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| 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 | Electromagnet ism | 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 Niefien, 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 | ||
| at the 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..........Ill | ||
| 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 /xs 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 and Frank Brückner | 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 Huttner | 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 | |
