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Shakedown analysis of fiber-reinforced metal matrix composite under fusion-relevant thermomechanical loading [Elektronische Ressource] / Byoung Yoon Kim

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Published 01 January 2005
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Max-Planck-Institut für Plasmaphysik
Garching bei München









Shakedown Analysis of Fiber-Reinforced Metal Matrix Composite
under Fusion-Relevant Thermomechanical Loading





Byoung Yoon Kim












Vollständiger Abdruck der von der Fakultät für Maschinenwesen der Technischen
Universität München zur Erlangung des akademischen Grades eines

Doktor-Ingenieurs

genehmigten Dissertation.




Vorsitzender: Univ.-Prof. Dr.-Ing. H. Baier
Prüfer der Dissertation: 1. Hon.-Prof. Dr.-Ing., Dr.-Eng. (Japan) H. H. Bolt
2. Univ.-Prof. Dr. mont. habil. E. Werner


Die Dissertation wurde am 16. 12. 2004 bei der Technischen Universität München
eingereicht und durch die Fakultät für Maschinenwesen am 10. 03. 2005 ange-
nommen.


Abstract

The aim of thermonuclear fusion research is to confine a hot deuterium-tritium (D-T)
plasma long enough so that fusion reactions between hydrogen isotope ions occur, leading
to a commercial power generation. The successful operation of fusion devices depends on
the development of plasma facing components (PFCs) which can withstand the surface
2heat loads of up to 20 MW/m under quasi-stationary conditions. Copper alloys have been
considered as a structural material for the heat sink substrate of a PFC due to their
excellent thermal conductivity. However, insufficient high temperature strength and large
thermal expansion set the limitations to structural applications.

Fiber-reinforced metal matrix composites (FRMMCs) can be a candidate for a structural
material for the future PFCs due to the excellent high temperature strength. Since the
FRMMCs of the PFCs are exposed to thermal and mechanical loads, the resulting stress
fields in mesoscopic level is highly heterogeneous and often exceed the yield limit of the
matrix. The shakedown limit was investigated as the safety criterion of the FRMMCs
considering the fusion-relevant thermomechanical loads.

In principle, it is possible to determine the macro- and mesoscopic stress states by means
of finite element method (FEM), in which the real FRMMC architecture is modeled by
direct meshing. Surely, this is not a practical approach since it requires a high
computational cost. In this case, shakedown analysis can be an appropriate tool to estimate
structural safety. The shakedown theorems were formulated by several researchers. Further,
these could be combined with FEM and the large-scale nonlinear optimization program
and applied to complex system.

In this work, the shakedown formulation was extended to three-dimensional models. The
developed computational algorithm was verified by comparing with literature results. The
shakedown limits were determined for both lamina and laminate of FRMMC composite
system. The results showed that shakedown limits were dependent on geometrical factor
(fiber architecture and fiber volume fraction), loading direction, thermal loading, and
hardening effect. They were discussed based on the maximum value and the distribution of
von Mises stress.

The stress and temperature loading paths of FRMMC components were determined in the
fusion-relevant loading. The thermomechanical loading paths obtained were compared
with the shakedown limits. The results showed that the loading paths in the real operation
situation were only partly covered by the area of shakedown limit. It was interpreted that
the FRMMC layers may undergo low cycle fatigue. Kurzfassung

Das Ziel der thermonuklearen Fusionsforschung ist es, ein heißes Deuterium-Tritium (D-
T) Plasma lange genug einzuschließen, so dass Fusionsreaktionen zwischen
Wasserstoffisotopen stattfinden, so dass eine kommerzielle Elektrizitätserzeugung
ermöglicht wird. Der erfolgreiche Betrieb von Fusionsanlagen hängt von der Entwicklung
plasmabelasteter Komponenten (PFCs) ab, die einer Wärmelast von bis zu 20 MW/m² auf
ihrer Oberfläche unter quasistationären Bedingungen standhalten können. Als
Strukturmaterial für die Wärmesenkenträger einer PFC werden Kupferlegierungen wegen
ihrer exzellenten thermischen Leitfähigkeit in Betracht gezogen. Ungenügende
Hochtemperaturfestigkeit und starke Wärmeausdehnung setzen jedoch Grenzen in der
Strukturanwendung.

Faserverstärkte Metallmatrix-Kompositmaterialien (FRMMCs) können wegen ihrer
hervorragenden Hochtemperaturfestigkeit als Strukturmaterialien für künftige PFCs in
Frage kommen. Da die FRMMCs der PFCs mit ihrer heterogenen Mikrostruktur
thermischen und mechanischen Lasten ausgesetzt sind, sind die resultierenden
Spannungsfelder auf mesoskopischer Ebene stark heterogen und überschreiten oft die
Fließgrenze der Matrix. In dieser Arbeit wurden die Einspielgrenzen als
Sicherheitskriterien der FRMMCs unter Berücksichtigung fusionsrelevanter
thermomechanischer Lasten untersucht.

Es ist prinzipiell möglich die makroskopischen und mesoskopischen Spannungszustände
mit der Finite-Elemente-Methode (FEM) zu ermitteln, wenn der tatsächliche FRMMC-
Aufbau durch direkte Vernetzung modelliert ist. Das ist natürlich keine praktische
Näherung, da sie hohe Rechnerleistung erfordert. Alternativ kann eine Einspielanalyse ein
geeignetes Werkzeug zur Abschätzung der strukturellen Sicherheit sein. Die
Einspieltheoreme wurden von mehreren Forschern formuliert. Ferner können sie mit FEM
und großskaligen nichtlinearen Optimierungsprogrammen kombiniert und auf komplexe
Systeme angewandt werden.

In dieser Arbeit wurde die Einspielformulierung auf dreidimensionale Modelle erweitert.
Der entwickelte Rechenalgorithmus wurde durch den Vergleich mit Literaturergebnissen
überprüft. Die Einspielgrenzen wurden sowohl für Einzelschichten als auch für Laminate
von FRMMC-Kompositsystemen ermittelt. Die Ergebnisse zeigten, dass die
Einspielgrenzen von geometrischen Faktoren (Faseraufbau und Faservolumenanteil),
Belastungsrichtung, thermischer Last und Aufhärtungseffekten abhängen. Sie wurden
unter Berücksichtigung der maximalen von-Mises-Spannungen und ihrer Verteilungen
interpretiert.

Spannungs- und Temperaturlastkurven der FRMMC-Komponenten wurden für
fusionsrelevante Bedingungen bestimmt. Die gewonnenen thermomechanischen
Lastkurven wurden mit den Einspielgrenzen verglichen. Die Lastkurven decken im realen
Betrieb nur teilweise den Bereich der Einspielgrenzen ab. Dies lässt sich mit plastischer
zyklischer Ermüdung der FRMMC-Schichten interpretieren. Contents
Abstract..................................................................................................................................................................i
Kurzfassung........................................................................................................................................................ iii
Content..................................................................................................................................................................v
List of Symbols ................................................................................................................................................... ix
Abbreviation..................................................................................................................................................... xiii




1. Introduction..................................................................................................................................................1
1.1. Structural components for fusion application.........................................................................................1
1.1.1. Fusion reactor and the role of plasma facing components (PFCs) ..............................................1
1.1.2. Fiber-reinforced metal matrix composites (FRMMCs) for PFC application ..............................4
1.2. Why safety analysis of FRMMCs for fusion application? .....................................................................5
1.2.1. Structural problem of FRMMCs...................................................................................................5
1.2.2. safety assessment under cyclic loading .......................................................................7
1.2.3. Overview of shakedown analysis..................................................................................................8
1.3. Literature review....................................................................................................................................10
1.3.1. Conventional methods.................................................................................................................10
1.3.2. FEM-based shakedown analysis.................................................................................................11
1.4. Scope of the thesis.................................................................................................................................13

2. Theoretical Backgrounds..........................................................................................................................17
2.1. Shakedown, one of structural behaviors ...............................................................................................17
2.1.1. Elasticity......................................................................................................................................17
2.1.2. Instantaneous plasticity ...............................................................................................................17
2.1.3. Incremental plasticity (ratcheting)..............................................................................................18
2.1.4. Alternating plasticity (plastic shakedown) .................................................................................19
2.1.5. Shakedown ..................................................................................................................................20
2.2. Review of static shakedown theorem....................................................................................................21
2.2.1. Prerequisites from mechanics of elastic-ideal plastic solids ......................................................21
2.2.1. Static shakedown theorem ..........................................................................................................23


2.3. Extension of static shakedown theorem................................................................................................26 Contents
2.3.1. Shakedown theorem with thermal loading................................................................................. 26
2.3.2. Shakedown theorem with unlimited kinematic hardening (ULKH) model .............................. 27
2.3.3. Shakedown theorem for limited kinematic hardening (LKH) model........................................ 28
2.4. Review of kinematic shakedown theorem............................................................................................ 28

3. FEM-based Shakedown Formulation .................................................................................................. 31
3.1. Introduction to FEM ............................................................................................................................. 31
3.2. Procedure of FEM-based shakedown formulation............................................................................... 33
3.2.1. Discretization of initial loading space........................................................................................ 33
3.2.2. Discretization of elastic stresses................................................................................................. 34
3.2.3. Discretization of residual stress field .........................................................................................35
3.2.4. FEM discretization of shakedown formulation.......................................................................... 36
3.3. Procedure of large-scale nonlinear optimization.................................................................................. 38
3.3.1. Description of large-scale nonlinear optimization problem ...................................................... 38
3.3.2. Optimization technique using augmented Lagrangian method ................................................. 39
3.3.2.1. Minimize the augmented Lagrangian function ................................................................ 39
3.3.2.2. Find approximate minimizer ............................................................................................ 39
3.3.2.3. Application to FEM-based shakedown formulation........................................................ 40
3.4. Summary of FEM-based shakedown formulation ............................................................................... 40

4. Verification Tests ...................................................................................................................................... 43
4.1. Two-dimensional plate with a hole ...................................................................................................... 43
4.2. ensional model of FRMMC................................................................................................... 46
4.3. Experimental result of FRMMC........................................................................................................... 48

5. Description of Problem............................................................................................................................. 51
5.1. Analysis objective................................................................................................................................. 51
5.2. Materials properties for analysis........................................................................................................... 52
5.2.1. FRMMC for shakedown analysis............................................................................................... 52
5.2.2. PFC related materials for incremental analysis ......................................................................... 52
5.3. Scope of shakedown analysis ............................................................................................................... 53
5.4. Scope of incremental analysis .............................................................................................................. 55
5.4.1. Thermal loading history and geometry of PFCs........................................................................ 55
5.4.2. Methodology of incremental analysis ........................................................................................ 56

6. Results of Shakedown Analysis ............................................................................................................... 59
6.1. Shakedown analysis of FRMMC lamina..............................................................................................59
vi Contents
6.1.1. Shakedown analysis of FRMMC lamina for in-plane loading...................................................59
6.1.2. Shakedown analysis of FRMMC lamina for out-of-plane loading............................................63
6.2. Comparison between 2D analysis and 3D analysis ..............................................................................68
6.3. Shakedown analysis of FRMMC laminate ...........................................................................................72
6.3.1. Shakedown analysis of FRMMC laminate for in-plane loading................................................72
6.3.2. Shakedown analysis of FRMMC laminate for out-of-plane loading.........................................75
6.4. Shakedown analysis with hardening effect...........................................................................................79
6.5. Remarks of shakedown results..............................................................................................................82

7. Application of Shakedown Analysis ........................................................................................................87
7.1. Loading parameters of PFC...................................................................................................................87
7.2. Diagram of thermomechanical loading paths .......................................................................................87
7.3. Shakedown limits and loading paths.....................................................................................................90
7.4. Remarks for fusion application .............................................................................................................98

8. Summary...................................................................................................................................................101

9. Appendix...................................................................................................................................................103
Appendix A. Thermonuclear Fusion .............................................................................................................103
A.1. Nuclear fusion......................................................................................................................................103
A.2. Magnetic plasma confinement ............................................................................................................105
A.3. International thermonuclear experimental reactor (ITER) .................................................................107
A.4. PFC and related material questions.....................................................................................................109
Appendix B. CMat3D: Fortran Code for C-Matrix ....................................................................................111
Appendix C. An Example of Standard Input File (SIF) .............................................................................119
Appendix D. Shakedown Limits in Cooling..................................................................................................127
Appendix E. Shakedown Limits and Loading Paths ...................................................................................131

10. References.................................................................................................................................................137


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