Crystal plasticity modelling of thermomechanical fatigue in ITER … · 2019-10-21 · UNIVERSITÉ...

UNIVERSITÉ CATHOLIQUE DE LOUVAIN

ECOLE POLYTECHNIQUE DE LOUVAIN

Octobre 2019

Crystal plasticity modelling of thermomechanical fatigue

in ITER relevant tungsten

Dissertation présentée par

Aleksandr Zinovev

Ingénieur,

pour l’obtention du grade de

docteur en Sciences de l’Ingénieur

Composition du jury

Pr. Laurent DELANNAY (promoteur) UCLouvain, Belgique

Pr. Thomas PARDOEN UCLouvain, Belgique

Dr. Dmitry TERENTYEV SCK•CEN, Belgique

Pr. Aude SIMAR UCLouvain, Belgique

Pr. Marc SEEFELDT KU Leuven, Belgique

Pr. Reinhard PIPPAN ÖAW, Autriche

Pr. Hervé JEANMART (Président du jury) UCLouvain, Belgique

The research was performed at:

Belgian Nuclear Research Centre SCK•CEN

Institute for Nuclear Materials Science Boeretang 200

2400 Mol, Belgium https://sckcen.be/

Université catholique de Louvain Institute of Mechanics, Materials

and Civil Engineering Av. Georges Lemaître 4

1348 Louvain-la-Neuve, Belgium https://uclouvain.be/

The work was supported by:

EUROfusion Boltzmannstr. 2

85748 Garching, Germany https://www.euro-fusion.org/

SCK•CEN Academy for Nuclear Science and Technology

Boeretang 200

2400 Mol, Belgium https://academy.sckcen.be/

Email: [email protected]

https://sckcen.be/

https://uclouvain.be/

https://www.euro-fusion.org/

https://academy.sckcen.be/

i

Acknowledgments

I wish to express my deepest gratitude to my SCK•CEN mentor Dr. Dmitry

Terentyev for giving me the opportunity to carry out this joint Ph.D. project,

invaluable help and support from day one, introduction to the world of fusion,

fruitful scientific discussions and encouragement to keep on going.

I gratefully thank my university promoter Prof. Laurent Delannay for a solid

support, clear guidance and regular meetings in person, his scientific passion, a

lot of patience and immense knowledge that he readily shared, motivating me to

grow as a researcher. The software developed by him played a key role in the

CPFEM simulations that I carried out.

My acknowledgements go to SCK•CEN and the SCK•CEN Academy for providing

the opportunity to do this doctoral research and for the funding of participation

in conferences, which played a significant role in disseminating my scientific

results and building the collaboration network.

The project has been carried out within the framework of the EUROfusion

Consortium. I acknowledge the funding received from the Euratom research and

training programme 2014–2018 and 2019-2020 under grant agreement No

633053.

I would also like to address my sincere gratitude to Prof. Reinhard Pippan, Prof.

Marc Seefeldt, Prof. Thomas Pardoen, Prof. Aude Simar and Prof. Hervé Jeanmart,

who have kindly accepted to review my thesis, for their questions and feedback,

which have helped me greatly in improving the manuscript.

Special thanks to Odette Wouters, Kirsten Huysmans, Els Jansen, Astrid Leduc,

Laurence Bertrand, Emilie Brichart and Christine Vergeynst for handling the

practical matters and paperwork with the highest efficiency.

Many people from SCK•CEN have significantly contributed to my work, and I am

grateful to Dr. Giovanni Bonny, Dr. Nicolas Castin, Dr. Alexander Bakaev, Dr.

Serguei Gavrilov, Dr. Petr Grigorev, Dr. Andrii Dubinko, Dr. Konstantza

Lambrinou, Dr. Anastasiia Bakaeva and Chao Yin for showing me around,

especially in the beginning of my project, for the scientific (and not-so-scientific)

discussions we have had, for efficient team-work related to our publications and

for a positive environment to work in, which truly contributed to my pleasure to

come to work every day.

ii

My sincere thanks goes to Prof. Sergey Dmitriev from Institute for Metals

Superplasticity Problems (Russia) for leading me into the realm of

computational science during my master’s studies and for the wise advice to do

an internship in this field in Belgium, which was crucial in my final decision to

enrol in the Ph.D. program at SCK•CEN and UCLouvain.

Last but not least, this work would certainly be impossible without the support I

was receiving from my family, the love and endless patience of my dear Izas, and

the fun my friends were sharing during all these not so easy years. Thank you all

very much for giving me confidence in my endeavour.

Mol, October 2019

Aleksandr Zinovev

iii

Abstract

This work contributes to a better understanding of the micromechanics of

tungsten during cyclic heat loads of plasma-facing components in the ITER fusion

reactor. Colossal energy will have to be extracted from the reaction chamber and

the temperature of the walls will oscillate together with periodical changes of

plasma intensity. This research addresses thermomechanical fatigue, i.e.

microscopic cracking due to repeated cycles of thermal expansion and

compression. The effect of neutron irradiation damage is not accounted for, but

the proposed model applies to thermal shock tests, which are used for the

qualification of ITER-relevant tungsten grades.

The computational model relies on crystal plasticity theory in order to account

for the anisotropy of individual grains. The sensitivity of the mechanical

response to strain rate and temperature is reproduced by considering thermally

activated dislocation slip. The influence of internal stresses during repeated

elastic-plastic transients is investigated using a simplified modelling of the

kinematic hardening due to dislocation pile ups at grain boundaries. The model

is implemented as a user-defined material law for the Abaqus finite element

code, allowing crystal plasticity based finite element modelling (CPFEM). Fatigue

indicators are defined to predict the onset of damage whereas cohesive elements

are used to simulate the propagation of intergranular cracks.

Based on an inverse finite element analysis, it is shown that experimental tensile

test data can be reproduced with a limited set of parameters and then applied to

as-received and recrystallized tungsten up to 1700 °C. Recrystallized tungsten

shows significant asymmetry in the mechanical response during the heating and

cooling phases. CPFEM predicts substantial inter- and intragranular

heterogeneity and the fatigue indicator values probed at grain boundaries

depend largely on the amplitude of backstresses. The occurrence of

intergranular cracks is influenced by grain shape as well as the orientation with

regard to the thermal flux. Experimentally observed foil delamination is properly

reproduced only when accounting for crystalline anisotropy. The model

predictions agree qualitatively with the outcome of thermal shock tests.

v

Contents

Acknowledgments ..................................................................................... i

Abstract ...................................................................................................... iii

Contents ....................................................................................................... v

List of publications ................................................................................. ix

Publications on this topic ........................................................................... ix

Publications on other topics ...................................................................... ix

List of acronyms ...................................................................................... xi

List of figures .......................................................................................... xiii

List of tables ............................................................................................ xxi

Chapter 1. Introduction ..................................................................... 1

1.1 Context .................................................................................................... 1

1.1.1 Energy demand in the world ............................................................................. 1 1.1.2 Nuclear fusion .......................................................................................................... 3

1.1.2.1 Fusion in a nutshell ........................................................................................ 3 1.1.2.2 Commercially attractive fusion reaction and fuel ............................ 5 1.1.2.3 How to achieve fusion on Earth? ............................................................. 6 1.1.2.4 Achievements and records in fusion ...................................................... 8

1.1.3 ITER operational conditions .............................................................................. 9 1.1.3.1 The role of divertor ........................................................................................ 9 1.1.3.2 Selected plasma scenario and the plasma-wall interaction ...... 11

1.2 Tungsten as a material for fusion reactors .............................. 12

1.2.1 ITER-specification tungsten ........................................................................... 13 1.2.1.1 Tungsten in the recrystallized state .................................................... 15

1.2.2 Mechanical properties under monotonic loading ................................. 16 1.2.3 Mechanical response of tungsten under cyclic loading ...................... 18

1.2.3.1 Overview of heat load tests ..................................................................... 18 1.2.3.2 High heat flux test ........................................................................................ 20 1.2.3.3 Thermal shock test ...................................................................................... 20

vi

1.3 Overview of techniques ................................................................... 22

1.3.1 Numerical analyses of heat load tests ........................................................ 22 1.3.2 Consideration of cyclic deformation under J2 plasticity theory ...... 24

1.3.2.1 The Bauschinger effect .............................................................................. 24 1.3.2.2 J2 plasticity model ........................................................................................ 24 1.3.2.3 Isotropic and kinematic hardening ...................................................... 25

1.3.3 Deformation of polycrystals at the microscale ....................................... 28 1.3.3.1 What is crystal plasticity? ........................................................................ 28 1.3.3.2 Relevant application of CPFEM in literature ................................... 30 1.3.3.3 Kocks-Mecking model of plasticity ...................................................... 31

1.3.4 Tensile tests ........................................................................................................... 32 1.3.5 Scanning electron microscope and electron backscatter diffraction

..................................................................................................................................... 35 1.3.6 Transmission electron microscope ............................................................. 35

1.4 Outline of the thesis and the addressed questions ............... 36

Chapter 2. Applicability of the Kocks-Mecking model to

temperature- and strain rate-sensitive uniaxial deformation of

ITER-specification tungsten ............................................................... 39

2.1 Introduction ........................................................................................ 39

2.2 Plasticity model and numerical techniques ............................ 41

2.2.1 Temperature- and strain-rate dependent hardening law ................. 41 2.2.2 Finite element simulation of tensile test beyond the onset of

deformation instability ..................................................................................... 44 2.2.3 Optimisation algorithm ..................................................................................... 45

2.2.3.1 Basics of downhill simplex method ..................................................... 45 2.2.3.2 IFEA based on the downhill simplex method .................................. 46

2.3 Results ................................................................................................... 49

2.3.1 Experimental tensile tests ............................................................................... 49 2.3.2 Results of inverse finite element analysis ................................................ 52

2.3.2.1 As-received tungsten.................................................................................. 52 2.3.2.2 Recrystallized tungsten ............................................................................. 54

2.4 Discussion ............................................................................................ 59

2.4.1 Strain-rate sensitivity ........................................................................................ 59 2.4.2 Deformation stage IV ......................................................................................... 60

2.4.2.1 Deviation from the Voce equation ........................................................ 60 2.4.2.2 What is deformation stage IV? ............................................................... 61 2.4.2.3 Evidence of deformation stage IV in tungsten ................................ 63

2.4.3 Individual curve fitting with the iterative algorithm ........................... 65

2.5 Conclusion ............................................................................................ 70

vii

Chapter 3. Development of a crystal plasticity model

accounting for backstresses ............................................................... 71

3.1 Introduction ........................................................................................ 71

3.2 Context .................................................................................................. 73

3.3 Workflow of the suggested two-scale simulation scheme .. 74

3.4 Crystal plasticity framework......................................................... 75

3.4.1 What is crystal plasticity theory? ................................................................. 75 3.4.2 Traditional crystal plasticity for polycrystals ......................................... 76 3.4.3 Mathematical formulation of the grain-level plasticity law ............. 77

3.5 Identification of material parameters ....................................... 81

3.6 Results ................................................................................................... 85

3.6.1 Identification of model parameters based on uniaxial tensile tests

..................................................................................................................................... 85 3.6.1.1 As-received tungsten.................................................................................. 85 3.6.1.2 Recrystallized tungsten ............................................................................. 87

3.6.2 Isothermal mechanical cycles ........................................................................ 89 3.6.2.1 As-received tungsten.................................................................................. 89 3.6.2.2 Recrystallized tungsten ............................................................................. 90

3.6.3 Macroscopic thermal shock simulation ..................................................... 90 3.6.4 Microscopic thermal shock simulation ...................................................... 93

3.6.4.1 As-received tungsten.................................................................................. 93 3.6.4.2 Recrystallized tungsten ............................................................................. 94

3.7 Discussion ............................................................................................ 96

3.7.1 Effect of kinematic hardening ........................................................................ 96 3.7.2 Note on the mixed hardening case............................................................... 98

3.8 Conclusion ......................................................................................... 100

Chapter 4. Damage evolution and failure of tungsten at the

microscopic level in thermomechanical fatigue conditions. . 101

4.1 Introduction ..................................................................................... 101

4.1.1 The need of CPFEM .......................................................................................... 101 4.1.2 Overview of CPFEM theory .......................................................................... 102 4.1.3 Indicators of fatigue ........................................................................................ 102 4.1.4 Overview of the chapter ................................................................................ 105

4.2 FEA setup ........................................................................................... 106

4.2.1 Crystal plasticity — standalone and CPFEM ........................................ 106 4.2.2 Cohesive elements............................................................................................ 107 4.2.3 Analysis of mesh sensitivity ......................................................................... 109

viii

4.2.4 Mesh of polycrystals used in the work.................................................... 114

4.3 Analysis of fatigue damage in recrystallized tungsten using

cohesive elements .......................................................................... 115

4.3.1 Results ................................................................................................................... 116 4.3.2 A note on the stress in cohesive elements ............................................. 122 4.3.3 Conclusions of the section ............................................................................ 123

4.4 Analysis of fatigue damage in as-received tungsten using the fatigue indicator ...................................................................... 124

4.4.1 Results ................................................................................................................... 124

4.5 The effect of crystalline anisotropy on grain delamination

............................................................................................................... 129

4.6 Discussion ......................................................................................... 134

4.6.1 Taylor-based CP and CPFEM simulations of thermal shock .......... 134 4.6.2 Cohesive elements and fatigue indicator ............................................... 134 4.6.3 The use of tensile tests to derive the threshold fatigue indicator

.................................................................................................................................. 135

4.7 Conclusion ......................................................................................... 136

Chapter 5. Conclusions and perspectives ................................ 139

5.1 Summary and conclusions .......................................................... 139

5.2 Perspectives ..................................................................................... 141

Bibliography ......................................................................................... 143

ix

List of publications

Publications on this topic

1) A. Zinovev, D. Terentyev, L. Delannay, Finite element analysis of heat

load of tungsten relevant to ITER conditions, Phys. Scr. T170 (2017) 014002.

doi:10.1088/0031-8949/2017/T170/014002.

2) A. Zinovev, D. Terentyev, A. Dubinko, L. Delannay, Constitutive law for

thermally-activated plasticity of recrystallized tungsten, J. Nucl. Mater. 496

(2017) 325–332. doi:10.1016/j.jnucmat.2017.09.044.

Publications are under preparation about

the use of inverse finite element analysis to process experimental stress-

strain curves with low uniform elongation (Chapter 2)

improvement of the CP model to account for continuously evolving

temperature, non-zero athermal stress of BCC metals and the effect of

backstresses (Chapter 3)

application of CPFEM to investigate the damage accumulation under

conditions of thermal shock (Chapter 4)

Publications on other topics

3) D. Terentyev, A. Zinovev, G. Bonny, Displacement cascades in Fe-Ni-Mn-

Cu alloys: RVP model alloys, J. Nucl. Mater. 475 (2016) 132–139.

doi:10.1016/j.jnucmat.2016.04.005.

4) G. Bonny, D. Terentyev, J. Elena Gonzales, A. Zinovev, B. Minov, E.E.

Zhurkin, Assessment of hardening due to dislocation loops in bcc iron: Overview

and analysis of atomistic simulations for edge dislocations, J. Nucl. Mater. 473

(2016) 283–289. doi:10.1016/j.jnucmat.2016.02.031.

5) A. V. Zinovev, M.G. Bapanina, R.I. Babicheva, N.A. Enikeev, S. V. Dmitriev,

K. Zhou, Deformation of nanocrystalline binary aluminum alloys with

segregation of Mg, Co and Ti at grain boundaries, Phys. Met. Metallogr. 118

(2017) 65–74. doi:10.1134/S0031918X16110144.

6) P. Grigorev, A. Zinovev, D. Terentyev, G. Bonny, E.E. Zhurkin, G. Van

Oost, J.-M. Noterdaeme, Molecular dynamics simulation of hydrogen and helium

trapping in tungsten, J. Nucl. Mater. 508 (2018) 451–458.

doi:10.1016/j.jnucmat.2018.05.052.

x

7) D. Terentyev, J. Riesch, S. Lebediev, T. Khvan, A. Zinovev, M. Rasiński, A.

Dubinko, J.W. Coenen, Plastic deformation of recrystallized tungsten-potassium

wires: Constitutive deformation law in the temperature range 22–600 °C, Int. J.

Refract. Met. Hard Mater. 73 (2018) 38–45. doi:10.1016/j.ijrmhm.2018.01.012.

8) X. Xiao, D. Terentyev, A. Ruiz, A. Zinovev, A. Bakaev, E.E. Zhurkin, High

temperature nano-indentation of tungsten: Modelling and experimental

validation, Mater. Sci. Eng. A. 743 (2019) 106–113.

doi:10.1016/j.msea.2018.11.079.

9) A. Bakaeva, V. Makhlai, D. Terentyev, A. Zinovev, S. Herashchenko, A.

Dubinko, Correlation of hardness and surface microcracking in ITER

specification tungsten exposed at QSPA Kh-50, J. Nucl. Mater. 520 (2019) 185–

192. doi:10.1016/j.jnucmat.2019.04.008.

10) X. Xiao, D. Terentyev, A. Bakaev, A. Zinovev, A. Dubinko, E.E. Zhurkin,

Crystal plasticity finite element method simulation for the nano-indentation of

plasma-exposed tungsten, J. Nucl. Mater. 518 (2019) 334–341.

doi:10.1016/j.jnucmat.2019.03.018.

11) A. Bakaev, A. Zinovev, D. Terentyev, G. Bonny, C. Yin, N. Castin, Yu.

Mastrikov, E. Zhurkin, Interaction of carbon with microstructural defects in a W-

Re matrix: an ab initio assessment, J. Appl. Phys. 126 (2019) 075110.

doi:10.1063/1.5094441.

12) A. V. Zinovev, A. M. Iskandarov, S. V. Dmitriev, A. I. Pshenichnyuk,

Criteria of instability of copper and aluminium perfect crystals subjected to

elastic deformation in the temperature range 0 – 400 K, Lett. Mater. 9(3) (2019)

265–269. doi:10.22226/2410-3535-2019-3-265-269.

xi

List of acronyms

ASTM standards Standards developed by ASTM International, an

organization formerly known as American Society for

Testing and Materials

BCC Body-Centred Cubic (lattice type)

BR2 The Belgian Reactor 2

CE Cohesive Element

CP Crystal Plasticity

CPFEM Crystal Plasticity based Finite Element Modelling

CRSS Critical Resolved Shear Stress

CZ Cohesive Zones

DBTT Ductile-to-Brittle Transition Temperature

DFR Double-Forged Recrystallized (tungsten grade)

dpa Displacements Per Atom

EBSD Electron Backscatter Diffraction

EDM Electrical Discharge Machining

ELM Edge-Localised Mode (plasma instability type)

ESS European Spallation Source

FCC Face-Centred Cubic (lattice type)

FEA Finite Element Analysis

FEM Finite Element Method

FIP Fatigue Indicator Parameter

GB Grain Boundary

HPT High Pressure Torsion

IGP ITER-Grade Plansee (tungsten grade)

IFEA Inverse Finite Element Analysis

ITER (“The way” in Latin) is the largest tokamak under

construction, formerly known as “International

Thermonuclear Experimental Reactor”

JET Joint European Torus

xii

MRI Magnetic Resonance Imaging

PFC Plasma-Facing Components

QSPA Quasistationary Plasma Accelerator

RVE Representative Volume Element

S-S curve Stress-Strain curve

SCK•CEN Studiecentrum voor Kernenergie•

Centre d’étude de l’Energy Nucleaire

SEM Scanning Electron Microscopy

SOL Scrape-Off Layer

TEM Transmission Electron Microscopy

TSL Traction-Separation Law

UMAT A User-defined MATerial subroutine for the standard

finite element solver Abaqus

UTS Ultimate Tensile Strength

XFEM eXtended Finite Element Method

XRD X-Ray Diffraction

xiii

List of figures

Figure 1.1. Nuclear binding energy per nucleon as a function of the atomic mass

[1]. ............................................................................................................................................................ 4

Figure 1.2. Potential energy as a function of the distance between two charged

nuclei [6]. .............................................................................................................................................. 5

Figure 1.3. Fusion cross-section of various fusion reactions as a function of

kinetic energy of an incident D or p+ on a stationary target [7]. ................................. 6

Figure 1.4. Magnetic fields in a tokamak. From left to right: toroidal component,

poloidal component and resulting helical magnetic lines [10]. ................................... 7

Figure 1.5. a) A schematic picture of ITER’s vacuum chamber [10,19]. Tungsten

is shown with red colour. Black lines represent an example of magnetic field lines

in the core plasma and a separatrix. The light pink region between the separatrix

and the walls of the chamber represents the divertor plasma layer. b) A divertor

cassette and its parts [20]. c) A plasma-facing unit: tungsten monoblocks

assembled on a cooling pipe [21]. .......................................................................................... 10

Figure 1.6. The monoblock geometry, the cross-section of the CuCrZr cooling

pipe and the Cu interlayer. Note the bevelling of 0.5 mm introduced to reduce

thermal load on the right edge of the monoblock by shadowing it with a

neighbouring monoblock to the right. Dimensions are in mm [22]. ....................... 10

Figure 1.7. Schematic representation of the normal operation with temperature

excursions during transient heat loading. Lower temperature range is defined by

the DBTT, while the upper range is limited by the recrystallization temperature

[1]. ......................................................................................................................................................... 11

Figure 1.8. Schematic view of a typical W monoblock indicating its

microstructure and typical dimensions [31]. .................................................................... 13

Figure 1.9. a) A fabricated bar of tungsten with indicated macroscopic directions;

b) a TEM image of microstructure of as-received tungsten; c) an EBSD map of as-

received tungsten; d) an EBSD map of recrystallized tungsten. ............................... 14

Figure 1.10. Metallographic cross-section in the centre of tungsten mock-up

subjected to 20 MW m-2 in a high heat flux test, showing the layer recrystallized

due to high thermal load [34]. .................................................................................................. 15

Figure 1.11. Divertor mock-up after exposure to a high heat flux [34]. ................ 19

Figure 1.12. Damage mapping of tungsten samples exposed to ELM-like thermal

shocks. Plain-coloured symbols stand for samples with 100 thermal shock

xiv

pulses. Symbols with a black centre stand for samples exposed to 1000 pulses

[26]. ...................................................................................................................................................... 21

Figure 1.13. Illustration of the Bauschinger effect in reversed loading. The yield

stress in compression is lower than the stress achieved prior the load reversal.

[90]. ...................................................................................................................................................... 24

Figure 1.14. Effect of strain hardening on the yield locus. The cross-section of the

locus in the x – y plane is shown. The isotropic model (a) corresponds to an

expansion of the locus. The kinematic hardening model (b) corresponds to a

translation of the locus in the direction of the loading path [89]. ............................ 26

Figure 1.15. Linear arrays of edge dislocations piled-up against barriers under

an applied shear stress [100]. ............................................................................................. 27

Figure 1.16. Comparison between a strain controlled fatigue test and a

monotonic tensile test of polished forged tungsten at 480 °C [59]. ........................ 28

Figure 1.17. (a) Sketch of a miniaturized dog-bone-shape tensile sample, and (b)

photo of a sample before and after tensile test. ............................................................... 33

Figure 1.18. A sample in the furnace of the Instron machine. The thermocouple

wires are attached to the gauge section of the sample. ................................................ 33

Figure 1.19. Examples of two main types of stress-strain curves encountered in

the present work: a) low uniform elongation, “early necking”, b) high uniform

elongation. The thick lines highlight the range of applicability of the standard

approach to extract true stress and strain.......................................................................... 34

Figure 2.1. A tensile sample used in FE analysis a) in the non-deformed state; b)

at the onset of deformation instability. Colours represent the von Mises stress,

ranging from 0 (in the grips) to 400 MPa (in the centre). ........................................... 45

Figure 2.2. Flowchart of the iterative algorithm for the plasticity model fitting to

multiple stress-strain curves simultaneously. .................................................................. 47

Figure 2.3. Illustration for the evaluation of the objective function. The sum of

squares of differences is calculated at a discrete set of strain values, marked with

circles. Both the engineering stress (a) and the stress-strain slope (b) contribute

to the total value of the objective function. The arrows indicate that as the

simulation result approaches the target, the objective function decreases. ....... 48

Figure 2.4. The evolution of a) input true S-S curves, b) output engineering S-S

curves in several iterations of the search algorithm. ..................................................... 49

Figure 2.5. Engineering stress-strain curves for as-received IGP tungsten. Solid

lines highlight the range of uniform deformation. .......................................................... 50

Figure 2.6. Engineering stress-strain curves for the recrystallized tungsten. .... 51

Figure 2.7. a) Yield stress and ultimate tensile strength as well as b) uniform

elongation and total elongation of both grades of tungsten tested at 6·10-4 s-1. 51

xv

Figure 2.8. The best fit of equation (2.1) to the yield stress of as-received IGP

tungsten at different temperatures and strain rates...................................................... 52

Figure 2.9. True stress-strain curves of as-received IGP tungsten obtained with

the help of IFEA of tensile test.................................................................................................. 53

Figure 2.10. Engineering stress-strain curves obtained in the last iteration of the

IFEA of tensile test for as-received IGP W, compared to the experimental data.

................................................................................................................................................................ 53

Figure 2.11. Yield stress of recrystallized W. Symbols represent experimental

data, and lines highlight its evolution with temperature at given strain rates,

obtained using two different fitting functions. ................................................................. 55

Figure 2.12. Experimental true stress-strain curves of recrystallized tungsten.

................................................................................................................................................................ 56

Figure 2.13. Hardening rate derived from experimental true stress-strain curves

of recrystallized IGP tungsten .................................................................................................. 56

Figure 2.14. Experimental and fitted true stress-strain curves for recrystallized

IGP tungsten at strain rate 6·10-4 s-1. ..................................................................................... 58

Figure 2.15. Experimental and simulated engineering stress-strain curves at

500 °C and 600 °C for recrystallized tungsten. ................................................................. 58

Figure 2.16. Illustration of rate sensitivity of tungsten (other than IGP) in a wide

range of strain rate, showing the true stress at 4% of strain as a function of strain

rate [146]. .......................................................................................................................................... 59

Figure 2.17. Strain rate sensitivity for Cu deformed in compression at 25°C,

shown as the stress at 15% of strain as a function of strain rate. [101]. .............. 60

Figure 2.18. Hardening rate as a function of true stress indicates the presence of

deformation stages III and IV (solid line). Hardening rate derived from the

standard Voce equation is shown by the dashed line for clarity. ............................. 61

Figure 2.19. (a) Engineering stress-strain curves and (b) hardening rate for

different grades of recrystallized tungsten obtained by uniaxial tensile tests at

500 °C [1,26]..................................................................................................................................... 62

Figure 2.20. TEM micrographs showing dislocation tangles (a, b) and dislocation

cells (c), probed by TEM at 20% (a, b) and 28%(c) of plastic strain in the DFR

tungsten. The grain refinement was observed in the tests performed at 500 °C

and 600 °C after 20% plastic strain. Zones I in the images show examples of the

grain interior, while zones II correspond to dislocation tangles where the

dislocation density was measured [120]............................................................................. 64

Figure 2.21. Dislocation density predicted by the model superimposed on the

experimental data from [115]. ................................................................................................. 65

Figure 2.22. Flowchart of the iterative algorithm for the Voce equation fitting to

a single stress-strain curve. ....................................................................................................... 66

xvi

Figure 2.23. Engineering stress-strain curves, as a result of FE simulation of a

tensile test, using the best constitutive laws obtained with the help of individual

curve fitting. Dashed lines show corresponding experimental results. ................ 67

Figure 2.24. The Voce parameters 0 (a) and sat (b) as a function of

temperature, obtained in the IFEA of individual experimental S-S curves. ......... 68

Figure 2.25. The Voce parameters 0 (a) and sat (b) as a function of

temperature, obtained in the IFEA of multiple experimental S-S curves. ............ 69

Figure 3.1. Stress-strain loops obtained in low cycle fatigue tests of forged

tungsten at 500 °C (other than IGP) [59] and the results of fitting of a J2

Armstrong-Frederick model of kinematic hardening. ................................................... 72

Figure 3.2. Schematic representation of the three types of samples used in the

thermal shock experiment [26]. .............................................................................................. 73

Figure 3.3. On the evaluation of the objective function. The squares of differences

between experimental and simulated values of stress are summed over three

points per curve and then are summed over all 15 stress-strain curves. ............. 83

Figure 3.4. Illustration of construction of the three cases of hardening. .............. 84

Figure 3.5. True deformation curves obtained in simulations of tensile test with

the crystal plasticity model and the best parameter sets for three hardening

cases. .................................................................................................................................................... 86

Figure 3.6. Evolution of dislocation density in the simulated polycrystal of as-

received tungsten with the best parameter sets for three hardening cases. ...... 87

Figure 3.7. True stress-strain curves obtained in uniaxial tension with the CP

model of recrystallized tungsten (dashed) superimposed on the experimental

curves (solid). .................................................................................................................................. 88

Figure 3.8. True stress-strain curves at high temperature and the limiting values

of strain rate, demonstrating the applicability domain of the model of

deformation of recrystallized tungsten. ............................................................................... 88

Figure 3.9. Stress-strain loops obtained in simulations of mechanical isothermal

cyclic loading of as-received tungsten. ................................................................................. 89

Figure 3.10. Stress-strain loops obtained in simulations of mechanical isothermal

cyclic loading of recrystallized and as-received tungsten. .......................................... 90

Figure 3.11. FE mesh of the sample studied in JUDITH1 facility for a macroscopic

thermo-mechanical simulation. The colour indicates temperature field at the end

of heat flux exposure, with the blue colour corresponding to the base

temperature, 400 °C, and the red - to the maximal attained temperature ~750 °C.

................................................................................................................................................................ 91

Figure 3.12. Evolution of temperature and the von Mises strain in the centre of

the heated area exposed to an ELM-like thermal shock of 190 MW m-2. .............. 91

xvii

Figure 3.13. Normal stress components in the centre of the sample exposed to

380 MW m-2 as a function of (a) time and (b) temperature. The only non-zero

components are S11 and S22, indicating a plane-stress state. .................................. 92

Figure 3.14. Schematic view of the simulation box in the initial and deformed

state. The thermal strain is denoted as . .......................................................................... 93

Figure 3.15. The mechanical response of as-received tungsten (isotropic

hardening only) to thermal shocks of different power density in terms of the in-

plane stress vs. the out-of-plane strain. ............................................................................... 94

Figure 3.16. The mechanical response of recrystallized tungsten to thermal

shocks of different power density in terms of the in-plane stress vs. the out-of-

plane strain. ...................................................................................................................................... 95

Figure 3.17. Stress-strain loops recorded in the as-received tungsten for the

three considered material laws. .............................................................................................. 96

Figure 3.18. Evolution of the in-plane tensile stress amplitude obtained in the as-

received tungsten for the three considered material laws. ......................................... 97

Figure 4.1. Representation of polycrystals in CPFEM framework. a) Simplified 3D

grain assembly [186], b) advanced grain assembly obtained with the help of the

Voronoï tessellation [187]. ..................................................................................................... 107

Figure 4.2. a) A diagram illustrating the use of cohesive elements in a crack

propagation problem. b) Example of a traction-separation law, normalised by the

maximal cohesive stress 0T and maximal cohesive separation 0 [197]. ........ 108

Figure 4.3. Maximal principal stress distribution in “bricks” with different level

of refinement: coarse (a, b), medium (c, d) fine (e, f), and element type: C3D8 in

“simple bricks” (a, c, e) and C3D10 in “Gmsh-generated bricks” (b, d, f) ........... 109

Figure 4.4. A boxplot shows schematically five main reference points of a smooth

distribution of stress in cohesive elements. .................................................................... 110

Figure 4.5. Distribution of normal stress in CZ (open boxes) and maximal

principal stress in grain interior (filled boxes) in a) “simple bricks” and b) “Gmsh-

generated bricks” with different mesh refinement at 30% of tensile strain. Each

colour of the filled boxes corresponds to one of the 13 grains. .............................. 112

Figure 4.6. Effect of cohesive element stiffness on the stress distribution in

cohesive elements and grain interior of a Gmsh-generated brick with medium-

size elements. Each colour of the filled boxes corresponds to one of the 27 grains.

............................................................................................................................................................. 113

Figure 4.7. The finite element meshes used in the simulations of thermal fatigue

with CPFEM, a) transversal with respect to the heat flux, b) longitudinal, c)

equiaxed. ......................................................................................................................................... 114

Figure 4.8. Stress-strain loops as a result of simulation of thermal shock on the

surface of the recrystallized tungsten (equiaxed grains).......................................... 116

xviii

Figure 4.9. Stress-strain loops as a result of simulation of thermal shock on the

surface of the recrystallized tungsten (elongated grains). ....................................... 117

Figure 4.10. The evolution of normal stress distribution in the GBs of polycrystal

with equiaxed grains in the first two thermal shocks. ............................................... 118

Figure 4.11. The evolution of normal stress distribution in the GBs of polycrystal

with elongated grains in the first three thermal shocks. ........................................... 119

Figure 4.12. The evolution of shear stress distribution in the GBs of polycrystal

with elongated grains in the first two thermal shocks. .............................................. 120

Figure 4.13. The evolution of normal stresses in the first two thermal pulses in

recrystallized tungsten exposed to 380 MW m-2 thermal power density at base

temperature 27 °C. a) Elongated grains, b) equiaxed grains. .................................. 121

Figure 4.14. Distribution of the angle between cohesive elements and the heated

surface in the three considered meshes. .......................................................................... 122

Figure 4.15. Correlation between normal stress in cohesive elements and their

orientation. .................................................................................................................................... 122

Figure 4.16. The average in-plane stress and out-of-plane strain in the

polycrystal subjected to thermal shocks at different base temperature and

absorbed power density. ......................................................................................................... 125

Figure 4.17. The average FIP in the whole polycrystal with the L orientation

tested at different base temperature and applied power density. ........................ 126

Figure 4.18. Distribution of microscopic FIP in two selected grains at the end of

ten thermal shocks of 380 MW m-2. .................................................................................... 127

Figure 4.19. FIP distribution in individual grains after ten thermal shocks with

power density 380 MW m-2 at base temperature 1000 °C in polycrystal with

mixed hardening.......................................................................................................................... 128

Figure 4.20. Distribution of normal stress in the cohesive elements within the

first two thermal shocks of 380 MW m-2 at base temperature 1000 °C applied to

the material with the mixed hardening............................................................................. 128

Figure 4.21. Distribution of normal stress in cohesive zones (open boxes) and

maximal principal stress in grain interior (filled boxes) in three meshes of the

polycrystal with mixed hardening after exposure to ten thermal shocks at base

temperature 1000 °C and power density 380 MW m-2. ............................................. 129

Figure 4.22. a) EBSD maps of pure 0.1 mm tungsten foil projected on a cuboid to

better visualize the pancake-like microstructure. Black lines denote the initial

notch [40]. b) An example of the foil delamination occurred at 400 °C [40]. c) A

polycrystal composed of prismatic grains to represent the microstructure of the

foil in CPFEM simulations. Colour highlights separate grains. d) Embedded

cohesive elements which have different strength, according to their role: GBs

(green) are weaker than the transgranular crack plane (highlighted with red).

............................................................................................................................................................. 130

xix

Figure 4.23. Maps of normal strain inside cohesive elements (LE33) in

simulations with the J2 (a, c, e) and CP (b, d, f) models. The strength of GBs

increases from top to bottom: 50 MPa (a, b), 250 MPa (c, d) and 550 MPa (e, f).

............................................................................................................................................................. 132

Figure 4.24. Distribution of the out-of-plane stress S33 in the grain interior in

simulations with different GB strength. ............................................................................ 133

Figure 4.25. The density of energy dissipated in tensile tests of the as-received

IGP tungsten. ................................................................................................................................. 136

xxi

List of tables

Table 1.1. Impurity content of the IGP tungsten. ............................................................. 13

Table 2.1. Set of parameters reproducing the yield stress of as-received IGP

tungsten. ............................................................................................................................................. 52

Table 2.2. The fitted parameters of the plasticity model for as-received IGP W.

................................................................................................................................................................ 54

Table 2.3. The best fitted parameter set of equation (2.1) to describe yield stress

of recrystallized tungsten. .......................................................................................................... 55

Table 2.4. The best fitted parameter set of equation (2.16) to describe yield stress

of recrystallized tungsten. .......................................................................................................... 55

Table 2.5. The fitted parameters of the plasticity model for recrystallized

tungsten. ............................................................................................................................................. 57

Table 3.1. The CP model parameters independent of hardening type for as-

received tungsten. .......................................................................................................................... 85

Table 3.2. The CP model parameters for the isotropic, kinematic and mixed

hardening cases of as-received tungsten............................................................................. 85

1

Chapter 1.

Introduction Equation Cha pter 1 Section 1

1.1 Context

1.1.1 Energy demand in the world

Thanks to the improvement of the average standard of living, the average power

consumption per person steadily increases, especially in the developing

countries. However, it represents a global challenge as well, amplified by the

population growth. During the last century the total amount of energy used per

year has increased by a factor of ten and the primary source was fossil fuel [1]. A

study by the European climate foundation (ECF) published in 2010 showed that

the yearly power demand of the European countries will increase from 3250

terawatt-hour (TWh) in 2010 to 4800 TWh in 2050 [2].

The increased energy demand requires new sources of clean and sustainable

energy. In 2016 the estimated global final energy demand was provided by three

main energy sources: fossil fuels (79.5%), renewables (18.2%) and nuclear

energy (2.2%) [3]. Fossil fuels, such as petroleum, coal and gas, are non-

renewable, and, even though they are still sufficiently available to satisfy the

constantly increasing energy consumption, they might run out in our lifetime or

their extraction would become economically unfavourable. In 2000 a study

estimated that at the current consumption rates, oil reserves will run out within

25–40 years, coal within 70 years and natural gas reserves in 200 years [1].

Importantly, fossil fuels are found locally on our planet, having major economic

and political consequences, such as significant influence of supplying countries

on the energy cost, political instability, international disagreements and

conflicts. Also, they represent a highly useful mixture of complex chemical

substances, which should efficiently be used in chemical industry for production

of plastic, textiles, composite and advanced materials, instead of being burned.

Energy production from fossil fuels is accompanied by emission of huge amounts

of greenhouse gases, and, in case of poor filtration and processing of exhaust,

toxic substances including dust, leading to climate change, environmental

pollution, deterioration of areas around power plants. In other words, the need

for alternatives is a consequence of the drawbacks of the use of fossil fuels in

energy production.

1. Introduction

2

The two main alternatives used nowadays are renewable energy and nuclear

power. The most important renewable energy source is hydroelectricity that

supplies 3.7% of the energy (or 16.6% of electricity) in the world, but the dams

drastically change the environment and their use should be limited. Other

renewable sources, which include geothermal, solar, tide, wind, wood, biomass

and waste, account for merely several percent of the total energy production [3].

Most of them are indigenous, widespread, and enhance a country’s

independence from external supplies of fossil fuels. However, they suffer from

intermittency (the source is not continuously available) and lack of “on demand”

availability.

Another alternative to fossil fuels, nuclear power generated by fission of heavy

nuclei, does not have this disadvantage. It is a quite efficient, climate friendly and

relatively safe energy source, which produces around 10% of the world’s

electricity nowadays [4]. The energy released by a fission reaction is a million

times larger than that released by a chemical reaction (such as burning) due to

different fundamental forces involved.

The electromagnetic force that holds the atoms together in molecules is much

weaker than the strong nuclear force that binds the protons and neutrons in an

atomic nucleus. Thus, the mass of fuel needed to generate a given amount of

energy in a nuclear power plants is a small fraction of fuel required by a coal or

gas power plant.

Unfortunately, nuclear energy is not free of problems, the most undesirable of

which is long-lived nuclear waste, spent fuel, that has to be securely stored in

special mines for hundreds of thousands of years. All of the used fuel ever

produced by the commercial nuclear industry since the late 1950s would cover

a football field to a depth of less than nine meters. That might seem like a lot, but

coal plants generate that same amount of waste every hour [5].

Another possibility is to reprocess the spent fuel in order to filter out the fission

products and extract the remaining uranium that can further be used as fuel. The

generation IV fission power plants will produce less radioactive waste and new

methods for waste treatment are under development.

A possible solution of this problem is offered by an alternative source of energy

proposed in the 1950s, which is the nuclear fusion, a reaction that powers stars,

including the Sun, in which light nuclei coalesce (“fuse”) into heavier ones,

releasing a colossal amount of energy.

It is more efficient than fission (in terms of the amount of energy released per

gram of fuel), is seen as virtually unlimited alternative energy for the future and

produces neither greenhouse gases nor long-term radioactive waste. The fusion

reaction selected from a range of several theoretically possible reactions, and

anticipated in commercial fusion reactors, does indeed produce high-energy

neutrons as a by-product, which induce radioactivity in the surrounding matter.

However, its negative effect can be significantly minimized by a clever choice of

1. Introduction

3

structural and other materials, thereby only short-lived radioactivity is

produced.

Historically this reaction was first implemented in a hydrogen bomb and since

then the efforts of researchers have been focussed on finding the means to

harness the energy release. Controlled fusion has already been achieved in

numerous devices, the most powerful of which being JET (Joint European Torus),

located in the UK.

The ultimate goal of fusion research is to construct a device where the extraction

of fusion energy would be durable, stable and commercially justified. One of the

key issues to be addressed in the development of a commercial fusion power

plant is the selection of materials, durable enough to enable profitable

exploitation of the device, able to withstand the severe environment imposed by

hot fusion plasma, cyclic high heat load and irradiation with high energy

neutrons.

1.1.2 Nuclear fusion

1.1.2.1 Fusion in a nutshell

Nuclear energy can be gained in two ways, fission and fusion.

Fission is in essence splitting a heavy nucleus into two medium-size

nuclei initiated by an incident neutron. This reaction is exploited in

current nuclear power plants to produce electricity using uranium as

fuel.

Fusion, on the contrary, is merging two light nuclei together into heavier

ones, such as fusing hydrogen into helium.

Fine-scale measurements revealed that the mass of a nucleus is always smaller

than the sum of masses of protons and neutrons constituting the nucleus.

The energy E released in a nuclear reaction corresponds to the difference in

the nuclear binding energy, which in turn corresponds to the mass difference

m according to Einstein’s relation 2E mc . Figure 1.1 shows the nuclear

binding energy per nucleon (proton or neutron), demonstrating that light nuclei

(lighter than iron) release energy in fusion reactions, while the heavier nuclei do

so in fission reactions.

1. Introduction

4

Figure 1.1. Nuclear binding energy per nucleon as a function of the atomic mass [1].

Different mechanisms of the two reaction types lead to different requirements

imposed on the media conditions.

A fission reaction is initiated by a neutron which freely reaches a heavy nucleus

of fuel, destabilizes it, such that the nucleus splits into parts, repelling from each

other with high kinetic energy due to their positive electric charge and releasing

more neutrons which hit other nuclei of fuel thus maintaining the chain reaction.

In order to fuse, two nuclei have to approach each other close enough for the

forces of strong nuclear interaction, which are responsible for the intra-nuclear

processes, to start acting, and this happens only when the nuclei are 10-14–

10-

15 m apart, comparable to the size of a nucleus itself. However, their approach

is significantly hindered by the Coulomb repulsion due to the positive electric

charge of both nuclei.

Figure 1.2 schematically shows the interaction energy between two charged

nuclei vs. the distance between them, demonstrating the Coulomb potential

barrier which has to be overcome in order for these nuclei to undergo a nuclear

reaction, i.e. fusion. A classical estimation of the barrier height results in

tremendous several hundreds of keV which corresponds to the average kinetic

energy of gas molecules heated up to several billion Kelvin! Such temperature is

impossible to reach on Earth in a volume of matter reasonable for commercial

applications. Luckily, quantum mechanics provides the possibility of tunnelling

through the barrier, for which a kinetic energy of colliding nuclei of ~10 keV is

sufficient.

1. Introduction

5

Figure 1.2. Potential energy as a function of the distance between two charged nuclei [6].

1.1.2.2 Commercially attractive fusion reaction and fuel

In stars like the Sun the dominant reaction is the fusion of protons ( 1

1H nuclei)

into 4

2 .He It has a too low rate for commercial application on Earth, given a very

limited volume of plasma available in laboratories or in fusion reactors

compared to the enormous volume of the Sun.

There are several fusion reactions involving hydrogen isotopes such as

deuterium (D) or tritium (T) considered for the use in a fusion reactor, for

instance,

2 2 3 1

1 1 1 11.01MeV 3.02MeVD D T H (1.1)

2 2 3 0

1 1 2 0.82MeV 2.45MeVD D He n (1.2)

2 3 4 1

1 2 2 13.6MeV 14.7MeVD He He H (1.3)

2 3 4 0

1 1 2 3.05MeV 14.1MeVD T He n (1.4)

A measure of the reaction probability is the reaction cross section, which is

shown in Figure 1.3 as a function of the particles kinetic energy. The DT reaction

(1.4) is clearly identified as the most favourable choice with the highest reaction

cross-section and rate at the given conditions.

1. Introduction

6

Figure 1.3. Fusion cross-section of various fusion reactions as a function of kinetic energy of an incident D or p+ on a stationary target [7].

This reaction liberates a neutron of ~14 MeV which escapes from the plasma,

and a 3.5 MeV alpha particle which transfers its kinetic energy to the fuel nuclei

in the plasma, maintaining its temperature.

The fuel components are the hydrogen isotopes, deuterium (D) which is

naturally present in water, and tritium (T), an unstable isotope that has to be

produced artificially. A commercial fusion power plant will be self-sufficient

from the point of view of production of T. High-energy neutrons resulting in the

DT reaction will be used to bombard lithium, placed around the reactor vessel,

to generate more tritium according to the following reactions:

7 0 4 3 0

3 2 12.5MeVnLi n E He T n (1.5)

6 0 4 3

3 2 1 4.8MeVLi n He T (1.6)

Thus the basic fuels for nuclear fusion are lithium and water, both being readily

and widely available on Earth.

1.1.2.3 How to achieve fusion on Earth?

The fusion reaction in the core of the Sun takes place at ~15 MK, where the

hydrogen plasma is compressed by the immense solar gravity up to a density

above 150 g cm-3 [8], whereas the average density of Earth is 5.5 g cm-3, i.e. 27

times lower, [9]. The same approach, gravitational compression, is impossible on

Earth, and other means of the plasma confinement have to be applied, keeping in

mind that no material can withstand the colossal temperature involved.

1. Introduction

7

Two main options are used to solve the problem. One of them is the inertial

confinement, where the fusion plasma is created for fractions of a second from

fuel pellets with the help of powerful laser shots and is confined by its own

inertia only. Another option is provided thanks to the temperature required for

the fusion reaction: at ~100 MK the atoms of hydrogen isotopes are already

completely ionised, i.e., split into separate components - electrons and nuclei.

Due to their electric charge, the particles can be controlled by magnetic fields,

whose strength is comparable to that in medical magnetic resonance imaging

(MRI) scanners. The high-energetic charged particles would follow the magnetic

field lines in spiral trajectories due to the Lorentz force exerted on them.

Consequently, the plasma can be held at a distance from the physical walls of the

reactor vessel, preventing them from evaporating.

Significant technological experience has been obtained using the magnetic

confinement, and a number of ideas has been tried in devices of various design.

The most advanced magnetic confinement concept is implemented in a device

called a tokamak. The name is derived form a Russian acronym that stands for

"toroidal chamber with magnetic coils". The magnetic coils are used to create a

toroidal magnetic field around the torus. A poloidal magnetic field (orthogonal

to the toroidal direction) is created by the toroidal plasma current, which in turn

is induced by changing current in the central solenoid. The resulting magnetic

field lines are spiralled around the torus, that is required for achieving plasma

equilibrium. Toroidal, poloidal and resulting magnetic fields in a tokamak are

schematically shown in Figure 1.4.

Due to the need of changing the current in the central solenoid, tokamaks can

operate in cyclic mode only. In a different design option, called a stellarator, a

poloidal magnetic field is created by external coils, and the device does not

require the plasma current. The calculation of parameters of magnetic coils of

stellarators is onerous and could efficiently be done with the help of

supercomputers only. That is why the development of stellarators has not been

as rapid as that of tokamaks. The advantage of stellarators is the possibility to

work in continuous mode, which significantly reduces the issue of thermal

fatigue of plasma-facing components (PFC). Scientific feasibility of both types of

magnetic confinement has been demonstrated by devices JET (UK) and

Wendelstein 7-X (Germany).

Figure 1.4. Magnetic fields in a tokamak. From left to right: toroidal component, poloidal component and resulting helical magnetic lines [10].

1. Introduction

8

1.1.2.4 Achievements and records in fusion

The experiments with a toroidal configuration of fusion chambers started in

1956 in the USSR with a device called "TMP". The first tokamaks were very small

with the major radius of around 40 cm. Already at that time theoretical design

estimations predicted that a thermonuclear reactor should be as large as

~1000 m3 of plasma volume, corresponding to the major radius of 12 m.

The increase of the device size turned out to be necessary for the improvement

of the three components characterizing the quality of the magnetic confinement

in a fusion reactor: plasma temperature, plasma density and confinement time

E , being the ratio of the energy stored in the plasma plasmaW and the rate of

energy loss lossP .

plasma

E

loss

W

P (1.7)

Many tokamaks have been built since then throughout the world to continually

advance the plasma performance. Their operation is marked with a number of

records, one by one paving the way to a first commercial fusion power plant.

Joint European Torus (JET) located in the Culham Centre for Fusion Energy in

Oxfordshire, UK, is the biggest tokamak in operation nowadays, with the major

radius of 2.96 m. It is renowned for having performed the first controlled energy-

releasing DT reaction in the world, [11] and holds a record of fusion power of

16 MW achieved in 1997 [12].

The longest plasma discharge in a tokamak (6.5 minutes) was recorded in Tore

Supra in December 2003 in France [13]. The tokamak has undergone an update

and is currently known as WEST.

EAST (Hefei, China) achieved 100 s world record of steady-state high

performance plasma (H-mode) in July 2017 [14], followed by a record 100 s

plasma discharge in the Wendelstein 7-X stellarator, in November 2018 [15]. The

H-mode is the operational scenario envisaged in ITER, which would provide

advanced energy output.

The construction of the ITER tokamak in Cadarache (France) is “one step to

DEMO”, a planned demonstration fusion power plant which would generate

electric energy. The main goal of ITER is to produce more fusion energy, than is

used to “ignite” the plasma; a milestone which has not been achieved so far in

existing devices.

Other important goals include development and testing of key technologies

necessary for an efficient and safe future power plant, such as different concepts

of tritium blanket modules (TBMs), which are needed to enable tritium self-

sufficiency of DEMO and future commercial power plants.

Finally, ITER experience is absolutely necessary for the improvement of the

plasma confinement and diagnostics, to ensure that a long pulse, required by the

1. Introduction

9

DEMO balance of plant, would become feasible. The closure of the ITER cryostat

in late 2024 will be an important milestone marking the end of the first ITER

assembly phase (machine core assembly) and the start of integrated

commissioning. The next step will be to evacuate the vacuum vessel and cryostat

enclosures for a first round of leak testing at room temperature [16], finally

followed by ITER’s first plasma as planned for December 2025 [17].

1.1.3 ITER operational conditions

The aim of the present project is to develop a physically-based model of

thermally-induced plastic deformation of tungsten exposed to operational

conditions of ITER and to understand the process of accumulation of damage,

eventually leading to fatigue cracking.

1.1.3.1 The role of divertor

Plasma in tokamaks is heated in toroidal vacuum vessels surrounded by

magnetic coils. The air is evacuated from the vessel and is then replaced with

low-pressure gaseous fuel, deuterium, which is further ionized to create plasma.

Magnetic confinement cannot be perfect, and a small fraction of plasma does

interact with the vessel walls. In this way the plasma can become contaminated

with other elements, impurities, which have a negative effect on the efficiency of

the fusion reaction: they cool down the plasma due to electromagnetic radiation

and can, in the worst case, shut down the reaction.

A solution was put forward already at the dawn of the fusion research. Lyman

Spitzer at Princeton proposed a method for mitigating the consequences of the

plasma-wall interaction by diverting a part of magnetic lines at the edge of the

plasma into a separate chamber, where the interaction can take place. The

system was called a divertor. The produced impurities would be trapped in the

small chamber; from which they could be evacuated with a pump system [18].

The magnetic lines would be arranged in such a way, so as to separate the core

high-temperature plasma, where the fusion reaction takes place, from the outer,

much colder, layer (called the scrape-off layer, SOL) which can interact with the

vessel walls. As can be seen from Figure 1.5a, the closed magnetic field lines form

a D-shape in the middle of the chamber where the core plasma is located,

separated from the SOL. The first non-closed magnetic field line is called

separatrix and marks the border between the core plasma and the SOL

highlighted with pink colour. Thus, impurities introduced in the SOL remain

there, and are swept to the divertor region.

The divertor of ITER will be constructed of 54 cassette assemblies (one of them

is shown in Figure 1.5b). Tungsten monoblocks will be used as plasma-facing

components (Figure 1.5c) in the most thermally-loaded part of the divertor —

the vertical targets. The monoblocks will be assembled on a cooling pipe made

of a CuCrZr alloy using a copper interlayer to reduce thermal stresses at the joint

caused by the difference in the thermal expansion coefficient of W and CuCrZr.

1. Introduction

10

Figure 1.5. a) A schematic picture of ITER’s vacuum chamber [10,19]. Tungsten is shown with red colour. Black lines represent an example of magnetic field lines in the core plasma and a separatrix. The light pink region between the separatrix and the walls of the chamber represents the divertor plasma layer. b) A divertor cassette and its parts [20]. c) A plasma-facing unit: tungsten monoblocks assembled on a cooling pipe [21].

Figure 1.6. The monoblock geometry, the cross-section of the CuCrZr cooling pipe and the Cu interlayer. Note the bevelling of 0.5 mm introduced to reduce thermal load on the right edge of the monoblock by shadowing it with a neighbouring monoblock to the right. Dimensions are in mm [22].

1. Introduction

11

An individual monoblock and its dimensions are shown in Figure 1.6 together

with the cooling pipe and the interlayer. The divertor monoblocks have to

withstand high heat loads, especially at the strike point, where the plasma flow

gets in contact with the divertor surface, and the power density reaches

10 MW m-2 in the steady state of plasma and 20 MW m-

2 during slow transients

(which last less than ten seconds), resulting in a surface temperature of

~1000 °C or ~2000 °C correspondingly [20]. Several scenarios to reduce the

heat load power density have been developed, including:

• Aligning the divertor surface at a small angle to the magnetic field

(around 3°), to increase the effective area of the wall subjected to the

heat load

• Flux expansion of the field lines as they approach the divertor

• Periodical sweeping of the strike point over a considerable distance in

the poloidal direction

Because copper cannot withstand such high temperatures, the coolant

temperature at the inlet is chosen as 70 °C [23], corresponding to the lower

boundary of the monoblock temperature between plasma discharges.

1.1.3.2 Selected plasma scenario and the plasma-wall interaction

The plasma-wall interaction will allow for extraction of the generated heat from

the plasma, and for its transfer to the coolant, which will generate steam and spin

electric alternators, like in any ordinary power plant. A typical fusion power

plant would generate about 1000–1500 MW of electricity [1].

Commercially profitable long-term operation of fusion reactors requires the use

of a special plasma scenario, called the high-confinement (H-mode). In this case

the plasma density, temperature and pressure in the core are increased

compared to previously used low-confinement scenario, making the H-mode

favoured for the fusion reaction. This mode of operation is proposed as the

standard operating scenario for ITER [24].

Figure 1.7. Schematic representation of the normal operation with temperature excursions during transient heat loading. Lower temperature range is defined by the DBTT, while the upper range is limited by the recrystallization temperature [1].

1. Introduction

12

However, the H-mode is associated with the occurrence of plasma instabilities

called “edge-localized modes” (ELMs) [21,25] responsible for transient heat

loads depositing significant power density, ~1–10 GW m-2, at millisecond time

scale [26]. Their high repetition rate, around 1 Hz, results in approximately 1

million events during the lifetime of ITER. The heat flux due to ELMs is 2–3

orders of magnitude higher than that during steady plasma. These instabilities

are thus one of the main limiting factors for the lifetime of plasma-facing

components of the divertor. Typical temperature excursions at the surface of the

ITER divertor during normal operation are illustrated in Figure 1.7.

The main effect of repetitive ELM heat loads is thermal fatigue, which will cause

irreversible damage and shorten the divertor lifetime. Due to the high heat fluxes

impinging onto the first wall and the divertor, and simultaneous cooling by the

heat sink, large thermal gradients will occur synchronously with the ELMs. The

resultant thermal gradients will give rise to pulsed stresses, which will lead to

the thermal fatigue damage of the components: surface roughening and

formation of a surface crack network. While the formation of cracks is nearly

unavoidable, their propagation can be detrimental for the safe and stable

operation of ITER, and has to be limited by various measures.

Another concern is due to the residual stresses in the joints between the PFCs

and the heat sink due to different thermal expansion coefficients, which could

lead to detachment and worsening the thermal contact. This implies that testing

of tungsten components subjected to cyclic heat loads and thermal shocks are

essential for the qualification of the ITER device.

1.2 Tungsten as a material for fusion reactors

Tungsten (chemical element designation — W) is a transition element from

group VI in the periodic table, with atom number 74 and the atomic mass 183.85

[27] which defines its high density of 19.254 g cm-3 [28]. Phase α, body-centred

cubic (BCC) structure, is the only stable crystalline phase of W with lattice

constant 0.3165 nm at room temperature.

The selection of tungsten as the plasma-facing material in ITER is determined by

its highest melting temperature among metals (3422 °C), advantageous thermal

conductivity, its resistance to sputtering, high strength at elevated temperature

and moderate activation in neutron fields [29].

In the course of material selection for ITER these qualities overweighed the

perspectives of plasma contaminations by heavy atoms of tungsten. As

mentioned above, a divertor will remove impurities from the outer layer of

plasma in ITER, where they could initially be generated. In 2007 the ASDEX-

Upgrade tokamak became the world’s first full-tungsten fusion device, where

tungsten was used in both divertor and the first wall. Experiments in this

tokamak demonstrated the compatibility of a full-tungsten reaction chamber

1. Introduction

13

with the H-mode scenario, while preserving the plasma sufficiently clean of

impurities [30].

1.2.1 ITER-specification tungsten

The ITER Organisation has imposed requirements for the fabrication of tungsten

for plasma-facing divertor components in terms of purity, grain size, hardness

and density [20]. An important requirement is the elongation of grains, such that

the monoblocks are fabricated with the grains elongated towards the plasma-

facing surface as shown in Figure 1.8. In this case the intergranular cracks

appearing due to cyclic thermal load would propagate perpendicular to the

heated surface which is considered less detrimental for the operation of ITER

than delamination of grains at the surface leading to the plasma contamination

and consequent cooling.

Figure 1.8. Schematic view of a typical W monoblock indicating its microstructure and typical dimensions [31].

Table 1.1. Impurity content of the IGP tungsten.

A commercial tungsten grade compliant with the ITER requirements, produced

by the Austrian company Plansee AG (denoted as “IGP” standing for “ITER-grade

Plansee”) was investigated in the present work. It is polycrystalline tungsten of

>99.97% purity with the residual impurities specification listed in Table 1.1.

The material was supplied as a bar with a square cross-section of 36×36 mm2 as

shown in Figure 1.9a. The bar was fabricated by hammering on both sides,

1. Introduction

14

resulting in needle-like grains elongated along the bar axis. Prior to cutting the

samples, the bar was annealed at 1000 °C for one hour in inert environment for

degassing. According to an Electron Backscatter Diffraction (EBSD) analysis it is

a single-phase material with a grain size of 5–20 µm and 10–100 µm, normal to

and along the bar axis, respectively (Figure 1.9c). The initial dislocation density

is about (4–8)·1012 m-2, depending on particular sub-grain, with an average of

4.5·1012 m-2 measured by analysis of Transmission Electron Microscopy (TEM)

images [32], one of which is presented in Figure 1.9d.

The investigated tungsten grade is in line with the ITER specification: it adheres

to the regulation for chemical composition, density, hardness, grain size and

microstructure. More information on the ITER specification can be found in [33].

a) b)

c) d)

Figure 1.9. a) A fabricated bar of tungsten with indicated macroscopic directions; b) a TEM image of microstructure of as-received tungsten; c) an EBSD map of as-received tungsten; d) an EBSD map of recrystallized tungsten.

1. Introduction

15

1.2.1.1 Tungsten in the recrystallized state

Recrystallization of most metals and alloys starts from 0.4Tm upwards, where Tm

is the melting temperature, thus, it begins in tungsten at 1200–1400 °C. As

shown above, the surface temperature of the tungsten divertor will reach

1000 °C in the steady-state plasma discharges with infrequent rises up to

2000 °C during slow transients and even higher, up to the melting point, for

milliseconds-long pulses, due to ELMs.

It is expected that the top layer of divertor monoblocks will be recrystallized up

to a few mm depth, as registered after high heat flux tests (Figure 1.10). Surface

melting of divertor plasma-facing components can be prevented by ELM

mitigation techniques, though. Conservative finite element analysis (FEA) of the

stress state in tungsten monoblocks may require the knowledge of mechanical

properties of recrystallized tungsten.

Recrystallized tungsten has large grains depleted of dislocations, which reduces

its strength. The process of recrystallization is extended in time, hence,

intermediate — partially-recrystallized — microstructure will be observed in

the tungsten monoblocks in operational conditions most of the time. An exact

analysis of the material strength (and microstructure) evolution during

operation would require knowledge of the distribution of fraction of

recrystallized grains as a function of time (or the number of plasma discharges),

necessitating the use of recrystallization models as well.

To avoid such complicated analysis, we investigate two limiting cases of tungsten

from the point of view of microstructure, relevant for the application in ITER:

“as-received” (or “as-fabricated”) and “recrystallized”. To that end the same

mechanical tests have been carried out for the as-received material and for a

number of samples recrystallized at 1600 °C for 1 h. Recrystallization resulted in

equiaxed grains with the size around 50–100 µm, as measured by EBSD (Figure

1.9d) and reduced dislocation density down to 1011 m-2.

Figure 1.10. Metallographic cross-section in the centre of tungsten mock-up subjected to 20 MW m-2 in a high heat flux test, showing the layer recrystallized due to high thermal load [34].

1. Introduction

16

1.2.2 Mechanical properties under monotonic loading

Even though tungsten is not a structural material in ITER, divertor monoblocks

will experience significant thermal stresses during operation, thus the

knowledge of mechanical properties of tungsten is essential. For the purpose of

ITER they are mainly characterized with the help of two types of mechanical

tests: tensile and fracture toughness tests. Due to anisotropy of mechanical

properties caused by texture and microstructure with elongated grains, set as a

requirement by ITER, experiments have to be performed on samples machined

with different orientation.

It is only possible to give a general overview of mechanical properties of

tungsten, like of any material, speaking of single crystals of high purity.

Mechanical properties of commercial, polycrystalline material, which are of

technological importance, strongly depend on its purity, microstructure and

processing route.

Elastic properties of tungsten are close to isotropic and vary with temperature:

an experimental work by Lowrie and Gonas reports that in the range from room

temperature to 1800 °C Young’s modulus E decreases from 410 GPa to 315 GPa,

and Poisson’s ratio — from 0.31 to 0.28 [35,36].

Tungsten has a high ductile-to-brittle transition temperature (DBTT), in the

range 250–650 °C, depending on the alloying elements, purity and processing

route [37]. Neutron irradiation will further increase the DBTT, as it is a general

characteristic of BCC metals. Anyways, pure bulk polycrystalline tungsten is

brittle at room temperature, while single crystals, thin foils or wires of tungsten

demonstrate considerable plasticity [38–40].

Fracture toughness tests of commercially pure tungsten carried out in the

temperature range between -196 °C and 1000 °C [41] revealed that the

conditional fracture toughness QK is quite low at low temperature and does not

exceed 10 MPa m0.5 below 500 °C. However, it increases significantly up to

~80 MPa m0.5 at 800 °C indicating that the crack propagation was accompanied

by considerable plastic deformation. Investigation of fracture surfaces showed

intergranular as well as transgranular fracture of all tested materials, although

intergranular fracture behaviour was dominant in most cases [41,42].

Intergranular fracture has been observed in tungsten with impurities as well as

in highly pure tungsten, with no traces of impurities on grain boundaries (GB).

Thus, the strain incompatibility between grains is believed to be the deciding

factor of limited ductility of tungsten [42,43].

A series of experiments on another tungsten grade, a rolled rod fabricated by

Plansee Metall GmbH (Austria), was consistent with the observations by

Gludovatz at lower temperature, however, showing much lower values of QK at

527–927 °C which did not exceed 20 MPa m0.5 [44]. Besides, a loading rate effect

on the value of fracture toughness has been observed.

1. Introduction

17

Forged tungsten with flat grains has been investigated up to 1000 °C [45]. The

crack propagation was predominantly intergranular for longitudinal orientation

up to 600 °C, whereas transgranular cleavage was observed at low test

temperatures for radial and circumferentially oriented specimens. At

intermediate test temperatures the change of the fracture mode took place for

radial and circumferential orientations. Above 800 °C all three specimen types

showed large ductile deformation without noticeable crack advancement. Again,

the conditional fracture toughness QK barely exceeded 20 MPa m0.5 at 400 °C,

but the results at higher temperature were considered invalid from the point of

view of validity criteria according to the ASTM 399 standard [46].

Fracture toughness JcK of ITER-specification tungsten, recently measured

according to ASTM 1921 and referring to ASTM 1820, was around 10 MPa m0.5

at low temperature, but reached 50 MPa m0.5 at 600 °C, indicating the

importance of plastic deformation at high temperature, expected at the surface

of the ITER divertor [47].

Two main ways to increase ductility and fracture toughness of tungsten have

been identified: alloying with rhenium (Re) and grain refinement down to

nanocrystalline microstructure [43], which still have to be investigated.

Tungsten-rhenium alloys outclass other tungsten-based materials in terms of

ductility improvement. The positive effect on fracture toughness has been

confirmed in a series of tests up to 900 °C, where a W-26%Re alloy demonstrated

fracture toughness of ~50 MPa m0.5 even at room temperature [48]. W-Re alloys

with 3–10% Re are used in modern rotating X-ray anodes thanks to

improvement of mechanical properties and suppression of crack formation

[49,50]. Re is an expensive and rare element, hence alloying with Re is not

considered for the ITER divertor, but can be needed for future power plants.

However, the feasibility of using W-Re alloys in fusion devices is still

undetermined, because of an extreme brittleness of Re-rich second phase

precipitates which are formed under neutron irradiation [51]. Besides, the

presence of Re decreases thermal conductivity of the alloy [52], which can lead

to overheating and melting.

The positive effect of grain refinement on fracture toughness has been

demonstrated in experiments of pure, ultrafine grained 100 μm tungsten foil in

samples with different crack orientation relative to the rolling direction in the

temperature range from -196 °C to 800 °C [38,40]. The grain shape anisotropy

and a strong rotated cube texture have been demonstrated to be the decisive

factors for anisotropic fracture properties. The pronounced transition in failure

mode was observed going from brittle, transgranular fracture at -196 °C towards

pronounced delamination at intermediate temperatures and to ductile fracture

at highest temperatures.

Ultra-fine grained tungsten fabricated with the help of High Pressure Torsion

(HPT) showed fracture toughness increased up to 30 MPa m0.5 even at room

1. Introduction

18

temperature. The grain size decreases down to ~300 nm as a result of severe

plastic deformation [53].

Beneficial changes of microstructure during thermomechanical processing of

tungsten include elimination of pores, which may serve as stress concentrators,

decrease of grain size and build-up of dislocation density. Grain refinement as a

result of deformation is favourable for ductility of tungsten thanks to increased

density of GBs, which serve as dislocation sources. They significantly contribute

to the density of edge and mixed dislocations, which are more mobile than screw

dislocations in BCC metals, resulting in lower energy needed for deformation.

Moreover, it leads to the increase of the fraction of low-angle GBs, which are

easily penetrable by dislocations. Consequently, recrystallization eliminates the

favourable factors and reduces the material plasticity [43,54,55].

The dominance of intergranular fracture mode in tungsten enables performing

finite element analysis of deformation of tungsten polycrystal, in which special

cohesive elements are placed along GBs, as potential crack propagation path

(will be discussed in Chapter 4). The importance of plastic deformation for

fracture at elevated temperature necessitates understanding and

parameterization of plasticity of tungsten, which is in the focus of Chapter 2.

Before the specification of tungsten suitable for ITER has been established,

mechanical properties of several grades fabricated by a few companies have

been studied, among which are a sintered block, a heavily-deformed rod, a rolled

plate, single-forged and double-forged blanks, as well as tungsten doped with

potassium (K) and La2O3, summarised in the work of Uytdenhouwen [1], where

the test temperature reached 2000 °C. Tensile tests of double-forged tungsten

and parameterization of a dislocation density-based plasticity model have been

reported up to 2000 °C as well [56]. Finally, a recent overview of plastic

properties of baseline and advanced tungsten grades covered the temperature

range up to 600 °C [57].

As mentioned above, many properties of tungsten (and all materials in general)

are mainly determined by its microstructure, which in turn depends on the

fabrication route. Numerous tensile tests of different grades of tungsten show

significant differences in yield stress, ultimate tensile strength, uniform and total

elongation and fracture mode in materials with different microstructures. That

is why for the purpose of the present project a series of tensile tests of a single

selected material, IGP, have been carried out and used as a base for the

parameterization of a plasticity model.

1.2.3 Mechanical response of tungsten under cyclic loading

1.2.3.1 Overview of heat load tests

In order to ensure the structural integrity of the divertor during operation of

ITER, its monoblocks have to be tested in thermal cycles of high amplitude.

Various methods, such as beams of electrons, ions, or neutral particles, laser and

1. Introduction

19

plasma shots, can be used to simulate the thermal conditions in a fusion reactor.

Two types of tests are mainly used:

High heat flux test, in which a mock-up of a divertor (several

monoblocks assembled on a cooling pipe, Figure 1.11) is subjected to

cycles of heat load lasting usually for seconds. Coolant flows through the

pipe at speed and temperature envisaged in the real operation of ITER.

Thermal loads of 10 MW m-2 or 20 MW m-

2 are applied to simulate the

effect of long stationary plasma discharges and slow transients,

correspondingly. In order to pass the tests, the monoblocks have to

survive 5000 or 300 cycles (of low and high deposited energy density

correspondingly), preserving the thermal contact with the cooling pipe

and without formation of macroscopic surface cracks (called “self-

castellation”) [31,58].

Thermal shock test, in which a tungsten sample is subjected to

microsecond-long pulses of significant thermal power, amounting to

several hundreds of MW m-2, to mimic the effect of ELMs. The opposite

face of the sample is maintained at constant temperature. The

temperature profile caused by such short transient heating does not

spread deep into the material. That is why the surface temperature

excursions due to short transients are almost unaffected by the choice

of coolant temperature.

Figure 1.11. Divertor mock-up after exposure to a high heat flux [34].

The applied thermal flux should be uniform over the exposed area in order to be

able to compare the experimental results with the heat load expected in ITER

and theoretical predictions. Besides, the initiation and growth of cracks under

cyclic heat loads can also be affected by the power density profile. However,

generation of an electron beam, uniformly distributed over a large area is a

challenging task. Instead, beam rastering is used in order to deposit high heat

flux over a wide area uniformly. The rastering frequency should be high enough

(more than tens of kHz) to smooth the pulsations of deposited energy at a given

point of the surface.

Other high-temperature applications of tungsten, where it is subjected to cyclic

thermal load, include the European Spallation Source (ESS) and X-ray sources.

ESS is a facility under construction in Lund, Sweden, where tungsten will be used

1. Introduction

20

as a rotating spallation target. The surface will experience cyclic thermal loads,

like the ITER divertor, but the expected operation temperature is much lower,

500 °C maximum [59]. That is why the mechanical characterization of tungsten

for ESS has been carried out within this temperature window in tension and low

cycle fatigue [59,60]. The material was brittle up to 200–250 °C, and

demonstrated early necking but high total elongation at higher temperature.

Rotating X-ray anodes are also made of tungsten and tungsten-based alloys

[49,61–64]. A continuous high-energy electron beam is directed onto the rotated

anode, hence, every given unit of anode area experiences cyclic thermal shocks,

as it passes through the focal spot during rotation at frequency up to 250 Hz [65].

The temperature can rise by several hundred or a thousand of °C [50]. It has to

be noted that numerical analyses of thermomechanical conditions in X-ray

anodes have rarely been published [65], thus the results of the present thesis

might be useful for this application of tungsten as well.

1.2.3.2 High heat flux test

Numerous tests of both types have already been performed. Monoblocks

fabricated of different grades of tungsten, using various techniques of joining the

monoblock with the cooling pipe were qualified in machines like ion beam

facility GLADIS [66], linear plasma device Pilot-PSI [67] and electron beam

facilities IDTF [68] and FE200 [34]. Later, the monoblock dimensions and shape

were fine-tuned, by thermo-mechanical FEA of a monoblock subjected to a

steady-state plasma discharge and simultaneous optimization of the

temperature field.

The main outcome of the full-tungsten qualification program (in the unirradiated

state) is that the tested mock-ups and full scale prototypes (280 monoblocks in

total) successfully demonstrated their performance, with just 30% of them on

average, showing self-castellation, i.e. formation of a crack network, in the

tungsten armour [58].

Some monoblocks withstood critical heat flux tests during 1000 heat cycles at

20 MW m−2, by far exceeding the threshold of 300 cycles imposed by the ITER

Organisation [69]. Accompanying simulations confirmed that appearance of

large thermal stresses at the centre of loaded surface was consistent with the

location of macro-cracks in monoblocks [69].

1.2.3.3 Thermal shock test

First thermal shock tests in electron beam devices designed to characterize the

response of tungsten to ELMs involved merely a single shot of thermal power

[70]. Recently, tests with one million thermal shocks were performed to

investigate the fatigue and long term behaviour of the material [71], that

coincides with the expected number of ELMs in ITER’s operational lifetime.

1. Introduction

21

Figure 1.12. Damage mapping of tungsten samples exposed to ELM-like thermal shocks. Plain-coloured symbols stand for samples with 100 thermal shock pulses. Symbols with a black centre stand for samples exposed to 1000 pulses [26].

The goal of these tests was to identify the threshold conditions of thermal shocks,

which do not lead to surface crack formation after a given number of heat pulses.

The base temperature, the pulse length and the power density of one shot were

varied. The tests have been performed in quasistationary plasma accelerator

QSPA Kh-50 [72] or electron-beam device JUDITH [26,71] for the material in

unirradiated state. As a result, maps of base temperature vs. absorbed power

density were obtained, with Figure 1.12 serving as example.

The evolution of surface morphology with the increase of either cycle number or

heat power density:

1. Initial surface

2. Surface still not changed after a certain number of cycles

3. Roughened surface

4. Micro cracks on the surface

5. Propagated cracks

Rapid increase of surface temperature leads to expansion of the subsurface

layers with strain rate up to 8 s-1 [1], i.e., several orders of magnitude higher than

the strain rate in conventional tensile tests. Tungsten is known to demonstrate

rate-dependent plasticity [1,52], meaning that in order to extrapolate the

experimentally-obtained mechanical properties to higher strain rate and higher

temperature, aiming at simulation of thermal shocks due to ELMs, a physically-

based model of plasticity has to be employed.

1. Introduction

22

1.3 Overview of techniques

The present project relies on computational analysis almost entirely, but the

input for the simulations was obtained in experiments. Tensile tests were used

to find out the constitutive law of the investigated tungsten grade, i.e. the stress-

strain curves in a range of temperature and strain rate, while its microstructure

was characterized with the help of scanning and transmission electron

microscopes. The obtained constitutive laws were used in the J2 finite element

analysis of tungsten exposed to ITER-relevant thermal shocks, and had to be

translated to the microscopic level in order to be used in the crystal plasticity

framework. A short overview of numerical and experimental techniques

involved in the project is presented below, whereas more detailed discussion

will be provided in the corresponding chapters.

1.3.1 Numerical analyses of heat load tests

The finite element method (FEM) is the most widely used universal simulation

tool in structural mechanics nowadays. It originates from the work of Courant

[73], and was significantly enriched by Zienkewicz and Taylor [74,75]. It is a

numerical method to solve nonlinear partial differential equations, and has to be

provided physical laws in order to construct a model.

The idea behind it is to discretize a simulated body into smaller simpler parts,

called finite elements. The entire mechanical problem is then addressed by

solving a large system of equations, corresponding to the tessellated elements.

The elements are connected at the exterior nodes only, which can be displaced

by the applied load and, in turn, drag the connected elements along.

Displacements of any point in the elements is an interpolation of the nodal

displacements. Strain tensor in elements is calculated from the nodal

displacement, while stress tensor in elements is derived from the forces exerted

on nodes. Interpolation of the values calculated at nodes is carried out with the

help of polynomial shape functions, which can be of first-, second- and higher

order. Standard first-order elements (linear) are essentially constant strain

elements, while the second-order elements (quadratic) are capable of

representing all possible linear strain fields [76].

Variational methods are applied to find a solution by minimizing an associated

error function. The relation between stress and strain in an element is governed

by a constitutive law. FE analysis requires the three following general procedural

steps which are illustrated by the most typical tasks involved in them [77]:

1. Pre-processing

Generation of the mesh, including node coordinates and the choice

of element type

Specification of loading conditions

Input of material (constitutive) law

1. Introduction

23

2. FE analysis

The solution is obtained incrementally in time; it has to converge at

every time increment, meaning that the obtained fields of stress and

strain tensors must satisfy the conditions on compatibility,

equilibrium, boundary conditions and the constitutive law

simultaneously

3. Post-processing

Calculation of element data at integration points

Analysis of temporal evolution of a given quantity

Graphical representation of obtained solution such as isoline

images, animation

Among the variety of software for finite element analysis, Abaqus 6.17-1 by

Simulia was chosen for the present research, as a popular and powerful tool,

flexible enough so as to incorporate user-defined material behaviour, necessary

for the analysis at the grain level with the help of crystal plasticity.

The problem of crack formation in tungsten monoblocks due to plastic

deformation under cyclic heat loads is important and is investigated not only

with the help of qualification tests discussed above, but also with the help of FEA.

It requires mechanical properties of investigated material as input (tungsten in

our case). The lack of a database of mechanical properties of tungsten in the

temperature range corresponding to the operational conditions of ITER, as well

as a big number of investigated tungsten grades and dependency of plastic

properties on microstructure and fabrication route forces the use of simplified,

linear-elastic and ideal-plastic constitutive laws in many cases.

Such approach has widely been used for simulation of both stationary plasma

discharges [78–82] and the effect of ELMs [83] to assess the evolution of stress

and strain in tungsten under cyclic heat loads. These simulations have been

performed at the macroscopic scale, using J2 plasticity model and materials were

defined as perfect elasto-plastic solids with the yield stress and Young’s modulus

decreasing with temperature.

Engineering investigations of design options to mitigate deep cracking in

monoblocks relied on macroscopic J2 FEA [80]. Also, the concept of thermal

break has been evaluated in this way [84]. It is an “obstacle” in the monoblock

design that impedes the thermal flux towards the cooling pipe and increases the

temperature in tungsten above its DBTT.

Thermo-mechanical analysis of heat-exposed monoblocks in two limiting cases

of microstructure — as-received W and recrystallized W — was reported in [85]

and used an elastic-linear plastic constitutive law with kinematic hardening,

however, little information was provided regarding its parameterization.

1. Introduction

24

1.3.2 Consideration of cyclic deformation under J2 plasticity

theory

1.3.2.1 The Bauschinger effect

An interesting phenomenon was discovered by Johann Bauschinger, who

installed the largest testing machine of the period and made very accurate strain

measurements with a mirror extensometer. It turned out that materials pre-

strained in tension, generally exhibit a lower yield stress in subsequent

compression, and vice versa. The effect named after its discoverer has been

observed in many, especially structural, materials such as steels [86,87] or TiAl

alloys [88], not only in tension-compression cycles, but under conditions of any

significant change in loading path. [89].

An example of the Bauschinger effect is schematically demonstrated in Figure

1.13, conventionally denoting tensile stress and strain as positive, and

compressive stress and strain as negative. The solid black lines show the flow

curves in tension and consequent compression. The material loaded in tension

reaches the flow stress 1t , is elastically unloaded and then compressed. But the

plastic yield begins at compression stress 2c which is lower than 1t . The

stress drop is evident when comparing the dashed blue line, which is the point

reflection of the compression curve, and the forward flow curve.

Figure 1.13. Illustration of the Bauschinger effect in reversed loading. The yield stress in compression is lower than the stress achieved prior the load reversal. [90].

1.3.2.2 J2 plasticity model

The von Mises constitutive model of plasticity is available in Abaqus by default

and is widely used for macroscopic FEA of metallic materials where the

anisotropic properties of individual grains are averaged thanks to their large

number. In order to investigate the mechanical response of material at the

1. Introduction

25

microscopic level, where plasticity is governed by the interaction between grains

and the mutual orientation of slip planes, in which dislocation glide takes place,

we will use crystal plasticity constitutive model discussed in Chapter 3.

The von Mises model assumes that yielding of metal is independent of the

hydrostatic stress H , which is the sum of the diagonal components of the stress

tensor ijσ .

3

1

H ii

i

(1.8)

The model states that yielding begins when the equivalent stress (referred to as

“von Mises stress”) eq reaches the value of yield stress recorded in a uniaxial

tensile test of the given material

3

:2

eq σ σ (1.9)

where σ is the deviatoric stress tensor

H σ σ (1.10)

As the von Mises stress is proportional to the second invariant of the deviatoric

stress tensor 2J ,

eq 23 J (1.11)

the von Mises constitutive model is called “J2 plasticity”.

1.3.2.3 Isotropic and kinematic hardening

In order to better understand the Bauschinger effect, we should refer to the

concept of a yield criterion. It is a mathematical expression of the stress states

that will cause yielding or plastic flow, which is especially useful for studying

material yielding under complex stress states. The most general form of a yield

criterion is [89]

, , , , ,x y z xy xz yzf const (1.12)

This expression defines a closed surface in the stress space, called “yield locus”.

A point in this space corresponds to a particular local stress state at a material

point. Stress states inside the yield locus generate merely elastic deformation,

while those on the surface cause plastic flow.

There are two main models of the evolution of yield loci during deformation [89].

According to the isotropic hardening model, the effect of strain hardening is to

expand the yield locus in an isotropic manner without changing its shape (Figure

1.14a). It means that the stress for yielding in reverse loading would remain

equal to the stress reached by the end of direct loading.

1. Introduction

26

An alternative model is kinematic hardening, according to which plastic

deformation shifts the yield locus in the direction of the loading path without

changing its shape or size (Figure 1.14b). The models represent the limiting

cases of material hardening, whereas their combination at different proportion

is observed in reality.

Isotropic hardening is due to accumulation of forest dislocations, which pose as

obstacles for gliding dislocations by pinning them. One of the models of

dislocation density evolution is the one developed by Kocks and Mecking [91],

where the density is controlled by two terms, the multiplication rate (thanks to

the dislocation sources activated during deformation) and the dynamic recovery

rate.

Kinematic hardening is observed in material where stress heterogeneity occurs

at the microscopic level during plastic deformation [92,93]. Examples include

composites and multiphase materials where constituents have different strength

or stiffness [88,94] and precipitate hardened alloys in which the reverse flow is

facilitated thanks to the Orowan loops formed during direct loading [95–97].

Formation of directional dislocation substructures (such as cell walls) also

contributes to kinematic hardening.

Figure 1.14. Effect of strain hardening on the yield locus. The cross-section of the

locus in the x – y plane is shown. The isotropic model (a) corresponds to an

expansion of the locus. The kinematic hardening model (b) corresponds to a translation of the locus in the direction of the loading path [89].

In single-phase materials it emerges mostly due to the formation of dislocation

pile-ups at GB [98,99], and we will focus on this mechanism in the present work.

When dislocations gliding in a given slip system one after another hit a GB, the

leading dislocation stops, as the stress necessary to overcome the GB is too high.

Dislocations of the same sign repel each other, thus, the whole sequence of

dislocations stops. Thus, a pile-up creates a back stress on the dislocation source

so that the stress to continue to operate the source must rise.

1. Introduction

27

The distance between the dislocations is defined by the balance of the applied

stress, pushing the dislocations towards the GB and the backstress (Figure 1.15).

Upon unloading the pile-up dissolves into individual dislocations whose reverse

motion is facilitated by the backstress, effectively reducing the external stress

required for the yield.

Figure 1.15. Linear arrays of edge dislocations piled-up against barriers under an applied shear stress [100].

The stress experienced by the leading dislocation in a pile-up composed of n

dislocations is [89]

lead appliedn (1.13)

The same stress is exerted locally on the GB, in the vicinity of the pile-up. It means

that the presence of dislocation pile-ups (and, thus, the ability of material to

exhibit kinematic hardening) leads to a more severe stress state at GBs,

compared to a material with purely isotropic hardening, and can provoke

fracture if GBs are the weakest link in the material.

Having identified two major differences between materials exhibiting purely

kinematic and isotropic hardening, namely

the Bauschinger effect

more severe stress state at GBs

we raise a question relevant for specific ITER conditions: what is the impact of

the potential presence of kinematic hardening in ITER-specification tungsten on

the stress state and damage accumulation in a monoblock exposed to ELMs and

being under conditions of cyclic thermal stress?

Information on the Bauschinger effect, and consequently, on kinematic

hardening, cannot be inferred from uniaxial tensile tests, the main source of

constitutive laws for FEA. In order to quantitatively estimate the proportion of

kinematic hardening relative to isotropic hardening, cyclic mechanical loading

tests have to be performed, in which a sample is periodically loaded in tension

and compression.

Low cycle instrumented fatigue tests to study the Bauschinger effect have

scarcely been carried out for tungsten due to its brittleness and associated

technical difficulties in the manufacturing of samples of complex geometry,

needed for the tests. In Ref. [59] the authors performed strain-controlled low-

1. Introduction

28

cycle fatigue tests of different tungsten grades with three levels of strain

amplitude at 480 °C. The obtained stress-strain loops reproduced in Figure 1.16

clearly demonstrate a significant Bauschinger effect in forged tungsten under

given experimental conditions: tungsten, loaded up to 450 MPa in tension, yields

at ~100 MPa in compression. However, further analysis of available literature

has not yielded other works devoted to instrumented fatigue tests, based on

which the Bauschinger effect could have been assessed in various tungsten

grades. Furthermore, works on fatigue tests in tantalum have not been found

either, disabling the possibility to construct a plasticity model of tungsten based

on another refractory material with similar properties.

Figure 1.16. Comparison between a strain controlled fatigue test and a monotonic tensile test of polished forged tungsten at 480 °C [59].

1.3.3 Deformation of polycrystals at the microscale

1.3.3.1 What is crystal plasticity?

Finite element analysis of macroscopic samples or structural components can

generally be performed assuming constitutive laws of isotropic homogeneous

continuum media and the J2 plasticity model. Despite the fact that macroscopic

metallic bodies are represented by aggregates of grains, i.e. anisotropic single

crystals, their properties are averaged thanks to the abundance of grains and

thus the anisotropy of a macroscopic body is eliminated. The outcome of the J2

(Mises) plasticity simulations depends on the accumulated equivalent plastic

strain only, and is not affected by the anisotropy of grains.

On the other hand, consideration of deformation of an individual crystal has to

account for its anisotropy, which originates from the features of deformation

mechanisms in crystal lattice. The most important deformation mechanism is

dislocation glide, which can take place only in certain atomic planes, due to the

anisotropy of individual crystals.

1. Introduction

29

The model that predicts which slip systems will be activated and which level of

strain will be reached in them, given the applied macroscopic stress, is crystal

plasticity (CP). It operates with quantities, such as stress components, strain

components, dislocation density, averaged over individual grains. The CP model

also accounts for the interaction of every grain with its neighbours during plastic

deformation and can be applied to polycrystals. Coupled with FEM, and referred

to as “CPFEM” in this case, it is able to predict the stress and strain heterogeneity

within every grain, because in this case the quantities like stress are calculated

locally in a given finite element, rather than in a given grain, and can help identify

the most loaded locations where, for instance, crack initiation is possible.

On top of that, prediction of the build-up of internal stresses and associated

damage in real materials at the microscopic level requires the knowledge of an

accurate strain path, i.e. the evolution of all the components of the strain tensor

at a given material point. Indeed, different strain paths (possibly leading to the

same final deformation) will induce different crystal lattice rotations and

different heterogeneity of hardening throughout the polycrystal, affecting the

final material state.

Simplified approaches to describe plastic deformation of a polycrystal such as

the Sachs or the Taylor approach, assume that either stress or strain

(respectively) in every grain coincides with the macroscopic applied value. This

simplification has its drawbacks: according to the Sachs model the strain

compatibility condition is violated, i.e. the grains in the deformed polycrystal

have to overlap or detach from each other, whereas stress equilibrium condition

is violated at GBs in Taylor’s polycrystal [101]. Both drawbacks being unphysical,

the latter is nevertheless seen as a more appropriate assumption, and the Taylor

model matches better with experiments [102]. However, grains in real materials

show both stress and strain heterogeneity during deformation due to the grain

anisotropy. This effect can be captured by CPFEM.

The effect of kinematic hardening has rarely been considered in

thermomechanical simulations of tungsten exposed to thermal shocks, whereas

it can strongly affect the rate of build-up of internal stresses in cyclic loading,

compared to purely isotropic hardening model. As long as one of the origins of

kinematic hardening is the build-up of dislocation pile-ups at strong obstacles,

which create backstresses assisting the reverse deformation, its effect must be

incorporated in a microscale model.

Dislocation motion in plastically-deformed metallic polycrystalline materials is

hindered by obstacles; either weak, such as forest dislocations, irradiation-

induced voids and dislocation loops, or strong, such as GBs. Having stopped at

strong obstacles, which require high stress in order to be overcome, dislocation

cores merge into larger discontinuities, the predecessors of cracks. As long as

cracks primarily initiate at the GBs and triple junctions, where favourable

conditions are provided by stress concentration, the knowledge of stress and

strain heterogeneity in grains is essential for accurate prediction of cracking.

1. Introduction

30

Consecutive thermal loads and dilatation-contraction cycles caused by them lead

to the amplification of stress/strain heterogeneity.

Plus, the effect of neutron irradiation on material behaviour is reflected at the

micro level. First, irradiation-induced defects, serving as obstacles, modify the

dislocation glide law and alter the grain response to mechanical loading. Second,

plastic deformation of neutron irradiated material is reported to cause such

microstructure changes as dislocation channelling. Channels correspond to thin

strips of material depleted of irradiation-induced defects, where the dislocation

glide is eased as compared to the surrounding material, rich in defects. Due to

intensive dislocation glide, plastic deformation accumulates rapidly in these

regions, leading to significant stress concentration and crack formation.

Thus the problem to be solved in the frame of the present project requires

detailed knowledge of stress and strain heterogeneity at the grain level, which

will amplify due to cyclic strain and neutron irradiation. Application of the J2

plasticity constitutive model at the grain level cannot predict any heterogeneity,

and is thus not appropriate for the present problem. CPFEM is used here to

predict damage accumulation and potential cracking in tungsten plasma-facing

components in ITER.

1.3.3.2 Relevant application of CPFEM in literature

CPFEM is a widely used method which has been applied in investigation of grain

size effects, grain interaction, surface roughening, high-temperature

deformation, nanoindentation and many other phenomena [75]. It is flexible

enough to introduce the effect of neutron irradiation on mechanical properties.

Works on crystal plasticity were mostly devoted to face-centred cubic (FCC)

materials, while BCC materials were basically represented by pure iron or iron-

based alloys and steels [103], that is why we had to modify the typically used

model equations in order to take into account a non-zero athermal stress typical

for BCC metals. A short review below is focused on the applications of the CPFEM

where dislocations were the main carrier of plastic deformation.

The importance of heterogeneity of deformation at the grain level on damage

nucleation in polycrystals was discussed in [104]. Listing 19 hypotheses

regarding the damage nucleation, the work points out that some of them are

incompatible with others. Nevertheless, we have selected one of the criteria,

more suitable to our problem in our opinion, which is discussed below.

Crystal plasticity has been applied to model the cyclic mechanical behaviour of a

polycrystalline nickel-based alloy at a constant temperature [105], ignoring,

however, the work hardening.

A more advanced work of low cycle fatigue of a Ni-based superalloy [106]

resulting in a good agreement with the experimental results for stress–strain

loops, cyclic hardening behaviour and stress relaxation. Accumulated plastic

strain was used as a fatigue indicator parameter (FIP) and the evolution of

backstresses was tracked, so that the Bauschinger effect was captured on the

1. Introduction

31

stress-strain loops. It was a 2D model with plane strain conditions, well suitable

for simulation of a fracture toughness compact tension sample.

CPFEM has also been applied to study thermal fatigue caused by periodical

temperature excursions in P91 steel [107], however, not many papers are

devoted to this topic. The FE model with periodic boundary conditions in both

in- plane and out-of-plane directions used regular 1st order cubic elements. It

was first parameterised using isothermal low cycle fatigue experiments. Similar

to the previously mentioned work, the evolution of backstress was considered,

however, fatigue life was predicted with a more complicated FIP, determined by

the accumulated plastic strain, stress triaxiality and temperature.

1.3.3.3 Kocks-Mecking model of plasticity

The Taylor equation is one of possible links between material’s strength and

microstructure, in terms of dislocation density :

c disb h (1.14)

where c is the critical resolved shear stress (CRSS),

dish is the dislocation

strength, is the shear modulus and b is the magnitude of the Burgers vector.

A model derived by Kocks and Mecking [91] on the basis of various experimental

observations prescribes the storage of in deformed material as concurrent

multiplication and annihilation (dynamic recovery) of dislocation:

1 r

r r

dL N

d b

(1.15)

The first term is responsible for hardening due to dislocation multiplication and

is purely of geometric or statistical nature, thus, athermal. The rate of dislocation

multiplication is defined by the mean free path swept by dislocations which

can be assumed to be proportional to the mean dislocation spacing 1 . The

second term describes the annihilation of dislocations, proportional to rN , the

number of dislocation segments per unit volume involved, their mean length rL

and the effective rearrangement rate r , rendering the process thermally-

activated. Since r rL N is proportional to , equation (1.15) can be rewritten,

relying on two parameters 1k and

2k , the latter being rate- and temperature-

sensitive [108]:

1 2k kt t

(1.16)

Later, numerous modifications and improvements of the model were suggested

[108–111]. Nowadays the model is commonly used to describe plasticity of BCC

and FCC materials, with many types of microstructure [112,113]. Using the laws

on the temperature- and rate-dependency of 2k proposed in [114], it was

1. Introduction

32

possible to apply the Kocks-Mecking model to characterize plasticity of different

tungsten grades in a range of temperature and strain rate, exploited

experimentally [115]. These laws will be applied in the present work to

extrapolate the mechanical properties of tungsten obtained in tensile tests to

higher temperature and higher strain rate ranges.

First of the questions addressed in the present research is whether the Kocks-

Mecking model can successfully be applied to and is sufficient for

characterization of plasticity of the ITER-specification tungsten in both as-

received and as-recrystallized states in the range of temperature and strain rate,

in which the mechanical tests were carried out.

1.3.4 Tensile tests

Tensile tests are the most widely performed mechanical tests, which allow one

to find out the response of material to mechanical deformation (elongation) and

extract several technologically-important characteristics, such as

- yield stress, the load at which plastic deformation commences

- uniform elongation, the deformation at which plastic instability takes

place

- ultimate tensile stress, the load at which plastic instability takes place

- fracture strain, or total elongation which material can sustain

- fracture stress, the maximal value of load that material can sustain

- reduction of area, a characteristics of plasticity that compares the cross-

section area before and after the test

Part of the samples was scheduled for irradiation in the BR2 material test reactor

to derive the material mechanical properties in the irradiated state, forcing the

use of miniaturized samples in order to fit the irradiation channels. Samples for

tensile tests can have circular or rectangular cross-section. Even though

cylindrical samples produced by turning or grinding are usually preferred in

mechanical testing due to axial symmetry, their fabrication from tungsten is

expensive and challenging, since the material is brittle at room temperature.

Electrical discharge machining (EDM) is a more appropriate tool for fabrication

of tungsten samples, as it introduces less damage compared to conventional

machining techniques. Since samples with rectangular cross-section use the

space in irradiation channels more efficiently and are easier for fabrication with

the help of EDM than round ones, this geometry, also called "dog-bone" (shown

in Figure 1.17) was adopted for the present project.

1. Introduction

33

a) b)

Figure 1.17. (a) Sketch of a miniaturized dog-bone-shape tensile sample, and (b) photo of a sample before and after tensile test.

Figure 1.18. A sample in the furnace of the Instron machine. The thermocouple wires are attached to the gauge section of the sample.

An Instron pull bench was used to perform displacement-controlled tensile tests

in air, hence the temperature was maintained below 600 °C to reduce oxidation

of tungsten. A sample placed in the furnace for testing in the Instron machine is

shown in Figure 1.18.

1. Introduction

34

A sample is pulled by the grips with a given speed, while its elongation and the

applied force are recorded. Then they are normalized by the sample dimensions

in order to be comparable with other tests.

Engineering stress is the ratio of the applied force F and the initial area of the

reduced cross-section IA

I

F

A (1.17)

Engineering strain eng is the ratio of elongation

0L L and the initial gauge

length 0L

0

0

eng

L L

L

(1.18)

The whole curve obtained in tensile tests serves as input in finite element

simulations, but in order to do so, true stress and true strain have to be extracted

from the experimental data, because the engineering quantities do not take large

strain into account, when the cross-section area significantly decreases

compared to the initial one.

The standard approach to extract true stress and strain is based on the

assumption of uniform deformation and is, thus, applicable up to the onset of

deformation instability only.

a)

b)

Figure 1.19. Examples of two main types of stress-strain curves encountered in the present work: a) low uniform elongation, “early necking”, b) high uniform elongation. The thick lines highlight the range of applicability of the standard approach to extract true stress and strain.

Two types of load-displacement curves have been obtained for the considered

tungsten grades. Recrystallized tungsten demonstrated high plasticity at high

temperature, with high uniform elongation (Figure 1.19a). The as-received

1. Introduction

35

material, however, is characterized by low uniform elongation, which is almost

absent at high temperature (500 °C, 600 °C) and is referred to as “early necking”

(Figure 1.19b). Significant part of the curve demonstrates softening. Thus, even

though the total elongation is high, indicating high plasticity, the standard

method of extraction is unable to obtain this information, being limited within

the range of strain, marked with the thick red line. An alternative method to

extract true stress-strain curve from engineering data with early necking,

involving reverse finite element analysis, is discussed in Chapter 2.

1.3.5 Scanning electron microscope and electron backscatter

diffraction

Scanning electron microscopy (SEM) was applied in order to investigate polished

surface of the investigated material. SEM JEOL 6610 located at SCK•CEN with

secondary electron detector (accelerating voltage 15 kV and working distance of

10–11 mm), which is based on the interaction of a focused monochromatic

electron beam with material. The details about this technique can be found in

[116]

Electron backscatter diffraction (EBSD) allows one to study grain size and

crystallographic orientation. It is based on recording and indexing of so-called

Kikuchi lines that are produced by the backscattered electrons that are

subsequently diffracted by the crystal planes. The detailed description of EBSD

technique can be found in [117]. A Field Emission Gun Quanta-450 FEI SEM with

20 kV acceleration voltage located in the Department of Materials, Textiles and

Chemical Engineering of Ghent University was used to scan the samples. The step

size was 1.2–2.0 µm. The specimen was mounted with 70° tilt in order to reduce

the path length of the backscattered electrons and to increase the intensity of

electron backscatter patterns. During the image post-processing using OIM

Analysis software, points with Confidence Index lower than 0.1 were removed.

The SEM investigation and EBSD analysis of tungsten for the present project has

been reported in [118].

1.3.6 Transmission electron microscope

Transmission electron microscope (TEM) JEOL 3010 EX, with acceleration

voltage 300 kV was used to investigate the microstructure of tungsten, namely,

the size of grains and subgrains, their aspect ratio and dislocation density, which

are relevant for the numerical models used in the present research.

The set-up of a TEM is comparable with that of an optical microscope. A beam of

electrons is generated by a hot cathode and is accelerated by high voltage. Then,

a combination of magnetic lenses focuses the electron beam on the sample. To

avoid undesirable interaction with gas molecules, the electron microscope is

operated under ultra-high vacuum conditions. After the interaction with the

specimen, the transmitted electrons pass through a second combination of

1. Introduction

36

magnifying lenses. Finally, the electrons hit a fluorescent screen producing an

enlarged image of the sample microstructure. The transparency of a TEM

specimen is acceptable up to thickness about 200 nm.

The TEM investigation of tungsten for the present project has been reported in

[32].

1.4 Outline of the thesis and the addressed

questions

ELMs, periodic plasma instabilities accompanying the plasma discharges in the

selected operational scenario of ITER, will deposit pulses of heat energy on the

divertor surface, causing cyclic thermal strain, the rate of which can reach 8 s-1,

while the temperature of divertor surface will exceed 1000 °C or even approach

the melting point. On the contrary, conditions in tensile tests of samples in a

mechanical lab are limited in terms of both temperature and strain rate.

Consequently, the applicability domain of constitutive laws directly derived from

tensile test results is merely a small fraction of the whole domain of ITER

operation conditions.

Willing to overcome this limitation and aiming at the analysis of mechanical

response of W to ELMs with the help of a finite element solver, we need to

extrapolate the experimental results using a physically-based model of plasticity.

The model developed by Kocks and Mecking has been chosen to this end, as it

predicts the evolution of microstructure of material (dislocation density) during

mechanical loading and links it with the material strength. The question whether

the temperature- and rate-sensitive plasticity of ITER-specification tungsten in

uniaxial loading can be successfully described by the Kocks-Mecking model in

the whole range of temperature and strain rate anticipated in ITER (or, how the

model should be modified in order to do so) is addressed in Chapter 2.

The Bauschinger effect and the associated kinematic hardening can substantially

change the mechanical response of material under cyclic loads, but they have

scarcely been studied in tungsten experimentally. Consequently, there was not

enough input for simulations of mechanical response of tungsten to cyclic heat

loads, necessary to estimate the conditions of crack formation. In the absence of

experimental results on mechanical cyclic loading of tungsten, we consider

limiting cases in the present project. In Chapter 3 we hypothesise how the

kinematic hardening should be incorporated in the model of plasticity, having

the results of uniaxial tensile tests as reference, and apply the modified model to

estimate the effect of the presence (or absence) of the kinematic hardening on

thermal fatigue of tungsten exposed to ELM-like thermal conditions. On top of

that we discuss how the macroscopic model of plasticity should be translated to

the microscopic level to be applied in the crystal plasticity framework. It is

necessary because microstructure of tungsten, including dislocation density,

crystallographic texture as well as grain size and shape, is an important factor

1. Introduction

37

that defines its mechanical properties and must be incorporated in the model, as

we cannot rely on the macroscopic, averaged stress.

Finally, in Chapter 4, we discuss the application of the obtained CPFEM model to

address the effect of microstructure and the kinematic hardening on mechanical

response of tungsten polycrystal subjected to ITER-relevant thermal shocks.

The work is devoted to the analysis of damage accumulation at the microscopic

level, which eventually leads to crack formation, in tungsten subjected to thermal

shocks due to ELMs. It is a challenging goal, even without consideration of

neutron irradiation defects in the material, that is why we limit the scope of the

thesis to non-irradiated tungsten. Thus, the present work is relevant for the first

years of operation of ITER, where numerous non-nuclear plasma experiments

will precede a short nuclear phase, during which the divertor is expected to

receive a limited neutron damage, ~0.1 dpa (displacements per atom) [20,119].

The advantage of the CP model is that it can further be extended to take account

of the modification of mechanical properties induced by neutrons.

39

Chapter 2.

Applicability of the Kocks-Mecking model

to temperature- and strain rate-sensitive

uniaxial deformation

of ITER-specification tungsten Equation Cha pter 2 Section 2

2.1 Introduction

In this chapter we address the question whether the Kocks-Mecking model can

be applied to characterise plasticity of ITER-specification tungsten by

reproducing the experimental stress-strain curves, as well as to extrapolate them

to high temperature and strain rate, expected in ITER. Part of this chapter

devoted to recrystallized tungsten has been published [120].

Plasma-facing components in ITER will host a very heterogeneous temperature

field, inducing large thermal strains. As a result of the combined action of

stationary heat loads and ELMs, the surface temperature can reach 2000–

3000 °C [121] and the strain rate can be as high as several s-1 [1]. The much lower

temperatures experienced at some distance below the surface make the material

stronger and less ductile. While aiming to assess the stress state inside plasma-

facing components during ITER operation, we need to rely on a constitutive law

valid in such wide ranges of temperature and strain rate. As temperature- and

strain-rate field continuously evolve in time, the constitutive law should be a

continuous function of the two quantities. The mechanical response is of course

also influenced by irradiation. This, however, is not accounted for in the present

work.

Tensile tests involving significant ductility of ITER relevant tungsten grades have

so far been reported for the grade labelled as M184 — in the recrystallized state

after heavy deformation — [1] and for IGP tungsten [26]. They reported

significant temperature and strain-rate sensitivity of plastic deformation. This

applies also to the ITER-specification tungsten fabricated by Plansee AG [57].

Zhu et al. [122] applied the Johnson-Cook model to characterize the work

hardening of commercially pure tungsten fabricated by AT&M in the

temperature range 20–700 °C and revealed a big difference between dynamic

compression and quasi-static compression of tungsten due to strain rate

sensitivity.

2. Kocks-Mecking model for ITER-specification tungsten

40

Deformation of W above 0.15Tm is primarily controlled by the motion of

dislocations, their multiplication and interaction with GBs [123]. Several

previous studies relied on this fact, using models based on the Kocks-Mecking

equation which links plasticity with thermally-activated dislocation-dislocation

interaction. Work hardening of double-forged commercially pure tungsten [115]

and potassium-doped tungsten [56], both fabricated by Plansee AG, was

described in this way at high temperature, reaching 1000 °C and 2000 °C

correspondingly.

The Kocks-Mecking plasticity model was successfully applied to other refractory

BCC materials, such as tantalum-tungsten (Ta-W) alloys [124], to simulate

material response to high strain rate loading. An Arrhenius model of plasticity

based on thermally-activated dislocation motion was used to numerically

characterize deformation of Ta polycrystal in a wide range of temperature and

strain rate, up to 104 s-1 [125]. It indicates potential applicability of this class of

models to tungsten at high temperature and strain rate as well.

Wang et al. [126] reviewed the applicability of six constitutive equations based

on Arrhenius, Johnson-Cook, modified Johnson-Cook, Zerilli-Armstrong,

modified Zerilli-Armstrong, and Khan-Huang-Liang models to the description of

hot deformation of powder metallurgy tungsten. The Arrhenius model was

shown to have the highest accuracy, since it considers the coupled effects of

strain, strain rate and deformation temperature on plasticity.

In this chapter we adapt and assess the applicability of a Kocks-Mecking

modelling of thermally-activated dislocation slip, to characterise mechanical

response of tungsten compliant with the ITER specification as well as

recrystallized tungsten. Tensile tests performed at SCK•CEN in the temperature

range 250–600 °C provide information for the parameterisation of the model. At

higher temperatures, experimental tensile tests are more difficult to carry out as

they require heat-resistant equipment and measures against rapid oxidation of

samples, such as reduction environment. Inspired by some successful previous

applications of the Kocks-Mecking model at high temperatures [56,115], we

expect that the present model will be valid beyond the temperature range of the

experimental curves used for the parametrisation. From a practical viewpoint,

the model should in any case be most accurate in the conditions favouring

damage development of IGP tungsten, which is not the highest operational

temperature range.

The constitutive law proposed here is suitable for FEA. In the next chapters, it is

used to study the mechanical response of tungsten subjected to complex thermo-

mechanical loads, e.g. cyclic heat loads resulting from the plasma instability

during operation of fusion devices.

The chapter is organized as follows. Section 2.2 describes the hardening law, the

first finite element model, and the mathematical algorithm used to fit material

parameters. Results of tensile tests and inverse finite element analysis (IFEA) are

presented in Section 2.3, followed by a discussion in Section 2.4.


41

2.2 Plasticity model and numerical techniques

2.2.1 Temperature- and strain-rate dependent hardening law

Whereas the experimental tensile tests presented in this chapter were

performed at a fixed temperature and strain rate, ITER operating conditions

involve varying conditions of deformation. Hence, the goal of the present section

is to select a strain hardening law enabling us to perform numerical analysis of

tungsten in wide range of temperature and strain rate with a single set of

material parameters. This was achieved by combining and adapting various

material laws. In particular, the widely used macroscopic Voce law, which

predicts that the true stress saturates at large strains, was enriched by

considering that the sensitivity of its parameters to temperature and strain rate

results from thermally activated dislocation slip.

The initial yield stress 0 depends on temperature T and strain rate according

to a phenomenological formula proposed by Voyiadjis et al. [127]. A constant

term 0

VS is introduced to account for non-zero lattice friction stress at high

temperature in BCC metals, material constants V and 1

V define the shape of

the curve, and superscript “V” stands for “Voyiadjis”.

11

1

0 01 ln

pV q

V V V

V

TS S

(2.1)

The increase of strength during plastic deformation (work hardening) follows

Taylor [108] according to whom the strength is proportional to the square root

of dislocation density . Since the initial yield stress is defined by an

independent formula, it is added in Equation (2.2) as a separate term.

Subtracting the square root of the initial dislocation density 0 in the equation

ensures that the equation produces the correct initial stress:

0 0A (2.2)

The parameter A defines the strength of dislocation interaction whose

evolution with temperature might be properly described by an exponential

decay function. But to simplify the optimization problem, it was assumed to

decrease linearly with temperature, in line with a publication devoted to another

grade of tungsten [115].

0 sA A AT (2.3)

The dislocation density is updated based on the Kocks-Mecking equation [108],

discussed in Section 1.3.3.3:


42

1 2M M k k

(2.4)

The Taylor factor M is the ratio between macroscopically measured equivalent

tensile stress eq and an average shear stress in slip systems, as well as the ratio

between an average shear in slip systems and the macroscopically measured

equivalent tensile strain eq . It usually takes values around 2.5–3, depending on

the crystal symmetry, on the loading mode and on texture [125]. It was assumed

to be 2.5 for as-received tungsten.

eq

eq

M

(2.5)

According to equation (2.4), saturation of the dislocation density (and,

consequently, the stress, according to equation (2.2)) occurs when the rate of

dislocation multiplication due to their mutual pinning is balanced by their

annihilation with other dislocations and GBs. These two processes are effectively

parameterized by coefficients 1k and

2k . Integrating equation (2.4) we obtain an

expression for the dislocation density as a function of the accumulated plastic

strain:

2

10

2 1

2 2exp 0.5

k

k k

Mk k

(2.6)

Combining this with equation (2.2) leads to the standard Voce law [128]

(representative of Stage III):

0

0

expVoce sat sat

(2.7)

where 0 is the yield stress calculated using equation (2.1), is the

accumulated plastic strain, sat is the saturation stress:

1

0 0

2

sat

kA

k

(2.8)

and the characteristic strain 0 is related to the initial hardening rate:

0

2

2

Mk (2.9)

It will be shown later in this chapter that experimental curves are much better

reproduced when considering that the strain hardening, rather than flow stress,

saturates to a constant value, which has been referred to as Stage IV hardening.

Inspired from the work of Prinz et al. [129] who considered a cellular dislocation


43

structure in which the cell walls had non-saturating dislocation density, and

anticipating non-saturating stress-strain curves of the studied grade of tungsten,

we added a linear term KMH to equation (2.6) in order to let the total

dislocation density increase beyond the saturation value

2

1

2

k

k

at large strain:

2

10

2 1

2 2exp 0.5KM

k

k kH

Mk k

(2.10)

Then, the stress-strain relationship becomes:

10

2 1

0 0

2 2exp 0.5KM

k

k kA H

Mk k

(2.11)

For mathematical convenience, linear hardening is effective already at the onset

of plasticity but its contribution becomes predominant only after attaining large

strain, when the saturation is achieved. The evolution of hardening rate with

strain can be derived from equation (2.11):

12 0

2

2

0.5

exp 0.5KM

kMk

kA H

Mk

(2.12)

By matching equations (2.7) and (2.11) we find the link between parameters of

the Voce equation and the Kocks-Mecking model:

0 0

1

0

2 sat Ak

AM

(2.13)

As proposed in [115], 1k is constant, whereas

2k accounts for the fact that

dislocation recovery is a thermally activated process [114], introducing the

temperature- and rate-sensitivity of the model:

2 1 2

0

1 lnk C C T

(2.14)

Here 1C ,

2C and 0 are constants. The dislocation annihilation parameterized

by 2k , becomes more significant at higher temperatures.


44

2.2.2 Finite element simulation of tensile test beyond the onset of

deformation instability

There exists a standard approach to extract true stress-strain (S-S) curves,

representative of the actual capacity of material to resist mechanical loads, from

the engineering S-S curves measured experimentally, as discussed in Chapter 1.

However, its application is limited up to the onset of deformation instability —

necking. Hence, when the material exhibits low uniform elongation, a significant

part of the experimental results is unsuitable for characterisation with the

standard approach. An alternative method is necessary in order to extract the

true material response at strains beyond uniform elongation.

One group of alternative methods relies on the measurement of the sample

cross-section during tension, which can be done using video recording and

consequent image processing [130,131].

Another approach is to use inverse finite element analysis (IFEA). In the

conventional direct FEA, a known true S-S curve (constitutive law) is used to

predict the mechanical response of a component with an intricate shape or a

non-uniform loading. On the contrary, IFEA identifies iteratively the constitutive

law which leads to the best reproduction of a known mechanical response,

usually in a specimen with simple shape and loading. Such numerical approach

has been applied not only for extraction of a constitutive law [132–135], but also

for geometry optimisation [136,137]. In this chapter we develop an algorithm

which allows extracting true material properties beyond the uniform elongation

range.

The physically-based model of plasticity described in the previous section was

implemented in the form of a UHARD, a user-defined subroutine for the standard

FE solver Abaqus. Both elasticity and plasticity are assumed isotropic (J2 theory)

and the UHARD defines only the strain hardening. The routine is called at every

increment and every iteration of the simulation in every integration point of the

finite element mesh. Thus, a non-uniform strain rate field across a sample can be

properly processed, being especially important after the onset of deformation

instability. Indeed, the necking region, where strain rates are largest,

experiences more hardening, hindering a fast strain localisation.

In order to take account of the decrease of the axial force after the onset of diffuse

necking, necessary to construct a constitutive law at large strain, we have

performed displacement-controlled FE simulations of the tensile tests with the

help of the commercial finite element solver Abaqus 6.17-1. The sample

geometry shown in Figure 2.1 represented the experimental specimen with a

gauge length 5.2 mm and cross-section dimensions 1.6×1.8 mm2. The sample

was meshed into about 3100 second-order tetrahedral elements varying in size

from 0.2 mm in the gauge middle area to 2.0 mm in the grips with the help of

Gmsh software [138].


45

A small tapering (geometry imperfection) of 1 m was introduced in the central

cross-section of the mesh to ensure a stable development of plastic instability

(necking) in the gauge. Such imperfections are permitted in tensile experiments

according to the ASTM standard E8M-16a [139] and are widely used in FEA of

tensile tests, usually measuring 10-4–10-3 of the sample thickness [135,140,141].

An example of von Mises stress distribution in the sample is presented in Figure

2.1b. The snapshot was taken at the initiation of plastic instability, when the

stress field in the specimen gauge became non-uniform.

The force applied to the sample in the axial direction and the sample elongation

were recorded and recalculated into engineering S-S curves with the help of a

Python script.

Figure 2.1. A tensile sample used in FE analysis a) in the non-deformed state; b) at the onset of deformation instability. Colours represent the von Mises stress, ranging from 0 (in the grips) to 400 MPa (in the centre).

2.2.3 Optimisation algorithm

2.2.3.1 Basics of downhill simplex method

The best set of material parameters is determined while considering all tensile

tests performed on a given W grade and by applying an advanced optimisation

method.

The Nelder-Mead method (also known as “downhill simplex method”, “amoeba

method”, or “polytope method”) is a numerical method used to solve

optimization problems, i.e. problems where the minimum or maximum of an

objective function has to be found [142]. A usual meaning of an objective function

is the measure of the deviation of the results predicted by the model from the

corresponding experimental results, typically expressed as the sum of squares of

their difference.

The algorithm operates in a multidimensional parameter space, and, being a

direct search method, does not require the calculation of the objective function

derivatives (contrary to the steepest descent method and other gradient


46

methods). As will be shown further, the objective function designed for the

purpose of finding the constitutive law, involved calling an external software —

a finite element solver. It made impossible the calculation of the objective

function derivative, justifying the choice of the optimization algorithm.

The method uses the concept of a simplex, which is a generalisation of a

polyhedron in an n-dimensional space, with n+1 vertices. The n-dimensional

parameter space is formed by the n independent parameters in the optimization

problem. The search begins from a predefined initial point, and the search

domain should be limited to mitigate possible numerical issues. The algorithm

evaluates the objective function at every vertex, and extrapolates its behaviour

in order to replace the “worst” vertex (where the objective function is the

largest) with a new one.

The downhill simplex method foresees a few possible scenarios of defining the

new vertex in as few attempts as possible, expecting the objective function to

eventually be lower; but discussion of the exact scenarios is out of scope of the

present work. In this manner, iteration after iteration, the algorithm probes

numerous parameter sets and eventually reaches the objective function

minimum. It means reaching the best possible match between the experimental

and simulated data if the objective function is a measure of deviation between

the two.

2.2.3.2 IFEA based on the downhill simplex method

An iterative algorithm based on the downhill simplex method and IFEA has been

coded in the R programming language and implemented as a shell that combines

the input-output interface, the objective function and the call to the downhill

simplex method. The simplex method implemented in the “neldermead” package

has been selected [143].

The flowchart of the algorithm is shown in Figure 2.2. It receives a set of N

engineering “target” S-S curves (obtained experimentally in a range of

temperatures and strain rates) and an initial set of the plasticity model

parameters, that form a point in the 7- dimensional parameter space (0 ,A ,SA

1,C 2 ,C

0 , 1,k

KMH ). Then a simplex with eight vertices is constructed around

this point. The objective function is evaluated at all these points, followed by

identification of the worst vertex and its replacement with a better one. The

objective function is then evaluated at the new point in the following iteration,

and the process repeats in a loop until the objective function minimum is found.

The finally obtained parameter set (0 ,fA ,f

SA 1 ,fC

2 ,fC 0 ,f

1 ,fk f

KMH ) yields a

constitutive law, which, being provided as input to a simulation of tensile test,

would return the closest match between the simulated and experimental

engineering S-S curves in the whole range of temperature and strain rate

simultaneously.


47

Figure 2.2. Flowchart of the iterative algorithm for the plasticity model fitting to multiple stress-strain curves simultaneously.

Every evaluation of the objective function involved launching N simulations of

tensile test in Abaqus with the mesh shown in Figure 2.1 and the plasticity model

discussed above as the constitutive law. It has to be noted that the simulated

engineering S-S curves were observed to deviate from the target ones beyond

certain strain even in the last iterations of the algorithm, hence the range of

strain for the objective function evaluation was capped at 24%–40% identified

for each curve individually depending on temperature. In spite of that, the range

of strain available for fitting was much larger than the uniform elongation,


48

summarized in Figure 2.7b. Besides, if an engineering S-S curve featured a yield

drop, then only the part beyond the yield drop was taken as a target curve, given

that the plasticity model is applicable to monotonically increasing curves only

(in terms of true stress).

The objective function was calculated as the sum of squares of differences

between the engineering stress , obtained in a simulation iteration, and the

target engineering stress (obtained experimentally) plus the sum of squares of

differences between their derivatives with respect to engineering strain (i.e.,

stress-strain slope ), according to Equation (2.15). The derivatives were

calculated with the finite difference method. A weight coefficient = 5·10-3 had

to be introduced to tune the contribution of both quantities to the total value of

the objective function. The differences were evaluated at a set of P discrete points

separated by strain spacing , whose optimal value was found to be 2%.

2 2

exp exp

1 1 1

N P Psim sim

i i i i

j i i

(2.15)

a) b)

Figure 2.3. Illustration for the evaluation of the objective function. The sum of squares of differences is calculated at a discrete set of strain values, marked with circles. Both the engineering stress (a) and the stress-strain slope (b) contribute to the total value of the objective function. The arrows indicate that as the simulation result approaches the target, the objective function decreases.

Figure 2.3 illustrates the calculation of the objective function for a single S-S

curve from the two contributions: a) the S-S curve itself and b) the stress-strain

slope. As mentioned above, only the part of the experimental S-S curve indicated

with the solid line is used for the search. The dashed line shows the complete

experimental S-S curve for clarity. The sum of squares of differences between the

experimental (target) and the simulated curves was evaluated at a discrete set

of strain values (shown by circles).

Figure 2.4 illustrates the evolution of the input constitutive law (a) and the

produced output engineering S-S curves compared with the target one (b) in a

few iterations of the search process for a single combination of temperature and


49

strain rate. The change of shape of the input S-S curve was due to the change of

the plasticity model parameter set from iteration to iteration, guided, in turn, by

the downhill simplex algorithm. The corresponding engineering S-S curve might

occasionally deviate from the target when the simplex algorithm substitutes the

“worst” vertex with a “not-so-good” one; such directions of search in the

parameter space are discarded. At certain iteration the tensile test simulation

returns an engineering S-S curve perfectly matching the target within defined

range of strain, and the search stops.

a)

b)

Figure 2.4. The evolution of a) input true S-S curves, b) output engineering S-S curves in several iterations of the search algorithm.

2.3 Results

2.3.1 Experimental tensile tests

Tensile tests were performed at SCK•CEN (Belgium) in the temperature range

250–600 °C and strain rates 6·10-5–6·10-3 s-1 on commercially pure W produced

by Plansee AG according to the ITER specification [20], referred to as “IGP” and

described in Chapter 1. The material was not subjected to neutron irradiation

and was tested in two states: as-received and recrystallized. However,

investigation of tungsten after neutron irradiation is planned for the future, thus


50

the miniaturized design of tensile samples has been selected to comply with the

dimension limitations in the irradiation channels in the material testing reactor

BR2 (Belgium). For instance, fusion-relevant irradiation of tungsten (up to ~1

dpa) can be performed inside fuel elements, where the channel is merely 15 mm

in diameter.

Experimentally obtained engineering stress-strain curves for as-received IGP

tungsten are shown in Figure 2.5 with dashed lines. This tungsten grade has a

high ductility even at 250 °C and high yield stress, 500–700 MPa.

Figure 2.5. Engineering stress-strain curves for as-received IGP tungsten. Solid lines highlight the range of uniform deformation.

It is known that the maximum of engineering stress (ultimate tensile strength

UTS) corresponds to the onset of deformation instability. Necking causes a rapid

decrease of the central cross-sectional area of a sample. The deformation ceases

to be uniform and the standard approach to extract true stress and strain

becomes invalid from that moment. The solid lines in Figure 2.5 correspond to

the uniform deformation range and show that significant part of the

experimental results cannot be processed with the standard approach. That is

why an advanced method based on IFEA is applied to parameterize the plasticity

model for as-received IGP tungsten.


51

Figure 2.6. Engineering stress-strain curves for the recrystallized tungsten.

a) b)

Figure 2.7. a) Yield stress and ultimate tensile strength as well as b) uniform elongation and total elongation of both grades of tungsten tested at 6·10-4 s-1.

The experimental engineering stress-strain curves for the recrystallized

tungsten are plotted in Figure 2.6. It has low ductility at low temperature, which

nevertheless increases with the temperature increase. The yield stress is much

lower than in the as-received W, but the work hardening capacity is much higher

in recrystallized tungsten. The values of yield stress and UTS as well as uniform

and total elongation of the two grades of tungsten are summarized in Figure 2.7.


52

2.3.2 Results of inverse finite element analysis

2.3.2.1 As-received tungsten

The parameters describing the evolution of the yield stress values as a function

of temperature and strain rate (equation (2.1)) were fitted before considering

the strain hardening. Following [144], the values of p and q were fixed at 0.5 and

1.5, respectively. The other parameters, identified with the help of a non-linear

least-squares optimisation method, are summarised in Table 2.1. A comparison

of the experimentally measured yield stress values and the predicted ones is

shown in Figure 2.8 for three different strain rates.

Table 2.1. Set of parameters reproducing the yield stress of as-received IGP tungsten.

VS , MPa V , K 1

V , s-1 0

VS , MPa p q

2886 1.91·104 2.8·107 495 0.5 1.5

Figure 2.8. The best fit of equation (2.1) to the yield stress of as-received IGP tungsten at different temperatures and strain rates.

In order to find the strain hardening parameters, the IFEA of the tensile test

combined with the iterative algorithm discussed above was applied to N = 15

experimental engineering S-S curves. The initial parameter set, i.e. the starting

point for the search, was roughly estimated, based on the typical order of

magnitude of the quantities involved in equations (2.2)–(2.12). The initial

dislocation density was set equal to 4.5·1012 m-

2, as measured by TEM[32].


53

Figure 2.9. True stress-strain curves of as-received IGP tungsten obtained with the help of IFEA of tensile test.

Figure 2.10. Engineering stress-strain curves obtained in the last iteration of the IFEA of tensile test for as-received IGP W, compared to the experimental data.


54

The finally obtained true S-S curves for all the tested samples of as-received IGP

tungsten are summarised in Figure 2.9. Note that the algorithm considered

responses far beyond uniform elongation, and the final plastic strain reached

values of 0.8–1.0, corresponding to localized plastic strain in the thinnest cross-

section of a sample, i.e. in the necking region.

The accuracy of the theoretical constitutive law for as-received IGP tungsten can

be checked by comparing the engineering S-S curves at the last iteration of the

IFEA. This is done in Figure 2.10. As mentioned above, the range of strain for the

curve fitting has been capped. Nevertheless, a perfect match has been achieved

for all the curves in the whole tested range of temperature and strain rate well

beyond uniform elongation, thus covering the mechanical response of material

after the onset of deformation instability.

The model parameters fitted to characterise plasticity of IGP tungsten in the IFEA

are summarised in Table 2.2. The coefficient at the linear term HKM is nearly zero,

indicating the validity of the standard Voce equation for as-received tungsten.

Table 2.2. The fitted parameters of the plasticity model for as-received IGP W.

0

fA ,

MPa·m-1

f

sA ,

MPa·m-1 K-1 1

fC 2

fC ,

K-1 0

f ,

s-1 1

fk ,

m-1

f

KMH ,

m-1

4.38·10-4 ≈0 1.17·10-4 0.348 5.1·107 2.64·106 ≈0

2.3.2.2 Recrystallized tungsten

This subsection contains material published by our team in [120], corrected and

improved afterwards.

Similarly to the analysis of as-received tungsten, yield stress of recrystallized

tungsten was fitted first with the help of equation (2.1) using a non-linear least

squares optimisation method. The values of p and q were fixed at 0.5 and 1.5 as

well. Unfortunately, even after trying various starting points (to avoid ending the

search at a local minimum instead of the global one) and after setting different

limits in the parameter space, this equation did not adequately reproduce the

experimental yield stress. Figure 2.11 summarises the experimental yield stress

and the curves corresponding to the closest obtained match (dashed lines).

An alternative to equation (2.1) was proposed by Beyerlein and Tomé [114], who

observed an exponential decay of the initial yield stress with temperature in low-

symmetry metals, including zirconium, magnesium, beryllium and -uranium

[145]. An enhanced equation version was used to describe the evolution of yield

stress in tantalum-tungsten alloys [124] and another grade of tungsten in [115].

Superscript “B” stands for “Beyerlein” and distinguishes the parameters

corresponding to equations (2.1) and (2.16), respectively.

0 0

1

exp ln 1B B B B

B B

TS K S

(2.16)


55

BS , B , BK , 1

B and 0

BS are constants defining the shape of the curve. This

equation showed a better match with the experimental data, after being fitted

with the help of the nonlinear least-squares optimization method, even though

the chemical composition of the two tungsten grades is identical. The difference

seems to originate from different grain size and GB types: a larger fraction of low-

angle boundaries is present in the as-received tungsten. Corresponding curves

are shown in Figure 2.11 with solid lines. The fit of yield stress with equation

(2.16) is used further in the text.

Figure 2.11. Yield stress of recrystallized W. Symbols represent experimental data, and lines highlight its evolution with temperature at given strain rates, obtained using two different fitting functions.

Table 2.3. The best fitted parameter set of equation (2.1) to describe yield stress of recrystallized tungsten.

VS , MPa V , K 1

V , s-1 0

VS , MPa p q

2464 7.61·103 9.48 149 0.5 1.5

Table 2.4. The best fitted parameter set of equation (2.16) to describe yield stress of recrystallized tungsten.

BS , MPa B , K K 1

B , s-1 0

BS , MPa

657 278 0.44 1.22·10-6 90.6


56

Figure 2.12. Experimental true stress-strain curves of recrystallized tungsten.

Figure 2.13. Hardening rate derived from experimental true stress-strain curves of recrystallized IGP tungsten


57

The set of material parameters of equations (2.1) and (2.16) allowing the best fit

of recrystallized tungsten are summarised in Table 2.3 and Table 2.4

respectively.

Given the large uniform elongation of recrystallized tungsten at high testing

temperatures, it would have been possible to derive true S-S curves with the help

of the standard approach (Figure 2.12). Nevertheless, parameterization of the

strain hardening parameters accounted for the material response beyond the

necking point.

The hardening rate, derived from the true S-S curves, is shown in Figure 2.13. It

highlights the deviation from the standard Voce equation, and the necessity of an

extra linear term in equation (2.11).

The iterative algorithm was used to perform the IFEA of the tensile test and

parameterize the hardening law. As mentioned in Chapter 1, the initial

dislocation density 0 measured in recrystallized W with the help of TEM was

1011 m-2 [32]. The parameter set, that provides the best match between the

model prediction and the experimental engineering S-S curves, is summarised in

Table 2.5, where 1C ,

0 and 1k were adopted from a paper on a similar grade of

tungsten [115]. The parameters 1k and

0 could be incorporated directly,

whereas comparing equation (2.14) in the present work and Eq. (9) in [115]

leads to

1

1

k bC

g

(2.17)

where is an interaction parameter, b is the Burgers vector and g is the

normalized activation energy; all three values being reported in the cited paper.

Table 2.5. The fitted parameters of the plasticity model for recrystallized tungsten.

0

fA ,

MPa·m-1

f

sA ,

MPa·m-1 K-1 1

fC 2

fC ,

K-1 0

f ,

s-1 1

fk ,

m-1

f

KMH ,

m-1

6.37·10-6 2.7·10-9 4.2·10-5 20.75 107 4.8·108 6.4·107

Figure 2.14 shows both experimental (solid) and modelled (dashed) true stress-

strain curves for recrystallized tungsten deformed at different strain rates and

temperatures. The stress-strain curves for 300 °C and 400 °C show that the

quality of fit is reduced at low temperature due to the narrow range of strain, i.e.

the low total elongation of recrystallized tungsten below 400 °C. However, the

match between the model prediction and experimental curves at 500 °C and

600 °C is excellent.


58

Figure 2.14. Experimental and fitted true stress-strain curves for recrystallized IGP tungsten at strain rate 6·10-4 s-1.

Figure 2.15. Experimental and simulated engineering stress-strain curves at 500 °C and 600 °C for recrystallized tungsten.

Figure 2.15 compares engineering S-S curves obtained in experiments (solid)

and in the last iteration of simulations (dashed). They are grouped according to

the test temperature and the strain rate. A good match is observed between them

up to the onset of plastic instability, shortly after which the model tends to

overestimate strength and fracture strain. The latter is most probably due to

ductile damage accumulation taking place in the material, but not accounted for

in the model. For the same reason as for the as-received tungsten, the strain

range within which the search of true S-S curve is performed, should be capped.


59

2.4 Discussion

2.4.1 Strain-rate sensitivity

Speaking of the applicability of the parameterized plasticity model to the

simulations of thermal loads relevant to ITER, the major concern is the strain

rate, which differs by orders of magnitude between quasi-static tensile tests and

the dilatation-contraction cycles caused by ELMs. So far tensile tests of IGP

tungsten have been performed at low strain rate, 10-3–10-5 s-1, thus no data are

available to calculate reliable parameters of the model at high strain rate.

However, an insight into strain-rate sensitivity of tungsten at high rates can be

gained from literature analysis.

Figure 2.16. Illustration of rate sensitivity of tungsten (other than IGP) in a wide range of strain rate, showing the true stress at 4% of strain as a function of strain rate [146].

Mechanical tests on another tungsten grade, an extruded tungsten rod from

Teledyne Firth-Sterling by the Army Research Laboratory, USA, have been

reported [146]. The experiments have been performed in a wide strain rate

range and included quasi-static compression (10-3–100 s-1), compression Kolsky

bar tests (103–104 s-1) and pressure-shear plate impact tests (104–105 s-1).

To detect strain rate sensitivity of that tungsten grade, the value of true stress

recorded at 4% of true strain was plotted as a function of strain rate, as

illustrated in Figure 2.16. There is a clear linear trend for both recrystallized and

as-received tungsten states in the range of strain rate up to 102 s-1 (shown with


60

dashed lines). This result indicates that the rate sensitivity of other tungsten

grades measured in tensile tests at low strain rate might as well be extrapolated

to higher rates, including 8 s-1 expected in the divertor monoblocks exposed to

ELMs. This way to show rate sensitivity was also applied in [101], where stress

measured at 0.15 strain was plotted against strain rate for copper single crystals

and revealed a linear trend up to strain rate around 103–104 s-1 as shown in

Figure 2.17.

Figure 2.17. Strain rate sensitivity for Cu deformed in compression at 25°C, shown as the stress at 15% of strain as a function of strain rate. [101].

2.4.2 Deformation stage IV

2.4.2.1 Deviation from the Voce equation

The true S-S curves obtained for the recrystallized IGP tungsten deviate from the

standard Voce equation, and require an extra non-zero linear term in it. Also, the

standard Voce equation prescribes the hardening rate to be a decreasing linear

function of true stress, which reaches zero at the saturation stress sat (Figure

2.18). The hardening rate calculated for recrystallized tungsten does not follow

this trend and approaches zero asymptotically with the increase of stress (Figure

2.19b).


61

Figure 2.18. Hardening rate as a function of true stress indicates the presence of deformation stages III and IV (solid line). Hardening rate derived from the standard Voce equation is shown by the dashed line for clarity.

A similar behaviour was detected in S-S curves reported in other works on

recrystallized tungsten. Figure 2.19 collects the engineering S-S curves obtained

at 500 °C (a) and the calculated hardening rate (b) from two other works apart

from the present project. Labels “U1” and “U2” correspond to tungsten

recrystallized at 1750 °C during 2 h in vacuum tested at different deformation

speed, 0.2 mm min-1 and 42 mm min-1 correspondingly and reported by

Uytdenhouwen [1], and label “W1” shows the results of IGP tungsten

recrystallized at 1600 °C during 1 h in vacuum tested by Wirtz et al. [26]. The

hardening rate calculated from tensile curves for these samples approaches zero

asymptotically as well, being a feature of deformation stage IV.

2.4.2.2 What is deformation stage IV?

Deformation stage IV in FCC and BCC metals and alloys was observed in tests

where large deformation can be attained, such as torsion, where the

development of deformation instability in a sample is postponed [147,148].

Instead of reaching saturation at the end of stage III, the true stress maintained

basically linear increase within a wide strain range (up to ~10 in Fe, and ~5 in

Ni at room temperature) [148].

The occurrence of stage IV hardening is often attributed to the formation of a

cellular dislocation structure inside individual grains. Several approaches have

been proposed to understand the cellular structure formation and its impact on

the subsequent stress-strain evolution [129,149,150]. Although the dislocation

density in the cell interiors does indeed saturate with strain, in accordance with

the Kocks-Mecking model, the dislocation density in the cell walls keeps growing

and thus contributes to further hardening.

Notably, most of those works dealt with FCC metals due to their superior

capacity for plastic deformation as compared to BCC metals. However, the

cellular structure was also observed in BCC tungsten deformed in compression

in the range from room temperature up to 600 °C [151,152]. The microstructure


62

of single crystal tungsten and dilute W-based alloys was investigated in a wide

temperature range from -123 °C (0.04Tm) up to 317 °C (0.16Tm) under

compressive deformation [153].

a)

b)

Figure 2.19. (a) Engineering stress-strain curves and (b) hardening rate for different grades of recrystallized tungsten obtained by uniaxial tensile tests at 500 °C [1,26].

The Kocks-Mecking–like models, modified in order to account for cellular

structures, were successfully used to describe deformation stage IV in FCC

metals with the help of a second state variable associated with lattice

incompatibility [109] as well as in BCC metals and alloys including pure tungsten

[112] and Ta-10W alloy by introduction of a separate dislocation substructure

density [124].


63

2.4.2.3 Evidence of deformation stage IV in tungsten

The microstructure evolution during plastic deformation in ITER-specification

tungsten was investigated by our team based on TEM examination after

interrupted tensile tests at 600 °C [152]. Due to the lack of recrystallized IGP

tungsten at our disposal, these observations were performed on double-forged

tungsten grade samples (provided by Plansee AG as well) after recrystallization

(and denoted as DFR, standing for “double-forged recrystallized”). Although the

annealing of both DFR and IGP grades was performed under the same conditions

(1600 °C during 1 h), the values of dislocation density in the samples did not

coincide and amounted to 5·1012 m-2 and 1011 m-2 correspondingly. Figure 2.20

demonstrates that samples deformed up to 20% contained dislocation tangles

(precursors of dislocation walls) and those deformed up to 28% showed the

formation of cells, resulting in grain refinement [152].

Figure 2.21 provides a comparison between the experimentally measured

dislocation density and the prediction of the present model, revealing that the

model overestimates the dislocation density. The initial dislocation density used

in this particular calculation was assumed to be equal to 5·1012 m-2,

corresponding to the experimental value measured in the DFR tungsten, for

which the interruptive tests were performed [115]. It should, however, be noted

that the experimental measurements excluded the dislocation density in the cell

walls, which were considered as precursors of low angle GB. Whereas the

dislocation density in the present model should be thought of as a weighted

average of dislocation density in the bulk and that in the cell walls. This

contributes to the discrepancy between the model results and the experimental

data.

Exploring the existing TEM images we can already assume the value 3·1015 m-

2

as a preliminary estimation of dislocation density in the dislocation tangles.

Zones I and II shown in Figure 2.20 correspond to the grain interior and

dislocation tangles, where the dislocation density was identified. This value is

much higher than the bulk dislocation density, such that the weighted average

dislocation density in tungsten after 20% deformation appears to be in the range

between 3·1014 m-2 and 1015 m-2, which is closer to the model prediction.

The discussed approach was selected as a quick and simple way to parameterize

the tensile S-S curves of recrystallized tungsten, which are important for FEA of

stress state in PFCs. It would be possible to fit the plasticity model starting from

experimentally measured dislocation density, provided it is known as a function

of strain, test temperature and strain rate. Note that the “weighted” dislocation

density would be necessary, which accounts for dislocation storage in both cell

interior and walls.

Then, the plasticity model would have to be fitted in two stages: first to

reproduce the dislocation density, adjusting 1k ,

1C , 2C and

0 , and second, to

reproduce tensile S-S curves, adjusting the remaining model parameters only.


64

a)

b) c)

Figure 2.20. TEM micrographs showing dislocation tangles (a, b) and dislocation cells (c), probed by TEM at 20% (a, b) and 28%(c) of plastic strain in the DFR tungsten. The grain refinement was observed in the tests performed at 500 °C and 600 °C after 20% plastic strain. Zones I in the images show examples of the grain interior, while zones II correspond to dislocation tangles where the dislocation density was measured [120].

This approach would result in a more accurate extrapolation of mechanical

properties to high temperature, but would require a number of interrupted

tensile test series, followed by laborious TEM analysis.

Observation of dislocation cells in the recrystallized tungsten is in line with the

detection of deformation stage IV on the S-S curves. Dislocation cells have also


65

been observed in TEM images of as-received IGP tungsten [152], but stage IV has

not been detected during the analysis of deformation curves, and the use of the

standard Voce equation leads to a good match with the experimental curves.

The apparent discrepancy can be explained by the difference in grain size.

Formation of numerous dislocation cells in large grains of recrystallized

tungsten significantly hinders the dislocation motion, leading to an increased

hardening and pronounced stage IV. On the contrary, dense GB network in as-

received tungsten is already a strong obstacle for dislocation motion, and the

fraction of newly formed cell walls is nearly negligible from the point of view of

dislocation motion.

Figure 2.21. Dislocation density predicted by the model superimposed on the experimental data from [115].

2.4.3 Individual curve fitting with the iterative algorithm

The proposed optimization algorithm can also be used to extract constitutive law

from individual engineering stress-strain curves. In this case the plasticity model

reduces to the Voce equation (2.7), and the algorithm searches for the best set of

the Voce parameters (0 , sat ,

0 ). The question is, whether the fitting of

individual curves would reveal the temperature- and rate-dependency of these

parameters.


66

Figure 2.22. Flowchart of the iterative algorithm for the Voce equation fitting to a single stress-strain curve.

The flowchart of this reduced algorithm is shown in Figure 2.22. It is different

from the algorithm for multiple curve fitting, outlined in Figure 2.2, in the sense

that the search is performed in a 3-dimensional parameter space (accelerating

the search), the simplex has 4 vertices, and the objective function is calculated

for a single engineering S-S curve, as long as N = 1 in equation (2.15).


67

Figure 2.23. Engineering stress-strain curves, as a result of FE simulation of a tensile test, using the best constitutive laws obtained with the help of individual curve fitting. Dashed lines show corresponding experimental results.

Engineering S-S curves obtained in the experiments and those found with the

help of IFEA of individual curves are summarized in Figure 2.23. The match

between the experimental and simulated results has improved, compared to

Figure 2.10. Strain range well beyond the onset of deformation instability has

been covered as well.

Given such high quality of fit, one would expect that the obtained parameters of

the Voce equation for individual curves could reveal trends, being plotted against

temperature and strain rate. To check that, the values of the characteristic strain

0 and the saturation stress sat are plotted in Figure 2.24 (a) and (b)

correspondingly. However, no clear trend is observed for neither quantity; the

values are quite scattered and even increase around 600 °C, whereas a

monotonic decrease is expected according to the plasticity model outlined above.

Thus, this approach does not have predictive power: the results cannot be

extrapolated to higher temperature and strain rate.

Another noticeable feature is the correlation between 0 and

sat . It means that

the Voce equation is less sensitive to simultaneous proportional change of the

two parameters, compared to their separate variation (especially considering

the fact that the simulation deals with a part of the Voce curve, limited up to true

strain ~1). In this case many pairs of 0 and sat located in the vicinity of the


68

found pair (0

f , f

sat ) could result in almost identical true S-S curves, thus,

forming a set of solutions which the IFEA of individual engineering S-S curves

cannot distinguish within the tolerance defined by the downhill simplex method.

Effectively it means quite large vertical error bars in Figure 2.24.

a)

b)

Figure 2.24. The Voce parameters 0 (a) and sat (b) as a function of temperature,

obtained in the IFEA of individual experimental S-S curves.

To complete the comparison of the two approaches, a similar analysis should be

made for the model fitting to multiple curves. The material constitutive

behaviour at every combination of temperature and strain rate can be described

with the Voce equation, whose parameters can be calculated using the plasticity

model and the found set of parameters (0

fA , f

sA , 1

fC , 2

fC , f

o , 1

fk , f

KMH ).

The characteristic strain 0 and the saturation stress

sat , being a result of

plasticity model fitting to multiple experimental curves, are shown in Figure 2.25

(a) and (b) correspondingly, as a function of temperature. Both quantities clearly

follow the theoretically prescribed trends, and can be extrapolated to higher

temperature and strain rate, relevant for ITER. They are quite large compared to


69

usual values, for instance 0 being typically in the range 0.1–0.5. The difference

is that the Voce equation is applied here to a material with low uniform

elongation and large total elongation, which means a sooner fulfilment of the

Considère criterion and later attaining the saturation stress. Consequently, both

sat and 0 must be larger than in materials with high uniform elongation, what

results in true constitutive laws with low curvature, as seen in Figure 2.9. Even

though it resembles the non-saturating constitutive law, observed in the

recrystallized tungsten, the IFEA returned a negligibly small value of the linear

coefficient KMH for as-received tungsten. Since fewer degrees of freedom result

in an enhanced fit accuracy, it is beneficial for the parameterization.

a)

b)

Figure 2.25. The Voce parameters 0 (a) and sat (b) as a function of temperature,

obtained in the IFEA of multiple experimental S-S curves.


70

2.5 Conclusion

In this chapter we have assessed a hardening law assuming dislocation-mediated

plasticity and accounting for the effect of temperature and strain rate on the

dislocation density evolution. The model has been parameterised for two grades

of tungsten: as-received IGP tungsten and recrystallized grade, relying on

experimental data obtained in tensile tests in a range of temperature and strain

rate. Numerical optimisation with the help of IFEA has been employed to fit the

model to multiple experimental curves at once and to obtain, thus, a bird-eye

view of the trends of temperature- and rate-sensitivity of the material.

Even though the parameterisation has been performed in the temperature range

of 250–600 °C, which represents the low temperature application window for

tungsten as divertor material in fusion reactor ITER, we assume that the

prediction of the model is valid at higher temperature, expected at the surface of

divertor, at least in the range where deformation by dislocation slip dominates.

The classical Kocks-Mecking approach is suitable for the characterisation of

plasticity of as-received tungsten. An extra term, linear with respect to applied

strain, has to be added in the evolution of the dislocation density of recrystallized

tungsten, to take account of what appears to be deformation stage IV, i.e. the

absence of saturation of dislocation density and true stress with increase of

strain.

Deformation-induced grain refinement by means of the build-up of dislocation

walls and cells as well as low-angle GBs has been confirmed by TEM

investigations and is assumed to induce stage IV hardening.

Finally, in order to proceed to investigation of mechanical state of tungsten

under ITER-relevant thermal conditions at the level of grains, the developed J2-

plasticity model needs to be implemented in the crystal plasticity framework,

which will be covered in the next chapter.

71

Chapter 3.

Development of a crystal plasticity model

accounting for backstresses Equation Cha pter 3 Section 3

3.1 Introduction

Thermal shock tests were identified as one of main qualification techniques for

the ITER project, where the plasma-facing components have to sustain a

predefined number of cycles, experiencing cyclic thermal strain and stress.

In order to investigate the conditions leading to intergranular crack formation in

tungsten exposed to thermal shocks, which mimic the plasma instabilities in a

fusion environment, we need to construct a plasticity model at the microscopic

level, where the anisotropy of individual grains causes microscopic

heterogeneity of stress and strain, in particular along GBs and near triple

junctions.

It is the constrained thermal expansion of the polycrystal which gives rise to

thermal stresses. Hence, the model has to face two main challenges: first, the

material strength continuously evolves following the oscillations of

temperature; second, the cyclic character of deformation can involve kinematic

hardening, i.e. the repeated elastic-plastic transients are expected to be

influenced by internal stresses due to the piling up of dislocations against

obstacles (such as GBs).

Predicting microscopic stresses based on crystal plasticity theory appears to be

a suitable choice. Such models are designed to take account of plastic anisotropy,

i.e. orientation of slip planes with respect to applied load. A temperature-

dependent resistance of crystal lattice to shear, which defines the material

strength, can be enforced to allow the simulation of mechanical response under

evolving temperature. Kinematic hardening can be organically incorporated in

the model, since it is attributed to a microscopic process occurring at the scale of

dislocation slip systems.

Crystal plasticity simulations presented in this work were all carried out using a

numerical tool previously developed at UCLouvain. In the context of the present

work, the crystal plasticity law has been improved in two regards. First, the

representation of thermally activated dislocation slip has been adapted from

room-temperature deformation of FCC metals [154] to high-temperature

deformation of BCC metals. Second, an original idea has been proposed and

3. Crystal plasticity model accounting for backstresses

72

tested in order to account, in a simplified way, for kinematic hardening resulting

from the accumulation of dislocation pile ups at GBs and their dissolution upon

load reversal.

In the present chapter, crystal plasticity relies on simplified “mean-field”

assumptions in order to relate the micro and macro scales. This means that strain

and stress are assumed uniform inside each grain. More advanced modelling of

the microscopic heterogeneity of mechanical fields is postponed to the next

chapter where crystal plasticity based finite element modelling (CPFEM) will

allow accounting also for intra-granular heterogeneity.

We have to acknowledge the scarcity of experimental data on kinematic

hardening in tungsten, due to high cost and technical difficulty associated with

fatigue tests, especially at high temperature. As shown in Figure 3.1 reproduced

from [59], there is a significant Bauschinger effect during low-cycle fatigue test

of forged tungsten at 480 °C.

Figure 3.1. Stress-strain loops obtained in low cycle fatigue tests of forged tungsten at 500 °C (other than IGP) [59] and the results of fitting of a J2 Armstrong-Frederick model of kinematic hardening.

The evolution of backstresses b , causing kinematic hardening, is usually

described by the Armstrong-Frederick family of models [155,156], which can be

written in the simplest one-dimensional case (tension in the direction 1) as

11 11

b bA B (3.1)

where A and B are the model constants and strain rate is denoted by .


73

To illustrate the applicability of the Armstrong-Frederick model to tungsten, we

coupled it with a J2 implementation of the hardening model presented in Chapter

2. As shown in Figure 3.1, the Bauschinger effect was reproduced in all three

experimental stress-strain loops when the saturating value of the backstress

(A/B) amounted to about 30% of the saturating value of the flow stress.

Implementation of the Armstrong-Frederick model in the crystal plasticity

framework is discussed further in the text.

The amplitude of kinematic hardening cannot be probed experimentally under

monotonic loads such as the uniaxial tensile tests reported in the previous

chapter. Hence, we do not know, at this stage, whether the phenomenon remains

important at higher temperatures and the present chapter explores several

hypothetical scenarios. Two limiting cases were parameterised for the

simulations, encompassing the absence of kinematic hardening and the absence

of isotropic hardening, respectively. An intermediate case was considered too, in

which the initial yield stress was partly due to kinematic hardening (i.e.

accumulation of dislocation pile-ups at GBs prior to macroscopic yield), and

isotropic hardening was responsible for the deformation strengthening.

3.2 Context

The thermal shock test is one of the qualification tests of plasma-facing

components accepted for the ITER project. Numerous tests have been carried out

on several baseline and advanced grades of tungsten, as discussed in Chapter 1.

In this chapter we aim to simulate experiments reported in [26], since they were

performed on IGP tungsten in a range of base temperature and thermal power

density, looking for the effect of the experimental conditions on the damage

formation in the material.

a) Transversal

b) Longitudinal

c) Recrystallized

Figure 3.2. Schematic representation of the three types of samples used in the thermal shock experiment [26].

Samples cut out of as-received IGP tungsten in two orientations (denoted as “T”

and “L” for “transversal” and “longitudinal” correspondingly) as well as out of

recrystallized tungsten (“R”) were tested. The top surface of blocks

12×12×5 mm3, shown schematically in Figure 3.2, was exposed to an electron

beam depositing thermal power of 190 MW m-

2 or 380 MW m-2 in pulses 1 ms

long. The temperature of the bottom surface was kept constant at 27 °C, 400 °C

or 1000 °C. 100 or 1000 thermal shocks were delivered, after which the state of

the surface indicated the material resistance to ELM-like thermal conditions.


74

Based on the experiments, it is possible to find out the damage threshold, which

is the value of thermal power density at a given pulse length, below which no

cracks appear on the surface after a given number of pulses. The damage

threshold decreases with the increase of base temperature, and the effect of

pulse frequency and the number of cycles alone might also play a role. Damage

threshold for the T orientation and recrystallized tungsten is much lower than

for the L orientation [71]. Not only IGP, but also double-forged tungsten in the T

orientation and recrystallized tungsten were found more susceptible to cracking

than the L orientation [157].

According to [157],

The general mechanisms of material degradation under electron

beam thermal shock loading are subdivided in two phases:

1) single pulse: formation of temperature gradients and related

thermal stresses, i.e. compressive stresses during heating due

to expansion of the volume below the loaded area constrained

by the ‘‘cold and rigid’’ bulk material and formation of tensile

stresses during cooling [158],

2) multiple pulses: thermal fatigue damage due to repetitive

loading.

Simulations dedicated to thermal shock tests are not reported frequently. Most

of them relied on simplified constitutive laws, such as ideally plastic [79,159], in

which, however, distributions of temperature, plastic strain and stress were

obtained. In an advanced technique “eXtended FEM” (XFEM) the authors could

calculate the J-integral for inserted pre-cracks in different positions and could

identify the damage threshold in terms of thermal power density, which agreed

roughly with experimental observations. Kinematic hardening was taken into

account in the work of Du [121], in a J2 analysis of thermal shock with the help of

a linear-plastic material law. The effect of coupled high-heat flux and thermal

shocks was investigated in a numerical analysis [83] using a J2 plasticity model

as well.

In this project we develop a crystal plasticity model, applicable to high-

temperature cyclic deformation of tungsten and incorporating the effect of

kinematic hardening to investigate the mechanical state and damage

accumulation in tungsten exposed to ELM-like thermal shocks.

3.3 Workflow of the suggested two-scale

simulation scheme

On the one hand, thermal shocks generate complex geometry-dependent stress

and strain states in the subsurface region of a sample, rapidly evolving in time.

In order to capture such a strain field, finite element modelling is required at the


75

macroscopic scale, i.e. at a scale much larger than characteristic microstructural

dimensions such as the grain size.

On the other hand, the CP approach is computationally demanding, and it is most

relevant at the scale of individual grains. Simulations involving large numbers of

grains have been reported too, but they require extra efforts and techniques to

reduce the computational time, such as homogenisation over a number of

neighbouring grains.

Balancing between feasible computational time and appropriate physical

description of the model, we have selected an original approach, combining the

macro- and micro-scale simulations.

The originality of the approach is to perform a thermomechanical simulation of

the ITER-relevant heat load at the macroscopic scale (a 12×12×5 mm3 tungsten

block, illustrated in Figure 3.11) under the assumption of isotropic J2 plasticity,

whereas crystal plasticity simulations are performed subsequently at the most

critical positions in the sample, using as input the strain and temperature history

which was predicted in the macroscopic simulation. The latter simulation uses

the macroscopic constitutive law, which was derived from tensile test results in

Chapter 2.

The macroscopic simulation was itself decoupled into a purely thermal

simulation followed by a mechanical simulation with a known temperature

history. The preliminary thermal simulation was performed to find the evolution

with time of the temperature field in the sample. It used the thermal properties

of tungsten available in literature [159]. Then, the obtained temperature field

served as input in a thermomechanical simulation on the same mesh, where the

bottom face was fixed and stresses emerged due to thermal expansion of the

heated area. The temperature- and rate-dependent constitutive law obtained for

the recrystallized tungsten in Chapter 2 was applied in the form of a user-defined

subroutine UHARD.

Finally, the strain path and the temperature profile were recorded at a point of

interest of the macroscopic specimen, and they were then applied at the

microscopic level to a polycrystal, whose mechanical response assumed that all

grains undergo uniform strain (Taylor mean-field crystal plasticity model).

3.4 Crystal plasticity framework

3.4.1 What is crystal plasticity theory?

The main mode of plastic deformation in polycrystalline materials is shear in

individual grains, which is carried out by dislocations gliding in slip systems.

Owing to the crystallographic anisotropy, the deformed grains change their

shape as well as the crystallographic orientation. Consequently, a

comprehensive theory of deformation of crystals must include the

crystallographic and anisotropic nature of dislocation glide, as well as take


76

account of the crystal orientation with respect to the external loading, or to the

neighbouring crystals [102].

Crystal plasticity is one of such models, stemming from the pioneering

contribution by Taylor [155]. The theory of CP is based on the assumption that

macroscopic plastic deformation is the superposition of crystalline slip in all

activated slip systems [160], whereas a slip system becomes active if the

resolved shear stress in it reaches a threshold value. The main conceptual

advantage of CP models is that they can combine a variety of mechanical effects,

which are direction-dependent owing to the underlying crystalline structure,

and can take account of dislocations as the main carriers of plastic deformation.

3.4.2 Traditional crystal plasticity for polycrystals

Prediction of deformation in single crystals is straightforward with the above-

mentioned assumption. Transitioning to polycrystals can be carried out by

considering that single crystal plasticity is valid within each grain, and by making

assumptions about how macroscopic constraints apply to individual grains in

the polycrystal [161]. For many years, this micro-to-macro transition has relied

on simplified assumptions after Sachs or Taylor, according to which all grains

undergo correspondingly either the same stress, or the same deformation as the

whole macroscopic polycrystal, hence the name “mean field” crystal plasticity.

These simplifications make it impossible to ensure strain compatibility and

stress equilibrium at GBs simultaneously. Grains in real deformed polycrystals

acquire non-uniform stress and strain due to the interaction with neighbours.

Such heterogeneities appear in the form of grain-scale orientation gradients as

well as a subdivision of the grain into volumes undergoing distinct deformation

[161].

On top of assuming uniform distribution of strain, stress and other quantities

within individual grains, the standard CP disregards potential effects of grain

shape, relative size and arrangement, thus the deformation of a given grain is

defined solely by its initial orientation. These simplifications can partly explain

the discrepancies between experimental results and the predictions of

simulations with Taylor-type models [161].

Nevertheless, the simulations of the present chapter assume a uniform

deformation of all grains, which has the effect of maximizing the heterogeneity

of stress resulting from the crystallites plastic anisotropy. Even though such

assumptions are less realistic than the CPFEM presented in the next chapter, they

rely on the same mathematical modelling of dislocation slip and allow us to

capture the main macroscopic trends.


77

3.4.3 Mathematical formulation of the grain-level plasticity law

This section was written with the help of my supervisor.

One originality of the model developed at UCLouvain is to rely on a simplified

treatment of elasticity in order to simplify the mathematical representation of

finite strains and to allow an efficient and fully implicit time integration of crystal

plasticity equations. The consistent tangent operator allows rapid convergence

of iterative solution schemes, for instance when the code serves as a user-defined

material law (UMAT) in the Abaqus finite element code.

As commonly assumed in crystal plasticity theories, the deformation gradient

tensor is multiplicatively decomposed: el p F R F F . In this expression, elF

and pF correspond to the elastic and plastic parts of the deformation, whereas

R is an orthogonal matrix representing the rotation of the crystal lattice from

its initial to its final orientation. The total deformation gradient F maps the

initial (undeformed) configuration onto the current (deformed) configuration.

The multiplicative decomposition of F defines two fictitious, intermediary

configurations [162]. The first one is obtained by mapping the initial

configuration using only pF , which amounts to consider that the material

deforms solely due to dislocation glide. The mapping el pF F defining the second

intermediary configuration includes also the elastic deformation of the lattice.

The latter fictitious configuration is sometimes called “unrotated configuration”

because it lacks only the crystal rotation. In metals, since elastic strains are small

as compared to plastic strains, the two intermediary configurations are very

close to one another.

Besides, when temperature fluctuates, the thermal strain is accounted for before

proceeding to the integration of crystal plasticity equations. The multiplicative

decomposition of the deformation gradient includes a supplementary isotropic

expansion, th T F I , where the value of thermal expansion coefficient

= 5.0·10-6 K-1 was taken as an approximate average of the temperature-

dependent value from [159] in the studied range of temperature.

Computing the velocity gradient tensor L requires time derivatives, which are

denoted with a dot:

1 11

1

T el el T el p p el T

L v F F

R R R F F R R F F F F R (3.2)

This expression is simplified by accounting for the fact that elastic strains are

infinitesimal so that el elF I ε with el 1ε , leading to:

1

1

T el p p T

T el p p T

p

( )

L R R R ε F F R

R R R ε F F R

LΩ

(3.3)


78

We have adjoined a tilde on top of three (second-order) tensors to highlight the

fact that they apply to the “unrotated configuration” (the second fictitious

intermediary configuration). In the latter configuration, the velocity gradient is

additively decomposed into a skew symmetric tensor Ω called the lattice spin

(i.e. rate of crystal rotation), a symmetric elastic strain rate elε and a plastic

velocity gradient p

L . These three tensors are most conveniently expressed in a

coordinate frame that rotates with the crystal frame. We will hence call them “co-

rotational” tensors.

The anisotropic elastic stiffness operator C then allows computing the co-

rotational Cauchy stress, el:σ C ε , whereas the plastic velocity gradient is equal

to:

p

sym skw

L b n M M M (3.4)

Here is the dislocation slip rate along the th slip system which is

characterized by the unit vectors b , parallel to the Burgers vector, and

n ,

normal to the slip plane. The Schmid tensor

M , dyadic product of the two

vectors, is subdivided into a symmetric sym

M and a skew-symmetric skw

M part.

The material response is computed incrementally by time integration under a

prescribed velocity gradient L D W where D is a symmetric strain rate

tensor and the skew-symmetric W tensor involves both the lattice spin and the

plastic spin. A set of nine equations is formed by separating symmetric and skew-

symmetric tensors:

Tsym

Tskw

:

R

σ

Ω

C R D R M

R W M

(3.5)

The dislocation slip rates are computed using a thermally activated viscoplastic

law which is discussed in the next section. Supplementary to σ , which is the

average microscopic stress, mobile dislocations are subjected to a backstress bσ

due to the pile up of other dislocation loops already immobilized at the GBs. The

backstress evolves according to a saturating Armstrong-Frederick law [155]:

symb b whereA B

Mσ σ (3.6)

A and B are material parameters. In case of monotonic loading, the tensor bσ

progressively aligns with the plastic strain rate (symmetric part of p

L ) and it

saturates at an amplitude scaling with /A B . The parameter A defines the initial

rate of increase of the backstress. Note that the model treats the backstresses


79

phenomenologically and does not consider dislocation pile-ups as objects. The

pile-ups are assumed not to contribute to the total dislocation density variable

. In this way the model is seen more versatile, since not only dislocation pile-

ups can generate backstresses in materials. Other sources include variation of

strength between material’s constituents, as discussed in Section 1.3.2.3.

Time integration of the material law is fully implicit and it is solved incrementally

following a Newton-Raphson scheme. As the increments of dislocation slip

t depend on the stress, there remain 14 independent unknowns in the

set of non-linear equations: the six components of the co-rotational stress

increment σ , the five independent components of the increment of backstress

bσ , which is deviatoric, and the three independent components of the skew-

symmetric tensor t ΩΩ representing the increment of lattice rotation:

sym

skw

symb b

:

/ (1 )

|

|

t t

t

t

t

t

t

A B B

σ C D M

Ω W M

σ M σ

(3.7)

The fully implicit time integration is simplified by accounting for the fact that

each time increment involves a sufficiently small increment of lattice rotation to

ensure that:

T

T

|

| | |

| | | |t t t t t t

t t t t

t

D R D R

R D R Ω D D Ω

D

(3.8)

|t tW is computed in the same way, thus depending only on the lattice rotation

at the beginning of the time step T |tR and on the increment of rotation Ω

which is determined in the Newton-Raphson solution.

The model now needs to be completed by expressions prescribing shear rate and

critical resolved shear stress as functions of state variables. The shear rate is

usually formulated as a function of the resolved shear stress, , and the critical

resolved shear stress,

, cf (3.9)

Due to work hardening and also to strain rate sensitivity, the critical resolved

shear stress evolves as a function of the total shear, , and the shear rate, .

,c g (3.10)


80

Thus, a system of equations with tightly interrelated variables is obtained, which

has to be solved for every iteration of the solution of the non-linear equation set

(3.7). This means that two Newton-Raphson schemes are imbedded.

In the present research, a thermally-activated law for dislocation slip [154] was

adapted in order to include the kinematic hardening mechanism as well as the

athermal stress characteristic of BCC metals at high temperature. To start, jumps

of dislocations over the obstacles are predicted using a usual equation of the

Arrhenius-type [163,164]:

0 exp 1 exp b c

B B

G ab

k T k T

(3.11)

Here kB, T, 0 and a are the Boltzmann constant, the temperature, the reference

strain rate and the area swept out for the reverse jump. Thus, the product ab

corresponds to the activation volume.

Here we had to account for the presence of athermal stress in BCC metals c , and

the CRSS at 0 K is denoted as ˆc , being the maximal value of thermal stress.

The athermal stress evolves as a function of dislocation density

0c y disb h (3.12)

where 0y may be chosen to depend on grain size.

G is the free enthalpy allowing a mobile dislocation to overcome short-range

obstacles [165], where 0F , p=0.5 and q=1.5 [144] are constants:

0 1ˆ

qp

b cG F

(3.13)

Thermally activated plasticity is properly modelled as long as

ˆc b c

. For lower stresses the response is purely elastic. For larger

effective stresses (which might appear at high temperatures), the plastic strain

rate is limited by viscous drag of dislocations, rather than by their mutual

pinning

0

ˆ

b c

(3.14)

The dislocation density is assumed to evolve according to the Kocks-Mecking

model. The product 2 plays the role of

2k used in Equation (2.4), where 2 is

constant and the temperature- and rate-effects are accounted by the factor

1 2k (3.15)


81

1/1/

0

0

1 ln

pq

Bk T

F

(3.16)

This factor also affects the value of B in equation (3.6) in order to enforce the

congruence between isotropic and kinematic hardening evolution:

0B B (3.17)

where 0B is a constant determined by fitting. In this way, similar patterns of

evolution of initial yield stress, initial work hardening and the saturating stress

on temperature and strain rate in all three materials (purely isotropic, purely

kinematic and mixed) are fulfilled, and the stress-strain curves in all three cases

of hardening can match each other in the whole investigated range of

temperature and strain rate.

In the first publication [120], a different formulation of viscoplasticity was used

to model recrystallized tungsten. The old formulation is not well adapted to very

high temperatures but provides similar predictions to those obtained with the

new formulation in the range of temperatures that was considered then. Slip

rates were defined following Rice, Hutchinson and Peirce [75] applicable not

only for FCC, but for BCC metals at high temperature [115]. Equation (3.9) took

the form of

1

0 sgnm

c

(3.18)

where the strain rate sensitivity exponent m was set equal to 0.02.

Expression for slip resistance was obtained from equations (2.7) and (2.10) with

the help of the Taylor factor, assumed to be equal to that in as-received tungsten,

namely 2.5. Essentially, the equation was rewritten in terms of critical resolved

shear stress c vs. accumulated shear strain in a slip system, preserving the

linear term with respect to shear 2

IVH

M

0

2

0

expsat sat IV

c

H

M M M M M

(3.19)

The equations were embedded in the user-defined subroutine UMAT containing

the algorithm of the crystal plasticity model.

3.5 Identification of material parameters

Before performing meaningful simulations e.g. of tungsten exposed to thermal

shocks, the constitutive law for the CP model had to be parameterized. As it is

impossible to derive exact parameters of kinematic hardening from tensile tests


82

alone, we had a certain freedom in the choice of parameters: different ratios

between kinematic and isotropic hardening lead to the same experimental

stress-strain curve. Hence, we have considered three cases of hardening: purely

isotropic, purely kinematic and mixed.

The parameters for the three hypothetical materials had to be fitted so as to

reproduce experimental tensile curves of as-received IGP tungsten (after

extraction of true stress as discussed in Chapter 2). The model parameterization

was performed with the help of an iterative optimization algorithm, similar to

the one discussed in Chapter 2. As long as this approach involves many iterations,

it is more efficient to rely on the less time-consuming crystal plasticity

implementation — the mean-field Taylor-type crystal plasticity, rather than

CPFEM.

The model polycrystal was composed of 100 randomly-oriented grains, enough

to form a representative volume element, whose plasticity was controlled by the

developed CP model. Every iteration of the optimisation algorithm involved a

group of N = 15 simulations (corresponding to the 15 tensile experiments

performed at different temperature and strain rate). They were launched for a

given set of model parameters (starting off from an initial guess). The objective

function was set as the total sum of squares of differences between the

simulated ,i n

s and the experimental ,i n

e true stress values at three

reference points, as illustrated in Figure 3.3, over all the 15 experimental stress-

strain curves,

3

2, ,

1 1

Ni n i n

e s

n i

(3.20)

where 𝑛 = 1–15 denotes the ID of the simulation in the group and 𝑖 = 1–3

corresponds to the selected reference points (0.02%, 0.2% and 0.5% of plastic

strain). Such a narrow range of strain for fitting was chosen in order to better

represent the very beginning of the deformation curve, as long as the thermal

strain achieved in a single cycle was not expected to exceed 1% [166]. The

hardening rate at 1% of plastic strain was also monitored to match the

experimental value. Different parameter sets were probed at every iteration,

until the objective function minimum was found, denoting the best match with

the experimental points. The known material parameters were fixed: the Young

modulus E = 388 GPa, Poisson ratio ν = 0.28, initial dislocation density

0 = 4.5·1012 m-2 [32] and Burgers vector b = 0.274 nm.


83

Figure 3.3. On the evaluation of the objective function. The squares of differences between experimental and simulated values of stress are summed over three points per curve and then are summed over all 15 stress-strain curves.

First, the parameterization of the purely isotropic hardening case was

performed. In order to parameterise the other two hypothetical materials, some

model parameters were manually adjusted, as summarized below:

Purely kinematic hardening

The dislocation strength hdis was reduced to suppress isotropic hardening, 0y

was increased to compensate the suppressed contribution to the CRSS from the

initial dislocation density, the rest of the parameters were left unchanged. The

objective function was minimized by searching for A and B.

Mixed case

was reduced to provide room for kinematic hardening in the very beginning

of plastic deformation, dislocation strength hdis was assigned the same value as

in the purely isotropic hardening case to provide the same work hardening rate

as in the isotropic case. The values of A and B were found using the iterative

algorithm.

Figure 3.4 shows how the three investigated hypothetical hardening cases were

constructed, illustrating the assumed decomposition of an experimental stress-

strain curve. The dotted lines in Figure 3.4 indicate the macroscopic stress, which

arises due to 0 only. When the resolved shear stress in a slip system exceeds

this level, the deformation is considered plastic and the hardening mechanisms

are activated. The dashed lines indicate the sum of 0 and the isotropic

hardening contribution, which increases with plastic strain due to the increase

of dislocation density. The solid lines correspond to the sum of all the

contributions, replicating the experimental stress-strain curve.


84

Figure 3.4. Illustration of construction of the three cases of hardening.

Thus, the difference between the dashed and the solid lines of the same colour

for the mixed case is due to kinematic hardening and is described by the

Armstrong-Frederick law. It means that a part of the initial yield stress is caused

by the accumulation of dislocation pile-ups at GBs prior to macroscopic yield,

while for the pure kinematic case the accumulation of pile-ups commences

merely after reaching the initial yield stress, is extended in time and develops

with much lower rate than in the mixed case.

The numerical stability of the developed constitutive law had to be ensured in

the ranges of temperature and strain rate that are relevant for the thermal shock

experiments. Following [26], the upper boundary for temperature can be

estimated as 1700 °C, and the largest strain rate is expected to reach order of

magnitude of several s-1 [1]. Besides, the selected equations require a limit on the

minimal allowed strain rate, chosen in this work quite arbitrarily to be 10-6 s-1.

0.00 0.02 0.04

200

300

400

500

600

Kinematic

hardening

Work hardening

due to dislocations

Kinematic

hardening

Tru

e s

tre

ss [M

Pa

]

Flow stress

Base stress

a) Isotropic

Initial dislocation

hardening

Work hardening

due to dislocations

0.00 0.02 0.04

True strain

Flow stress

Dislocation

contribution

Base stress

b) Mixed

0.00 0.02 0.04

Flow stress

Base stress

c) Kinematic


85

3.6 Results

3.6.1 Identification of model parameters based on uniaxial tensile

tests


Applying the fitting algorithm outlined above, we have parameterized the crystal

plasticity model for as-received IGP tungsten in the three considered cases of

hardening: purely isotropic, purely kinematic and mixed. Some of the

parameters were set to be independent of the hardening type, because they

describe the rate of dislocation accumulation and the activation energy. Such

parameters are listed in Table 3.1. The other parameters, which were varied

manually or fitted, are grouped in Table 3.2. Note that, since the parameters of

the Armstrong-Frederick model are responsible for hardening above the dotted

lines in Figure 3.4, the value of A is much higher for the mixed hardening case.

Table 3.1. The CP model parameters independent of hardening type for as-received tungsten.

k1, m-1 2 F0 a

0 , s-1

1.16·108 10.7 0.53𝜇𝑏3 100𝑏2 6.64·104

Table 3.2. The CP model parameters for the isotropic, kinematic and mixed hardening cases of as-received tungsten.

Hardening A , GPa B 0y , MPa , MPa hdis

Isotropic - - 17.86 260.7 0.129

Kinematic 1.1 5.36 28.57 260.7 0.013

Mixed 3400 750 17.86 167.9 0.129

Figure 3.5 shows the corresponding fitted true S-S curves for temperature and

strain rate used in tensile tests (300 °C and 600 °C and 6·10-4 s-1), strain rate

expected when tungsten is exposed to thermal shocks (6 s-1) as well as for the

model applicability limit, 1700 °C and 10-6 s-1. The curves obtained for the three

hardening modes coincide well with experiments at all combinations of test

conditions. This confirms that monotonic tensile tests alone do not suffice in

order to probe the amplitude of backstresses.


86

Figure 3.5. True deformation curves obtained in simulations of tensile test with the crystal plasticity model and the best parameter sets for three hardening cases.

The evolution of dislocation density is shown in Figure 3.6 for the same

combinations of test conditions. As expected, it is identical for the three

considered hardening modes, whereas it depends on temperature and strain

rate. Note that the model does not treat dislocation pile-ups as separate objects

and does not take into account their density. Besides, this figure illustrates the

existence of the model applicability domain: at a certain combination of

temperature and strain rate (1700 °C and 10-6 s-1 in this case), the derivative of

dislocation density w.r.t. strain diminishes, and becomes negative outside the

domain, provoking numerical instability of simulations. The applicability domain

was defined beforehand, anticipating the conditions in future simulations.


87

Figure 3.6. Evolution of dislocation density in the simulated polycrystal of as-received tungsten with the best parameter sets for three hardening cases.


Simulations of uniaxial tension of recrystallized tungsten were also performed

to validate the direct use of parameters obtained in the macroscopic model

fitting. The obtained stress-strain curves are compared with their experimental

counterparts in Figure 3.7. The rate sensitivity of the simulated material

decreases with the increase of temperature, as the contribution of the thermal

stress component decreases.

Figure 3.8 highlights the applicability domain of the model, showing true stress-

strain curves obtained in uniaxial tension simulations at high temperature and

at the limiting values of strain rate as discussed above.


88

Figure 3.7. True stress-strain curves obtained in uniaxial tension with the CP model of recrystallized tungsten (dashed) superimposed on the experimental curves (solid).

Figure 3.8. True stress-strain curves at high temperature and the limiting values of strain rate, demonstrating the applicability domain of the model of deformation of recrystallized tungsten.


89

A good match between the simulated and experimental mechanical response to

uniaxial tension indicates the possibility to use the developed microscopic

constitutive law for simulations of a complex load, such as the thermal strain and

stress caused by the thermal shocks.

3.6.2 Isothermal mechanical cycles


Figure 3.9. Stress-strain loops obtained in simulations of mechanical isothermal cyclic loading of as-received tungsten.

Before using the developed model in simulations of thermal fatigue, it was first

tested in isothermal cyclic loading, i.e. tension-compression. A series of

simulations was performed in a range of temperature, covering the whole

applicability domain, and strain rate 6·10-4 s-1. The obtained stress-strain loops

are summarized in Figure 3.9 for the three studied hardening modes. Slow

gradual increase of stress amplitude is seen for the isotropic hardening case,

because in this model the multiplication of dislocations takes place in both direct

and reverse loading. The loops corresponding to kinematic hardening are stable

because the backstresses accumulated during direct loading diminish and

change sign during reverse loading. A small Bauschinger effect is visible even in

the purely isotropic hardening case, originating from grain-to-grain variations of


90

strength, caused by plastic anisotropy. Deformation of polycrystals is similar to

that of composites, as discussed in Subsection 1.3.2.2.


Simulations of isothermal mechanical cyclic loading, i.e. tension-compression

cycles, of recrystallized tungsten revealed a more rapid increase of stress

amplitude than in as-received tungsten, as compared in Figure 3.10. It is caused

by two reasons: the large work hardening capacity of recrystallized tungsten and

the presence of the linear term in the equation for slip resistance. Note that the

rate of increase of stress diminishes, as the temperature approaches the model

applicability limit of 1700 °C.

Figure 3.10. Stress-strain loops obtained in simulations of mechanical isothermal cyclic loading of recrystallized and as-received tungsten.

3.6.3 Macroscopic thermal shock simulation

The simulations replicated the experiment carried out at Forschungszentrum

Jülich in the JUDITH1 facility, in which box-shaped W samples were subjected to

ELM-like heat load induced by an electron beam [26]. A thermal power density

of 190 MW m-

2 or 380 MW m-2 was applied at the centre of the top surface of the

W block 12×12×5 mm3 by an electron pulse, 1 ms long, while the bottom surface

was kept at a constant base temperature 27 °C, 400 °C or 1000 °C.


91

Figure 3.11 depicts the FE mesh used in the macroscopic simulation and the

temperature field at the end of exposure to the heat flux 190 MW m-2, showing

the heated central area (4×4 mm2). Deformation of the sample due to high heat

load has been magnified by a factor of 10 for illustrative purpose. The yellow dot

in the centre indicates the element for which the strain state was recorded. High

stress concentration and initiation of intergranular cracks are expected in this

region.

Figure 3.11. FE mesh of the sample studied in JUDITH1 facility for a macroscopic thermo-mechanical simulation. The colour indicates temperature field at the end of heat flux exposure, with the blue colour corresponding to the base temperature, 400 °C, and the red - to the maximal attained temperature ~750 °C.

Figure 3.12. Evolution of temperature and the von Mises strain in the centre of the heated area exposed to an ELM-like thermal shock of 190 MW m-2.

For the microscopic simulations we need to know the evolution of temperature

at a selected point, as well as stress or strain state. Figure 3.12 shows an example

of the evolution of the temperature and of the von Mises strain with time in the

selected element, where one can see a rapid increase of both values during the


92

heating period (the first millisecond). Upon cooling down, the temperature fades

out significantly. However, von Mises strain decreases only slightly. That is

caused by the irreversible nature of plastic strain and its ability to accumulate

after a number of heat load cycles, provoking further build-up of stress and

consequent crack initiation. The most significant drop of temperature is

observed in the first millisecond after switching off the heating (i.e. between time

marks 1–2 ms).

Components of stress tensor were measured in the centre of the heated area (the

yellow dot in Figure 3.11). The shear components turned out to be zero while the

normal components are shown in Figure 3.13 as functions of time (a) and

temperature (b). Only two non-zero components are present, identical thanks to

symmetry of the problem: S11 and S22, which are normal stresses along the

heated surface, revealing the plane stress state in that location. The most

significant excursion of the in-plane stress components is observed during

heating and the first millisecond of cooling. These findings allow us to apply

simplified boundary conditions of plane stress in the microscopic simulations,

instead of following the exact recorded strain path. This approach can help

identify main trends in the mechanical response of material under thermal

shocks and significantly reduce the required computational time.

The value of normal stress in the subsurface region is close to residual stresses

in tungsten exposed to ELM-like thermal shocks measured with the help of X-ray

diffraction (XRD) [72,167]. The neutron diffraction method has been used for

detection of internal stresses in fabricated monoblocks [168] and can be applied

to measure residual stress in tungsten after thermal shocks.

a) b)

Figure 3.13. Normal stress components in the centre of the sample exposed to 380 MW m-2 as a function of (a) time and (b) temperature. The only non-zero components are S11 and S22, indicating a plane-stress state.


93

3.6.4 Microscopic thermal shock simulation

The developed CP model was used in mean-field simulations of the mechanical

response of a tungsten polycrystal to ELM-like thermal shocks, defined by

triangular waveform excursions of temperature between the base and the peak

values under the conditions of plane stress. 1000 or 100 thermal pulses were

simulated for every combination of base temperature and power density for both

as-received and recrystallized tungsten, and the values of thermally-induced in-

plane stress and out-of-plane strain were recorded.

The simulation box represented a polycrystal composed of 100 randomly

oriented grains, which is schematically shown in Figure 3.14. The thermal

expansion (shown by the orange colour) was allowed only in the direction Z,

whereas the other two directions were constrained.

Figure 3.14. Schematic view of the simulation box in the initial and deformed state. The thermal strain is denoted as .


Thermally-induced stress and strain recorded in simulations of thermal shocks

in as-received tungsten with isotropic hardening are shown in Figure 3.15 (other

material laws will be reviewed in Section Discussion). Due to the cyclic nature of

deformation, the plots have the form of loops.

Application of the lower power density, 190 MW m-2, leads mostly to cyclic

elastic deformation in as-received tungsten, which begins with 1·10-3–2·10-3 of

plastic strain, thus the material is in the state of elastic shakedown. That explains

the absence of cracks on its surface in experiments [26].

Significant plastic deformation and non-degenerate stress-strain loops are

observed in the simulations of higher heat loads (380 MW m-2). The average

strain (the loop centre) and the strain amplitude have increased by the factor of

2–3 as compared to the lower power density. The shape of the loops does not

change significantly from cycle to cycle, due to quite low increment of work

hardening per cycle and due to kinematic hardening originated from the stress


94

heterogeneity in neighbouring grains. Macroscopically observed yield stress

corresponds to yielding in the softest grain, whereas all the grains reach the

plastic plateau soon after that. It indicates that the state of elastic shakedown in

as-received tungsten might not be reached even after a large number of cycles

and the material is under conditions of low cycle thermal fatigue.

Figure 3.15. The mechanical response of as-received tungsten (isotropic hardening only) to thermal shocks of different power density in terms of the in-plane stress vs. the out-of-plane strain.


The evolution of stress-strain loops recorded in simulations of thermal shocks in

recrystallized tungsten is shown in Figure 3.16 for the material with isotropic

hardening only. The loops progressively change their dimensions from one cycle

to another along with material hardening.

The first cycle begins with accumulation of plastic deformation in compression

during heating: 0.2–0.3% and ~0.8% at low and high power density

correspondingly. Tungsten exposed to low power density 190 MW m-2 at base

temperature 27 °C experiences elastic shakedown, i.e. the absence of plastic

deformation in cyclic loading, from the second cycle, as its response has not

changed from cycle 1 to cycle 10. This observation is in line with the experiment

[26] where no damage and no surface modification was observed in


95

recrystallized tungsten at room base temperature at low power density. The

contribution of plastic strain decreases with the increase of stress amplitude.

And, since the total strain is fixed, the loops degenerate and elastic shakedown is

eventually reached by the 1000th cycle in tungsten exposed to 190 MW m-2 at

400 °C and 380 MW m-2 at 27 °C.

Figure 3.16. The mechanical response of recrystallized tungsten to thermal shocks of different power density in terms of the in-plane stress vs. the out-of-plane strain.

The loops are asymmetric due to the difference between yield stress at maximum

and minimum temperature during cyclic heating, which is especially pronounced

at lower base temperature. It means that as the sample is cooled down, its

strength continuously increases, resulting in a higher recorded stress. As the

base temperature increases, the stress amplitude decreases; since the material

loses its strength. In contrast with the as-received tungsten, low power density

causes plastic deformation in recrystallized tungsten at high base temperature

due to its reduced strength.

The maximal strain does not exceed 0.5% and 1% at low and high power density

correspondingly similar to the as-received tungsten, but the stress amplitudes

differs significantly in the two states of tungsten. Plastic strain accumulation is

accompanied by the increase of stress amplitude and ratcheting, i.e. the increase


96

of the mean plastic strain with the number of pulses. It might be caused by low

initial yield stress, high work-hardening capacity and the linear term in the

constitutive law for recrystallized tungsten. The latter might still be excessive, as

the amplitude of stress reaches unreasonable values, and the range of strain in

which its presence is physically-sound has still to be checked experimentally by

applying large plastic deformation to tungsten or by cyclic mechanical tests.

Another important point is to introduce a fracture criterion which would limit

the number of cycles sustained by tungsten, probably way before reaching the

elastic shakedown.

3.7 Discussion

3.7.1 Effect of kinematic hardening

Figure 3.17. Stress-strain loops recorded in the as-received tungsten for the three considered material laws.

Stress-strain loops for all hardening cases of as-received tungsten are grouped

in Figure 3.17, and the tensile stress amplitude (measured at the end of every


97

pulse) is plotted as a function of cycle number in Figure 3.18. The stress-strain

loops obtained in different hardening cases generally look similar, but a more

pronounced Bauschinger effect for the mixed hardening, and a slower increase

of stress amplitude for the pure kinematic hardening are observed.

Figure 3.18 reveals that the stress amplitude decreases with the increase of base

temperature for the three cases, showing the tendency of the material to sustain

a larger number of thermal pulses at high temperature, or to reduce the density

of surface cracks after a given number of pulses, which agrees with the

experiments.

Figure 3.18. Evolution of the in-plane tensile stress amplitude obtained in the as-received tungsten for the three considered material laws.

The presence of isotropic hardening, i.e. higher dislocation strength and their

continuous multiplication, leads to a fast expansion of the yield surface,

increasing the maximal tensile stress. The difference between the isotropic and

kinematic cases is negligible at low power density 190 MW m-2, as deformation

is mostly elastic and the rate of accumulation of dislocation density is low.

On the other hand, at high power density, the application of purely kinematic

hardening model returns the lowest macroscopic stress amplitude among the


98

explored cases. Indeed, kinematic hardening corresponds to the reversal of

microscopic internal stresses in every cycle, whereas isotropic hardening is

associated with continuous dislocation multiplication.

We should, however, not conclude from the observation of stress amplitude

evolution that the existence of kinematic hardening has a positive impact on the

fatigue resistance of tungsten.

When dealing with kinematic hardening, it is important to keep in mind that the

backstress amplitude actually measures the heterogeneity of stresses witnessed,

respectively, by the mobile dislocations accommodating plasticity and by

dislocations already piled up against GBs. It is a repulsive stress which tends to

separate the two types of dislocations. During monotonic loading, mobile

dislocations progress against the backstress (“they have wind in the face”) since

the backstress makes it more difficult to reach the GBs and extend the already

existing backstresses. When the macroscopic load is reversed, it adds up to the

pre-existing backstress. Mobile dislocations now move in the opposite direction

(“with the wind in the back”). This causes early plasticity (the Bauschinger effect)

while the dislocation pile-ups are progressively dissolved.

In GB regions, the effect of the backstress is to push immobilized dislocations

against the GB (making the pile-up denser). Hence, the stress amplitude in the

vicinity of GBs, i.e. where damage is expected to initiate, is larger than the stress

measured macroscopically in experiments. When assessing the strength of a

material, kinematic hardening is likely to favour the onset of damage at GBs.

Here we do not consider the possibility of dynamic recrystallization in hot

tungsten, which can reduce its strength and the rate of accumulation of

backstresses. It would affect both hypothetical materials, with isotropic and

kinematic hardening, by increasing the grain size and reducing dislocation

density. The proposed model assumes that the grain size remains constant, and

the dislocation density changes due to deformation only. The main application of

the model might be to analyse stress state at a small depth of a tungsten PFC,

where temperature is not sufficient for recrystallization, but periodic stresses,

transmitted from the surface, are significant to cause formation of fatigue cracks.

3.7.2 Note on the mixed hardening case

The results of simulations of the mixed hardening case (the presence of both

isotropic and kinematic hardening) are different from the pure cases: a rapid

accumulation of internal stresses is observed in the first few cycles, especially at

low power density. It contradicts the intuitive expectations of the mixed case

being somewhere between the pure cases. However, it might indeed represent

the reality. Experimental validation would be necessary to find out the actual

response of material to cyclic loading, by means of instrumented strain-

controlled fatigue tests and thermomechanical fatigue tests.


99

The first conclusion that could have been made is that the synergy between

isotropic and kinematic hardening in the material would increase the risk of

crack formation during thermal shocks compared to material exhibiting pure

isotropic hardening. Even though piling up of dislocations at GBs in material with

kinematic hardening is known to increase stress concentration, and provoke

damage, the simulation observations should be treated carefully due to the

following point.

Deformation in the present model is considered plastic (and, consequently,

hardening mechanisms are activated), when resolved shear stress exceeds the

value of 0 (the dotted lines in Figure 3.4). In the purely kinematic case

0

corresponded to the CRSS, but in the mixed case 0 was purposely lowered. As

long as a fraction of the CRSS is due to the Hall-Petch effect [115], related to the

build-up of dislocation pile-ups at GBs, the kinematic hardening mechanism (the

Armstrong-Frederick model) was set to define this portion of CRSS in the mixed

case (whereas in the purely isotropic case the contribution of the Hall-Petch

effect was taken into account in 0 ).

From the physical point of view, the mixed case interprets deformation of

material as if the multiplication of mobile dislocations commenced

simultaneously with the initial formation of dislocation pile-ups prior to their

saturation and consequent macroscopic flow. Hence, in the mixed case, a larger

portion of deformation loops is considered plastic, compared to the pure

hardening cases, and, consequently, a higher increment of dislocation density

and stress is achieved in every thermal cycle.

An alternative approach would be to assume that deformation switches from

elastic to plastic when stress exceeds the initial CRSS (at zero strain), which in

the mixed case is the sum of 0 and the saturated backstress. It would mean that

at the onset of plastic deformation dislocations start gliding and form pile-ups,

but do not multiply until the backstress saturates. Up to our best knowledge

there is no clear information in literature whether the multiplication of mobile

dislocations takes place simultaneously with the initial accumulation of

dislocation pile-ups prior to macroscopic yield, and thus we test only one of the

two possibilities in the present work.

Moreover, the present formulation of the mixed hardening will be beneficial in

the next chapter, since the presence of dislocation pile-ups in material is known

to aggravate stress concentration at GBs, and to promote damage. Thus, what we

called “mixed hardening” here should be more appropriate for the description of

kinematic hardening, than what we denoted as “pure kinematic hardening”. For

this reason, we will investigate the effect of kinematic hardening on the

mechanical response of tungsten to thermal shocks, considering two material

laws — isotropic hardening and mixed hardening in Chapter 4.


100

3.8 Conclusion

An existing CP model of deformation of polycrystals has been enhanced i) to

correctly capture the existence of athermal stress in BCC metals, and ii) to take

into account backstresses to properly work with kinematic hardening, which is

necessary for simulations of cyclic loading.

The model can be used in two modes — standalone and coupled with the Abaqus

finite element solver (CPFEM mode). The former has been used in the present

chapter, while the latter will be considered in Chapter 4, to perform a thorough

investigation of damage accumulation during loading at the microscopic level.

Three hypothetical material laws with different share of kinematic hardening

have been parameterized to represent mechanical properties of ITER-

specification tungsten. On top of that, recrystallized tungsten was also

parameterized, considering the presence of isotropic hardening only.

The main purpose of this chapter was to check the applicability of the model to

simulation of cyclic loading of tungsten with different modes of hardening, which

was proven in analysis of both isothermal mechanical loading and thermal shock

loads mimicking the fusion environment. Qualitative agreement with the

thermal shock experiments has been obtained, linking the stress state during

cyclic loading and the damage of the exposed surface.

In addition, our analysis shows that the presence of the initial ELM-induced

plastic deformation (attained in the first cycle) does not allow the material to

return to its initial deformation-free state at the end of the heat pulse. This means

that the post-heated area tends to remain elevated due to a non-zero out-of-

plane strain component. Such observation explains the origin of surface

roughening, which is regularly observed experimentally as early as after a few

pulses and is well known to be a precursor of crack nucleation at later stages of

ELM exposure.

101

Chapter 4.

Damage evolution and failure of tungsten

at the microscopic level

in thermomechanical fatigue conditions. Equation Cha pter 4 Section 4

4.1 Introduction

4.1.1 The need of CPFEM

Finally, we have come to the central question of the project: understanding the

origin of crack formation in tungsten exposed to thermal shocks. As the fatigue

cracks in tungsten appear at GBs and, especially, at triple junctions, fine-scale

processes at the grain level are significant, making the J2 plasticity model

inapplicable for analysis.

Understanding the initiation of fatigue damage requires accounting for

heterogeneity of stress and strain not only from one grain to another

(intergranular heterogeneity) but also inside every grain (intragranular

heterogeneity), which could not be captured by the Taylor-type CP model used

in Chapter 3. The intragranular heterogeneity originates due to complex

boundary conditions at every grain in real deformed polycrystals, where both

stress equilibrium and strain compatibility have to be fulfilled simultaneously.

Plus, given the fact that the microstructure heterogeneity will further increase

during neutron irradiation in fusion devices, as a result of plastic flow instability

observed in irradiated material experimentally, the heterogeneity of strain in

grains will amplify as well. Another important, experimentally discovered point,

is the effect of the grain shape and the orientation of the elongated direction with

respect to the exposed surface on the formation of cracks in tungsten. Defining

grain shapes and mapping their neighbours is not possible in the Taylor-type CP

simulations.

That is why a thorough investigation of mechanical response of tungsten to

thermal shocks requires an appropriate computational tool, able to describe

material plasticity at the grain level, to take account of the grain shape and

interaction with neighbours, and not to neglect the strain and stress gradients

inside grains.

A full potential of crystal plasticity is unlocked in an advanced technique called

“crystal plasticity based finite element modelling” (CPFEM), which is used in this

4. Thermomechanical fatigue of tungsten at the microscopic level

102

chapter for exact predictions of fatigue damage in polycrystalline tungsten

subjected to thermal shocks due to ELMs. In the CPFEM framework every grain

of a polycrystal is tessellated into finite elements, in which stress and strain are

defined by a complex interaction with the surrounding elements, controlled by

the CP model, relying on dislocation glide in slip systems.

4.1.2 Overview of CPFEM theory

A brief quotation from a comprehensive overview of the CPFEM approach

[75,102] is provided here for the convenience of the reader, highlighting its

features and differences from the mean-field CP approach.

The CPFE method has evolved as an attempt to employ some of the extensive

knowledge gained from experimental and theoretical studies of single crystal

deformation and dislocation motion to inform the further development of

continuum field theories of deformation. The general framework supplied by

variational crystal plasticity formulations provides an attractive vehicle for

developing a comprehensive theory of plasticity that incorporates existing

knowledge of the physics of deformation processes [169–171] into the

computational tools of continuum mechanics [74] with the aim of developing

advanced and physically based design methods for engineering applications

[172]. The first CPFE simulations were performed by Peirce et al. in 1982 [173].

One main advantage of CPFE models lies in their ability to solve crystal

mechanical problems under complicated internal and/or external boundary

conditions. This aspect is not a mere computational advantage but is an inherent

part of the physics of crystal mechanics since it enables one to tackle those

boundary conditions that are imposed by inter- and intra-grain micromechanical

interactions [174]. This is not only essential to study in-grain or grain cluster

mechanical problems but also to better understand the often quite abrupt

mechanical transitions at interfaces [175].

CPFE simulations can be used both at microscopic and macroscopic scales [176].

Examples for small-scale applications are inter- and intra-grain mechanics,

damage initiation, mechanics at interfaces, simulation of micromechanical

experiments (e.g. indentation, pillar compression, beam bending), or the

prediction of local lattice curvatures and mechanical size effects. Consequently,

CPFE methods gain momentum for the field of small-scale material testing where

the experimental boundary conditions are difficult to control and/or monitor. In

such cases the experimental results may sometimes be hard to interpret without

corresponding CPFE simulations that allow an experimentalist to simulate the

effects of details in the contact and boundary conditions.

4.1.3 Indicators of fatigue

This subsection is inspired by analysis of a review paper by Korsunsky [177] and

the references therein.


103

To increase reliability of fatigue durability prediction, the used finite element

model must possess a constitutive law, validated in a wide range of temperature

and loading conditions, rely on relevant experimental data and utilize an

adequate relationship linking deformation history and damage [177]. But,

generally speaking, very large accumulated strain, exceeding the experimental

limit, can be attained in finite element analysis, if solely a plasticity law is

provided, since the model is not informed about the fracture strain. It is not

physically-sound, and the model requires a criterion to limit the possible

deformation range, or to indicate the moment of fracture.

In order to predict crack initiation in mechanical fatigue, a number of fatigue

indicator parameters (FIP) has been proposed and found useful in different

applications. Generally, an FIP can be written as a function of cyclic strain or

stress. When it reaches a predefined critical threshold, the sample is considered

to break.

In specimens of simple shape this indicator can be calculated as a single number

for the whole sample (evolving with time), but in objects of complex shape

subjected to fatigue, the indicator must be evaluated locally, showing a non-

uniform spatial distribution which would indicate the most probable location

and moment of material failure.

Deformation history can be accounted for if is taken as a functional, calculated

including the combination of stress and strain values at all data points in a stress-

strain loop [177]. An example of a functional criterion is the energy dissipation

criterion [178,179], which may be written as

dt

t t t (4.1)

where the integral can be taken over a fatigue cycle or another time interval t of

interest.

According to [177],

“Skelton [180] suggested that the dissipated energy per saturated

cycle could be considered as a material constant expressing the

necessary energy to propagate a crack by the distance equivalent to

the size of the process zone, and proposed a way in which the

lifetime of the specimen was related to a crack propagation law

driven by dissipated energy.”

In the case of non-saturated stress-strain cycle, the evolution of the loop shape

and dimensions has to be taken into account in the durability prediction, or the

FIP has to be calculated incrementally in every cycle.

Whereas many criteria were obtained for uniaxial cyclic stress, the choice of the

exact mathematical expression in the case of multiaxial stress is challenging, and

can be solved, for instance, by choosing an appropriate “norm” of the involved

tensors, such as the von Mises stress and strain or a tensor operation resulting

in a scalar. Not only dissipated energy density was tested as an FIP, but such


104

individual quantities as accumulated slip [181,182] or slip rate [183]. Some

works have not identified any particular approach to give invariably better

predictions than others [177]. Others clearly prefer the energy density [103]

claiming that individual quantities have not been found to correlate with

experimentally observed number of cycles to nucleation of crack.

Considering the reasoning above, the dissipated energy density was selected as

an FIP in the present work. The increment of dissipated energy at a given

material point experiencing multiaxial stress state can be written at the

microscopic level based on shear stresses and shear rates in the -th

crystallographic slip system [177], which can be calculated either within the CP

model

1

d dN

t

(4.2)

or at the macroscopic level, being the product of the Cauchy stress σ and the

plastic component of velocity gradient pL [184].

pd : dM t σ L (4.3)

Subscripts and M stand for “microscopic” and “macroscopic”

correspondingly. The chosen kinds of stress and strain rate must be power-

conjugate in order to represent a meaningful product.

Simulations of fatigue crack formation with the help of CPFEM involving an FIP

have already been performed and reported [185].

It is still unclear, however, and has scarcely been reported in literature whether

the emerging backstresses have an effect on the fatigue life. Do the dislocation

pile-ups (the main cause of kinematic hardening), moving periodically back and

forth due to external cyclic load, contribute to the damage accumulation? If yes,

how should the backstresses be included in the calculation of FIPs?

Most of the conventional fatigue indicators assume that damage initiates inside

the bulk of grains whereas damage is expected at GB in this material. Hence, we

define a new fatigue indicator which is specifically designed for GB cracking

(although it could be justified also in particle strengthened metals). Using CPFEM

the fatigue indicator is computed everywhere inside the grain in order to

demonstrate that it's value is maximized at GB. Based on the anisotropy of

adjacent grains, CPFEM allows predicting which are the weak spots along the GB

according to the chosen fatigue indicator.

We hypothesized that since dislocation pile-ups produce stress field

concentrated at GBs, either boundary cracks or yielding in the adjacent grains

can nucleate [100]. It means that the ability of material to experience kinematic

hardening due to the presence of dislocation pileups may promote damage

initiation. In comparison, purely isotropic hardening assumes that dislocation


105

density increases proportionally to the accumulated strain irrespective of the

location in a grain; be it next to or far away from a GB (or another obstacle).

In this work, we assumed that, in the presence of kinematic hardening, the FIP

should scale with the stress predicted ahead of dislocation pile ups rather than

the stress experienced by mobile dislocations. The relevant stress value is thus

the sum of (rather than the difference between) the resolved shear stress and

the backstress b

in slip system . This takes account of the stress-magnifying

nature of dislocation pile-ups.

1

d dM

b t

(4.4)

An important point is to determine the threshold value, which corresponds to

the fracture of material as a result of exhausting the deformation capacity.

Usually in literature this is done on the basis of fatigue experiments, knowing the

energy dissipated in one cycle [177]. The threshold should depend on the

material microstructure, and even orientation of grains of a given material, since

a high density of GBs at the sample surface promotes percolation of initial cracks

and reduces the fatigue life. In the absence of fatigue tests on IGP tungsten we

attempted to calculate the threshold based on energy dissipated in tensile tests,

as shown in section Discussion.

The ability to capture stress- and strain heterogeneity in CPFEM models implies

that the FIP distribution is also heterogeneous, and its maximum indicates the

location of a potential crack formation. It can be thought of reaching such

dislocation configuration at high strain, where every new coming dislocation

contributes to the formation of a void, which in turn serves as a seed for a crack.

The fact that the studied tungsten is a single-phase uniform material eliminates

a potential difficulty of treating in a different way regions close to a second-phase

inclusion, where cracks can form even at low plastic strain due to the mismatch

of stiffness or strength.

4.1.4 Overview of the chapter

In this chapter we set the goal to find the difference between the mechanical

response of tungsten exposed to ELM-like thermal shocks, assuming prevalence

of either isotropic or kinematic hardening of the material.

We extend the workflow proposed in Chapter 3, and apply the strain path

recorded in the macroscopic simulations of thermal shock, to a finite element

mesh representing a polycrystal, in which crystal plasticity model developed in

Chapter 3 serves as the constitutive law. The details of the FE simulation setup

are discussed in Section 4.2.

To obtain a clear view and single out the effect of grain shape alone on the

damage accumulation and on the intra-granular heterogeneity of strain and

stress, each constitutive law is applied here to both of the observed grain shapes


106

— elongated and equiaxed ones — as if we had two hypothetical materials on

top of the two existing tungsten grades. Different quantities are used to assess

damage in the polycrystals — stress at GBs and FIP, linked to the dissipated

energy. They are applied to recrystallized and as-received tungsten grades

correspondingly in Sections 4.3 and 4.4.

Another application of the CPFEM model is considered in Section 4.5, where a

delamination of tungsten foil is simulated, inspired by an experimental work

dedicated to the measurement of fracture toughness of a ultrafine-grained foil

[40].

The obtained results are discussed in Section 4.7. Part of this chapter has been

included in our published paper [166].

4.2 FEA setup

4.2.1 Crystal plasticity — standalone and CPFEM

The CP model discussed in Chapter 3 can be used in two modes:

i) In the standalone mode — it is a Taylor-based crystal plasticity, in

which individual grains are assigned the same deformation as imposed

to the whole polycrystal. Grains cannot be visualized in this approach;

they actually do not have exact location, shape and neighbours. Such

simulations of an “abstract polycrystal” are presented in Chapter 3.

ii) In the CPFEM framework — it is coupled with finite element solver

Abaqus, in which a representative volume element (RVE) is tessellated

into several grains, and every grain is tessellated into finite elements

using a FE mesh generator (Figure 4.1). Then every finite element is

characterized by a given grain orientation and a set of state variables,

whose evolution is controlled by the CP model. This type of simulations

is discussed in the present chapter.

Grains in the CPFEM framework are usually tessellated into simple shapes, such

as cubes, as shown in Figure 4.1a. However, grains in real polycrystals have

complex shape, which is best represented in simulations with the help of the

Voronoï tessellation (Figure 4.1b). Meshing of complex polyhedral can be done

with the help of triangulation, forcing the use of tetrahedral elements, whereas

simple cubic grains are easily tessellated into hexahedral elements. Below we

compare the use of the two element types.

Apart from being initially integrated into commercial finite element codes,

CPFEM approach can be implemented as a user-defined subroutine, making it

more accessible to researchers and available on many platforms [102].


107

a) b)

Figure 4.1. Representation of polycrystals in CPFEM framework. a) Simplified 3D grain assembly [186], b) advanced grain assembly obtained with the help of the Voronoï tessellation [187].

UMAT is a user-defined material subroutine for the standard finite element

solver Abaqus, which provides sufficient flexibility to parameterize the

mechanical response of virtually any material, including anisotropic, and to

incorporate any physics behind. Hence, UMAT was considered as a tool of choice

to implement the crystal plasticity model and couple it with Abaqus.

The model presented in the thesis is of local type. This means that the response

of a material point is assumed independent of the state at neighbouring points

[188]. In reality, the local amplitude of backstresses should depend on the

shortest distance to the GB, and the strength of pile ups should depend on the

permeability of GBs to dislocations. In the present model, grain size is accounted

for in a very simplified way. In a nonlocal model, the plastic strain gradient

contributes to hardening while introducing a material length scale. There is thus

exchange of information between neighbouring material points [189–194].

Another simplified approach would be to add an extra state variable at every

material point, which is the distance to the nearest GB, and modify the model

equations as well, to hinder the dislocation motion in the direction of the nearest

GB, mimicking the repulsive effect of backstresses.

4.2.2 Cohesive elements

Cohesive elements (CE), also called cohesive zones (CZ), are 2D finite elements

that can be used as a layer connecting neighbouring 3D elements. As the name

suggests, they are placed along surfaces inside the FE mesh which are initially

joined but are likely to separate in the course of deformation. The cohesive force

must be representative of the physics of the bonding of the two surfaces. The

latter surfaces may belong to two distinct components (e.g. coating on substrate,

or a layer of glue) or be the crack faces in a single component (Figure 4.2a). CE

are typically used in simulations of delamination of fibre-reinforced composite

laminates, crack growth at the macroscopic level [195] and in polycrystals at the


108

microscopic level [193,194,196].Note that even at the lower scale cohesive

elements are applied under assumption of continuous media and cannot capture

effects originating at the level of atoms or dislocations, where such models as

molecular dynamics, kinetic Monte-Carlo or dislocation dynamics would be

more appropriate. If together with placing CE along GBs the strain gradient

plasticity model is applied in a FEA, grain-size-dependent strength and plasticity

of polycrystals could be investigated [193,194].

Akin to the volume finite element, CE are characterised by a specific constitutive

law, called “traction-separation law” (TSL), which defines the stress applied to a

cohesive element as a function of the associated separation of the two surfaces.

Various formulations of TSL exist for different applications, most of which

enforce zero strength beyond a critical separation 0 (Figure 4.2b), meaning that

the given element cannot bear any (tensile) load and is considered “broken” or

“opened”. The slope of the initial linear region is the element stiffness, and the

area under the TSL is the cohesive energy 0 , or the energy spent per unit area

of crack extension. For brittle materials this value represents the fracture

toughness, but for predominantly plastic material, such as tungsten at high

temperature, the TSL must be different and include the effect of the plastic zone

ahead of the crack tip in real materials.

In this chapter, the cohesive elements are first used in a rather unconventional

way: they are used as strain gauges in order to measure the predicted stress

values exactly along the GBs. For this, they are assigned a very large stiffness

(much larger than the elastic stiffness of tungsten) and an unattainable strength,

so that the opening remains infinitesimal and reversible. In the last section of the

chapter, CE are used while allowing progressive damage in order to predict the

interplay of intra-granular and inter-granular cracking in tungsten as reported

experimentally in [40].

a) b)

Figure 4.2. a) A diagram illustrating the use of cohesive elements in a crack propagation problem. b) Example of a traction-separation law, normalised by the

maximal cohesive stress 0T and maximal cohesive separation 0 [197].


109

4.2.3 Analysis of mesh sensitivity

Before we can simulate thermal shock tests in the CPFEM framework, we should

identify the mesh features: element type, order, mesh density. To do so, a

benchmark of uniaxial tension of a polycrystal composed of 3×3×3 cubic grains

has been performed, using the crystal plasticity UMAT as the material law.

a)

b)

c)

d)

e)

f)

Figure 4.3. Maximal principal stress distribution in “bricks” with different level of refinement: coarse (a, b), medium (c, d) fine (e, f), and element type: C3D8 in “simple bricks” (a, c, e) and C3D10 in “Gmsh-generated bricks” (b, d, f)

The grains have been meshed assuming either a linear or a quadratic

displacement filed inside each element. Hexahedral elements of the 1st order

(referred to as C3D8 in Abaqus and “simple bricks” here) were generated

manually. Then, using the Gmsh software [138], the same polycrystal was

meshed again using tetrahedral elements of the 2nd order (referred to as C3D10


110

in Abaqus and “Gmsh-generated bricks” here). It is known that the use of

tetrahedral elements of the 1st order should be avoided under large plastic

strains, as they possess unphysically high stiffness when incompressibility is

approached [75]

Three levels of mesh density were selected for both types of meshes:

coarse, around 2200…2500 nodes

medium, 11000…12000 nodes

fine, 22000…26000 nodes

The polycrystal composed of randomly-oriented grains was loaded under

uniaxial tension in the X direction at 400 °C and 6·10-4 s-1 up to 30% of strain.

Periodic boundary conditions were enforced so that grains on the edge behave

as if they were embedded in the bulk of a polycrystal.

Figure 4.3 collects the images of the deformed 3×3×3 polycrystals at the end of

deformation, where the colour maps the distribution of the maximal principal

stress. Notably, the global shape change is always the same and the pattern of

typical maxima and minima of stress is repeated in both mesh types, and

especially in the simple bricks.

It would be meaningful to compare the whole distribution of stress in the grain

interior and in cohesive elements, to characterize intergranular and

intragranular heterogeneity. As the crack formation is defined by the maximal

tensile stress value, analysis of stress distribution is important from the point of

view of damage accumulation assessment and is used in the further text. An

excessive scatter of the values, being unphysical, could indicate an inadequate

mesh.

Figure 4.4. A boxplot shows schematically five main reference points of a smooth distribution of stress in cohesive elements.


111

We use so-called “boxplots” in this chapter to easily visualize whole distributions

of a quantity in question (stress in GBs in this case). The distribution is

characterized by five most important reference points, facilitating comparison of

several distributions in one figure.

As Figure 4.4 illustrates, the filled boxes represent the middle 50% of all the

stress values, taken between the first and the third quartiles of the distribution

(denoted in the figure as Q25 and Q75 correspondingly). Contrary to usual

convention, we use the thin vertical “whiskers” to highlight the range between

the 5th and 25th percentiles (the lower whisker) and between the 75th and 95th

percentiles (the upper whisker). A horizontal black bar in the middle of the box

corresponds to the distribution median, which is the second quartile or the 50th

percentile (Q50).

To characterize the stress state in several meshes quantitatively, distributions of

normal stress in the cohesive zones are plotted in Figure 4.5 with open boxes

grouped by their orientation with respect to the coordinate axes. Due to the

geometry simplicity, the cohesive elements can be split into three groups, normal

to one of the three coordinate axes (X, Y, Z), denoted as CZX, CZY, CZZ

respectively. Filled boxes show the distribution of maximal principal stress in the

grain interior, measured in 13 out of 27 grains (to reduce the number of boxes in

one plot). The boxplots are grouped by the element size, from coarse to fine.

The plot provides the following evidence. Distribution of stress in cohesive

elements is very wide, spanning the range beyond 1 GPa in both compression

and tension. Even though the median of stress distribution in CZX is higher than

in other groups (implying predominantly tensile nature of stress in CZX), all the

distributions overlap significantly, and this effect does not diminish with the

mesh refinement.

Both intragranular (the range of individual filled boxes in Figure 4.5a) and

intergranular heterogeneity (the difference between neighbouring filled boxes)

of maximal principal stress in the grain interior of simple bricks is much lower

than the range of distribution of stress in CZ and looks reasonable. This indicates

that linear hexahedral elements might be suitable for simulations with the help

of CPFEM, but not together with cohesive elements, which demonstrated

excessively wide range of stress, compared to that in the 3D elements.

A similar analysis of stress state in the Gmsh-generated bricks (Figure 4.5b)

shows much narrower distribution of normal stress in the cohesive elements.

Here the central 50% of stress recorded in CZX and the other two groups do not

overlap. Besides, the stress distribution in CZY and CZZ is symmetric with

respect to zero, in line with the uniaxial stress state of the polycrystal.


112

a)

b)

Figure 4.5. Distribution of normal stress in CZ (open boxes) and maximal principal stress in grain interior (filled boxes) in a) “simple bricks” and b) “Gmsh-generated bricks” with different mesh refinement at 30% of tensile strain. Each colour of the filled boxes corresponds to one of the 13 grains.


113

The density of Gmsh-generated mesh affects the intragranular heterogeneity of

stress in the grain interior, which is high in the coarse mesh, compared to the

medium and the fine ones, and slightly higher compared to that in the simple

bricks. Also, just like in the case of simple bricks, the stress in the grain interior

is higher than in the CZ normal to the tensile axis.

As mentioned above, we first used cohesive elements with unrealistic stiffness

(million times higher than the Young modulus) to serve as “stress gauges” only.

Can the choice of element stiffness play a role in the appearance of the gap

between stress in cohesive elements and the grain interior, observed in Figure

4.5a-b? To check that, the same Gmsh-generated brick with the medium mesh

density was loaded in tension, while the stiffness of cohesive elements was

reduced to 400 GPa, close to the value of the Young modulus of tungsten.

Distribution of stress in cohesive elements and in grains turned out to be almost

identical, irrespective of the stiffness, as shown in Figure 4.6.

Figure 4.6. Effect of cohesive element stiffness on the stress distribution in cohesive elements and grain interior of a Gmsh-generated brick with medium-size elements. Each colour of the filled boxes corresponds to one of the 27 grains.

Thus, higher stress in the grain interior compared to the GBs is a feature of

models of this type, and is not an undesirable artefact. Note that the maximal

principal stress in the grain interior was compared to the stress normal to

cohesive elements, and these quantities are not necessary measured in the same

coordinate system. As Figure 4.3 shows, the cohesive elements rotate with

deformation, and the norm to their surface deviates from the tensile direction.


114

Consequently, the stress state in rotated cohesive element changes: the shear

component of stress appears, and the normal stress component reduces.

To conclude, the mesh with tetrahedral 2nd order elements shows better

performance, making this element type favourable for the consequent CPFEM

simulations. It avoids unphysically wide distribution of stress in cohesive

elements even despite the fact that cohesive elements of 1st order were used with

both 3D element types. Assigning high stiffness to cohesive elements does not

have adverse effect on the stress heterogeneity neither in the grain interior, nor

in the cohesive elements themselves, but suppresses their opening, undesirable

when the CZ are used as stress gauges at GBs. Besides, to avoid symmetric and

ordered structure of GBs, which can bias the distribution of normal stress in

cohesive elements, polycrystals with polyhedral grains obtained with the help of

the Voronoï tessellation, are preferred to those with cubic grains (bricks).

4.2.4 Mesh of polycrystals used in the work

Periodic finite element meshes which conform to GBs in a polycrystalline

aggregate made of periodic Voronoï cells were generated with an automatic

method [187]. The algorithm allows the user to set up the grain number, the

average aspect ratio in the polycrystal, and the crystallographic orientation of

every grain. Cohesive elements were placed along all the GBs to serve as stress

gauges.

a) b) c)

Figure 4.7. The finite element meshes used in the simulations of thermal fatigue with CPFEM, a) transversal with respect to the heat flux, b) longitudinal, c) equiaxed.

In accordance with the experimentally obtained EBSD grain orientation maps of

tungsten in both as-received and recrystallized conditions, two FE meshes were

created with, respectively, elongated and equiaxed grains. The aspect ratio of the

elongated grains in the longitudinal, normal and transverse directions (“L”, “N”

and “T”) was chosen equal to 4:2:1. To comply with the thermal shock

experiments, where the tungsten samples were oriented either in the

longitudinal or in the transverse direction with respect to the heated surface

(that is, the elongated grains were parallel, or perpendicular to the heated


115

surface), the mesh with elongated grains was oriented in the two possible

directions, too, as shown in Figure 4.7.

The meshes in the present simulation consisted of eight grains for both equiaxed

and elongated polycrystals, tessellated into approximately 10 000 2nd order

tetrahedral elements and the same number of cohesive elements, connecting the

neighbouring grains. It is to be noted that the mesh with elongated grains could

not be obtained by mere stretching the mesh with equiaxed grains, otherwise the

initially distorted elongated finite elements would cause numerical issues during

simulations. It means that in order to obtain a mesh with elongated grains, the

mesh with equiaxed grains was constructed first, then only the surfaces defining

the GBs were stretched, and both the GBs and the grain interior were re-meshed

with equiaxed finite elements.

Despite the fact that the elongated grains are a feature of the as-received

tungsten, and the equiaxed grains are a result of recrystallization, both mesh

types were probed in simulations with constitutive laws for the as-received and

as-recrystallized tungsten in order to single out the effect of grain shape alone.

4.3 Analysis of fatigue damage in recrystallized

tungsten using cohesive elements

The thermomechanical effect of the exposure to thermal shocks was simulated

at the microscopic level in three polycrystalline meshes shown in Figure 4.7. The

aggregate of equiaxed grains was also oriented in two different directions with

respect to the heated surface (labelled as “X” and “Z” orientation). Even though

the two orientations of the mesh with equiaxed grains are identical from the

physical point of view, such procedure allows one to estimate the sensitivity of

the output to the grain arrangement and the mesh variation. For the same reason

a random crystallographic texture was assigned to all polycrystals.

Similar to the analysis done in Chapter 3, the polycrystals were restrained in all

directions, except the normal to the heated surface, thus enforcing plane stress

conditions. The temperature evolved following a triangular waveform with 1 ms

of heating and 1 ms of cooling to mimic a thermal shock. The build-up of

intergranular stresses was predicted with the help of the CP model, while

accounting for the elastic anisotropy of tungsten.


116

4.3.1 Results

Since CPFEM analysis is time-consuming, four thermal pulses were simulated for

every test condition and polycrystal. First quantities recorded in the simulations

were the in-plane stress and the out-of-plane strain averaged over the whole

polycrystal. The stress-strain loops obtained in the thermal shock simulations

and demonstrated in Figure 4.8 and Figure 4.9 for equiaxed and elongated grains

correspondingly, generally repeat the patterns observed in the corresponding

standalone simulations in Chapter 3.

Comparing Figure 4.8 and Figure 4.9, no effect of the selected mesh and its

orientation with respect to the exposed surface on the average mechanical

response of recrystallized tungsten can be found.

Figure 4.8. Stress-strain loops as a result of simulation of thermal shock on the surface of the recrystallized tungsten (equiaxed grains).


117

Figure 4.9. Stress-strain loops as a result of simulation of thermal shock on the surface of the recrystallized tungsten (elongated grains).

The distributions of normal stresses in cohesive elements are shown in Figure

4.10 and Figure 4.11 for the equiaxed and elongated grains correspondingly. The

panels of the figures correspond to various base temperature and thermal flux.

Each panel shows two thermal pulses which consisted of 1 ms heating and 1 ms

cooling each, in which compressive stresses dominate in the heating phase, and

tensile stresses are observed in the cooling phase.

These figures give an overview of the trends in the mechanical response of

tungsten to thermal shocks. Similar to the stress-strain loops, the stress in GBs

decreases with the base temperature, as material becomes softer. The median

tensile stress at the end of cooling is higher than the median compressive stress

at the end of heating, due to the asymmetry of the stress-strain loops shown in

Figure 4.8 and Figure 4.9.


118

Figure 4.10. The evolution of normal stress distribution in the GBs of polycrystal with equiaxed grains in the first two thermal shocks.

The choice of orientation (“X” or “Z”) of the equiaxed grains does not influence

significantly the distribution of normal stresses in GBs, confirming the model

stability to variations of grain arrangement. However, the choice of orientation

of the elongated grains with respect to the exposed surface impacts the stress

state drastically (Figure 4.11). Stresses in the boundaries of grains in the “T”

orientation (perpendicular to the exposed surface) are much larger in tension as

well as in compression. Hence, the majority of cohesive elements experiences

tensile stress and are prone to the intergranular crack formation and

propagation. Thus, the microscopic simulation results conform with the

experimentally observed premature cracking of samples with transversal

orientation of grains [26,71].


119

Figure 4.11. The evolution of normal stress distribution in the GBs of polycrystal with elongated grains in the first three thermal shocks.

The distribution of shear stresses in the GBs was assessed as well. Figure 4.12

shows its evolution in time for the elongated grains. The shear stress is not

influenced by the choice of the polycrystal orientation and shows the same

trends with temperature and thermal power density as the normal stresses.

Finally, to compare the mechanical response of the material with equiaxed and

elongated grains, the distribution of normal stresses in GBs were analysed in the

first two thermal pulses to make the boxplots clearer. Figure 4.13 reveals that

tensile stresses developed in the boundaries of equiaxed grains are high and

compared to those in the elongated grains in the “T” orientation. This can be the

main reason why cracks are formed early in tungsten with equiaxed grains, as

observed in experiments.


120

Figure 4.12. The evolution of shear stress distribution in the GBs of polycrystal with elongated grains in the first two thermal shocks.


121

a)

b)

Figure 4.13. The evolution of normal stresses in the first two thermal pulses in recrystallized tungsten exposed to 380 MW m-2 thermal power density at base temperature 27 °C. a) Elongated grains, b) equiaxed grains.


122

4.3.2 A note on the stress in cohesive elements

The first impression is that the simulations confirm that cracks will appear in the

transversal orientation sooner than in the longitudinal orientation, as observed

by experiments [71]. However, the difference in normal stresses in these two

polycrystals can be purely geometrical, and, thus, should be treated with care.

Figure 4.14. Distribution of the angle between cohesive elements and the heated surface in the three considered meshes.

Figure 4.15. Correlation between normal stress in cohesive elements and their orientation.


123

To check that, the distribution of the angle between a cohesive element and the

heated surface is plotted in Figure 4.14. In the transversal orientation the

majority of cohesive elements are perpendicular to the heated surface

(maximum at 90°), while in the longitudinal orientation most of them are

misaligned to it by a small angle (maximum at 30°).

Moreover, normal stress in cohesive elements correlates with the angle between

cohesive elements and the heated surface, as illustrated in Figure 4.15, that shifts

the distribution of stresses plotted in Figure 4.11 and Figure 4.13 to higher

values for the transversally-oriented polycrystals.

However, the predominant orientation of GBs can indeed be the real origin of

sooner crack formation in the transversal samples. An alternative approach to

characterise the possibility of cracking under thermal shocks is considered in the

next section for the as-received tungsten.

4.3.3 Conclusions of the section

Experimentally, commercial grades of tungsten in the transversal orientation

show early fracture compared to the longitudinal orientation in thermal shock

tests [26,198], tensile tests [57] and fracture toughness tests [45,199].

The simulation of thermal shock tests performed in the present work, using an

approach that links the macro- and microscopic levels, demonstrates that this

difference can be explained by statistically higher tensile stresses in GBs in the

transversal orientation during the cooling phase (immediately after every

thermal pulse). As the maximal tensile stress increases with the increase of

thermal power density and with the decrease of the base temperature, it explains

the existence of the cracking threshold derived from the thermal shock

experiments, i.e. the minimal thermal power density at which cracks appear, as

a function of the base temperature.

This conclusion, however, could have been biased due to purely geometrical

reasons: the majority of cohesive elements in the simulations with longitudinal

samples are oriented in the stress-free direction, i.e. they do not experience

normal stresses, which drastically shifts the distribution to lower values. On the

other hand, the difference in the geometrical arrangement of grains can indeed

be the main reason of different strength of the material in transversal

orientation. An alternative approach to characterize development of damage in

thermo-mechanically loaded material in various orientations is discussed in the

next section devoted to the as-received tungsten.


124

4.4 Analysis of fatigue damage in as-received

tungsten using the fatigue indicator

Having finalised the analysis of thermomechanical response of recrystallized

tungsten to thermal shocks, here we estimate the accumulation of fatigue

damage under thermal shocks in the as-received IGP tungsten. Different grain

shape and their orientation with respect to the heated surface are considered in

the same manner as for the recrystallized tungsten. In contrast with the previous

section, an FIP was selected as an alternative way characterising the fracture

probability, instead of measuring the normal stress in cohesive elements.

Besides, as long as a commercial grade of tungsten demonstrated the

Bauschinger effect associated with kinematic hardening [59], we investigate its

effect on the damage accumulation.

Similar to the simulations of the recrystallized tungsten, the polycrystal was

restrained in two directions, and was free to deform in the third direction,

ensuring plane stress state. The temperature of the whole polycrystal evolved

following triangular waveform. Three values of the base temperature were used:

27 °C, 400 °C or 1000 °C, and the temperature pulse height was either 350 °C or

700 °C, corresponding to thermal power density 190 MW m-2 or 380 MW m-2 in

the experiments performed by the team of Wirtz [26]. Ten thermal cycles were

simulated to identify trends in FIP. In this series of simulations only two

hardening cases were considered: purely isotropic and mixed, as explained in

Section Discussion of Chapter 3.

4.4.1 Results

The average in-plane stress and out-of-plane strain in the polycrystal collected

in Figure 4.16, grouped by the thermal shock parameters, basically reproduce

the results obtained in the mean-field CP simulations in Chapter 3, showing that

deformation is mostly elastic at low power density 190 MW m-2. At high power

density 380 MW m-2, the plastic strain dominates, the presence of kinematic

hardening leads to a pronounced Bauschinger effect. The stress-strain loops for

the as-received tungsten have higher symmetry compared to the as-

recrystallized grade shown in Figure 4.8.


125

Figure 4.16. The average in-plane stress and out-of-plane strain in the polycrystal subjected to thermal shocks at different base temperature and absorbed power density.

The evolution of FIP averaged over the whole polycrystal during the ten thermal

shocks is shown in Figure 4.17. When kinematic hardening is active, the FIP

grows faster, as the stress at GBs is magnified due to the cumulative effect of

dislocation pile-ups. Hence, the eventual crack formation is delayed in the case

of pure isotropic hardening.

Damage is accumulated very slowly in samples exposed to the power density

190 MW m-

2, and we focus on simulations with the power density 380 MW m-2

in the further text, covering the case of significant plastic deformation and low

cycle fatigue. Little effect of the base temperature is seen, however.


126

Figure 4.17. The average FIP in the whole polycrystal with the L orientation tested at different base temperature and applied power density.

Attention must also be paid to the microscopic spatial heterogeneity of damage.

The distribution of FIP in two selected grains after ten thermal shocks at

380 MW m-2 is shown in Figure 4.18. Here the distribution is shown in the form

of boxplot, like in the section devoted to the recrystallized tungsten, but in the

traditional way, where the whiskers span the following range: 1.5 times the filled

box height, measured from the first quartile down and from the third quartile up.

The values outside of this range are considered “outliers” and are shown with

dots or small circles.

Figure 4.18 reveals that FIP has a lower median in the case of isotropic

hardening, in line with Figure 4.17, but it is difficult to make a conclusion

regarding the effect of kinematic hardening on the range of FIP. Equiaxed grain

1 shows the range of FIP comparable to that of elongated grain 1, but FIP in grain

4 is much larger with numerous outliers.


127

Figure 4.18. Distribution of microscopic FIP in two selected grains at the end of ten thermal shocks of 380 MW m-2.

It makes sense to check the distribution of FIP in all grains, hence we plotted it

in Figure 4.19 for the polycrystal with mixed hardening exposed to 380 MW m-2

at 1000 °C. Apparently, FIP is not affected by the orientation of the elongated

grains with respect to the heated surface. However, it is sensitive to the grain

shape, as wider boxes and significantly longer distribution tails are observed in

the equiaxed grains, showing that they are more prone to crack formation

(determined by the maximal attained value of FI), compared to elongated grains.


128

Figure 4.19. FIP distribution in individual grains after ten thermal shocks with power density 380 MW m-2 at base temperature 1000 °C in polycrystal with mixed hardening.

Even though we focus on the analysis of FIP in polycrystals exposed to thermal

shocks, distribution of normal stress in cohesive elements was also considered,

similarly to the recrystallized tungsten. The evolution of this distribution in

polycrystals in the first two thermal shocks of 380 MW m-2 at base temperature

1000 °С is illustrated in Figure 4.20 with boxplots for the material with mixed

hardening. Like for the recrystallized W, the highest stress is observed after

cooling of the transversal samples, followed by equiaxed and longitudinal ones.

Figure 4.20. Distribution of normal stress in the cohesive elements within the first two thermal shocks of 380 MW m-2 at base temperature 1000 °C applied to the material with the mixed hardening.


129

Figure 4.21. Distribution of normal stress in cohesive zones (open boxes) and maximal principal stress in grain interior (filled boxes) in three meshes of the polycrystal with mixed hardening after exposure to ten thermal shocks at base temperature 1000 °C and power density 380 MW m-2.

Besides, stress in CZ of the same mesh was also compared with the maximal

principal stress in the grain interior, revealing the same gap between their

average values, as in the preliminary simulations in Section “Analysis of mesh

sensitivity”.

The applicability of FIP to the understanding of damage accumulation has been

explored. It was found to be almost unaffected by the material microstructure

within the framework of CP (contrary to the normal stress in cohesive elements,

considered in the simulations of as-recrystallized tungsten). The presence of

kinematic hardening in the as-received tungsten was shown to increase the

accumulation of fatigue damage at GBs, indicating shorter fatigue life of divertor

components in this case.

4.5 The effect of crystalline anisotropy

on grain delamination

As a side problem, we use the CPFEM framework to analyse the propagation of

crack in a thin tungsten foil. As mentioned in Chapter 1, the typical fracture mode

of tungsten at intermediate temperature is intergranular cracking. It was

especially pronounced in experiments on fracture toughness of thin foils [40],

where the initial notch was perpendicular to thin, pancake-like grains, as shown

in Figure 4.22a. Being loaded in mode I, the foil experienced delamination of thin

grains (Figure 4.22b), indicating two main characteristics of this test:


130

i) GBs were weaker than the material in the transgranular plane ahead

of the notch,

ii) plane strain state switched to the plane stress, once constraints

were lost due to delamination. The change of loading state induced

relatively high fracture toughness.

a) b)

c) d)

Figure 4.22. a) EBSD maps of pure 0.1 mm tungsten foil projected on a cuboid to better visualize the pancake-like microstructure. Black lines denote the initial notch [40]. b) An example of the foil delamination occurred at 400 °C [40]. c) A polycrystal composed of prismatic grains to represent the microstructure of the foil in CPFEM simulations. Colour highlights separate grains. d) Embedded cohesive elements which have different strength, according to their role: GBs (green) are weaker than the transgranular crack plane (highlighted with red).


131

We tested the CPFEM framework trying to reproduce the key features of the cited

experiment. A simplified finite element mesh was created to represent three

layers of flat grains with periodic boundary conditions in all three directions.

Figure 4.22c shows the six grains with different colours. The crystal orientation

of the two grains in blue corresponded to the rotated-cube component

{001}<110>, which was dominant in the experimental texture. The other four

grains were assigned orientations representative of a BCC cold rolling texture

({111}<123>, {113}<110>, {111}<110> and {111}<112>).

Cohesive elements were embedded in the model to represent the crack path

ahead of the notch (the vertical plane in Figure 4.22d), with a small initial notch,

obtained by removing a narrow strip of cohesive elements at its front. GBs were

represented by cohesive elements as well (horizontal planes in Figure 4.22d),

whose strength was set lower than that of the transgranular crack.

The mesh was loaded in mode I, i.e. in the direction Y up to 2% of strain, and was

restrained in the direction Z to achieve plane strain condition. The strength of

GBs varied in a wide range, and that of the crack path was fixed at 950 MPa. The

goal of the simulation was to find out the dependency of delamination on GB

strength and whether delamination is promoted by the crystalline anisotropy.

To that end simulations were launched in pairs — either using the developed CP

model, or assuming the J2 plasticity, i.e. without consideration of crystalline

anisotropy.

Delamination of grains can be assessed by the value of normal strain in the

cohesive elements representing GBs. Colour maps of a few indicative simulations

grouped in Figure 4.23 confirm that if GBs are weak (50 MPa and 250 MPa),

delamination takes place in the CP simulations despite the fact that they were

parallel to the loading direction. The value of normal strain in the GBs became

comparable to the applied strain, almost 0.02 locally. The strain across strong GB

is very low (Figure 4.23f) and no delamination takes place.

Note that delamination was not observed in the J2 simulations even with the

weakest GBs. Instead, the strain accumulates in a thin strip at the crack tip in the

cohesive elements forming the crack plane, which were not able to open thanks

to higher strength. A few simulations with lower strength of the crack plane were

numerically unstable and halted before the sample reached 2% of strain. The

uneven distribution of normal strain in the weak GB in the CP simulations

indicates that local concentration of stress (due to the grain anisotropy)

promoted opening of a few cohesive elements locally, which was subsequently

spread further in the GB.


132

a) [GB] = 50 MPa; J2 b) [GB] = 50 MPa; CP

c) [GB] = 250 MPa; J2 d) [GB] = 250 MPa; CP

e) [GB] = 550 MPa; J2 f) [GB] = 550 MPa; CP

Figure 4.23. Maps of normal strain inside cohesive elements (LE33) in simulations with the J2 (a, c, e) and CP (b, d, f) models. The strength of GBs increases from top to bottom: 50 MPa (a, b), 250 MPa (c, d) and 550 MPa (e, f).


133

On top of that, to check whether the plane stress condition was achieved due to

delamination, the distribution of the out-of-plane stress (S33) in the 3D elements

of the grain interior was recorded at the end of deformation. The boxplot of the

distributions shown in Figure 4.24 provides the following insights:

A sudden drop of the out-of-plane stress takes place in the CP

simulations at the GB strength between 250 MPa and 350 MPa. The

distribution becomes symmetric with respect to zero, being considered

a clear indicator of achieving plane stress due to crystal anisotropy.

On the contrary, the decrease of the GB strength in the J2 simulations

gradually shifts the distribution of stress towards zero, keeping it

essentially tensile, even for the weakest GBs.

The scatter of stress is much lower in the J2 simulations, which is

expected in the homogeneous, isotropic material.

The difference between median stress observed at high GB strength can

be explained by low number of grains involved, as their mechanical

response is expected to differ from that of an RVE (around 100 grains)

which provided the constitutive law for the J2 simulations.

Figure 4.24. Distribution of the out-of-plane stress S33 in the grain interior in simulations with different GB strength.

The series of simulations clearly show the difference between J2 and CP models

at the grain level and the necessity of the latter when grain anisotropy is

essential. In the considered example the presence of weak GBs alone was not

sufficient to promote delamination of grains parallel to the loading direction.

However, when the crystal anisotropy was taken into account with the help of


134

the CP model, heterogeneous distribution of stress in grains built up in the

beginning of plastic deformation caused high stress in GBs locally, initiating the

delamination. This justifies the use of the CP model in the analysis of damage

accumulation at the grain level in tungsten exposed to thermal shocks, discussed

further.

4.6 Discussion

4.6.1 Taylor-based CP and CPFEM simulations of thermal shock

Chapters 3 and 4 are devoted to the mechanical response of IGP tungsten

subjected to ITER-relevant thermal shocks. The Taylor-based CP simulations in

Chapter 3 considered a “virtual” polycrystal where every grain deformed in the

same manner as the whole polycrystal, and development of intragranular stress

heterogeneity was not possible. Simulations of this type are fast, can predict the

stress-strain response and the evolution of crystallographic texture, but they

assume that every state variable is distributed uniformly in every grain. These

simulations were performed first of all to estimate the effect of the chosen

hardening model (isotropic or kinematic) on the shape of stress-strain loops, i.e.

the typical outcome of fatigue experiments. Without consideration of

heterogeneous distribution of stress, it was not possible to properly estimate

damage accumulation.

The framework of CPFEM was utilized in this chapter to go to a finer level, and

to take account of heterogeneous distributions of stress, strain, dislocation

density and other variables within grains, which occur in real polycrystals where

conditions of both strain compatibility and stress equilibrium have to be fulfilled

at GBs, thus avoiding unrealistic overlaps and gaps at GBs.

A microscopic quantity, an FIP, equal to the density of dissipated energy was

chosen to assess the probability of material cracking in the simulations with the

help of CPFEM, reported in this chapter. Its more rapid accumulation in case of

kinematic hardening indicates sooner reaching the threshold and lower fatigue

life of material with high share of kinematic hardening.

4.6.2 Cohesive elements and fatigue indicator

Along with the two approaches to characterise the impact of kinematic

hardening on the material resistance to thermal shocks, the influence of grain

shape was also assessed in two ways. Both of them were applied in the CPFEM

simulations, where the grain shape can be defined explicitly.

The distribution of FIP turned out to be almost unaffected by the grain shape and

their orientation with respect to thermal flux. However, experiments on thermal

shocks confirmed that cracks appear sooner in the samples with the transversal

orientation, i.e. when the grains are perpendicular to the heated surface. It

appeared that the heterogeneous distributions of stress and strain in grains


135

alone (which was taken into account in the CPFEM simulations) did not affect the

tendency to accumulate fatigue damage, and, consequently, could not predict the

difference in fracture of material with different grain orientation.

On the contrary, the simulations incorporating cohesive elements showed that

the normal stress at GBs (chosen as an indicator of the crack formation under

thermal shocks) significantly depended on the grain orientation with respect to

the heat flux. Initially considered as a purely geometrical effect, and even an

undesired artefact, which forced us to find another criterion for the crack

initiation, the difference in normal stresses at GBs appeared to be the actual

cause of the sooner formation of cracks in the tungsten with transversal

orientation of grains.

Thus, the conclusion can be made that the crack formation in tungsten exposed

to thermal shocks is controlled by the strength of GBs, rather than the strain

heterogeneity within a grain. And indeed, unfavourable orientation of grains, i.e.

unfavourable orientation of GBs, when most of them are perpendicular to the

heated surface and experience normal stress during the cooling phase of thermal

pulses, leads to much sooner material failure under plasma instabilities. Then, to

be able to predict the fatigue life of tungsten under thermal shocks, the TSL at

the grain level should be adjusted based on experimental data, introducing

critical stress and strain. Further enhancing the model, the TSL can be made

dependent on the value of FIP in a nearest 3D element, rendering the model

nonlocal.

4.6.3 The use of tensile tests to derive the threshold fatigue

indicator

We can also propose a way to estimate the number of thermal cycles, that the

material can sustain before cracking. It should be possible to estimate the

threshold FIP based on the energy density dissipated in tensile tests. Its

dependency on the tungsten grade and grain orientation can be taken into

account. An example of dissipated energy density for IGP tungsten is shown in

Figure 4.25. It was calculated as the area under true stress-strain curves

obtained in Chapter 2, in the range of strain limited by the reduction of sample’s

cross-section area measured at fracture experimentally. The scatter of individual

points in Figure 4.25 gives an idea of the uncertainty of the threshold FIP.

Taking this value as an FIP threshold, and relying on the rate of increase of FIP

obtained in simulations, we could estimate that IGP tungsten is able to sustain

~100–200 thermal shocks 1 ms long at 380 MW m-2 without cracking at the

grain level. This is a preliminary upper boundary, since the information on the

threshold dissipated energy is not available in the temperature range of thermal

shock tests, where it is expected to be lower. Experimentally, small cracks were

observed in IGP tungsten exposed to 100 thermal shocks at base temperature

above 400 °C, and crack network appeared on the surface of tungsten exposed at


136

room base temperature [26], where the threshold FIP tends to decrease

according to Figure 4.25.

Figure 4.25. The density of energy dissipated in tensile tests of the as-received IGP tungsten.

4.7 Conclusion

The crystal plasticity model developed in Chapter 3 was implemented in the

CPFEM framework in the present chapter, driven by the interest in intragranular

heterogeneity of stress in deformed material, which is considered responsible

for crack initiation in cyclically loaded material. The sensitivity analysis has been

performed, identifying the best choice of element type and mesh density,

necessary for simulations of thermal shocks, after which the CPFEM model was

benchmarked in uniaxial tension. The advantage of CPFEM model —

consideration of the grain anisotropy — was proven useful at the microscopic

level, where the two models (CP and J2) were compared in a FE analysis to

reproduce experimentally observed delamination of thin tungsten foil, loaded in

the direction parallel to the plane of grains. Delamination appeared only in the

CPFEM analysis, due to concentration of stress at GBs caused by the crystal

anisotropy.

In simulations of thermal shocks, we have found that the tungsten surface

exhibited different mechanical response depending on the proportion between

kinematic and isotropic hardening mechanisms. The presence of kinematic

hardening resulted in the accelerated growth of FIP in the material, thus, the

ability of material (under given loading conditions) to exhibit kinematic

hardening should have negative effect with respect to the accumulation of stress

concentration and eventual crack formation under cyclic thermal shocks.


137

As-received IGP tungsten exposed to thermal shocks 1 ms long with power

density 190 MW m-2 experienced elastic shakedown, and its lifetime should be

assessed from the point of view of high cycle fatigue. Whereas exposure to higher

power density 380 MW m-

2 clearly caused cyclic plastic strain.

The effect of orientation of elongated grains with respect to the heated surface

was not captured by the FIP, but was predicted by another characteristic of

fatigue damage — the normal stress at GBs, which nevertheless should be taken

with care due to geometrical considerations.

Consideration of kinematic hardening contribution is shown to be essential in

the FEA of stress state in tungsten, since the accumulation of dislocation pile-ups

at GBs reduces the lifetime of plasma-facing components under conditions of

ITER. The model with mixed hardening will suit the best to this end. The

proposed method provides a qualitative measure of the impact of hardening

modes, while rigorous calculations would require dedicated experimental data

on the actual value of kinematic hardening for the studied material. In addition,

the performed analysis helps to rationalize the conditions for the appearance of

surface roughness caused by the residual plastic deformation induced by ELM

pulses.

A possible link with the energy dissipated in tensile tests is discussed, giving a

preliminary estimate of the endurance of the as-received IGP tungsten as 100–

200 ELM thermal shocks before crack formation, which agrees with the

experiment by the order of magnitude.

139

Chapter 5.

Conclusions and perspectives

5.1 Summary and conclusions

Plasma-facing components in ITER — the largest fusion device under

construction nowadays — will have to withstand severe operational conditions:

high-amplitude oscillations of temperature due to plasma instabilities (ELMs)

and high-energy neutron flux. Appearance of surface cracks in plasma-facing

components made of tungsten is considered an issue from the point of view of

stable operation of ITER, thus the qualification of tungsten has to be performed

by numerous experiments, including thermal shock tests.

In these tests samples of tungsten are exposed to short pulses of particle flux

(such as electrons) depositing power density typical for the plasma instabilities,

while the bottom face of the sample is kept at constant temperature. A series of

tests, performed on ITER-specification tungsten and recrystallized tungsten in a

range of base temperature and thermal power density, was reported in literature

[26]. Appearance of cracks, roughness, or no damage at all was observed

depending on the experimental conditions.

In the present work we have set the goal to reproduce the experiments with the

help of finite element simulations, to rationalize the different experimental

observations of the surface modification, and to investigate the mechanisms at

the microscopic scale leading to the damage accumulation in cyclically loaded

tungsten.

The material of research was the ITER-specification tungsten fabricated by

Plansee AG (Austria). According to the specification, this grade of tungsten has

elongated refined grains. On top of that, recrystallized tungsten with increased

grain size and reduced dislocation density was tested as well.

A series of tensile tests of as-received and recrystallized tungsten provided their

mechanical properties, necessary for the planned finite element simulations. A

physically-based model was used to characterize the experimental deformation

curves of tungsten in as-received and recrystallized state, and to extrapolate the

mechanical response outside the temperature- and strain rate ranges used in

tensile tests to the ITER-relevant values. The model by Kocks and Mecking, in

which plasticity is controlled by thermally-activated motion of dislocations, was

used to this end. The typical form of the model was applicable to as-received

5. Conclusions and perspectives

140

tungsten, however, an extra term had to be added to the equations in order to

take account of non-saturating stress and dislocation density observed in the

recrystallized tungsten.

Non-saturating true stress-strain curves were observed in tensile tests of

recrystallized tungsten by other authors as well, where the linear growth of

stress began around 10% of strain. This feature can indicate the presence of

deformation stage IV. Dislocation tangles and cells were observed in

recrystallized tungsten with the help of TEM at 20%–28% of strain, which can

serve as microstructural evidence of stage IV at large strain, but in order to

confirm stage IV at the onset of linear stress growth, new interrupted tensile

tests and TEM investigation would be needed.

A new method has been proposed to obtain true material properties from

experimental curves of material with low uniform elongation and early onset of

deformation instability, like as-received tungsten. The method is based on

inverse finite element analysis and is used to find the parameters of the Kocks-

Mecking model by fitting to a set of experimental curves obtained at different

temperature and strain rate. This approach respected the theoretically defined

temperature- and rate-dependency of material plasticity and resulted in a single

set of parameters, valid in a wide range of temperature and strain rate.

Since the formation of cracks is defined by stress and strain heterogeneity at the

grain level, we have to use a microscopic model in simulations, operating at the

level of grains. Crystal plasticity model was selected to this end, as long as it

properly considers crystal anisotropy — the main cause of stress heterogeneity.

An existing CP model, previously developed at UCLouvain, was improved to

properly take into account athermal stress, typical for BCC materials. Another

significant improvement of the model is the consideration of backstresses, which

play a major role in cyclic deformation, causing, for instance, the Bauschinger

effect due to kinematic hardening.

Three hypothetical material laws with different share of kinematic hardening

were parameterized to represent experimental deformation curves of ITER-

specification tungsten. The model is shown to be applicable at strain rate up to

~10–100 s-1, and temperature up to ~1700 °C, where plasticity is controlled by

thermally-activated dislocation motion, thus, covering the conditions expected

in ITER in plasma-facing components including the surface region. An original

approach to simulate thermal shocks has been proposed, in which firstly

temperature- and strain field is assessed at the macroscopic scale, where the

mesh represents the whole tungsten sample exposed to thermal shocks.

Secondly, the temperature profile and strain path typical for the heated surface

are applied at the microscopic level to a polycrystal in order to find its

mechanical response and identify the influence of the chosen material law.

Full potential of the CP model was unlocked in CPFEM simulations, in which not

only intergranular, but also intragranular heterogeneity of stress and strain can

be captured due to grain anisotropy. The important role of consideration of grain


141

anisotropy in simulations at the microscopic level was proven in a FE analysis of

tungsten foil delamination, observed experimentally [40]. In this study,

delamination was provoked only when the crystalline nature of grains was taken

into account in CPFEM simulations, as opposed to the J2 plasticity model.

The three material laws were used in simulations of thermal shock with the help

of the crystal plasticity model coupled with the standard finite element solver

Abaqus by means of a user-defined material subroutine UMAT. The finite

element meshes represented polycrystals with either elongated or with

equiaxed grains, corresponding to as-received and recrystallized tungsten.

Correlation was found between the sample deformation state characterized in

the simulations and the surface features observed experimentally [26].

Permanent residual plastic deformation was observed in all conditions, which

could explain the appearance of surface roughness in exposed samples.

Dissipated energy density served as an FIP, whose evolution with the number of

thermal shocks was monitored. The rate of its growth provided an idea on the

material resistance to thermal shocks of a given power density. The ability of

tungsten to exhibit kinematic hardening was shown to have a negative effect on

its lifetime under conditions of cyclic thermal load, as it accelerated the growth

of fatigue indicator.

5.2 Perspectives

The ways to estimate the material resistance to thermal shocks, considered in

the present work, have their advantages and drawbacks. The fatigue indicator

can take account of backstresses, whose development is inevitable in

polycrystalline materials subjected to cyclic loading, but it fails to account for the

grain shape and orientation with respect to the thermal flux — the factors

playing an important role in thermal shock experiments. The normal stress at

grain boundaries, measured in cohesive elements placed along them, clearly

predicts a sooner failure of material with the transversal orientation of grains,

but lacks information on the backstresses due to dislocation pile-ups in the

current implementation. Developing an approach where both quantities would

play a role and interact with each other would bring the assessment of the

material resistance to thermal shocks to a more physically-justified level with a

possibility of quantitative characterisation.

It is also important to find an appropriate and physically-sound way to define the

threshold value of fatigue indicator, at which a crack is formed. As a first guess,

it should be based on the total mechanical energy dissipated in a uniaxial tensile

test, as shown in the end of Chapter 4. However, care must be taken when

comparing the energy dissipated in material in complex stress state and at

constantly evolving temperature to the value measured in isothermal uniaxial

test.


142

For both mechanical response and the threshold fatigue indicator the effect of

texture must be taken into account. As-received tensile samples tested in two

perpendicular orientations showed different stress-strain curves at 600 °C

which means that high-temperature mechanical response can also be different.

This effect has not been included in the present model, because as-received

tungsten in the transversal orientation broke in the brittle mode and we did not

have enough experimental data for the plasticity model parameterization.

However, the threshold fatigue indicator should be parameterized in the two

sample orientations in the whole range of temperature. The main factor affecting

the threshold is the difference of grain boundaries at the sample surface which

controls the chance of a critical crack formation which can propagate into the

sample and lead to the sample fracture.

An essential step is to characterize neutron-irradiated tungsten, which will

definitely have different mechanical properties and resistance to thermal shocks.

Tensile tests and thermal shock tests of tungsten irradiated up to 1 dpa are being

performed at the moment of finalization of the thesis and will provide

information for the construction of constitutive law, the plasticity model

parameterization and its validation in simulations of thermal shocks. It is

interesting to rationalize the shift of the threshold of fatigue indicator, or the

number of thermal cycles to crack formation in neutron-irradiated tungsten.

As a side application, the model might be found useful in finite element analysis

of rotating X-ray anodes, if provided with sufficient input in terms of mechanical

test results. The anodes experience similar thermomechanical load and similar

failure modes, including the formation of surface crack network.

The procedure developed to extract true material properties with the help of

inverse finite element analysis will especially be valuable for the irradiated

tungsten, in which the deformation instability occurs at even lower strain than

in non-irradiated material.

143

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