Computational Framework for Fracture of Graphite Bricks in an AGR Core

142 Pages·2017·6.79 MB·English

Kodsi, Costy (2017) Computational framework for fracture of graphite
bricks in an AGR core. PhD thesis.

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Computational Framework for Fracture
of Graphite Bricks in an AGR Core
Costy Kodsi
Submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy
School of Engineering
University of Glasgow
April, 2017
Abstract
Life-extensionofEDFEnergy’sexistingnuclearfleetisbasedonanassumptionofcontinued
safe operation. Potential fracture of graphite bricks in the nuclear reactor core of a power
stationrepresentsanunknownvariableintheequation. Anunderstandingofthenatureofthis
phenomenonandtheimpactonoperationofthepowerstationisdesired. Thisworkprepares
the way for the future study of fracture in graphite bricks in a reactor core subject to dy-
namicexcitation. Methodologytocoupleamulti-bodyfiniteelementcontactcodetoacrack
propagationcodeisthusdeveloped. Threeimportantscientificcontributionshavebeenmade:
(i) An optimisation problem formulated on a smooth manifold to yield the rotation re-
sponsible for infinitesimal rigid body motion. This involves an iterative scheme in the form
ofNewton’smethodthattakesintoaccountthegeometryoftheunderlyingparameterspace.
Therearenoissueswithsingularitiesoradditionalcomputationsineachiterationtoscalethe
solutionontothemanifold.
(ii) An energy consistent crack initiation criterion for brittle material where nucleation is
treated as a sudden and discrete rupture event at the macroscopic level. At the heart of
the criterion is the finite difference form of the energy release rate; an expression for the
characteristic length is derived and the change in total potential energy is obtained from an
asymptotic argument involving the topological derivative. The criterion can predict crack
onset at a sharp or blunt notch. Fracture toughness and material strength are the only input
requirements.
(iii) Algorithms related to the detection of sharp notches in a tetrahedral finite element
mesh and a general computational procedure for evaluation of non-local crack initiation cri-
teria. TheonlytoolintheimplementationofthesealgorithmsisC++11. Thereisnoneedfor
acomplexdatastructurestoringallincidenceinformation. Unorderedassociativecontainers
in the standard library are exploited in the design of these rather efficient algorithms, which
coversurfaceextractionandprovideconnectivityoftheedgesrepresentingasharpnotchtip.
Ameshre-generationroutineforpurposesofrefinementatthesharpnotchtipshasalsobeen
developed.
ii
Contents
Abstract ii
ListofTables viii
ListofFigures ix
ListofFunctions xii
Acknowledgments xiii
Declaration xiv
Nomenclature xv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Computationalmodellingoffracture . . . . . . . . . . . . . . . . . . . . . 3
iii
1.4.1 SOLFEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.2 MoFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.3 Couplingmethodology . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Thesisoutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Mathematicalprerequisitesandnotation . . . . . . . . . . . . . . . . . . . 8
2 RigidBodyMotionMitigation 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Problembackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Costfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Geometryof SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Optimisationonthemanifold SO(3) . . . . . . . . . . . . . . . . . . . . . 18
2.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 CrackInitiation: ANon-LocalEnergyApproach 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Problembackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Linearelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Perturbedproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 V-notchtipstressdistribution . . . . . . . . . . . . . . . . . . . . 29
3.3 Topologicalsensitivityanalysisofthetotalpotentialenergyproblem . . . . 32
3.3.1 Asymptoticexpansion . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.2 Topological-shapesensitivityanalysis . . . . . . . . . . . . . . . . 32
iv
3.3.3 Shapesensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.4 Topologicalderivative . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Fracturecriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Strengthcriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.2 Griffithenergycriterion . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 MinimumSEDcriterion . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.4 ModifiedMcClintockcriterion . . . . . . . . . . . . . . . . . . . . 41
3.4.5 Novozhilov-Seweryncriterion . . . . . . . . . . . . . . . . . . . . 42
3.5 Proposedtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5.1 Discretecrackpropagation . . . . . . . . . . . . . . . . . . . . . . 43
3.5.2 Characteristiclength . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.3 Energychange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5.4 Virtualholeradius . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.5 Fracturecondition . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.2 V-notchedspecimens . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6.3 Circularnotchedspecimens . . . . . . . . . . . . . . . . . . . . . 52
3.6.4 U-notchedspecimens . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 AutomaticSharpNotchandFractureDetection 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
v
4.2 Problembackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 Linearandaffinesubspaces . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.3 Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Sharpnotchtipdetection . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Typedeclarations . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.3 Extractionofmeshboundaryfacets . . . . . . . . . . . . . . . . . 75
4.3.4 Extractionofmeshboundaryedges . . . . . . . . . . . . . . . . . 79
4.3.5 Edgebelongingtosharpnotchtip . . . . . . . . . . . . . . . . . . 81
4.3.6 Edgeconnectivityonasharpnotchtip . . . . . . . . . . . . . . . . 82
4.3.7 Meshre-generation . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Crackonsetprediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.2 ImplementationinFreeFem++ . . . . . . . . . . . . . . . . . . . . 90
4.4.3 DoubleedgeV-notchedspecimenwith2β = 40◦ . . . . . . . . . . 91
4.4.4 L-shapedspecimen . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 SummaryandFutureWork 95
A VectorsandTensors 98
A.1 Vectorandtensoralgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.2 Changeofbasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
vi
A.3 Pseudovectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Gradientanddivergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.5 DivergencetheoremofGauss . . . . . . . . . . . . . . . . . . . . . . . . . 103
B ProofofR(a×b) = Ra×Rb 105
C V-notchStressDistributionAngularFunctions 106
D TopologicalDerivativeProofs 108
D.1 Holeextension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.2 Shapesensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
E CantorPairing 110
F LocalMeshRefinementinTetGen 112
References 116
vii
List of Tables
3.1 Specimen(PMMA)materialproperties. . . . . . . . . . . . . . . . . . . . 49
3.2 Non-localfracturecriteriaparameters. . . . . . . . . . . . . . . . . . . . . 50
3.3 Ratiosofgeneralisedstressintensityfactorstotensileandshearloads. . . . 51
3.4 PredictedpureMode-Ifractureloads. . . . . . . . . . . . . . . . . . . . . 52
3.5 Comparisonofnumericalresults. . . . . . . . . . . . . . . . . . . . . . . . 59
viii
List of Figures
1.1 (a)Fuelandinterstitialgraphitebricksand(b)inter-connectivitysystem. . . 2
1.2 Simplifiedgraphitecoresubjecttoseismicexcitation. . . . . . . . . . . . . 4
1.3 Crackpropagationingraphitebrickexample. . . . . . . . . . . . . . . . . 6
2.1 Testmodels: (a)cube;(b)quarterpartofagear;and(c)brain. . . . . . . . 22
2.2 Example of a cube subject to (a) pure rotation with v = (0,0,1)T and θ =
0.2094,(b)purestretchwith b = 0.1,and(c)pureshearwith c = 0.1. . . . 22
2.3 Rateoflocalconvergence. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Meanerrorfor(a)purestretchand(b)pureshear. . . . . . . . . . . . . . . 25
3.1 Perturbeddomain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Polarcoordinatesystemcentredatthetipofasharpnotch. . . . . . . . . . 29
3.3 Modesofdeformation(fromlefttoright): (i)Mode-I;(ii)Mode-II;and(iii)
Mode-III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Variationofexponents η and η withthewedgeangle. . . . . . . . . . . . 31
I II
3.5 Orthonormalcoordinatesystem (t,n) definedontheholeboundary ∂ω . . . 35
ξ
3.6 Acircularholeinaninfiniteplatesubjecttoremotestresses. . . . . . . . . 36
ix