Theory and numerics of non-classical thermo-hyperelasticity [Elektronische Ressource] / Swantje Bargmann
193 Pages
English

Theory and numerics of non-classical thermo-hyperelasticity [Elektronische Ressource] / Swantje Bargmann

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Published 01 January 2008
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Swantje Bargmann
Theory and numerics of
non-classical thermo-hyperelasticity
UKL/LTM T 08-02 Mai 2008
Lehrstuhl für Technische MechanikBibliografische Information Der Deutschen Bibliothek
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Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.ddb.de.
Farbig erschienen auf http://kluedo.ub.uni-kl.de
Herausgeber: Fachbereich Maschinenbau und Verfahrenstechnik
Lehrstuhl für Technische Mechanik
Postfach 3049
Technische Universität Kaiserslautern
D-67653 Kaiserslautern
Verlag: Technische Universität Kaiserslautern
Druck: Technische Universität Kaiserslautern
ZBT – Abteilung Foto-Repro-Druck
D-386
© by Swantje Bargmann 2008
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concern-
ed, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction
on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts there of
is permitted in connection with reviews or scholarly analysis. Permission for use must always be obtained
from the author.
Alle Rechte vorbehalten, auch das des auszugsweisen Nachdrucks, der auszugsweisen oder vollständigen
Wiedergabe (Photographie, Mikroskopie), der Speicherung in Datenverarbeitungsanlagen und das der
Übersetzung.
Als Manuskript gedruckt. Printed in Germany.
ISSN 1610-4641
ISBN 978-3-939432-77-7Theory and numerics of non-classical
thermo-hyperelasticity
vom Fachbereich Maschinenbau und Verfahrenstechnik
der Technischen Universit¨at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor–Ingenieur (Dr.–Ing.)
genehmigte Dissertation
von Dipl.-Math. Swantje Bargmann
aus Kaiserslautern
Hauptreferent: Prof. Dr.-Ing. habil. P. Steinmann
Korreferenten: Prof. Dr. rer. nat. B. Svendsen
Prof. Dr. V. Kalpakides
Vorsitzender: Prof. Dr. techn. H.-J. Bart
Tag der Einreichung: 17.12.2007
Tag der mundl.¨ Pr¨ufung: 09.05.2008
Kaiserslautern, Mai 2008
D 386Preface
The research presented in this thesis has been carried out at the Chair of Applied Mechanics
at the University of Kaiserslautern, Germany, during the period 2004 to 2007. The finan-
cial support of the DFG (Deutsche Forschungsgemeinschaft) within the project “Theory and
numerics of non-classical thermoelasticity” (STE 544/23-1-2) is gratefully acknowledged.
First of all, I would like to thank Professor Paul Steinmann for giving me the opportunity to
work in the fascinating field of non-classical thermoelasticity, for his support and for providing
me with inspiring advice through all these years.
Moreover, I would lke to express my appreciation to Professor Bob Svendsen and Professor
Vassilios Kalpakides for kindly accepting to be correferees of this thesis. Thanks for useful
suggestions, the careful reading and the interest in my work. Also, I’m very grateful to
Professor Hans-J¨org Bart for taking the chairmanship of the examination committee.
Moreover, I would like to express my gratitude to my fellow colleagues at the Chair of Applied
Mechanics. The discussions I have had with them have been fruitful, inspirational and have
facilitated my research. They have also helped a great deal to gain a better insight into the
world of mechanics. In addition, I benefited from engineers, mathematicians and physicists
who had impact on this work during research stays, conferences and personal discussions, for
example.
Last but not least, I would like to thank all those who have contributed in one way or another
in their very personal ways.
Kaiserslautern, May 2008 Swantje Bargmann
IContents
Nomenclature VIII
1. Introduction 5
1.1. ThetheoryofGreenandNaghdi .... ........... ......... 6
1.2. Structureofthepresentwork ...... 6
2. State of the art 9
3. Continuum mechanics 13
3.1. Kinematics ..... ........... ........... ......... 13
3.1.1. Spatialmotionproblem ..... 13
3.1.2. Materialmotionproblem 15
3.2. Mechanicalbalanceequations ...... ......... 16
3.2.1. Spatialmotionproblem ..... ........... 17
3.2.2. Materialmotionproblem 19
3.3. Materialmodeling . ........... ......... 19
4. Thermodynamic principles 21
4.1. Entropy principle of Mu¨llerandLiu ... ........... ......... 21
4.1.1. ThermodynamicalanalysisofGreen–NaghditypeI . . 23
4.1.2. ThermodynamicalanalysisofGreen–NaghditypeII.. 28
4.1.3. ThermodynamicalanalysisofGreen–NaghditypeIII . ......... 32
4.2. EntropyexploitationaccordingtoGreenandNaghdi ..... 34
4.2.1. TypeI ... ........... ........... 35
4.2.2. TypeII ... ......... 37
4.2.3. TypeIII... 38
5. Theory of non-classical heat conduction 41
5.1. Governingconstitutiveequations .... ........... ......... 41
5.1.1. TypeI ... ........... 41
5.1.2. TypeII ... 42
5.1.3. TypeIII... ......... 44
5.2. Finiteelementdiscretization ....... ........... 44
5.2.1. Spatialfiniteelementdiscretization .......... 45
5.2.2. Temporalfiniteelementdiscretization ......... ......... 47
5.2.2.1. Discontinuousfiniteelementmethod .... 49
IIIContents
5.2.2.2. Continuousfiniteelementmethod ..... ......... 51
5.3. Numericalresults.. ........... ........... 53
6. Theory of non-classical thermo-hyperelasticity 67
6.1. Governingconstitutiveequations .... ......... 67
6.1.1. TypeI ... ........... ........... 67
6.1.2. TypeII ... 68
6.1.3. TypeIII... ......... 69
6.2. Finiteelementdiscretization ....... 70
6.2.1. Spatialdiscretization ........... 70
6.2.1.1. Spatialdiscretization . ......... 72
6.2.2. Temporaldiscretization ..... 74
6.2.2.1. MixedGalerkintimefiniteelementmethod . 76
6.2.2.2. ContinuousGalerkintimefiniteelementmethod ....... 76
6.3. Newton–Raphsonsolutionmethod.... ........... ......... 77
6.4. Numericalexamples ........... 78
6.4.1. Secondsoundphenomenon ... 78
6.4.1.1. Bismuth(Bi) ..... ......... 79
6.4.1.2. SodiumFluoride(NaF) ........... 84
6.4.2. CryovolcanismonEnceladus ... 88
6.4.2.1. Set-upformodelingcryovolcanismonEnceladus ....... 91
6.4.2.2. Results......... ......... 93
6.4.3. Materialforcemethod ...... ........... 100
6.5. EnergyconservingfiniteelementdiscretizationintimefortypeII ....... 120
6.5.1. Energyconsistentalgorithmicstresstensormethod. . ......... 120
6.5.2. Numericalexample........ 123
7. Incremental variational formulation 127
7.1. Preliminaries .... ........... ........... ......... 127
7.1.1. Continuoussetting ........ 127
7.1.2. Incrementalsetting 128
7.1.2.1. TypeI ......... ......... 130
7.1.2.2. TypeII ........... 132
7.1.3. Numericalexample........ 133
7.1.3.1. TypeI ......... ......... 133
7.1.3.2. TypeII 138
8. On wave speed in Green–Naghdi heat conduction 141
8.1. Mathematicalformulation ........ ........... ......... 141
8.2. Analyticalsolutions ........... 142
8.2.1. TypeIII... 142
8.2.2. TypeII ... ......... 145
8.2.3. TypeI ... ........... 145
8.3. Results ....... ........... 147
IVContents
9. Conclusion 151
A. Mathematics 153
A.1. Notation ...... ........... ........... ......... 153
A.2. Definitions ..... ......... 153
A.3. Jumpconditions . . 155
A.4. Symmetryandskew-symmetryfortwo-pointtensors ..... 155
A.5. Transformations . . ........... ........... ......... 156
A.6. Calculationofabsoluteentropy ..... 156
A.7. Cauchy’sstresstheorem ......... 157
A.8. Lemmaonlinearalgebraicequationswithaninequalityconstraint ....... 157
A.9. Analyticalsolutions ........... ........... ......... 158
B. A glance at analogous fundamental laws 159
B.1. Fourier’slaw .... ......... 159
B.2. Fick’slaw ..... ........... ........... 160
B.3. Darcy’slaw..... 161
B.4. Hooke’slaw .... ......... 162
B.5. Maxwell’sequations 163
C. Albert E. Green and Paul M. Naghdi 165
C.1. AlbertE.Green .. ........... ........... ......... 165
C.2. PaulM.Naghdi .. 166
VNomenclature
VI