The Theory of Laser Materials Processing (eBook)

Prof. John Dowden was educated at Bedford School and Cambridge University, UK, where he graduated with a First Class degree in Mathematics in 1962. He became the first student of the new University of Essex obtaining a PhD in Mathematical Oceanography in 1967. He was appointed to the staff of the Mathematics Department of the university and subsequently changed his main research interests to the mathematics and physics of laser technology while retaining interests in mathematically related applications of heat and mass transfer. Before retirement he was Head of the university’s Department of Mathematical Sciences, a member of the Institute of Physics and of the Laser Institute of America. He is still a Fellow of the Institute of Mathematics and its Applications and is now an Emeritus Professor of the University.  Prof. Dr. Wolfgang Schulz studied physics at Braunschweig University of Technology. He graduated from the Institute for Theoretical Physics and received a postgraduate scholarship in 1986 on the topic of "Hot electrons in metals". In 1987, he accepted an invitation to the department Laser Technology at RWTH Aachen University. He received the "Borchers Medal" award in 1992 in recognition of his PhD thesis. In 1997, he joined the Fraunhofer Institute for Laser Technology in Aachen and, in 1999, received the "Venia Legendi" in the field "Principles of Continuum Physics applied to Laser Technology". His postdoctoral lecture qualification (habilitation) was awarded with distinction in 1999 with the prize of the Friedrich-Wilhelm Foundation at RWTH Aachen University. Since March 2005, he has represented the newly founded department "Nonlinear Dynamics of Laser Processing" at RWTH Aachen University and is the head of the newly founded department of "Modelling and Simulation" at the Fraunhofer Institute for Laser Technology in Aachen. Since 2007, he is the coordinator of the Excellence Cluster Domain "Virtual Production" at RWTH Aachen University. His current work is focused on developing and improving laser systems and their industrial applications by combination of mathematical, physical and experimental methods. In particular, he applies the principles of optics, continuum physics and thermodynamics to analyse the phenomena involved in laser processing. The mathematical objectives are modelling, analysis and dynamical simulation of Free Boundary Problems, which are systems of nonlinear partial differential equations. Analytical and numerical methods for model reduction are developed and applied. The mathematical analysis yields approximate dynamical systems of small dimensions in the phase space and is based on asymptotic properties such as the existence of inertial manifolds.

Preface 6
Contents 9
Contributors 16
1 Mathematics in Laser Processing 17
Abstract 17
1.1 Mathematics and Its Application 17
1.2 Formulation in Terms of Partial Differential Equations 19
1.2.1 Length Scales 19
1.2.2 Rectangular Cartesian Tensors 20
1.2.3 Conservation Equations and Their Generalisations 23
1.2.4 Governing Equations of Generalised Conservation Type 25
1.2.4.1 Flow of a Viscous Fluid 25
1.2.4.2 Viscous Heat Flow 27
1.2.4.3 Conservation of Electric Charge 28
1.2.4.4 Linear Thermo-Elasticity in a Moving Frame of Reference 28
1.2.5 Gauss’s Law 29
1.3 Boundary and Interface Conditions 30
1.3.1 Generalised Conservation Conditions 30
1.3.2 The Kinematic Condition in Fluid Dynamics 35
1.4 Fick’s Laws 36
1.5 Electromagnetism 37
1.5.1 Maxwell’s Equations 37
1.5.2 Ohm’s Law 39
References 40
2 Simulation of Laser Cutting 41
2.1 Introduction 42
2.1.1 Physical Phenomena and Experimental Observation 44
2.2 Mathematical Formulation and Analysis 47
2.2.1 The One-Phase Problem 49
2.2.2 The Two-Phase Problem 62
2.2.3 Three-Phase Problem 70
2.3 Outlook 84
References 85
3 Glass Cutting 89
Abstract 89
3.1 Introduction 90
3.2 Phenomenology of Glass Processing with Ultrashort Laser Radiation 90
3.3 Modelling the Propagation of Radiation and the Dynamics of Electron Density 92
3.4 Radiation Propagation Solved by BPM Methods 93
3.5 The Dynamics of Electron Density Described by Rate Equations 93
3.6 Properties of the Solution with Regard to Ablation and Damage 95
3.7 Electronic Damage Versus Thermal Damage 98
3.8 Glass Cutting by Direct Ablation or Filamentation? 102
Acknowledgements 103
References 103
4 Keyhole Welding: The Solid and Liquid Phases 105
Abstract 105
4.1 Heat Generation and Heat Transfer 105
4.1.1 Absorption 105
4.1.2 Heat Conduction and Convection 107
4.2 Steady State 3D-Solutions Based on Moving Point Sources of Heat 108
4.3 Steady State 2D-Heat Conduction from a Moving Cylinder at Constant Temperature 110
4.4 Sophisticated Quasi-3D-Model Based on the Moving Line Source of Heat 112
4.4.1 Surface Convection and Radiation 113
4.4.2 Phase Transformations 114
4.4.3 Transient and Pulsed Heat Conduction 115
4.5 Model for Initiation of Laser Spot Welding 115
4.5.1 Geometry of the Liquid Pool 117
4.6 Mass Balance of a Welding Joint 118
4.7 Melt Flow 119
4.7.1 Melt Flow Passing Around the Keyhole 120
4.7.2 Numerical 2D-Simulation of the Melt Flow Around a Prescribed Keyhole Shape 121
4.7.3 Marangoni Flow Driven by Surface Tension Gradients 123
4.7.4 Flow Redirection, Inner Eddies, Spatter and Stagnation Points 124
4.7.5 Humping Caused by Accumulating Downstream Flow 125
4.7.6 Keyhole Front Melt Film Flow Downwards, Driven by Recoil Pressure 125
4.8 Concluding Remarks 126
References 127
5 Laser Keyhole Welding: The Vapour Phase 129
Abstract 129
5.1 Notation 129
5.2 The Keyhole 131
5.3 The Keyhole Wall 135
5.3.1 The Knudsen Layer 135
5.3.1.1 Ablation Through the Knudsen Layer 135
5.3.1.2 Thermal Flux and Viscous Slip in the Knudsen Layer 138
5.3.2 Fresnel Absorption 139
5.4 The Role of Convection in the Transfer of Energy to the Keyhole Wall 140
5.5 Fluid Flow in the Keyhole 144
5.5.1 General Aspects 144
5.5.2 Turbulence in the Weld Pool and the Keyhole 145
5.5.3 Stability of the Keyhole Wall 147
5.5.4 Stability of Waves of Acoustic Type 147
5.5.5 Elongation of the Keyhole 151
5.6 Further Aspects of Fluid Flow 152
5.6.1 Simplifying Assumptions for an Analytical Model 152
5.6.2 Lubrication Theory Model 152
5.6.3 Boundary Conditions 153
5.6.4 Solution Matched to the Liquid Region 157
5.7 Electromagnetic Effects 158
5.7.1 Self-induced Currents in the Vapour 158
5.7.2 The Laser Beam as a Current Guide 163
5.7.2.1 Note on Cooling by Thermal Convection 165
References 165
6 Basic Concepts of Laser Drilling 168
6.1 Introduction 169
6.2 Technology and Laser Systems 169
6.3 Diagnostics and Monitoring for s Pulse Drilling 171
6.4 Phenomena of Beam-Matter Interaction 173
6.4.1 Physical Domains---Map of Intensity and Pulse Duration 174
6.4.2 Beam Propagation 180
6.4.3 Refraction and Reflection 182
6.4.4 Absorption and Scattering in the Gaseous Phase 183
6.4.5 Kinetics and Equation of State 184
6.5 Phenomena of the Melt Expulsion Domain 186
6.6 Mathematical Formulation of Reduced Models 188
6.6.1 Spectral Decomposition Applied to Dynamics in Recast Formation 189
6.7 Analysis 190
6.7.1 Initial Heating and Relaxation of Melt Flow 190
6.7.2 Widening of the Drill by Convection 192
6.7.3 Narrowing of the Drill by Recast Formation 193
6.7.4 Melt Closure of the Drill Hole 195
6.7.5 Drilling with Inertial Confinement---Helical Drilling 197
6.8 Outlook 199
References 200
7 Arc Welding and Hybrid Laser-Arc Welding 204
Abstract 204
7.1 The Structure of the Welding Arc 204
7.1.1 Macroscopic Considerations 205
7.1.2 Arc Temperatures and the PLTE Assumption 215
7.1.3 Multi-component Plasmas 221
7.2 The Arc Electrodes 224
7.2.1 The Cathode 224
7.2.2 The Anode 226
7.3 Fluid Flow in the Arc-Generated Weld Pool 227
7.4 Unified Arc and Electrode Models 230
7.5 Arc Plasma-Laser Interactions 233
7.5.1 Absorption 234
7.5.2 Scattering 239
7.5.3 Absorption Measurements 241
7.6 Laser-Arc Hybrid Welding 242
References 250
8 Metallurgy and Imperfections of Welding and Hardening 255
Abstract 255
8.1 Thermal Cycle and Cooling Rate 255
8.2 Resolidification 258
8.3 Metallurgy 259
8.3.1 Diffusion 259
8.3.2 Fe-Based Alloys 261
8.3.2.1 Low Alloy Steel 261
8.3.3 Model of the Metallurgy During Transformation Hardening of Low Alloy Steel 263
8.3.4 Non-Fe-Based Alloys 265
8.4 Imperfections 266
8.4.1 Large Geometrical Imperfections 267
8.4.2 Cracks 268
8.4.3 Spatter 269
8.4.4 Pores and Inclusions 270
References 274
9 Laser Cladding 276
Abstract 276
9.1 Introduction 276
9.2 Beam-Particle Interaction 283
9.2.1 Powder Mass Flow Density 283
9.2.2 Effect of Gravity on the Mass Flow Distribution 284
9.2.3 Beam Shadowing and Particle Heating 286
9.3 Formation of the Weld Bead 289
9.3.1 Particle Absorption and Dissolution 290
9.3.2 Shape of the Cross Section of a Weld Bead 291
9.3.3 Three-dimensional Model of the Melt Pool Surface 293
9.3.4 Temperature Field Calculation Using Rosenthal’s Solution 294
9.3.5 Self-consistent Calculation of the Temperature Field and Bead Geometry 296
9.3.6 Role of the Thermocapillary Flow 297
9.4 Thermal Stress and Distortion 300
9.4.1 Fundamentals of Thermal Stress 300
9.4.2 Phase Transformations 302
9.4.3 FEM Model and Results 304
9.4.4 Simplified Heuristic Model 305
9.4.5 Crack Prevention by Induction Assisted Laser Cladding 311
9.5 Conclusions and Future Work 314
References 316
10 Laser Forming 320
Abstract 320
10.1 History of Thermal Forming 322
10.2 Forming Mechanisms 323
10.2.1 Temperature Gradient Mechanism 324
10.2.2 Residual Stress Point Mechanism 331
10.2.3 Upsetting Mechanism 333
10.2.4 Buckling Mechanism 338
10.2.5 Residual Stress Relaxation Mechanism 342
10.2.6 Martensite Expansion Mechanism 343
10.2.7 Shock Wave Mechanism 344
10.3 Applications 345
10.3.1 Plate Bending 346
10.3.2 Tube Bending/Forming 347
10.3.3 High Precision Positioning Using Actuators 348
10.3.4 Straightening of Weld Distortion 349
10.3.5 Thermal Pre-stressing 350
References 351
11 Femtosecond Laser Pulse Interactions with Metals 354
Abstract 354
11.1 Introduction 354
11.2 What Is Different Compared to Longer Pulses? 356
11.2.1 The Electron-Electron Scattering Time 356
11.2.2 The Nonequilibrium Electron Distribution 359
11.3 Material Properties Under Exposure to Femtosecond Laser Pulses 361
11.3.1 Optical Properties 361
11.3.2 Thermal Properties 363
11.3.3 Electronic Thermal Diffusivity 365
11.4 Determination of the Electron and Phonon Temperature Distribution 366
11.4.1 The Two-Temperature Model 366
11.4.2 The Extended Two-Temperature Model 369
11.5 Summary and Conclusions 372
References 373
12 Meta-Modelling and Visualisation of Multi-dimensional Data for Virtual Production Intelligence 375
Abstract 375
12.1 Introduction 375
12.2 Implementing Virtual Production Intelligence 377
12.3 Meta-Modelling Providing Operative Design Tools 378
12.4 Meta-Modelling by Smart Sampling with Discontinuous Response 384
12.5 Global Sensitivity Analysis and Variance Decomposition 389
12.6 Reduced Models and Emulators 392
12.7 Summary and Advances in Meta-Modelling 393
Acknowledgements 393
References 394
13 Comprehensive Numerical Simulation of Laser Materials Processing 396
13.1 Motivation---The Pursuit of Ultimate Understanding 397
13.2 Review 398
13.3 Correlation, The Full Picture 404
13.4 Introduction to Numerical Techniques 405
13.4.1 The Method of Discretisation 405
13.4.2 Meshes 406
13.4.3 Explicit Versus Implicit 407
13.4.4 Discretisation of Transport pde's 408
13.4.5 Schemes of Higher Order 411
13.4.6 The Multi Phase Problem 413
13.5 Solution of the Energy Equation and Phase Changes 416
13.5.1 Gas Dynamics 419
13.5.2 Beam Tracing and Associated Difficulties 421
13.6 Program Development and Best Practice When Using Analysis Tools 423
13.7 Introduction to High Performance Computing 425
13.7.1 MPI 425
13.7.2 openMP 427
13.7.3 Hybrid 428
13.7.4 Performance 429
13.8 Visualisation Tools 430
13.9 Summary and Concluding Remarks 431
References 432
Index 437