Table of Contents
1 Mathematics in Laser Processing John Dowden 1
1.1 Mathematics and its Application 1
1.2 Formulation in Terms of Partial Differential Equations 3
1.2.1 Length Scales 3
1.2.2 Conservation Equations and their Generalisations 4
1.2.3 Governing Equations of Generalised Conservation Type 7
1.2.4 Gauss's Law 10
1.3 Boundary and Interface Conditions 11
1.3.1 Generalised Conservation Conditions 11
1.3.2 The Kinematic Condition in Fluid Dynamics 13
1.4 Fick's Laws 15
1.5 Electromagnetism 15
1.5.1 Maxwell's Equations 15
1.5.2 Ohm's Law 18
References 19
2 Simulation of Laser Cutting Wolfgang Schulz Markus NieBen Urs Eppelt Kerstin Kowalick 21
2.1 Introduction 22
2.1.1 Physical Phenomena and Experimental Observation 23
2.2 Mathematical Formulation and Analysis 26
2.2.1 The One-Phase Problem 29
2.2.2 The Two-Phase Problem 42
2.2.3 Three-Phase Problem 51
2.3 Outlook 64
2.4 Acknowledgements 65
References 65
3 Keyhole Welding: The Solid and Liquid Phases Alexander Kaplan 71
3.1 Heat Generation and Heat Transfer 71
3.1.1 Absorption 71
3.1.2 Heat Conduction and Convection 73
3.1.3 Surface Convection and Radiation 79
3.1.4 Phase Transformations 80
3.1.5 Transient and Pulsed Heat Conduction 80
3.1.6 Geometry of the Liquid Pool 82
3.2 Melt Flow 84
3.2.1 Melt Flow Passing Around the Keyhole 86
3.2.2 Marangoni Flow Driven by Surface Tension Gradients 89
3.2.3 Uncontrolled Violent Melt Motion and Drop Ejection Behind the Keyhole 89
3.2.4 Humping Caused by Accumulating Downstream Flow 90
3.2.5 Stagnation Point for Accelerated Flow, Causing Undercuts and a Central Peak 90
3.2.6 Interior Eddies, Driven by Vertical Downstream Flow atthe Keyhole's Rear Wall 91
3.2.7 Root Drop-out by Gravity and the Keyhole Front Film Ejected by Ablation Pressure 91
3.2.8 Concluding Remarks 92
References 92
4 Laser Keyhole Welding: The Vapour Phase John Dowden 95
4.1 Notation 95
4.2 The Keyhole 95
4.3 The Keyhole Wall 101
4.3.1 The Knudsen Layer 101
4.3.2 Fresnel Absorption 104
4.4 The Role of Convection in the Transfer of Energy to the Keyhole Wall 106
4.5 Fluid Flow in the Keyhole 109
4.5.1 General Aspects 109
4.5.2 Turbulence in the Weld Pool and the Keyhole 111
4.6 Further Aspects of Fluid Flow 113
4.6.1 Simplifying Assumptions for an Analytical Model 113
4.6.2 Lubrication Theory Model 113
4.6.3 Boundary Conditions 114
4.6.4 Solution Matched to the Liquid Region 118
4.7 Electromagnetic Effects 119
4.7.1 Self-Induced Currents in the Vapour 119
4.7.2 The Laser Beam as a Current Guide 123
References 126
5 Basic Concepts of Laser Drilling Wolfgang Schulz Urs Eppelt 129
5.1 Introduction 129
5.2 Technology and Laser Systems 130
5.3 Diagnostics and Monitoring for μs Pulse Drilling 132
5.4 Phenomena of Beam-Matter Interaction 134
5.4.1 Physical Domains - Map of Intensity and Pulse Duration 135
5.4.2 Beam Propagation 142
5.4.3 Refraction and Reflection 143
5.4.4 Absorption and Scattering in the Gaseous Phase 145
5.4.5 Kinetics and Equation of State 146
5.5 Phenomena of the Melt Expulsion Domain 148
5.6 Mathematical Formulation of Reduced Models 149
5.6.1 Spectral Decomposition Applied to Dynamics in Recast Formation 150
5.7 Analysis 151
5.7.1 Initial Heating and Relaxation of Melt Flow 152
5.7.2 Widening of the Drill by Convection 152
5.7.3 Narrowing of the Drill by Recast Formation 154
5.7.4 Melt Closure of the Drill Hole 156
5.7.5 Drilling with Inertial Confinement - Helical Drilling 159
5.8 Outlook 160
5.9 Acknowledgements 161
References 162
6 Arc Welding and Hybrid Laser-Arc Welding Ian Richardson 167
6.1 The Structure of the Welding Arc 167
6.1.1 Macroscopic Considerations 172
6.1.2 Arc Temperatures and the pLTE Assumption 176
6.1.3 Multi-Component Plasmas 182
6.2 The Arc Electrodes 185
6.2.1 The Cathode 186
6.2.2 The Anode 188
6.3 Molten Metal Flow 189
6.3.1 The Arc Generated Weld Pool 189
6.3.2 Metal Transfer 191
6.4 Unified Arc and Electrode Models 193
6.5 Arc Plasma - Laser Interactions 196
6.5.1 Absorption 197
6.5.2 Scattering 202
6.6 Laser-Arc Welding 203
References 210
7 Metallurgy of Welding and Hardening Alexander Kaplan 217
7.1 Thermal Cycle and Cooling Rate 217
7.2 Resolidification 219
7.3 Metallurgy 220
7.3.1 Diffusion 220
7.3.2 Fe-Based Alloys 221
7.3.3 Model of the Metallurgy During Transformation Hardening of Low Alloy Steel 224
7.3.4 Non-Fe-based Alloys 226
7.4 Defects 227
References 233
8 Laser Cladding Dietrich Lepski Frank Brüet;ckner 235
8.1 Introduction 235
8.2 Beam-Particle Interaction 241
8.2.1 Powder Mass Flow Density 241
8.2.2 Effect of Gravity on the Mass Flow Distribution 243
8.2.3 Beam Shadowing and Particle Heating 244
8.3 Formation of the Weld Bead 247
8.3.1 Particle Absorption and Dissolution 248
8.3.2 Shape of the Cross Section of a Weld Bead 249
8.3.3 Three-Dimensional Model of the Melt Pool Surface 251
8.3.4 Temperature Field Calculation using Rosenthal's Solution 253
8.3.5 Self-Consistent Calculation of the Temperature Field and Bead Geometry 255
8.3.6 Role of the Thermocapillary Flow 256
8.4 Thermal Stress and Distortion 259
8.4.1 Fundamentals of Thermal Stress 259
8.4.2 Phase Transformations 261
8.4.3 FEM Model and Results 263
8.4.4 Simplified Heuristic Model 265
8.4.5 Crack Prevention by Induction Assisted Laser Cladding 270
8.5 Conclusions and Future Work 274
References 276
9 Laser Forming Thomas Pretorius 281
9.1 History of Thermal Forming 281
9.2 Forming Mechanisms 284
9.2.1 Temperature Gradient Mechanism 285
9.2.2 Residual Stress Point Mechanism 292
9.2.3 Upsetting Mechanism 294
9.2.4 Buckling Mechanism 299
9.2.5 Residual Stress Relaxation Mechanism 303
9.2.6 Martensite Expansion Mechanism 304
9.2.7 Shock Wave Mechanism 305
9.3 Applications 306
9.3.1 Plate Bending 307
9.3.2 Tube Bending/Forming 308
9.3.3 High Precision Positioning Using Actuators 309
9.3.4 Straightening of Weld Distortion 310
9.3.5 Thermal Pre-Stressing 311
References 312
10 Femtosecond Laser Pulse Interactions with Metals Bernd Hüet;ttner 315
10.1 Introduction 315
10.2 What is Different Compared to Longer Pulses? 317
10.2.1 The Electron-Electron Scattering Time 317
10.2.2 The Nonequilibrium Electron Distribution 320
10.3 Material Properties Under Exposure to Femtosecond Laser Pulses 322
10.3.1 Optical Properties 322
10.3.2 Thermal Properties 325
10.3.3 Electronic Thermal Diffusivity 327
10.4 Determination of the Electron and Phonon Temperature Distribution 328
10.4.1 The Two-Temperature Model 328
10.4.2 The Extended Two-Temperature Model 330
10.5 Summary and Conclusions 334
References 335
11 Comprehensive Numerical Simulation of Laser Materials Processing Markus Gross 339
11.1 Motivation - The Pursuit of Ultimate Understanding 339
11.2 Review 341
11.3 Correlation, the Full Picture 348
11.4 Introduction to Numerical Techniques 348
11.4.1 The Method of Discretisation 349
11.4.2 Meshes 349
11.4.3 Explicit versus Implicit 350
11.4.4 Discretisation of Transport pde's 351
11.4.5 Schemes of Higher Order 354
11.4.6 The Multi Phase Problem 356
11.5 Solution of the Energy Equation and Phase Changes 359
11.5.1 Gas Dynamics 362
11.5.2 Beam Tracing and Associated Difficulties 364
11.6 Program Development and Best Practice when Using Analysis Tools 367
11.7 Introduction to High Performance Computing 368
11.7.1 MPI 369
11.7.2 openMP 371
11.7.3 Performance 372
11.8 Visualisation Tools 374
11.9 Summary and Concluding Remarks 375
References 375
Index 381
