o- It follows fromthe principles of superposition (see Fig. 1.13b) that the temperature during the cooling periodis given by:"

(1-37)

where 8"("(4) and 8"( 't4 - 1) are the temperatures calculated for the heat source and the heatsink, respectively, using equation (1-36).

Equations (1-36) and (1-37) have been solved numerically for different values of 0.'and 't4.The results are presented graphically in Fig. 1.14. Considering the contact section (0.'= 0), thetemperature increases monotonically with time during the heating period, in correspondancewith the relationship:

(1-38)


(a)

(b)

He~tingpenod

t:'-'" .-...~ ~

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 231.0

0.2

Cooling

0.8

10.6-


In this particular case, it is possible to check the accuracy of the calculations against in situthermocouple measurements carried out on friction welded components made under similarconditions. A comparison with the data in Fig. 1.15 shows that the model is quite successful inpredicting the HAZ peak temperature distribution. In contrast, the weld heating and coolingcycles cannot be reproduced with the same degree of precision. This has to do with the factthat the present analytical solution omits a consideration of the plastic straining occurringduring friction welding, which displaces the coordinates and alters the heat balance for thesystem.

1.9 Moving Heat Sources and Pseudo-Steady State

In most fusion welding processes the heat source does not remain stationary. In the followingwe shall assume that the source moves at a constant speed along a straight line, and that the netpower supply from the source is constant. Experience shows that such conditions lead to afused zone of constant width. This is easily verified by moving a tungsten arc across a sheet ofsteel or aluminium, or by moving a soldering iron across a piece of lead or tin. Moreover,zones of temperatures below the melting point also remain at constant width, as indicated bythe pattern of temper colours developed on welding ground or polished sheet.

It follows from the definition of pseudo-steady state that the temperature will not vary withtime when observed from a point located in the heat source. Under such conditions the tem-perature field around the source can be described as a temperature 'mountain' moving in thedirection of welding (e.g. see Fig. 15 in Ref. 11). For points along the weld centre-line, thetemperature at different positions away from the heat source (which for a constant weldingspeed becomes a time axis) may be presented in a two-dimensional plot as indicated in Fig.1.16. Specifically, this figure shows a schematic representation of the temperature in steelwelding from the base plate ahead of the arc to well into the solidified weld metal trailing thearc. If we consider a fixed point on the weld centre-line, the temperature will increase veryrapidl y during the initial period, reaching a maximum of about 2000- 2200C for positionsimmediately beneath the root of the arc. 11 When the arc has passed, the temperature will startto fall, and eventually (after long times) approach that of the base plate. In contrast, an ob-server moving along with the heat source will always see the same temperature landscape,since this will not change with time according to the presuppositions.

It will be shown below that the assumption of pseudo-steady state largely simplifies themathematical treatment of heat flow during fusion welding, although it imposes certain re-strictions on the options of the models.

1.10 Arc Welding

Arc welding is a collective term which includes the following processes":

- Shielded metal arc (SMA) welding.- Gas tungsten arc (GTA) welding.- Gas metal arc (GMA) welding.

*The terminology used here is in accordance with the American Welding Society's recommendations. 12


- Flux cored arc (FCA) welding.- Submerged arc (SA) welding.

The main purpose of this section is to review the classical models for the pseudo-steadystate temperature distribution around moving heat sources. The analytical solutions to thedifferential heat flow equations under conditions applicable to arc welding were first presented

(a) 800

I Heating period 1..-I600C,)0

PI:::J~ 400CDa.E~

200

00 2

(b)600

1o0

~- 500::J~Q)a.E$~


by Rosenthal.P'!" but the theory has later been extended and refined by a number of otherinvestigators.9,11,15-20

1.10.1 Arc efficiency factors

In arc welding heat losses by convection and radiation are taken into account by the efficiencyfactor 'Yl, defined as:

(1-40)

where qo is the net power received by the weldment (e.g. measured by calorimetry), I is thewelding current (amperage), and U is the arc voltage.

For submerged arc (SA) welding the efficiency factor ('Yl) has been reported in the rangefrom 90 to 98%, for SMA and GMA welding from 65 to 85%, and for GTA welding from 22to 75%, depending on polarity and materials. 11

A summary of ranges is given in Table 1.3.

1.10.2 Thick plate solutions

Model (after Rykalin9)According to Fig. 1.17, the general thick plate model consists of an isotropic semi-infinitebody at initial temperature To limited in one direction by a plane that is impermeable to heat.At time t = 0 a point source of constant power q0 starts on the surface at position 0 moving inthe positive x-direction at a constant speed v. The rise of temperature T - To in point P at timet is sought.

During a very short time interval from z'to t' + dt'the amount of heat released at the surfaceis dQ = q.dt'. According to equation (1-7) this will produce an infinitesimal rise of tempera-ture in P at time t:

2q dt' [(R1)2 ]dT= 0 exppc[ 41ta(t - t,]3/2 4a(t - t')

(1-41)

-2qo dt" [(R1)2 ]= pc( 41tat,,)3/2 exp - 4at"

where t" = t - t' is the time available for conduction of heat over the distance

R'=~(XO-1.)t')2+y;+Z; topointP.For a convenient presentation of the pseudo-steady state solution, the position P should be

referred to that of the moving heat source. This is achieved by changing the coordinate systemfrom 0 to 0' (see Fig. 1.17):

y=Yo' Z=Zo' x=xo-vtand

Hence, we may write:


Solidifiedweld metal

x=vtRelative position along weld centre ..line

Fig. 1.16. Schematic diagram showing weld centre-line temperature in different positions from the heatsource during steel welding at pseudo-steady state.

Table 1.3 Recommended arc efficiency factors for different welding processes. Data from Refs 11, 21.

Arc efficiency factor 11

Welding Process Range Mean

SA welding (steel) 0.91-0.99 0.95

SMA welding (steel) 0.66-0.85 0.80

GMA welding (CO2-steel) 0.75-0.93 0.85

GMA welding (Ar-steel) 0.66-0.70 0.70

GTA welding (Ar-steel) 0.25-0.75 0.40

GTA welding (He-AI) 0.55-0.80 0.60

GTA welding (Ar-AI) 0.22-0.46 0.40


-2qo dt" ['\)x R2 '\)2t,,]= pe( 41tat,,)3/2 exp - 2a - 4at" - ~

where R=~x2+y2+z2.

(1-42)

The total rise of temperature at P is obtained by substituting:

andm = vRl4a, m21u2 = vt''l4a

into equation (1-42), and integrating between the limits u = (R2/4at")112 and u = 00. This gives(after some manipulation):

T-To = 2!M( In)exP(-uX,2a)f exp(-u2-m2 /u2)duu

(1-43)

It is well-known that:

OOJ 2 2 2 j1t j1texp(-u -m lu )du=-" -exp(-2m)=-exp(-uRI2a)2 2

o

Hence, the general thick plate solution can be written as:

T-T =.!lsL.(~)eXp(-UXI2a)o 21tA R

x[exp( -uR / 2a) - In I exp( _u2 - m2 / u2 )du1If u is sufficiently small (i.e. when welding has been performed over a sufficient period), we

obtain the pseudo-steady state temperature distribution:

(1-44)

T-To =.!lsL.(!)exp[-~(R+X)]21tA R 2a

(1-45)

This equation is often referred to as the Rosenthal thick plate solution, 13, 14 in honour ofD. Rosenthal who first derived the relation by solving the differential heat flow equation di-rectly for the appropriate boundary conditions.

1.10.2.1 Transient heating periodItfollows from the above analysis that the pseudo-steady state temperature distribution is


tz

Fig. 1.17. Moving point source on a semi-infinite slab.

attained after a transient heating period. The duration of this heating period is determined bythe integral in equation (1-44).

Taking the ratio between the real and the pseudo-steady state temperature equal to K1, wehave:

(T - To) 2 ( uR )JU 2 2 2K} = =1- cexp - exp(-u -m /u )du

(T - To)p.s. 'V1t 2ao

(1-46)

Equation (1-46) can be expressed in terms of the following parameters:


uR0'3=-2a

(1-47)


(1-48)

Substituting these into equation (1-46) gives:

K1= 1- J1n exp( (53) f exp[ -( (53)2 / (2't) - 't /2] 't~;2't

(1-49)


Equation (1-49) has been solved numerically for a limited range of 0"3 and 'to The results arepresented graphically in Fig. 1.18. A closer inspection of Fig. 1.18 reveals that the duration ofthe transient heating period depends on the dimensionless radius vector 0"3' In practice thismeans that the Rosenthal equation is not valid during the initial period of welding unless thedistance from the heat source to the observation point is very small. It should be noted, how-ever, that a dimensionless distance 0"3 may be 'short' for one combination of welding speedand thermal diffusivity, while the same position may represent a 'long' distance for anothercombination of v and a. Similarly, the dimensionless time 't may be 'short' or 'long' at achosen number of seconds, depending on the ratio vl2a.

Example (1.5)

Consider stringer bead deposition on a thick plate of aluminium at a constant welding speed of5 mm S-I. Calculate the duration of the heating period when the distance from the heat sourceto the point of observation is 17 mm.

SolutionTaking a = 85 mm? S-I, the dimensionless radius vector becomes:

a = 5x17 =0.503 2x85

0.8

2 6 7 8

tu)

~ 0.6

~.........;:5'~ 0.4II

~

0.2

Fig. 1.lS. Ratio between real and pseudo-steady state temperature in thick plate welding fordifferent combinations of (J' 3 and T.


It is seen from Fig. 1.18 that the pseudo-steady state temperature distribution is approachedwhen 't = 3, which gives:

2a 2x85t =2't = --2-3s ~20s

U (5)

This corresponds to a total bead length of:

L2 = in = 5 X 20 = 100mm

The above calculations show that the Rosenthal equation is not valid if the ratio between Rand L2 exceeds a certain critical value (typically 0.15 to 0.30 for aluminium welds and 0.4 to0.6 for steel welds). This important point is often overlooked when discussing the relevance ofthe thick plate solution in arc welding.

1.10.2.2 Pseudo-steady state temperature distributionThe Rosenthal equation gives, with the limitations inherent in the assumptions, full informa-tion on the thermal conditions for point sources on heavy slabs. Accordingly, in order to obtaina general survey of the pseudo-steady state temperature distribution, it is convenient to presentthe different solutions in a dimensionless form. The following parameters are defined for thispurpose: 11


(1-50)


~= Ux2a

(1-51)

- Dimensionless y-coordinate:

\If = uy2a

(1-52)

- Dimensionless z-coordinate:

~= Uz2a

(1-53)

By substituting these parameters into equation (1-45), we obtain:e- = (1/ (j3)exp( -(j3 -~)n3

(1-54)

where e and (J 3 are the dimensionless temperature and radius vector, respectively (previouslydefined in equations (1-9) and (1-47.

Equation (1-54) has been solved numerically for chosen values of (J3 and . A graphicalpresentation of the different solutions is shown in Fig. 1.19. These maps provide a good


(a)

1.4

1.2

1.0

0.8Sp/n3

(t)

cas0.6

0.4

0.2

o~==~====~~==~~~----~~~~-..10 -8 6 ~4 -2 0 2 ~

Fig. 1.19. Dimensionless temperature maps for point sources on heavy slabs; (a) Vertical sections paral-lel to the -axis.

overall indication of the thermal conditions during thick plate welding, but are not suitable forprecise readings. Consequently, for quantitative analyses, the following set of equations canbe used;'!

Isothermal zone widthsThe maximum width of an isothermal enclosure is obtained by setting aln(fJln3)/afI3 = 0:

aln(Oln3) .~=oa~ d(J3From the definition of fI3 we have:

d~ _ 2~~2 +'V2 +S2d(J3 - 2~

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 33(b)

..10*=0.01

3

Fig. 1.19. Dimensionless temperature maps for point sources on heavy slabs (continued): (b) Isothermalcontours in the -'V-plane for different ranges of eln3.

Partial differentiation of the Rosenthal equation gives:2

a(lnS I n3) = __ 1_ -1-1. (J3m = O,i.e.~m = -( (J3m)a(j3 (j3m ~m (j3m + 1


and

8p 1 [ ]- = --exp -0"3m / (0"3m + 1)n3 0"3m

(1-55)

Equation (1-55) can be used for calculations of isothermal zone widths 'Vm and cross sec-tional areas At. From Fig. 1.20 we have:

(1-56)

and

(1-57)

A graphical presentation of equations (1-55), (1-56), and (1-57) is shown in Fig. 1.21.

Length of isothermal enclosuresReferring to Fig. 1.20, the total length of an isothermal enclosure ~t is given by:

(1-58)

where ~' and ~"are the distance from the heat source to the front and the rear of the enclosure,respectively.

Fig. 1.20.Three-dimensional graphical representation of Rosenthal thick plate solution (schematic).


10

2 2A =1L'I' =A~

12m 4a2

v(j3m= Rm2a

1- 0and

~"= In( n3K~~~")} ~"< 0

(1-89)

(1-90)


10

A2 = 2B'I'm = A v2/4a2

O'Sm = rm v/2a'I'm = Ym v/2a

t~Eiob

0.1

0.01 ~~~~~~~~~~ ~~~~~~~--~~~~~0.1 100 1000

Fig. 1.31.Dimensionless distance 0"5m' half width "'m and cross sectional area AiS vs ni9pS.

100

t 10uP-~~

0.1 ;t=~I_~1I

0.0 1 ~"""""-.a...&"""""'Lo....--~....a.-""'&'~""""""""l""'O ---~...---1"---'- &....&..1.01....00---1

n3/90 .

Fig. 1.32. Dimensionless distance from heat source to front (~') and rear (~") of isothermal enclosure vsni9S (thin plate welding).


A graphical presentation of equations (1-89) and (1-90) is shown in Fig. 1.32. Included isalso a plot of the total length of the enclosure ~t vs the parameter n3 18B.

Cooling conditions close to weld centre-lineFor points located on the weld centre-line behind the heat source (J5 = -~ = 'to When r is largerthan about 1, (i.e. t> 2alrr), it is a fair approximation to set Ko ('t) Z exp( -'t)~1t / 2't (see Fig.1.27). Hence, equation (1-83) reduces to:

8B ~-~-y1t/2'tn3

Equation (1-91) provides a basis for calculating the cooling time within a specific tempera-ture interval (e.g. from 81 to 92):

(1-91)

(1-92)

The dimensionless cooling time from 800 to 5000e is thus given by:

1t ~ 2[ 1~'t 8/5 = - (n3 I u) 22 (8500)from which the real cooling time is obtained:

MS/5 = [ 41t~PC I(500~ To)2(1-93)

(1-94)

Taking as average values A= 0.025 W mm' e-1, pc = 0.005 J mm=' e-1, and To = 200efor welding of low alloy steels, we have:

(1-95)

Similarly, the cooling rate at a specific temperature is obtained by differentiating equation(1-91) with respect to time:

(1-96)

By multiplying equation (1-96) with the appropriate conversion factor, we get:


(1-97)

For welding of low alloy steels, the cooling rate becomes:

7.85 X 10-10 (T _ T )3(T\E / d)2 0 (1-98)

Example (1.10)

Consider GTA butt welding of a 2mm thick sheet of cold-rolled aluminium (AI-Mg alloy)under the following conditions:

1= 110A, U=15V; v=4mms-l, 'Yl=0.6, To=20C

Sketch the contours of the fusion boundary and the Ar-isotherm in the ~-'" (x-y) plane atpseudo-steady state. The recrystallisation temperature Ar of the base material is taken equal to275C. Calculate also the cross sectional area of the fully recrystallised HAZ and the coolingrate at 275C for points located within this region.

SolutionReferring to Fig. 1.22(a) (Example (1.6 it is sufficient to calculate the coordinates in fourdifferent (characteristic) positions to sketch the contour of the fusion boundary. If we neglectthe latent heat of melting, the n3/8B ratio at the melting point becomes:

!!l- qo = 110x15xO.6 =0.84eo - 2n' a d (Hm - H0) 2n x 55 x 2 x 1.7

End-pointsThe end-points can be read from Fig. 1.32:

~' = 0.25 (x = 6.88mm)and

~" = - 0.90 (x = -24.75mm)

Maximum widthsThe maximum width of an isothermal enclosure can generally be calculated from equations(1-86) and (1-87) or read from Fig. 1.31. When n3/8pB = 0.84, we obtain:

"'m = 0.41 (ym = 11.41mm)and

gm = -0.29 (xm = -7.87mm)

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 55Intersection point with If/(y)-axisIn this case ~ = 0 and 0"5 = '1'. Hence, equation (1-83) reduces to:

which gives'I' = 0.37 (y = IO.I8mm)

Similarly, the contour of the Ar-isotherm can be determined by inserting n3/08 = 2.08 intothe same set of equations. Figure 1.33 shows a graphical representation of the calculatedisothermal contours.

Fig. 1.33. Calculated contours of fusion boundary and Ar-isotherm in GTA butt welding of a 2mm thickaluminium plate (Example 1.10).


Cross sectional area of fully recrystallised HAZIn general, cross sectional areas can be read from Fig. 1.31. Taking the n3/8p8 ratio equal to0.84 (8p = 1) and 2.08 (ap = 0.48), respectively, we have:

4x2M.2 =8(3-0.8)=--2.2=0.16

2x55which gives

4(55)2M=0.16--mm2 =121mm2

(4)2

Cooling rate at 275CThe cooling rate at a specific temperature can be calculated from equation (1-97). In the presentcase, we obtain:

c.R.rCs-I)= 21txO.149xO.0027 (275-20)3 Cs-I ~3Cs-I(110x15xO.6/4x2)2

1.10.3.3 Simplified solution for a fast moving high power source

Model (after RykaUn9)It follows from Fig. 1.29 that the isotherms behind the heat source become increasingly elon-gated as the 881n3 ratio decreases. In the limiting case the isotherms will degenerate intosurfaces which are parallel to the welding x direction, as shown in Fig. 1.34.

In a short time interval dt the amount of heat released per unit length of the weld is equal to:

dQ = qo dt =.!!.SLdA ddx d

(1-99)

According to the assumptions this amount of heat will remain in a rod of constant crosssectional area due to the lack of a temperature gradient in the welding direction. Under suchconditions the mode of heat flow becomes essentially one-dimensional, and the temperaturedistribution is given by equation (1-5):

q IudT - To = 0 112 exp( _y2 14at) (1-100)

pc( 41tat)

Equation (1-100) represents the simplified solution for a fast moving high power source * ina thin sheet, and is valid within a limited range of the more general Rosenthal equation for two-dimensional heat flow (equation (1-81.

By substituting the appropriate dimensionless parameters into equation (1-100), we obtain:

*Since the shape of a given isotherm in the x-y plane is determined by the q.Id ratio, the minimum welding speedwhich is required to maintain I-D heat flow increases with decreasing qJvd ratios. Hence, the term 'fast moving highpower source' is also appropriate in the case of the thin plate welding.


x

I,1,, _I,t _,,

II

II

Fig. 1.34. Fast moving high power source in a thin plate.

90 *' 2- = - exp( -'II /2't)n3 2't (1-101)The locus of the peak temperatures is readily evaluated from equation (1-101) by setting

dln(8B1n3)ld't = 0:

which gives

and

9po _ ~_ f1C_1n3 ~2et;;; ~2; 'IIm (1-102)

It is evident from the plot in Fig. 1.35 that the predicted width of the isotherms is alwaysgreater than that inferred from the general thin plate solution (equation (1-83)) due to theassumption of one-dimensional heat flow. However, such deviations become negligible atvery small 8pB1n3 ratios because of a small temperature gradient in the welding x directioncompared to the transverse y direction of the plate.

A general graphical representation of the weld thermal programme (similar to that shown inFig. 1.25 for a fast moving high power source on a heavy slab) can be obtained by combiningequations (1-101) and (1-102):

58

6

t 55'

4I~~E---9' 3,...;e

2


L----------=--=-:-::-=-=-:-:-=-=-- =_-=_-=_:-:- - - - ~~Y.!Y''p~o~e- - - -

o~~~--~~~~wu~--~--~~~~--~--~~~~~0.01 0.1 10

6po/n3

Fig. 1.35. Theoretical width of isotherms under 1-D and 2-D heat flow conditions, respectively at pseudo-steady state (thin plate welding).

1.0

0.8 . (T T )ale = 0P (Tp-TO)

0.60-

ct>'""


~ = ~ \If exp[- ('V m )2 ]8p ~~ m 2't

Equation (1-103) has been plotted in Fig. 1.36.

(1-103)

Example (1.11)

Consider butt welding of a 2mm thin plate of austenitic stainless steel with covered electrodes(SMAW) under the following conditions:

1= 80A, U = 25V, v = 5mm s', To = 20C

Calculate the retention time within the critical temperature range for chromium carbideprecipitation (i.e. from 650 to 850C) for points located at the 850C isotherm.

SolutionIf we use the melting point of the steel as a reference temperature, the parameter n3/8 becomes:

n3 = 80x25xO.8 =4.308 21tx4x7.4x2

A comparison with Fig. 1.35 shows that the assumption of 1-D heat flow is justified when8p ~ 1. Hence, the total time spent in the thermal cycle from 6 = 0.43 (T = 650C) to 8p = 0.56(Tp = 850C) and down again to 6 = 0.43 can be read from Fig. 1.36. Taking the ordinate 6/6pequal to 0.76, we obtain:

which gives

~'t = 6.6 ('Vm)2 = 6.6~(4.30)2 ~ 112r 2 4e 0.56

and

2a 2x4~t = -2 ~'t = --112s ~ 36s

r u r (5)2

1.10.4 Medium thick plate solution

In a real welding situation the assumption of three-dimensional or two-dimensional heat flowinherent in the Rosenthal equations is not always met because of variable temperature gradi-ents in the through thickness z direction of the plate.

Model (after Rosenthal!")The general medium thick plate model considers a point heat source moving at constant speedacross a wide plate of finite thickness d. With the exception of certain special cases (e.g.watercooling of the back side of the plate), it is a reasonable approximation to assume that the


plate surfaces are impermeable to heat. Thus, in order to maintain the net flux of heat throughboth boundaries equal to zero, it is necessary to account for mirror reflections of the sourcewith respect to the planes of z = 0 and z = d. This can be done on the basis of the 'method ofimages' as illustrated in Fig. 1.37. By including all contributions from the imaginary sources... 2q-2 , 2q-l , Zq, , 2q2 ,.. .located symmetrically at distances 2id below and above the uppersurface of the plate, the pseudo-steady state temperature distribution is obtained in the form ofa convergent series *:

T-T =~ exp(- ux)o 2nA. 2a

(1-104)

where R, =~x2 + y2 +(z-2id)2 .Note that equation (1-104) is simply the general Rosenthal thick plate solution (equation

(1-45)) summed for each source.

T~~----~~--~~--~

f2d

+2dy

Fig. 1.37. Real and imaginary point sources on a medium thick plate.

*The number of imaginary heat sources necessary to achieve the required accuracy depends on the chosen values ofs, and vd/2a.


By substituting the dimensionless parameters defined above into equation (1-104), we ob-tain:

(1-105)

where

It follows from equation (1-104) that the thermal conditions will be similar to those in athick plate close to the centre of the weld. Moreover, Rosenthal 13, 14 has shown on the basis ofa Fourier series expansion that equation (1-104) converges to the general thin plate solution(equation (1-81) for points located sufficiently far away from the source. However, at interme-diate distances from the heat source, the pseudo-steady state temperature distribution will de-viate significantly from that observed in thick plate or thin plate welding because of variabletemperature gradients in the through-thickness direction of the plate. Within this 'transitionregion', the thermal programme is only defined by the medium thick plate solution (equation(1-104)).

1.10.4.1 Dimensionless mapsfor heatflow analysesBased on the models described in the previous sections, it is possible to construct a series ofdimensionless maps which provide a general outline of the pseudo-steady state temperaturedistribution during arc welding. 20

Construction of the mapsThe construction of the maps is done on the basis of the medium thick plate solution (equation(1-105)). This model is generally applicable and allows for the plate thickness effect in aquantitative manner. Since the other solutions are only valid within specific ranges of thisequation, they will have their own characteristic fields in the temperature-distance or the tem-perature-time space. The extension of the different fields can be determined from numericalcalculations of the temperature distribution by comparing each of these models with the me-dium thick plate solution, using a conformity of 95% as a criterion.

Similarly, when the 95% conformity is' not met between the respective solutions, the fieldsare marked 'transition region'. Since any combination of dimensionless temperature, operat-ing parameter, and plate thickness locates a point in a field, it means that the dominating heatflow mechanism can readily be read off from the maps.

Peak temperature distributionThe variation of peak temperature with distance in the 'V(y )-direction has been numericallyevaluated from equation (1-105) for different values of the dimensionless plate thickness (8 =vd/2a). The results are shown graphically in Fig. 1.38(a) and (b) for the two extreme cases of~ = 0 (z = 0) and, = 8 (z = d), respectively.

An inspection of the maps reveals that the temperature-time pattern in stringer beadweldments can be classified into three main categories:

62 METALLURGICAL MODELLING OF WELDING(a)

0.1

....",,..,.,

0.010.01 0.1 10 100

'I'm .,(b)

100

('t)

c:'-c.

: 1-D heat flowIII

0.1

0.010.01 0.1 10 100

Fig. 1.38. Peak temperature distribution in transverse direction ('V= 'tim) of plate; (a) Upper plate surface(~ = 0), (b) Lower plate surface (~ = 8).


1. Close to the heat source, the thermal programme will be similar to that in a thick plate(Fig. 1.38(a)), which means that the temperature distribution is determined by equation (1-54). For large values of the dimensionless plate thickness, the mode of heat flow maybecome essentially two-dimensional. This corresponds to the limiting case of a fast mov-ing high power source in a thick plate (equation (1-74)). Under such conditions the slope ofthe 8p /n3-'Vm curves in Fig. 1.38(a) attains a constant value of -2.

2. With increasing distance from the heat source, a transition from three-dimensional totwo-dimensional heat flow may occur, depending on the dimensionless plate thickness andthe operational conditions applied. Considering the upper surface of the plate (Fig. 1.38(a)),the extension of the transition region is seen to decrease with increasing values of 8 as theconditions for thick plate welding are approached. The opposite trend is observed for thebottom plate surface (Fig. 1.38(b )), since a small dimensionless plate thickness generallyresults in a more rapid equalisation of the temperature gradients in the t(z) direction. Whenthe curves in Fig. 1.38(b) become parallel with the x-axis, the temperature at the bottom ofthe plate reaches its maximum value. Note that within the transition region, reliable predic-tions of the pseudo-steady temperature distribution can only be made from the mediumthick plate solution (equation (1-105)).

3. For points located sufficiently far away from the heat source, the temperature gradi-ents in the through-thickness direction of the plate become negligible. This implies that thetemperature distribution at the upper and lower surface of the plate is similar, and can becomputed from the thin plate solution (equation (1-83)). When the conditions for one-dimensional heat flow are approached (equation (1-101)), the slope of the 8p/n3-'Vm curvesin Fig. 1.38(a) and (b) attains a constant value of -1.

Cooling conditions close to weld centre-lineFigure 1.39 contains a plot of the cooling programme for points located on the weld centre-line('V= t = 0), as calculated from equation (1-105). A closer inspection of Fig. 1.39 reveals thatthe slope of the cooling curves increases gradually from -1 to -0.5 with increasing distancefrom the heat source. This corresponds to a change from three-dimensional to one-dimen-sional heat flow.

From Fig. 1.39 it is possible to read-off the cooling time within specific temperature inter-vals for a wide range of operational conditions. These results are also valid for positionsoutside the weld centre-line, since the cooling curves are virtually parallel in the transverse 'Vdirection of the plate. A requirement is, however, that the peak temperature of the thermalcycle is significantly higher than the actual temperature interval under consideration.

Retention times at elevated temperaturesThe retention time, Ll'tr' is defined as the total time spent in a thermal cycle from a chosenreference temperature 8 to the peak temperature 8p and down again to 8. This parameter canreadily be computed from equation (1-105) by means of numerical methods. The results ofsuch calculations (carried out in position t = 0) are shown graphically in Fig. 1.40 for 8 = 0.5Sp.

An inspection of Fig. 1.40 reveals a complex temperature-time pattern. In this case it is notpossible to determine the exact field boundaries between the respective solutions, since the


1000

Thin plate solution(2-D heat flow) !(1-D heat flow)

100

.i..

C')c:a, 1.0

0.1

't~

Fig. 1.39. Cooling programme for points located on the weld centre-line ('" = , =0).

10A'tr

Fig. 1.40.Total time spent in a thermal cycle from e through ep to e for a chosen reference temperature ofe = D.5ep-

0.1

1.0 100


mode of heat flow may vary within a single thermal cycle. Hence, the extension of the differ-ent fields is not indicated in the graph. The results in Fig. 1.40 provide a systematic basis forcalculating the retention time within specific temperature intervals under various welding con-ditions.

Isothermal contoursBecause of the number of variables involved, it is not possible to present a two-dimensionalplot of the isotherms without first specifying the dimensionless plate thickness. Examples ofcalculated isotherms in different planes are shown in Figs. 1.41 and 1.42 for 8 equal to 0.5 and5, respectively. It is evident that an increase in the dimensionless plate thickness from 0.5 to 5has a dramatic effect on the shape and position of the isothermal contours. However, in orderto explain these observations in an adequate manner, it is necessary to condense the results intoa two-dimensional diagram. As shown in Fig. 1.43, this can be done by plotting the calculatedfield boundaries in Fig. 1.38(a) at maximum width of the isotherms vs the parameters Op In3and vdl2a.

It is seen from Fig. 1.43 that a large plate thickness generally will favour three-dimensionalheat flow. With decreasing values of Op In3' the conditions for a fast moving high power sourceare approached before the transition from the thick plate to the thin plate solution occurs. Insuch cases the isotherms at the bottom of the plate will be strongly elongated in the welding ~direction and shifted to positions far behind the heat source. The opposite trend is observed atsmall values of vdl2a, since a rapid equalisation of the temperature gradients in the through-thickness direction of the plate will result in elliptical isotherms at both plate surfaces, locatedin an approximately equal distance from the heat source. In either case the temperature atwhich the cross-sectional isotherms approach a semi-circle or become parallel with the ~(z)-axis can be obtained from Fig. 1.43 by reading-off the intercept between the line for thedimensionless plate thickness and the respective field boundaries.

Limitations of the mapsSince the maps have been constructed on the basis of the analytical heat flow equations, it isobvious that they will apply only under conditions for which these equations are valid. Thesimplifying assumptions inherent in the models can be summarised as follows:

(a) The parent material is isotropic and homogeneous at all temperatures,and no phase changes occur on heating.

(b) The thermal conductivity, density, and specific heat are constant andindependent of temperature.

(c) The plate is completely insulated from its surroundings, i.e. there are noheat losses by convection or radiation from the boundaries.

(d) The plate is infinite except in the directions specifically noted.

(e) The electrode is not consumed during welding, and all heat is concen-trated in a zero-volume point or line.


-1 -0.5'I'o 0.5(a)

(b)S

-6 -5 -4 -3 -2 -1 0

I

I

21 ~.75~3~10

6/n3= 1.51 ~0.5

(c)

-6 -5 -4 -3 -2

~ .. -1 o

Fig. 1.41. Computed isothermal contours in different sections for 8 = 0.5;(a) Front view ('JI = 'JIm),(b) Side view ('JI = 0),(c) Top view (~ = 0) and bottom view (~ = 8).

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 67(a)

-10 -5\(I

o 5 10o

5

(b)

-20 -16 -12 -8 -4 o

(c)

~=O 9/n3 =0.05

~--~----~----~--~----~----~--~----~----~--~----~5-20 -16 -12 -4 o

Fig. 1.42. Computed isothermal contours in different sections for B = 5;(a) Front view ('V = 'Vm);(b) Side view ('V = 0);(c) Top view (~ = 0) and bottom view (~ = B).


1000

100Thick plate solution

(3-~ heat flow)

10

t('t) 1.0 Thin plate~ a. solutionc:D

0.1 \1-0 heat flow

Thick plate0.01 solution

(2-D heat flow)

0.0010.01. 0.1 1.0 10 100 1000

8=vd/2a~Fig. 1.43. Heat flow mechanism map showing calculated field boundaries in transverse direction (tV= tVm) of plate vs 6pln3 and 0 = udl2a.

(f) Pseudo-steady state, i.e. the temperature does not vary with time whenobserved from a point located in the heat source ..

In general, the justification of these assumptions relies on a good correlation between theoryand experiments. However, since the analytical solutions ignore the important role of arcenergy distribution and directed metal currents in the weld pool, predictions of the weld ther-mal programme should be restricted to positions well outside the fusion zone where sucheffects are of less importance (to be discussed below).

Example (1.12)

Consider stringer bead welding (GMAW) on a 12mm thick plate of aluminium (> 99% AI)under the following conditions:

I = 260A, U = 25V, 1) = 3mm s", To = 20C

Based on Fig. 1.43, sketch the peak temperature contours in the transverse section of theweld at pseudo-steady state.

HEAT FLOW AND TEMPERATURE DISTRIBUTION IN WELDING 69SolutionIf we neglect the latent heat of melting, the parameter n3 at the chosen reference temperature(T; = Tm) becomes:

260 x 25 x 0.8 x 3 . 1n3 = 4'1T(85)2(1.73) = 0.1, I.e. n3 = 10

Similarly, when v = 3mm S-1 and a = 85mm2 S-l we obtain the following value for the di-mensionless plate thickness:

0= 3x12 =0.212x85

Readings from Fig. 1.43 give:

8p Tp (OC) Model System Comments

0.50 --. 1.0 340 --. 660 Medium thick Heat flow in x and y directions,plate solution partial heat flow in z direction

0.17 --. 0.50 130 --. 340 Thin plate solution Heat flow in x and y directions,(2-D heat flow) negligible heat flow in z direction

(a)


WELD A1y(mm)

..30 ..20 ..10 0 10 20 300.-..E

T( E""-"N

12.5

WELD A2

y(mm)-30 -20 -10 0 10 20 30

0.-..EE"'-"N12.5

70

(b)

Fig. 1.45. Computed peak temperature contours in aluminium welding at pseudo-steady state (Case study1.1); (a) Weld AI, (b) WeldA2. Black regions indicate fusion zone.

Aluminium weldingIn general, the maximum width of the isotherms at the upper and lower surface of the plate canbe obtained from Fig. 1.38(a) and (b), although these maps are not suitable for precise read-ings. A comparison with the computed peak temperature contours in Fig. 1.45(a) and (b)reveals a strong influence of the welding speed on the shape and position of the cross-sectionalisotherms at a constant gross heat input of 1.5 kJ mrrr '. It is evident that the extension of thefusion zone and the neighbouring isotherms becomes considerably larger when the weldingspeed is increased from 2.5 to 5 mm s'. This effect can be attributed to an associated shiftfrom elliptical to more elongated isotherms at the plate surfaces (e.g. see Fig. 1.43), whichreduces heat conduction in the welding direction. It follows from Fig. 1.43 that the upper leftcomer of the map represents the typical operating region for aluminium welding.

Because of the pertinent differences in the heat flow conditions, the temperature-time pat-tern will also vary significantly between the respective series as indicated by the maps in Figs.1.39 and 1.40. Hence, in the case of aluminium welding the usual procedure of reporting arcpower and travel speed in terms of an equivalent heat input per unit length of the bead is highlyquestionable, since this parameter does not define the weld thermal programme. In general,the correct course would be to specify both q0' v and d, and compare the weld thermal historyon the basis of the dimensionless parameters n3 and 8.

Steel weldingIf welding is performed on a steel plate of similar thickness, the operating region will beshifted to the lower right comer of Fig. 1.43. Under such conditions, the isotherms adjacent tothe fusion boundary will be strongly elongated in the x-direction even at very low weldingspeeds (see Fig. 1.42). This implies that the thermal programme approaches a state where thetemperature distribution is uniquely defined by the net heat input l1E, corresponding to the


(a) WELD 81

y(mm)

~.EE.......N

(b) WELD 82

-20 -10y(mm)

o 20...-..r....,...-,.... ....................~ 0

10

........

EE......N

Fig. 1.46. Computed peak temperature contours in steel welding at pseudo-steady state (Case study 1.1);(a) Weld S I, (b) Weld S2. Black regions indicate fusion zone.

Table 1.4 Operational conditions assumed in Case study (1.1).

qo v d E n3 sMaterial Series (W) (mm s") (mm) (kJ mrrr ')

AI-Mg-Si Al 6000 5 12.5 1.5 0.36 0.50alloy A2 3000 2.5 12.5 1.5 0.09 0.25

Low alloy Sl 9600 8 12.5 1.5 32.6 ]0steel S2 4800 4 ]2.5 1.5 8.2 5


limiting case of a fast moving high power source. As a result, the calculated shape and widthof the fusion boundary and neighbouring isotherms are seen to be virtually independent ofchoice of qo and vas illustrated in Fig. 1.46(a) and (b).

1.10.4.2 Experimental verification of the medium thick plate solutionIt is clear from the above discussion that the medium thick plate solution provides a systematicbasis for calculating the temperature distribution within the HAZ of stringer bead weldmentsunder various welding conditions. In the following, the accuracy of the model will be checkedagainst extensive experimental data, as obtained from in situ thermocouple measurements andnumerical analyses of a large number of bead-on-plate welds.

Weld thermal cyclesExamples of measured and predicted weld thermal cycles in aluminium welding are presentedin Fig. 1.47. It is evident that the medium thick plate solution predicts adequately the HAZtemperature-time pattern under different heat flow conditions for fixed values of the peaktemperature. This, in tum, implies that the model is also capable of predicting the total timespent in a thermal cycle within a specific temperature interval as shown in Fig. 1.48.

Weld cooling programmeAt temperatures representative of the austenite to ferrite transformation in mild and low alloysteel weldments, the conditions for a fast moving high power source are normally approachedbefore the transition from thick plate to thin plate welding occurs (see Fig. 1.39). In suchcases, it is possible to present the different solutions for L\'t8/5 (at 'I' = ~= 0) in a single graph byintroducing the following gro

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Grong - Metallurgical Modelling of Welding