The Theory of Laser Materials Processing: Heat and Mass Transfer in Modern Technology

442 Pages·2017·12.003 MB·English

by  John Dowden, Wolfgang Schulz (eds.)

Springer Series in Materials Science 119
John Dowden
Wolfgang Schulz E ditors
The Theory of
Laser Materials
Processing
Heat and Mass Transfer in Modern
Technology
Second Edition
Springer Series in Materials Science
Volume 119
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⋅
John Dowden Wolfgang Schulz
Editors
The Theory of Laser
Materials Processing
Heat and Mass Transfer in Modern
Technology
Second Edition
123
Editors
John Dowden WolfgangSchulz
Mathematical Sciences Fraunhofer-Institut für Lasertechnik
University of Essex RWTH Aachen
Colchester, Essex Aachen
UK Germany
ISSN 0933-033X ISSN 2196-2812 (electronic)
SpringerSeries inMaterials Science
ISBN978-3-319-56710-5 ISBN978-3-319-56711-2 (eBook)
DOI 10.1007/978-3-319-56711-2
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Preface
Theuseoflasersinmaterialsprocessinghasbecomewidespreadinrecentyears,so
that an understanding of the nature of heat and mass transfer in this branch of
moderntechnologyisofincreasingimportance.Theaimoftheauthorsofthisbook
is to concentrate on the physical processes; these can be developed from a math-
ematicalpointofview,orfromdirectexperimentallyderivedobservation.Thetwo
approaches are complementary; each can provide insights and the synthesis
of the two can lead to a very powerful understanding of the processes involved.
Mathematical modellingofphysicalprocesseshashadanimportantrole toplayin
the development of technology over the centuries and particularly so in the last
150 years or so. It can be argued that it is more important today than ever before
sincetheavailabilityofhigh-speedcomputersallowsaccuratenumericalsimulation
of industrial processes at a fraction of the cost of the corresponding experiments.
Thisisoneaspectofmathematicalmodelling,highprofileandmuchvalued,butit
is not the only one.
Inthepastmathematicalmodellinghadtorelyonqualitativeinvestigation,very
special analytical solutions, or inaccurate and time-consuming calculations per-
formed with littleinthe way oftabulated ormechanical assistance.Log tablesand
slide rules are still remembered by people working today, though there are surely
few who regret their disappearance.
The value and distinctive function of methods based on the analytical approach
is now becoming much clearer, now that they are no longer expected to produce
detailed imitations of what happens in real experiments of industrial processes, a
function now fulfilled mostly by numerical methods, considered below. The
emphasis today is on their ability to confirm and extend our understanding of the
basicphysicalmechanismsinvolvedintheprocessesofinterest.Theseareessential
for any intelligent use of numerical simulation.
The argument about the value of teaching people how to do arithmetic them-
selves without the aid of a calculator seems to be passing into history, but it is an
important one and provides a simple analogy. If someone does not have a feeling
for numbers and the way arithmetic works, they will all too easily fail to spot an
error produced by a machine. Computers are not infallible—and neither are those
v
vi Preface
who build or program them. Computers are now taking on less mundane mathe-
maticaltasksandthesamecontroversiesareappearinginconnectionwithalgebraic
manipulation.Equally,andwithevengreaterpenaltiesintermsofcostintheevent
oferrors,thesameconsiderationsapplytonumericalsimulationofmajorindustrial
processes.Awarenessoftheanalyticalsolutionscanbeinvaluableindistinguishing
therightfromthewrong,i.e.forthepractitionertounderstandthebasisofthework,
and to have an idea of the kinds of outcomes that are plausible—and to recognise
those which are not.
Thephrasemathematicalmodellingis,however,ambiguous,perhapsmorenow
than it has ever been. There is an enormous amount of work done today on sim-
ulation based on the use of very powerful computer programs, and it is quite
correctly referred to as mathematical modelling. The programs are sometimes
constructedin-housebutareusuallycommercialpackages.Thisisanentirelyvalid
approach with specific (generally commercial) objectives. In general there are two
uses. The dominant objective is initially numerical agreement with a particular
experiment, leading subsequently to predictive commercial use. The second
objective is the clarification of physical mechanisms, aimed at the generation of
understanding of complex interconnected processes, rather than the exact repro-
duction of a particular experiment. It is sometimes overlooked that, with sufficient
care, a numerical approach is equally valid in the investigation of physical fun-
damentals. Numericalsimulation isnotacentraltopic ofthis book,butbecause of
itscrucialimportancetoeachofthetwousestowhichnumericalmodellingcanbe
put,itisvitalthatthecomputationalbasisoftheworkshouldbecompletelysound.
In addition, the level of process detail which can be considered by the numerical
approach usually exceeds what is possible with the analytical approach by a sig-
nificant amount, leaving little choice but torevert tothe numerical treatmentwhen
investigatingtheinterconnectionsbetweenprocesses.Itisforthesereasonsthatthe
book concludes with a chapter on comprehensive numerical simulation.
In many ways, the approach adopted here is complementary to the more phe-
nomenological approach. It is always important in a field which has very direct
industrialapplicationstobearinmindhowtechniquessuchasthosedescribedhere
will be used, but it is essential not to lose sight of the fundamentals. There are
serioussafetyimplications;therearecostimplications;therearemoralimplications;
there areconsiderationsoftheappropriatenessofthe technology totheapplication
underconsideration.Aproperrespectforalltheserequiresanunderstandingofthe
fundamentals.
This second edition has been revised and updated, and two extra chapters have
been added, one on the use of lasers to cut glass, and the other on the concept of
Meta-Modelling.
Itisoneoftheproblemsofmodelbuildinginscienceandtechnology,whatever
theactualapplicationofamodelistobe,thatthere canbeanuncomfortablylarge
gap between the theoretical background and the desired outcomes of the model.
There is a danger of the model becoming excessively complicated and unman-
ageable, when it should be possible to obtain what is desired more simply by the
intelligent use of model reduction. This can be the key to successful numerical
Preface vii
implementation.Theaimistoavoidanyunnecessarycomplexityandmakeiteasier
to control error of whatever kind. Any mathematical–physical model is capable of
suffering from unsatisfactory “reduction” from an excessively complex compre-
hensive description. Simplification can be achieved in a number of ways. These
tend to fall into the following categories:
(cid:129) phenomenological methods (e.g. rate equations with phenomenological
coefficients);
(cid:129) mathematical–physical methods (e.g. Buckingham’s Π-Theorem, asymptotic
analysis, singular perturbation);
(cid:129) numericalmethods(e.g.properorthogonaldecomposition,principalcomponent
analysis);
(cid:129) data-driven methods (e.g. meta-modelling, design of experiments).
The concept of meta-modelling is still rather unfamiliar, but it is a concept with
great potential in many fields and certainly not least in laser technology.
Wearealltoowellawarethatthisbookdoeslittlemorethanscratchthesurface
of the problems involved in a fundamental understanding of the phenomena
involved in laser materials processing and the ways in which theory and practice
can interact. If we have provided ideas and information that cause others to test
them experimentally or intellectually, agree with them or dispute them vigorously,
and develop them further, we will consider that we have achieved our aim.
Colchester, UK John Dowden
Aachen, Germany Wolfgang Schulz
Contents
1 Mathematics in Laser Processing... .... .... .... .... ..... .... 1
John Dowden
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 Rectangular Cartesian Tensors... .... .... ..... .... 4
1.2.3 Conservation Equations and Their Generalisations .... 7
1.2.4 Governing Equations of Generalised
Conservation Type............................... 9
1.2.5 Gauss’s Law .... .... .... .... .... .... ..... .... 13
1.3 Boundary and Interface Conditions.. .... .... .... ..... .... 14
1.3.1 Generalised Conservation Conditions . .... ..... .... 14
1.3.2 The Kinematic Condition in Fluid Dynamics .... .... 19
1.4 Fick’s Laws .. .... ..... .... .... .... .... .... ..... .... 20
1.5 Electromagnetism.. ..... .... .... .... .... .... ..... .... 21
1.5.1 Maxwell’s Equations.. .... .... .... .... ..... .... 21
1.5.2 Ohm’s Law..... .... .... .... .... .... ..... .... 23
References. .... .... .... ..... .... .... .... .... .... ..... .... 24
2 Simulation of Laser Cutting... .... .... .... .... .... ..... .... 25
Wolfgang Schulz, Markus Nießen, Urs Eppelt
and Kerstin Kowalick
2.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 26
2.1.1 Physical Phenomena and Experimental Observation ... 28
2.2 Mathematical Formulation and Analysis.. .... .... ..... .... 31
2.2.1 TheOne-PhaseProblem ............................ 33
2.2.2 The Two-Phase Problem... .... .... .... ..... .... 46
2.2.3 Three-Phase Problem.. .... .... .... .... ..... .... 54
2.3 Outlook.. .... .... ..... .... .... .... .... .... ..... .... 68
References. .... .... .... ..... .... .... .... .... .... ..... .... 69
ix
x Contents
3 Glass Cutting.. .... .... ..... .... .... .... .... .... ..... .... 73
Wolfgang Schulz
3.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 74
3.2 Phenomenology of Glass Processing with Ultrashort Laser
Radiation .... .... ..... .... .... .... .... .... ..... .... 74
3.3 Modelling the Propagation of Radiation and the Dynamics
of Electron Density. ..... .... .... .... .... .... ..... .... 76
3.4 Radiation Propagation Solved by BPM Methods ... ..... .... 77
3.5 The Dynamics of Electron Density Described
by Rate Equations.. ..... .... .... .... .... .... ..... .... 77
3.6 Properties of the Solution with Regard to Ablation
and Damage .. .... ..... .... .... .... .... .... ..... .... 79
3.7 Electronic Damage Versus Thermal Damage .. .... ..... .... 82
3.8 Glass Cutting by Direct Ablation or Filamentation?. ..... .... 86
References. .... .... .... ..... .... .... .... .... .... ..... .... 87
4 Keyhole Welding: The Solid and Liquid Phases... .... ..... .... 89
Alexander Kaplan
4.1 Heat Generation and Heat Transfer.. .... .... .... ..... .... 89
4.1.1 Absorption. ..... .... .... .... .... .... ..... .... 89
4.1.2 Heat Conduction and Convection .... .... ..... .... 91
4.2 Steady State 3D-Solutions Based on Moving Point Sources
of Heat.. .... .... ..... .... .... .... .... .... ..... .... 92
4.3 Steady State 2D-Heat Conduction from a Moving Cylinder
at Constant Temperature.. .... .... .... .... .... ..... .... 94
4.4 Sophisticated Quasi-3D-Model Based on the Moving
Line Source of Heat..... .... .... .... .... .... ..... .... 96
4.4.1 Surface Convection and Radiation.... .... ..... .... 97
4.4.2 Phase Transformations. .... .... .... .... ..... .... 98
4.4.3 Transient and Pulsed Heat Conduction .... ..... .... 99
4.5 Model for Initiation of Laser Spot Welding ... .... ..... .... 99
4.5.1 Geometry of the Liquid Pool.... .... .... ..... .... 101
4.6 Mass Balance of a Welding Joint... .... .... .... ..... .... 102
4.7 Melt Flow.... .... ..... .... .... .... .... .... ..... .... 103
4.7.1 Melt Flow Passing Around the Keyhole ... ..... .... 104
4.7.2 Numerical 2D-Simulation of the Melt Flow Around
a Prescribed Keyhole Shape .... .... .... ..... .... 105
4.7.3 Marangoni Flow Driven by Surface
Tension Gradients............................... 107
4.7.4 Flow Redirection, Inner Eddies, Spatter
and Stagnation Points . .... .... .... .... ..... .... 108
4.7.5 Humping Caused by Accumulating
Downstream Flow............................... 109