Analysis and numerics for a thermomechanical phase transition model in steel [Elektronische Ressource] / Daniela Stefanie Kern. Betreuer: Dietmar Hömberg

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Analysis and numerics for a thermomechanical
phase transition model in steel
vorgelegt von
Diplom-Mathematikerin
Daniela Stefanie Kern
aus Wuppertal
Von der Fakult at II { Mathematik und Naturwissenschaften
der Technischen Universit at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Martin Skutella
Betreuer / Gutachter: Prof. Dr. Dietmar Hom berg
Gutachter: Prof. Dr. Alfred Schmidt (Universit at Bremen)
Tag der wissenschaftlichen Aussprache:
26. Mai 2011
Berlin, 2011
D83Eidesstattliche Versicherung
Hiermit versichere ich an Eides statt, dass die vorliegende Dissertationsschrift mit
dem Titel Analysis and numerics for a thermomechanical phase transition model in
steel von mir selbstandig und ohne Verwendung anderer als der angegebenen Hilfs-
mittel angefertigt wurde. Ferner versichere ich, dass alle Stellen, die im Wortlaut
oder dem Sinn nach aus wissenschaftlichen Publikationen oder anderen Quellen ent-
nommen sind, von mir als solche kenntlich gemacht wurden.
Datum UnterschriftAbstract
This thesis is concerned with the thermomechanical modeling and numerical treat-
ment of metallurgical phase transitions in steel during quenching.
Based on the fundamental principles of thermodynamics, a coupled model con-
sisting of partial and ordinary di erential equations is derived. The constitutive
equations are adapted to the focus of interest which is deformation induced by phase
transitions considered on the macroscale. For the modeling of phase transitions a
mixture approach is chosen leading to a system of ordinary di erential equations.
The deformation caused by phase transitions is included into the model by di erent
expansion coe cients of the respective steel phases and an integral term accumu-
lating deviatoric stresses during transformation, which accounts for transformation
induced plasticity (trip).
Existence and uniqueness results are obtained utilizing xed point arguments
applied for a series of subproblems until nally the complete original equation system
pis solved. The xed point iterations take place in an L -setting, p > 4, to achieve
the self-mapping property of the operator by embedding theorems.
With respect to the numerical treatment, a scheme for a reduced model, which
still captures major e ects and includes transformation induced plasticity, is set up.
The implementation is done within the nite element framework provided by the
toolbox WIAS-pdelib. The single equations are solved sequentially for each time
discretization point, the time stepping is carried out by a semi-implicit approach.
The resulting code is applied to an experimental setup investigated within the
collaborative research center SFB 570 Distortion Engineering in Bremen. The e ect
of inhomogenous quenching strategies on the so-called "out-of-roundness" of roller
bearing rings is investigated and the outcome of the numerical computations is
compared to experimental observations. Motivated by the correspondence of locally
varying heat transfer coe cients and shape alteration of the workpiece, a strategy
for distortion compensation by means of a gradient method obtained from optimal
control theory is introduced.Zusammenfassung
Das Thema dieser Arbeit ist die thermomechanische Modellierung und numerische
Behandlung von metallurgischen Phasenumwandlungen in Stahl beim Abkuhlen.
Auf Grundlage der Hauptsatze der Thermodynamik wird ein gekoppeltes Modell
aus partiellen und gewohnlichen Di erentialgleichungen hergeleitet. Die Material-
gleichungen werden mit Hinblick auf makroskopische Verformungen, die von Pha-
senub ergangen ruhren, formuliert. Fur die Modellierung der Phasenub ergange wird
ein Mischungsansatz gewahlt, der zu einem System von gewohnlichen Di erential-
gleichungen fuhrt. Die Berucksichtigung der transformationsbedingten Verformun-
gen geschieht durch verschiedene Ausdehnungskoe zienten der jeweiligen Phasen
und durch einen Integralterm, der wahrend Phasenubergangen auftretende Scher-
spannungen akkumuliert. Dadurch wird umwandlungsbedingter Plastizitat (up oder
Englisch trip) Rechnung getragen.
Durch Verwendung von Fixpunktargumenten wird Existenz und Eindeutigkeit
fur eine Reihe von ansteigend komplexen Teilproblemen gezeigt bis schlie lich die
Losbarkeit des ursprunglichen Systems resultiert. Die Fixpunktiterationen werden in
pUnterraumen von L , p > 4, betrachtet, um die Selbstabbildungseigenschaft durch
Einbettungssatze gewahrleisten zu konnen.
Fur die numerischen Berechnungen wird ein iteratives Schema zu einem redu-
zierten Modell aufgestellt und mit Hilfe der Finite-Elemente-Werkzeuge des Softwa-
repakets WIAS-pdelib implementiert. Simulationen zu einem Experiment des Bre-
mer Sonderforschungsbereichs SFB 570 Distortion Engineering zur "Unrundheit"
bei Kugellagerringen zeigen die Korrespondenz von lokal variierenden Warmeub er-
gangskoe zienten und Formver anderungen des Werkstucks. Motiviert durch diese
Ergebnisse wird eine Strategie fur die Verzugskompensation im Sinne eines Abstiegs-
verfahrens mit Mitteln der Theorie optimaler Steuerprozesse entworfen.Acknowledgments
The research presented in thesis was supported by the outstanding research con-
ditions given at the Weierstrass Institute for Applied Analysis and Stochastics, at
the DFG research center Matheon, and by the research environment for applied
mathematics in Berlin as a whole.
My gratitude belongs to numerous colleagues who helped and motivated me com-
pleting this work. I am indebted to Prof. Dr. Dietmar H omberg who supervised
this thesis and taught me on mathematical techniques for coupled partial di eren-
tial equations. Apart from sharing his mathematical expertise, he encouraged the
attendance at international conferences and collaborations with di erent experts in
the eld of steel treatments and gave me the opportunity to fully participate in
scienti c life. Cordial thanks go to Prof. Dr. Alfred Schmidt, who did not hesitate
to be a referee for this thesis. I enjoyed the stays at the ZeTeM in Bremen very
much and received valuable hints from him and various colleagues, in particular Dr.
Michael Wol , Dr. Bettina Suhr, and S oren Boettcher. Moreover, I am grateful to
Dr. Thomas Lubb en and Dr. Friedhelm Frerichs from IWT Bremen for explana-
tions regarding experimental setups, the supply with material data, and interesting
discussions regarding engineering aspects of steel treatments. Further, I am obliged
to Prof. Dr. Krzysztof Che lminski from the Technical University of Warsaw for his
good ideas on solving thermomechanical systems for phase transitions.
I am particularly indebted to my colleagues at WIAS who were always available
for discussions. I thank Dr. Wolf Weiss and Oliver Rott, who introduced me into the
world of thermomechanics and its simulation, for their interest and their valuable
thoughts; Prof. Dr. Christian Meyer for sharing his broad mathematical knowledge
of existence and regularity theory, numerical treatment and optimal control of PDEs,
and the willingness and time to give concise and comprehensible explanations; Timo
Streckenbach for providing his expertise regarding pdelib, his practical help, and his
valuable remarks; Dr. Dorothee Knees and Dr. Johannes Elschner for their hints
on elasticity theory. Furthermore, I like to express my sincere gratitude to Thomas
Arnold, Anke Giese, Dr. Guanghui Hu, Norbert Kleemann, Dr. Chantal Landry, Dr.
Lucia Panizzi, Thomas Petzold, Dr. Andreas Steinbrecher, Nataliya Togobytska,
and all the other friendly sh in the pond for their help, fruitful discussions, and
the pleasant time.
Finally, my thanks belong to Heinz-Josef Kurschilgen and Thomas Bruc her, who
encouraged me to learn and study, to my father Bernhard Kern for his trust, and to
my mother Ulrike Kern for the freedom and continuous support of my aims.