Lattice-Modelling of Nuclear Graphite for Improved

Lattice-Modelling of Nuclear Graphite for

Improved Understanding of Fracture Processes

A thesis submitted to The University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

By

Craig N Morrison

2015

School of Mechanical, Aerospace and Civil Engineering

Contents

I Introduction and Background 16

1 Nuclear graphite 17

1.1 Introduction to graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 The use of graphite in the nuclear industry . . . . . . . . . . . . . . . . . . 18

1.3 Characterising the global behaviour of graphite . . . . . . . . . . . . . . . 20

1.4 Fracture Process Zone (FPZ) . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Modelling quasi-brittle material behaviour 24

2.1 Global approach to modelling material failure . . . . . . . . . . . . . . . . 24

2.1.1 Limitations of global fracture mechanics . . . . . . . . . . . . . . 25

2.2 Local approach to fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Discrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Generalised continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Project outline 29

3.1 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Report structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

II Review of Literature 31

4 Manufacture and microstructure of graphite 32

4.1 Manufacturing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Resulting microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Nuclear graphite grades . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.2 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.3 Fracture mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.4 Characterising the Fracture Process Zone . . . . . . . . . . . . . . 38

3

CONTENTS CONTENTS

4.3 Effects of radiation damage . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Fast neutron irradiation . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.2 Radiolytic oxidation . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Size effect of quasi-brittle structures 42

5.1 Statistical Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Power laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.2 Weakest Link Theory . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Deterministic Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Non-linear fracture mechanics 50

6.1 Cohesive zone/Discrete crack models . . . . . . . . . . . . . . . . . . . . 51

6.2 Smeared crack models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7 Local approach to material failure 56

7.1 Failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.1.1 Global failure criterion . . . . . . . . . . . . . . . . . . . . . . . . 56

7.1.2 Statistical failure criterion . . . . . . . . . . . . . . . . . . . . . . 59

7.2 Local constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.3 Modelling approaches for graphite . . . . . . . . . . . . . . . . . . . . . . 63

8 Discrete local approaches - lattice models 65

8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2 Variations of lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.2.2 Lattice network arrangement and choice of element . . . . . . . . . 68

8.2.3 Method for generating and incorporating heterogeneity . . . . . . . 69

8.2.4 Constitutive law and failure criterion . . . . . . . . . . . . . . . . . 71

8.3 Lattice models of note . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.4 Site-Bond lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9 Generalised Continuum 77

9.1 Couple stress theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4

CONTENTS CONTENTS

III Contribution to the field 80

10 Modelling and published work 81

10.1 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10.2 Microstructure-informed model . . . . . . . . . . . . . . . . . . . . . . . . 85

11 Conclusions 88

12 Further work 89

12.1 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

12.1.1 Improved relationship between bond deformation and energy . . . . 89

12.1.2 Couple with dual graph . . . . . . . . . . . . . . . . . . . . . . . . 92

12.2 Explore the effect of porosity on graphite failure energy at grain level . . . 92

12.3 Inclusions and validation of physical phenomena . . . . . . . . . . . . . . 93

12.4 Structural integrity assessment . . . . . . . . . . . . . . . . . . . . . . . . 94

IV Appendices 118

A Linear Elastic Fracture Mechanics (LEFM) 119

A.1 Energy approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.1.1 Stress concentration factor (Inglis) . . . . . . . . . . . . . . . . . . 119

A.1.2 Griffith approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.1.3 Modified Griffith approach . . . . . . . . . . . . . . . . . . . . . . 122

A.1.4 Energy release rate . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.1.5 The R curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.2 Stress intensity approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

A.3 Crack tip yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.3.1 Irwin’s plastic zone correction . . . . . . . . . . . . . . . . . . . . 127

A.3.2 Dugdale-Barenblatt cohesive zone/strip concept . . . . . . . . . . . 128

A.3.2.1 Dugdale strip yield model . . . . . . . . . . . . . . . . . 129

A.3.2.2 Barenblatt cohesive force model . . . . . . . . . . . . . 130

B Continuum Damage Mechanics (CDM) 132

B.1 Damage parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

B.2 Damage evolution and constitutive laws . . . . . . . . . . . . . . . . . . . 134

5

CONTENTS CONTENTS

C Meso-scale features and couple stresses in fracture process zone 136

D A meso-scale site-bond model for elasticity: Theory and calibration 147

E Lattice-spring model of graphite accounting for pore size distribution 153

F A discrete lattice model of quasi-brittle fracture in porous graphite 158

G Fracture energy of graphite from microstructure-informed lattice model 175

H Site-bond lattice modelling of damage process in nuclear graphite under bend-

ing 183

I Multi-scale modelling of nuclear graphite tensile strength using the Site-Bond

lattice model 194

6

List of Figures

1.1 The crystal structure of graphite . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 A scaled-down AGR core consisting of graphite moderator bricks . . . . . 18

1.3 High temperature reactor; (a) prismatic reflector; (b) graphite pebbles . . . 19

1.4 The stress-displacement graph for; (a) brittle (b) perfectly plastic/ductile

and (c) quasi-brittle materials under uniaxial tension . . . . . . . . . . . . . 20

1.5 A fracture process zone at the tip of a crack . . . . . . . . . . . . . . . . . 21

1.6 Zones of non-linear behaviour for; (a) linear elastic (b) non-linear plastic

(c) non-linear quasi-brittle materials . . . . . . . . . . . . . . . . . . . . . 22

1.7 A typical stress-displacement curve of concrete under uniaxial tension . . . 23

2.1 The global approach to modelling material failure . . . . . . . . . . . . . . 24

2.2 The local approach to fracture considering local material behavior . . . . . 27

4.1 Nuclear graphite production flow sheet . . . . . . . . . . . . . . . . . . . 33

4.2 A CCD image of the microstructure of Gilsocarbon. . . . . . . . . . . . . . 36

4.3 The turn-around for Gilsocarbon at varying temperatures for dimensional

change and relative Youngs Modulus . . . . . . . . . . . . . . . . . . . . . 40

5.1 (a) Geometrically similar structures of different sizes; (b) power scaling laws 43

5.2 Geometrically similar flaws in two components . . . . . . . . . . . . . . . 44

5.3 (a) A chain with links of distributed strength; (b) failure probability of a

given element; (c) a strucutre with a population of micro-cracks . . . . . . 45

5.4 An example of a Weibull plot for graphite; (a) tensile specimens (b) bend

specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.5 Size effect relations produced from elastic and plastic yield criteria, LEFM

and NLFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7

LIST OF FIGURES LIST OF FIGURES

6.1 The fictitious crack model for quasi-brittle materials; (a) the stress-displacement

response (b) the crack opening displacement at the ficticious crack tip . . . 52

6.2 Example softening curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3 (a) Smeared micro-cracking in a band of width h; (b) The inelasitc deform-

ation in the FPZ represented by an equivalent inelastic strain . . . . . . . . 54

7.1 A simple failure criteria where the bond fails at the tensile stress/strain con-

ditions, σt , εt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 (a) The Rankine failure envelope; (b) The Von-Mises and Maximum Shear

failure envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.3 The Mohr-Coulomb failure envelope. . . . . . . . . . . . . . . . . . . . . 58

7.4 An LEFM approximation (a) of a material with microstructure (b) . . . . . 60

7.5 The Rose-Tucker graphite failure model . . . . . . . . . . . . . . . . . . . 60

7.6 The Burchell graphite failure model . . . . . . . . . . . . . . . . . . . . . 61

7.7 An uncoupled local approach to fracture . . . . . . . . . . . . . . . . . . . 62

7.8 An uncoupled local approach to fracture with a continuous representation

of a crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.9 A coupled local approach to fracture . . . . . . . . . . . . . . . . . . . . . 64

8.1 Different lattice arrangements . . . . . . . . . . . . . . . . . . . . . . . . 67

8.2 A lattice superimposed onto a synthetic concrete microstructure . . . . . . 70

8.3 A centre particle lattice configuration . . . . . . . . . . . . . . . . . . . . . 70

8.4 A failure criterion accounting for tension softening with modified secant

elastic modulus for unloading . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.5 (a) Cellular representation of material; (b) the skeletal bond structure . . . 76

9.1 The stresses present on a 2D couple stress element under static load . . . . 78

10.1 Completed works during 0-12 months of the project . . . . . . . . . . . . . 82

10.2 Completed works during 12-24 months of the project . . . . . . . . . . . . 82

10.3 Completed works during months 24-36 months of the project . . . . . . . . 83

10.4 The 6 degrees of freedom represented by springs in the Site-Bond model . . 84

10.5 The displacement of a cantilever beam in the y-direction along the beam

length (z-direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.1 An elliptical hole in a flat plate . . . . . . . . . . . . . . . . . . . . . . . . 120

A.2 The strain energy released around a crack of length 2a . . . . . . . . . . . 121

8

LIST OF FIGURES LIST OF FIGURES

A.3 The prediction of the Griffith energy balance for energetically favourable

fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.4 (a) Flat R curve (b) rising R curve . . . . . . . . . . . . . . . . . . . . . . 125

A.5 Coordinate and element definition ahead of a crack tip . . . . . . . . . . . 125

A.6 The 3 modes of loading for a crack . . . . . . . . . . . . . . . . . . . . . . 126

A.7 Estimates of the plastic zone size for small-scale yielding . . . . . . . . . . 128

A.8 The strip-yield model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.9 The crack-opening force, P, acting at a distance x from the crack’s centre-line129

A.10 The cohesive force model . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

B.1 The concept of a fictitious undamaged state, on which the effective stress

principle is based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9

Abstract

University of Manchester

Craig Morrison

Doctor of Philosophy

Lattice-Modelling of Nuclear Graphite for Improved Understanding of Fracture

Processes

September 2015

The integrity of graphite components is critical for their fitness for purpose. Since

graphite is a quasi-brittle material the dominant mechanism for loss of integrity is crack-

ing, most specifically the interaction and coalescence of micro-cracks into a critically sized

flaw. Including mechanistic understanding at the length scale of local features (meso-scale)

can help capture the dependence on microstructure of graphites macro-scale integrity. Lat-

tice models are a branch of discrete, local approach models consisting of nodes connected

into a lattice through discrete elements, including springs and beams. Element properties

allow the construction of a micro-mechanically based material constitutive law, which will

generate the expected non-linear quasi-brittle response.

This research focuses on the development of the Site-Bond lattice model, which is con-

structed from a regular tessellation of truncated octahedral cells. The aim of this research

is to explore the Site-Bond model with a view to increasing understanding of deformation

and fracture behaviour of nuclear graphite at the length scale of micro-structural features.

The methodology (choice of element, appropriate meso length-scale, calibration of bond

stiffness constants, microstructure mapping) and results, which include studies on fracture

energy and damage evolution, are presented through a portfolio of published work.

10

Declaration

University of Manchester

PhD by published work Candidate Declaration

Craig Morrison

Faculty of Engineering and Physical Sciences

Lattice-Modelling of Nuclear Graphite for Improved Understanding of Fracture

Processes

Authorship for the presented published works is assigned according to size and significance

of contribution.

All work presented has been completed whilst registered at the University of Manchester.

No portion of the work referred to in this thesis has been submitted in support of an applic-

ation for another degree or qualification of this or any other university or other institute of

learning.

I can confirm that this a true statement and that, subject to any comments above, the sub-

mission is my own original work.

Signed: ......................................... Date: .........................................

11

Acknowledgements

Firstly I would like to thank my entire supervisory team; Dr Andrey Jivkov, Prof John Yates

(prior to retirement), Dr Chris Race, Prof James Marrow and Dr Andy Hodgkins. Particu-

lar thanks are directed to Dr Jivkov whose guidance, infectious enthusiasm and seemingly

boundless knowledge have made this endeavour both productive and enjoyable. I would

also like to thank the academics and researchers working on the QUBE project, in partic-

ular Mingzhong Zhang for useful discussion and collaborations. Further thanks go to my

fellow students based in the School of MACE for creating a vibrant working environment.

There are quite simply too many of them to name all here but special thanks go to Huw,

Tom, Andy, Dean, Alan, Adrian, Umair and Sophia who have been present throughout

the duration of my project. Moreover, I would like to acknowledge the Nuclear FiRST

Doctoral Training Centre and the EPSRC for project funding and general support provided

to me throughout this PhD. I would also like to thank the graphite team at AMEC Foster

Wheeler, Andy Hodgkins, Chris Jones and Owen Booler who allowed me to experience

the industrial context of my research while on a short placement. Finally, I am indebted to

Becky for her understanding especially during the final months of this project in addition to

my family and friends.

12

Copyright

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owns certain copyright or related rights in it (the “Copyright”) and s/he has given

The University of Manchester certain rights to use such Copyright, including for

administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic

copy, may be made only in accordance with the Copyright, Designs and Patents Act

1988 (as amended) and regulations issued under it or, where appropriate, in accord-

ance with licensing agreements which the University has from time to time. This

page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other intellec-

tual property (the “Intellectual Property”) and any reproductions of copyright works

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scribed in this thesis, may not be owned by the author and may be owned by third

parties. Such Intellectual Property and Reproductions cannot and must not be made

available for use without the prior written permission of the owner(s) of the relevant

Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and com-

mercialisation of this thesis, the Copyright and any Intellectual Property and/or Re-

productions described in it may take place is available in the University IP Policy

(see http://documents.manchester.ac.uk/DocuInfo.aspx? DocID=487), in any relev-

ant Thesis restriction declarations deposited in the University Library, The University

Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations)

and in The University’s policy on Presentation of Theses.

13

Published Works

1. Meso-scale features and couple stresses in fracture process zone

Craig N Morrison, Andrey P Jivkov, John R Yates

Proceedings of the 13th International Conference on Fracture (ICF13), June

2013, Beijing, China

2. A meso-scale site-bond model for elasticity: Theory and calibration

Mingzhong Zhang, Craig N Morrison, Andrey P Jivkov

Materials Research Innovations, vol 18, 2014, pp 982-986

3. Lattice-spring model of graphite accounting for pore size distribution

Craig N Morrison, Andrey P Jivkov, Gillian Smith, John R Yates

Key Engineering Materials, vol 592-593, 2014, pp 92-95

4. Discrete lattice model of quasi-brittle fracture in porous graphite

Craig N Morrison, Mingzhng Zhang, Andrey P Jivkov, John R Yates

Materials Performance and Characterization, vol 3(3), 2014, pp 414-428

5. Fracture energy of graphite from microstructure-informed lattice model

Craig N Morrison, Mingzhong Zhang, Andrey P Jivkov

Procedia Materials Science, vol 3, 2014, pp 1848-1853

6. Site-bond lattice modelling of damage process in nuclear graphite under bending

Craig N Morrison, Mingzhong Zhang, Dong Liu, Andrey P Jivkov

Proceedings of the 23rd Conference on Structural Mechanics in Reactor Tech-

nology (SMiRT23), August 2015, Manchester, UK

7. Multi-scale modelling of nuclear graphite tensile strength using the Site-Bond lattice

model

14

LIST OF FIGURES

Craig N Morrison, Andrey P Jivkov, Yelena Vertyagina, T James Marrow

Carbon, vol 100, 2016, pp 273-282

15

Part I

Introduction and Background

16

Chapter 1

Nuclear graphite

1.1 Introduction to graphite

Graphite is an allotrope of carbon which occurs naturally in a variety of states [1, 2]. At

the atomic scale, Figure 1.1, graphite is composed of single atom thick layers of carbon

(namely graphene sheets), where each carbon atom is covalently bonded to its three nearest

neighbours [3]. This creates a regular lattice arrangement of hexagonal carbon atom rings.

These layers form basal planes, with successive planes held weakly together by Van-der-

Waals forces to form a graphite crystallite, in an ABAB sequence (where the atoms in a

given layer are situated directly above/below the centre point of the hexagonal ring in the

below/above layer) although an ABCABC sequence is also possible [4].

a

c

a

Figure 1.1: The crystal structure of graphite [5]

Many of the properties of naturally occurring graphite, such as its inert nature and high

thermal and electrical conductivity, have led to a synthetic counterpart being manufactured

for many nuclear, aerospace and electromechanical applications. Synthetic graphite is man-

ufactured from suitable carbon-based constituents, generally petroleum cokes and a suitable

binder material. The features of the resulting polycrystalline microstructure - grain size,

pore size/density, are strongly influenced by the manufacturing process and the structure of

17

CHAPTER 1. NUCLEAR GRAPHITE

the coke and binder particles used [2]. The manufacturing process and subsequent micro-

structure will be described in more detail in Chapter 4.

1.2 The use of graphite in the nuclear industry

Graphite is an important material in the nuclear industry, having featured in over 100 nuc-

lear power plants, both commercial and research (comprehensive lists of graphite-moderated

reactors are given in [6, 7]). Its main functions within current and past reactors are to

thermalize (moderate) and reflect fast neutrons to sustain fission within the reactor core but

also act as a structural material, a notable advantage over other moderators such as water

or heavy water [2]. Examples of such reactors are the UK’s current fleet of Advanced Gas-

cooled Reactors (AGR), which were preceded by the Magnox reactors. In such designs the

reactor comprises of a construction of interlocking graphite moderator bricks with holes for

fuel element and control rod insertion and coolant flow, Figure 1.2.

Figure 1.2: A scaled-down AGR core consisting of graphite moderator bricks (Figure from[8] courtesy of British Energy)

Furthermore graphite’s high resistance to thermal shock, low coefficient of thermal ex-

pansion and increasing strength with temperatures up to 2500oC (assuming a non-oxidising

environment) have made it an important material in the design of a Generation IV Very

High Temperature Reactor (VHTR) and the predecessing High Temperature Gas Cooled

Reactors (HTGR). In such reactors, graphite is again used as a moderator and core struc-

18


tural material, such as the prismatic core of the High Temperature Test Reactor (HTTR) in

Japan, but can also be incorporated into the fuel itself, in the form of graphite pebbles, such

as the Thorium High Temperature Reactor (THTR) in Germany or the High Temperature

Reactor 10 (HTR-10) in China, Figure 1.3.

Figure 1.3: High temperature reactor; (a) prismatic reflector blocks undergoing machining[9]; (b) graphite pebbles with tennis ball to gauge scale [10]

The integrity of graphite, as with all structural reactor components, is critical for their

fitness for purpose. As a result, understanding the fracture behaviour of graphite is essential

for a number of reasons:

• Approving plant life extensions of the current UK AGRs. The loss of integrity in the

graphite bricks and other irreplaceable graphite components, such as the fuel sleeves,

could lead to disrupted coolant flow or control rod deployment, both of which can

lead to the overheating of fuel elements.

• Predicting in-service performance of current and future reactors. Typical graphite

bricks are subject to complex thermal and mechanical loading, the response to which

is time-dependent due to radiation damage. Component testing during service can be

complex so increasing the confidence in analytical and numerical models can reduce

required factors of safety and allow more structured maintenance planning.

• Deciding the best course of action for graphite legacy waste. The use of graph-

ite in UK and global reactors has left a significant irradiated graphite waste legacy,

currently over 230,000 tonnes worldwide [6]. Knowledge of the integrity of used-

graphite is essential for planning its recovery from reactors.

This task requires an understanding of the unirradiated (virgin) mechanical properties of

graphite, its response to complex structural loads and a grasp of how the properties change

with time when subject to the severe environment within the reactor, through fast neutron

19


irradiation and radiolytic oxidation. An extensive review of graphite and its use in the

nuclear industry is given by Burchell [7].

1.3 Characterising the global behaviour of graphite

The global response of brittle materials to an applied tensile or bending load is believed to

follow linear elasticity (LE), where a linear increase in load (stress) creates a linear exten-

sion (strain) before a sudden fracture, with very little plastic deformation (Figure 1.4(a)).

Unirradiated (or virgin) graphite’s similar fracture behaviour, with an initially linear re-

sponse and fracture occurring suddenly at low strains led to the assumption, that it too could

be fully described by LE [11]. There are however significant differences between graph-

ite’s response and that of a classically brittle material, such as an observable non-linearity

prior to peak load [12]. This bears similarities to that of an elastic-plastic or ductile mater-

ial (Figure 1.4(b)) where the response is linear elastic up to a point, defined as the elastic

limit or yield point. From this point onwards the response is non-linear with the material

yielding and undergoing plastic deformation prior to failure. This reduced stiffness, along

with other apsects of graphite’s global response, such as tension softening between the peak

load and fracture and a distinct size effect when considering strength and its relation to spe-

cimen size, are better understood when its behaviour is considered at a more local scale.

Distributed micro-cracking within an area ahead of the tip of the macro-crack, defined as

the fracture process zone (FPZ), have led to its characterisation as a quasi-brittle mater-

ial [12, 13], with ultimate failure occurring when distributed micro-cracks coalesce into a

critically sized flaw.

nonlinear

crack localization

(b)

l l

nonlinear

crack localization

(c)

l

crack localizationσ

σ

Δ Δ Δ

σ

nonlinear

(a)

Figure 1.4: The stress-displacement graph for; (a) brittle (b) perfectly plastic/ductile and(c) quasi-brittle materials under uniaxial tension [14]

20


Quasi-brittle is a term which has been used to describe many materials of heterogeneous

microstructure, e.g. concrete, rocks, sea ice, several ceramics, polymers [15], with much

work done on the behaviour of quasi-brittle materials coming from a desire to understand

the behaviour of concrete. Quasi-brittle materials appear to have a mixed response with

characteristics of both elastic and elastic-plastic materials (Figure 1.4(c)) with an initial

linear response corresponding to LE followed by a post-elastic limit non-linearity similar

to that from plasticity [16]. In fact quasi-brittle materials exhibit very little plasticity, with

the macro non-linear response that resembles plasticity being (most prominantly) due to the

formation and accumulation of distributed micro-cracks, prior to the ultimate failure point

of the material. These micro-cracks dissipate the elastic strain energy within the system

causing a local reduction in stiffness as this energy is no longer available to allow the crack

to propagate [16]. The response of the material at the micro-scale remains true to linear

elasticity throughout this region (i.e. the material still remains fundamentally brittle), but

the aggregate based, heterogeneous microstructure prevents sudden fracture. Instead cracks

are allowed to propogate progressively, along a microstructure-dependent fracture path [12].

The affect of graphites microstructure on its mechanical properties and fracture behaviour

are discussed in more detail in Chapter 4.

1.4 Fracture Process Zone (FPZ)

The FPZ is an important concept for modelling fracture of many varieties of materials, with

its size allowing insight into the type of material [16]. Figure 1.5(a) gives a basic example

of an FPZ, with the mechanistic processes at the micro-scale occurring over a distance xc.

Figure 1.5: A fracture process zone at the tip of a crack; (a) shown schematically (top); (b)shown mechanistically (bottom) [17]

21


The FPZ has not been clearly defined but the general definition given by Cottrell [18]

of an identified regime where the specific dissipated energy under steady state propagation

is constant, allows it to be distinguished from the plastic zone in relevant materials. Brittle

materials, are deemed to fail according to Linear Elastic Fracture Mechanics (LEFM - Ap-

pendix A), with fracture occurring suddenly, due to the existing flaws within the material

before the development of a significant damage zone, Figure 1.6(a). In such materials any

plastic zone at the crack tip will be due to small-scale yielding (Appendix A.3). This re-

gion can be defined as the FPZ, the size of which is negligible (of the order of micrometers

[12]) in comparison to the crack. Ductile materials cannot be described by LEFM, with

dislocations, void interaction and coalescence creating a significant damage zone around

crack tips. The failure is instead predicted using the yield strength criteria and a branch

of fracture mechanics corresponding to elastic-plastic behaviour (EPFM). Such materials

will exhibit a small FPZ (although 10− 100 times larger than LEFM [12]) surrounded by

a much larger zone of plastic deformation, Figure 1.6(b). In a quasi-brittle material there

is negligible plasticity, instead the FPZ occupies the entire region subject to non-linearity,

Figure 1.6(c).

F

N

L

F

N

L

F

N

L

Figure 1.6: Zones of non-linear behaviour for linear elastic (left), non-linear plastic(middle) and non-linear quasi-brittle materials (right) (adapted from [14] and [19]). Thecross hatched area, L denotes linear elastic material. N denotes the material subject tonon-linear behaviour in the form of plasticity. F denotes the material subject to non-linearbehaviour that does not include plasticity, i.e. the fracture process zone.

As mentioned, within this process zone are distributed micro-cracks, which individually

behave according to LEFM but prevent the storage of elastic strain energy expected of an

LEFM material at the macroscale. Figure 1.7 shows the typical stress-displacement curve

for a concrete specimen under uniaxial tension, where concrete’s ability to carry a residual

load after the material strength has been reached is evident. The pre-peak load non-linearity

and initial post-peak tension softening are considered to be due to micro-cracking, whereas

22


tension softening at the tail section is due to bridging, locking and frictional effects between

aggregate particles before the crack tip [16, 17], Figure 1.5(b). The characterisation of the

FPZ in graphite is discussed in more detail in Sections 4.2.3 and 4.2.4.

w

l

max

Str

ess,

σ

w

Elongation, Δ l

A

B

C

σ

σ

σ

σ σ

Figure 1.7: A typical stress-displacement curve of concrete under uniaxial tension [14]

23

Chapter 2

Modelling quasi-brittle material

behaviour

2.1 Global approach to modelling material failure

Strain constitutive

equations

Method of

calculation

Stress and strain

field histories

Critical

conditionsStructure

Loading

Initial conditions

Figure 2.1: The global approach to modelling material failure (adapted from [20])

Figure 2.1 illustrates the basic approach to modelling the global failure behaviour of a

material. If the loading and initial stress state are known, the stress and strain fields can

be evaluated. These fields can be formulated into boundary value problems, which can be

solved numerically using methods such as the Finite Element Method [21] for increasing

load or time steps until critical conditions are reached. There are essentially two methods

of globally defining the failure criterion of the material. The first is based on a strength

of materials approach, where the criterion is based on a critical value derived from the

classical continuum theories of elasticity or plasticity. In such a continuum the material

24

CHAPTER 2. MODELLING QUASI-BRITTLE MATERIAL BEHAVIOUR

is assumed to consist of one continuously distributed mass. Any point within this mass

are considered to have only 3 degrees of freedom (DOF), whereby their movement is fully

described by 3 components of displacement, ux1, ux2 and ux3 in a 3 dimensional x1, x2, x3

coordinate system. The displacement of these points leads to a symmetric stress tensor, and

the loads are described completely by a force vector. The second method is an extension of

these continuum theories into specifically derived failure theories, based on damage - Con-

tinuum Damage Mechanics (CDM), or the presence of a flaw - Fracture Mechanics, either

Linear-Elastic (LEFM), Elastic-Plastic (EPFM) or Non-Linear (NLFM). Fracture mechan-

ics is generally considered the preferred method for assessing and designing engineering

materials throughout academia and in many sectors of industry, with significant advances

since the seminal papers of Griffith [22] and Irwin [23]. Classical global fracture mechanics

essentially models the effect of introducing a discontinuity, in the form of a crack or flaw,

on a material under the assumption of a classical continuum. This assumption is generally

accurate and valid for component/macro scale behaviour predictions but there are several

limitations under other conditions.

Readers with no prior knowledge of LEFM and CDM are referred to Appendices A and

B respectively. NLFM aims to introduce into the continuum failure model the non-linear

processes occurring ahead of a crack that are evident in quasi-brittle materials. This will be

discussed more in Chapter 6. EPFM will not be covered in this thesis, but readers can refer

to [17] for more details.

2.1.1 Limitations of global fracture mechanics

Historically the assumption of a classical continuum has proved accurate and reliable in

cases which consider materials at their macro-scale. However, when exploring smaller

length scales, such as stress concentrations and discontinuities around cracks, notches,

micro-cracks and voids, material microstructure begins to have an effect on behaviour so

materials no longer behave according to classic continuum predictions [24]. The reasons

for this are the limitations to classical global fracture mechanics, both fundamentally and

when specifically applied to quasi-brittle materials. LEFM and EPFM can only be used in

the presence of an initial crack or flaw, which remains sufficiently far away from any bound-

ary, with the size effect relationship between failure load and component size described by

a power-law. Furthermore both follow the assumption that the structure lacks any charac-

teristic length, i.e. any fracture processes occurring ahead of the crack are located within

an insignificantly small region in comparison to the crack length, Figure 1.6. The signific-

25


ant “local” damage within a FPZ during quasi-brittle fracture renders such assumptions of

purely global behaviour invalid. Hence LEFM and EPFM can be used with a degree of con-

fidence for very large structures, but fail at smaller component sizes. Non-Linear Fracture

models, Chapter 6, have been developed to account for a non-negligible FPZ but even so

when the size becomes significant it is necessary to account for the actual processes which

dissipate energy in a local approach. An approximate guideline to the suitability of such

models for an FPZ length l, and specimen width D, given by Bažant [25], is shown in Table

2.1.

Length scale Most suitable approachD/l ≥ 100 LEFM

5≤ D/l < 100 NLFMD/l < 5 Local approaches

Table 2.1: Suitable analysis procedures for varying fracture process size

2.2 Local approach to fracture

Predicting the macro-response based on the processes occurring at the length scale of the

micro-structural features (meso-scale) in a so-called “local approach” could potentially be

more representative of the actual material response, rather than just material geometry as

in the “global approach” [20]. Understanding the mechanisms within the fracture process

zone of quasi-brittle materials and how these dissipate the strain energy ahead of the crack

tip is neccesary before these responses can be linked to the macroscopic properties of such

materials. Experimental techniques such as X-ray tomography and digital image/volume

correlation have allowed significant progress in this area [26].

There is no definitive way of introducing representative local behaviour into a frac-

ture model. Different approaches tend to involve either constructing constitutive equations

to account for such behaviour, e.g. accounting for accumulative damage by coupling the

strain constitutive equations with the aforementioned CDM, or assigning a local failure cri-

terion as a post-processing procedure. Developing the constitutive laws has tended to be

phenomenological with macroscopic experimental data used as a basis for parameter curve

fitting for individual loading cases and geometries [27, 28]. Basing the constitutive laws

on micro-structural mechanisms would provide a more realistic representation, with cur-

rent methods being developed including discrete models. A promising approach involves

finding a length scale at which a representative volume element (RVE) can be used to rep-

26


resent the micro-structural mechanisms within that volume. This length scale is called the

meso-scale, Figure 2.2.

Figure 2.2: The local approach to fracture considering local material behavior, either at themicro-scale or using a Representative Volume Element (RVE) at the meso-scale [29].

For the approach of assigning a local failure criterion, current methods generally rely

on weakest link (WL) assumptions, where size effect is still statistically modelled using a

power law [30]. Weakest link fracture, as described with relation to size effect in Section

5.1.2, assumes the failure behaviour of a material can be likened to a chain, with the fail-

ure stress of the chain dictated by the failure stress of its weakest link. This assumption

is thought to naturally model the statistical size effect due to the materials hetereogenity,

where larger structures have a higher probability of containing a critical flaw. This approach

fails to account for the micro-crack interactions apparent in quasi-brittle materials [31]. A

more extensive review of local approach methods is given in Chapter 7 with further details

given by Pineau [32].

2.2.1 Discrete models

The discrete approach, formulated by Cundall [33], is a promising method for develop-

ing micromechanical based constitutive equations. Lattice models are a branch of discrete,

local approach models, consisting of nodes connected into a lattice through discrete ele-

ments including springs [34] and beams [35]. Element properties allow a material response

according to actual mechanistic failure data. Unlike WL methods, lattice models are based

27


around a parallel statistical system, with load redistribution amongst remaining bonds once

a bond is broken. Such models have been developed for graphite [36–38] after initial de-

velopment for concrete and cementitious materials [35, 39, 40]. The focus of this work is

the development of the Site-Bond lattice model proposed by Jivkov and Yates [41]. A full

overview of lattice models is given in Chapter 8.

2.3 Generalised continuum

Local approaches attempt to use micro-structural mechanistic understanding to improve

predictions of fracture where the global approach and hence classical continuum assump-

tions break down. Further understanding of the microstructure-fracture relation can be

gained by considering generalized continuum theories [42]. Generalized continua essen-

tially form a local approach to continuum modelling, but strive to maintain continuity

whereas local approaches to fracture model the actual mechanisms of the discontinuities.

One such theory, couple stress theory [43] of which micropolar theory is a branch, uses ad-

ditional deformation measures to describe this relation such as the curvature tensor, defined

as the relative rotation between micro-structural features within the continuum with re-

spect to the distance between them. Within these theories the classical deformation energy,

arising from symmetric strains, is amended with curvature energy, naturally introducing a

microstructure-related length scale that is missing in classical fracture mechanics and study

of size effect. Including such length related terms requires the inclusion of the previously

ignored couple stress component of the stress tensor into the continuum theory. Although

generalised continua and couple stress theory offer a stand-alone approach to modifiying

a classical continuum to account for micro-structural affects, initial work in this project

aimed to use generalized continuum theory as complimentary to local fracture models, in

order to benefit from this internal length scale. In particular a micropolar continuum has

been used in lattice models where both displacements and rotations between lattice nodes

are allowed [44–47] as this allows for rotational invariance. In this manner realistic calib-

ration requires consideration of couple stress theory. An overview of couple stress theory

and other generalised continuum theories are given in Chapter 9.

28

Chapter 3

Project outline

3.1 Aims and objectives

The purpose of this project is to use microstructure-informed lattice-models to improve un-

derstanding of the fracture processes of graphite and hence its behaviour at an engineering

scale. The project has 2 main aims:

• Increase understanding of deformation and fracture behaviour of nuclear graphite

through application of lattice-models.

• Propose an improved methodology of graphite integrity assessment.

These aims are to be achieved through 3 objectives:

1. Develop a micro-structurally informed lattice model at the length scale of graphites

micro-structural features (meso-scale).

2. Validate the model against experimental data in its ability to reproduce elastic con-

stants, material properties and general quasi-brittle behaviour.

3. Use the comparison of model and experiment to explore methods for integrating

micro-structure informed models into engineering integrity assessment procedures.

3.2 Report structure

The thesis is structured as follows; in Part II a review of the literature is undertaken. The

literature surrounding the manufacture, microstructure and fracture properties of nuclear

29

CHAPTER 3. PROJECT OUTLINE

graphite is explored in Chapter 4. The variation of size effect and non-linear fracture mech-

anics are discussed with relation to quasi-brittle materials in Chapters 5 and 6 respectively.

A more comprehensive review is given of the local approach to fracture in Chapter 7 with

specific considerations of constitutive models and failure criterion for concrete, cement and

graphite. An overview of discrete local approaches is given in Chapter 8 with emphasis on

lattice models. The concept of generalised continuum theory is explored in Chapter 9.

This thesis is presented in the form of published or submitted work. The portfolio of

published works can be found in Appendices C-I. Part III outlines the published works

with a brief overview of each paper with accompanying discussion, Chapter 10, before

presenting overall conclusions, Chapter 11, and possible extensions with relation to the

Site-Bond model, Chapter 12.

30

Part II

Review of Literature

31

Chapter 4

Manufacture and microstructure of

graphite

4.1 Manufacturing process

The process of producing synthetic graphite was established in the late 19th Century, fol-

lowing the discovery by Edward Acheson that graphitic carbon, rather than amorphous

carbon, was left behind upon heating silicon carbide in a furnace [2]. The raw constituents

of synthetic graphite can be any graphitizing source of carbon with the general basis of

particulate filler material held together using a binder [1, 48]. Coke particles are used as

filler material due to their high carbon content. These are generally either petroleum cokes,

produced by delayed coking from by-products of petroleum oil distillation, or pitch cokes,

which are produced from coal-tar pitch [2, 4]. The resulting cokes vary in shape depending

on the initial feedstock, ranging in the extremes from high-aspect ratio needle cokes to more

spherical shot cokes [49]. Binder materials are usually distillation products from coal, such

as coal-tar pitch, which soften upon heating, allowing forming of the raw carbon article to

take place before hardening once cooled [4]. The choice of raw materials and the manu-

facturing route taken, particularly with regards to the method of forming, has a significant

effect on the microstructure and hence properties of the resulting grade of graphite.

The manufacturing process, as shown in Figure 4.1, begins with the preparation of the

filler particles. These are calcined in order to reduce the amount of volatile matter from

approximately 15% to below 0.5% [2, 4]. In addition this step serves to pre-shrink the

particles prior to a latter baking stage. Failure to do so may result in poor cohesion between

the filler and binder phases. The filler particles are then subject to milling/grinding before

32

CHAPTER 4. MANUFACTURE AND MICROSTRUCTURE OF GRAPHITE

Coke

Calcined coke

Coke flour Coke particlesBinder material

Green article

Baked article

Impregnated

Calcination

Milled & sized

Mixed

Formed

Cooled

Baked

Nuclear Graphite

Graphitized

Purified

Figure 4.1: Nuclear graphite production flow sheet

being sieved and graded into varying sizes. It is necessary to have a range of sizes from large

filler particles to small fragments, called flour, to allow tighter packing and hence higher

density in the final product [2, 4]. The binder material is then added to the preferred mixture

of particles of varying sizes. The mixture is used to form a solid billet through extrusion or

moulding (either block moulding, isostatic pressing or vibration moulding). The forming

method can introduce a directional bias into the resulting microstructure, depending on the

shape of the initial coke filler particles used. Extrusion tends to cause an alignment of

the long axes of filler particles, if present, to the direction parallel to that of the extrusion.

Moreover moulding can align the particles perpendicular to the direction of the moulding

force [2]. The formed “green article” is then baked at around 800oC to further remove

volatile substances and to carbonize the pitch. This results in a reduction in density and

increase in porosity to 25−35% in the baked article, which is combated by an impregnation

of molten pitch prior to the graphitization stage [2]. Graphitization involves heating the

33


carbon product at temperatures ranging from 2200− 3000oC, in order to restructure the

carbon article to form regions of crystallite graphite.

The baking and graphitization stages are effective at removing a large number of im-

purities, producing a final product suitable for many industrial applications. Nuclear-grade

graphite however, requires a further purification step to remove impurities which remain

due to their high boiling points [2, 4]. Boron is of particular concern in nuclear applications

due to its high neutron capture cross-section which is inhibitive to the moderating proper-

ties of graphite. A low boron content can be ensured with careful selection of raw materials

and the introduction of an additive into the graphitizing furnace which reacts with the boron

and allows it to be removed along with other volatile elements.

4.2 Resulting microstructure

The resulting product from this manufacturing process is a high purity graphite, the micro-

structure of which is three phase; relatively large filler particles (graphitized coke particles),

a matrix of graphitized binder (sometimes only partially so [50]) and various populations

of porosity. The structure varies significantly depending on the raw materials and manu-

facturing process used. In this section some comments will be made regarding the general

structure of graphite and resulting mechanical properties before considering the specific

structures of a selection of nuclear-grade graphites.

During the graphitization process the underlying structure within the filler particles will

change to form mosaic regions of small graphitic crystallites which can grow, reorientate

and coalesce into domains of longer-range order [51]. This longer-range order within the

filler particles is aligned predominantly in the direction of any bias of the particles, i.e. the

basal planes of the graphitic crystallite will become parallel to the direction of extrusion

[48, 52]. The binder or matrix phase consists of a continous mosaic of randomly orientated

graphitic crystallites [52], although studies have shown that this phase itself can consist of

a combination of graphite crystallites, quinoline insoluble (QI) particles (resulting from the

fractionation process which produced the pitch) and ungraphitized carbon [50].

The three main porosity/initial crack populations, ranging from nm to mm in size, total

approximately 20% of virgin graphite volume [52]. Gas evolution cracking occurs within

the matrix during the impregnation stage of manufacture as gas bubbles form when liquid

pitch boils during baking. As such these are found predominantly in the matrix phase [52].

Calcination and Mrozowski cracks form throughout the graphite due to uneven thermal ex-

34


pansion and shrinkage as the graphite heats and cools during calcination and graphitization

respectively. The produced cracks are the result of the differing thermal expansion coeffi-

cients of the a and c axis within graphites atomic structure, Figure 1.1, where upon cooling

the graphite shrinks at different rates in the two directions [53]. The porosity phase results

in a density of approximately 1.6−1.8g/cm3, a significant reduction from 2.26g/cm3, the

theoretical density of a perfect graphite crystal. Porosity and associated micro-cracking can

interact and coalesce [54], forming interconnected networks throughout the structure which

are both open and closed to the external environment.

4.2.1 Nuclear graphite grades

The many applications of graphite has led to the development of different varieties, with

different average grain sizes, ranging from coarse grained graphite with grains larger than

4mm to microfine grained graphite with grains smaller than 2µm. In nuclear graphite these

typically range from ultrafine (< 10µm) to medium grains (< 4mm) [37]. The variation in

grain size, alongside other micro-structural differences can result in different mechanical

properties [55]

Past and current reactors in the UK employ a selection of graphites grades. The earliest

generation of gas-cooled reactor in the UK, Magnox reactors, used Pile Grade A (PGA)

as moderator, an extruded graphite distinguishable by its coarse needle coke filler particles

with length in the region of 0.1−1mm [12]. Gilsocarbon, or IM1-24, is used as the mod-

erator and reflector in the UK Advanced Gas-cooled Reactors (AGRs) with VFT and later

Nittetsi graphites used as fuel element sleeves [56]. In Gilsocarbon the spherical filler

particles are derived from Gilsonite pitch coke and range from 0.3− 1.5mm in size with

layers analagous to those in an onion [57]. The microstructures of both Gilsocarbon [12, 58]

and PGA [12, 58, 59] have been well characterised.

Outside of the UK, there are different graphite grades in use. In particular grade IG110,

an ultrafine historical grade of graphite currently used in the Japanese High Temperature

Test Reactor (HTTR) [60], has attracted considerable research interest [61–63]. Further-

more several graphite grades are currently under consideration for possible future Genera-

tion IV high temperature reactor designs, such as PGX, PCEA and NBG-18 [61, 62]. A list

of candidate grades for the Generation IV reactor designs can be found in the NGNP graph-

ite selection report [9] in addition to the extensive list of current nuclear grade graphites

complete with origin, forming method and mechanical properties given by Burchell [7].

35


Figure 4.2: A CCD image of the microstructure of Gilsocarbon [57]. The filler particles arecircled for clarity.

4.2.2 Mechanical properties

The resulting macroscopic properties of graphite include; increasing strength up to 2500oC,

a low tensile strain and also low tensile strength and stiffness when compared to other

structural materials [4]. The material properties of graphite at the crystalline level are highly

anisotropic, reflected in derived elastic moduli [4], with a perfect crystal possessing low

shear strength between basal planes as a result of the weak Van-der-Vaals forces. It has been

well established that the material properties at the macroscale in graphite are dependent

on the prominent heterogneous microstructure, with the anisotropy of individual crystals

projected differently onto macroscopic properties according to the size and orientation of

the crystallites within the microstructure [4, 64].

Early studies set out to both empirically define the constitutive relationship of graphite

[65, 66] and relate the resultant macroscopic properties to characteristics of the constitu-

ent parts of the graphites, with density, Young’s modulus and flexural strength shown to

increase with decreasing particle size [2, 54, 59]. Moreover it was established that the man-

ufacturing process used had a significant affect on the resulting properties, most notably

by influencing the directional bias of properties introduced through the forming methods.

As mentioned previously, the long axis of a particle will align parallel with the extrusion

direction and this results in higher strength and Young’s modulus parallel to the extrusion

direction than perpendicular to it [48]. In this manner mechanical properties for affected

grades are stated as “with grain” (WG) or “against grain” (AG) within the literature. In

addition the failure stress has been shown to decrease with increasing pore volume fraction

[54, 59, 67] for reasons which will be discussed in the following sub-section.

36


With regard to specific graphite grades, the needle-like coke particles in extruded PGA

graphite strongly project the materials anisotropy onto the macroscopic properties such that

the WG Young’s modulus can be double the AG value [68]. Gilsocarbon, shows minimal

directional dependence, with the individual crystallites within the spherical filler particles

having a tendency to align circumferentially. This results in near isotropic mechanical prop-

erties [7], a preferred property of graphite. The high dependence of macroscopic proper-

ties on distributed micro-scale features means these properties can vary between measured

samples of the same grade, resulting in a scattered distribution of strength [37].

4.2.3 Fracture mechanisms

Failure mechanisms compete at the length-scales of the microstructure features (described

mechanistically by Tucker et al. [54]), with the dominant mechanism sometimes varying

both between grades and regions of the same grade/sample. Early studies, including Jenkins

[52], gave strong indications that localised micro-cracking occured in graphite before final

fracture, initiating as early as 13 of the final load or deflection of final fracture. These

suggestions of the non-linear quasi-brittle stress-strain behaviour eluded to in Section 1.3

were further supported by acoustic emission studies (summarised well by Burchell [11] and

Tucker and McLachlan [69]) and optical microscopy studies [54, 59].

From these early studies it was established that micro-cracks generally occur either by

cleavage along the weak basal plane of the graphite crystallites; in either the matrix or

filler particle phase, or along grain boundaries/interface between phases [54, 70, 71]. For

grades of graphite with a prefered orientation of crystallites within a particle, weak cleavage

planes across the particle were created leading to preferential fracture along the length of

the filler [52, 59]. This can also occur in regions of binder phase with a high degree of order

[54]. Porosity plays a significant role in these mechanisms, not only concentrating stress

leading to micro-crack initiation but also manipulating the stress-field in such a way that

other micro-failures are drawn towards it [7, 11, 54]. Conversely it can also provide an area

for crack arrest, which in itself may lead to secondary cracking [59].

More recent works have been aided by the progress made in advanced microscopy and

imaging techniques along with the benefit of increased computational capability for data

processing. Detailed micro-structural characterisation has been undertaken using improved

optical microscopy [62], Raman spectroscopy [58] and Transmission Electron Microscopy

[63] with techniques such as Small Angle Neutron Scattering (SANS) [72], X-ray tomo-

graphy [26, 73], helium pycnometry and mercury porosimetry [61, 74] used effectively

37


to measure pore size/shape distributions. There has also been focus on in-situ character-

isation of the microstructure, progressive damage and fracture characteristics [75], using

techniques such as strain mapping [57, 73], X-ray tomography [26, 73] and digital im-

age/volume correlation [26, 68, 76], with several techniques sometimes used in parallel.

The combined outcome of these works is a better understanding of the fracture mechan-

isms and how the microfailures interact and accumulate to form the fracture process zone

(FPZ) discussed previously in Section 1.4. Work by Joyce et al. [57] further supported the

view that damage initiates at porosity by showing that the initiating sites of strain localisa-

tions coincided with porosity in a diametral compression sample. Mostafavi and Marrow

[13] showed the same phenomena under flexural loading. Moreover Marrow et al. [73]

showed that the propagation of a crack occurs as a result of the coalescence of microfail-

ures in the FPZ, with Becker et al. [77] observing the same process using the double torsion

technique for stable crack propagation. Furthermore both of these works suggest mechan-

isms that may be increasing the resistance to propagation as the crack extends (R-curve

behaviour, see Appendix A), including micro-cracking and wakes effects such as crack

bridging.

4.2.4 Characterising the Fracture Process Zone

As discussed by Hodgkins et al. [12], the FPZ size and the resultant affect on global re-

sponse is dependent on geometry and applied load. In plain specimens (Hodgkins uses the

example of a beam) microfailures are generally initiated at areas of high stress, due to fea-

tures such as pores, or the loading. As such, damage is distributed over a relatively large

region. This large FPZ allows an increased amount of energy dissipating damage to occur

and hence increases the strain at which global failure occurs, i.e. a larger FPZ increases the

failure strain. When the specimen is notched, the region of high stress is intensified around

the notch reducing the FPZ size, restricting the volume in which damage can occur and

hence reduces energy dissipation and nonlinearity. Furthermore as the component volume

decreases, the considered length scale approaches that at which the microfailures occur and

there will be a size affect, with properties changing for geometrically identical specimens

of differing size. This is discussed further in Chapter 5.

Many attempts to characterise the FPZ size as a material property have been made,

beginning with Hillerborg et al. [78] who termed it the characteristic length, with more

recent attempts including Saucedo et al. [79, 80]. Hillerborg’s characteristic length is

discussed in relation to cohesive zone models in Section 6.1. Although some of these

38


characterisation attempts include reference to microstructure, the parameters controlling

the size of the FPZ are still not clearly understood. According to Aliha and Ayatollahi [81],

Awaji et al. [82] and Claussen et al. [83], the size of the FPZ for ceramics scales as a

function of the fracture toughness and tensile strength, according to variations of Equation

4.1.

r = A(

Kσc

B)2

(4.1)

where K is fracture toughness, A is a constant and B can be a constant or a function of

another parameter (e.g. a function of crack angle [83]). Conversely Ayatollah and Aliha

[84] have shown empirically that the FPZ size in ceramics is approximately 100 times

greater than the average grain size alone, while Bažant and Oh showed that FPZ width is 3

times the maximum aggregate size [19].

4.3 Effects of radiation damage

The demanding environment within a nuclear reactor can lead to considerable radiation

damage of structural components including graphite. In the following section the two main

mechanisms, fast neutron irradiation and radiolytic oxidation, and their effect on the micro-

structure and mechanical properties of graphite will be briefly discussed.

4.3.1 Fast neutron irradiation

Fast energetic neutrons can collide with carbon atoms within the graphite crystallite, displa-

cing the atom and hence damaging the lattice [2, 48]. This results in significant changes in

the dimensions of the graphite component and its mechanical properties, with these changes

being highly dependent on the temperature and the microstructure, specifically the dire-

citonal bias of crystallites [6].

In general as carbon atoms are displaced, the basal plane from which the atoms ori-

ginated suffers shrinkage. The displaced atoms can cluster between planes, which leads

to swelling in the c-direction (with reference to Figure 1.1) [48, 85]. However porosity

between basal planes can accommodate the swelling, resulting in a net volume shrinkage.

This porosity is termed “accommodation porosity” and includes Mrozowski cracks between

the basal planes. Eventually this porosity is filled so there is expansion in the c-direction,

occurring at a greater rate than the in-plane shrinkage, leading to a net-volume increase and

39


density decrease. The point at which this occurs is termed the “turn-around” as shown in

Figure 4.3. The changes in mechanical properties with neutron dose is strongly linked to

the corresponding change in volume [11]. Initially the net shrinkage and increase in density

results in an increase in strength and modulus [2, 86] coupled with linear elastic behaviour

as potentially energy dissipating pores are removed. Following turn-around the decrease in

density causes a reduction of strength as pores are created.

Figure 4.3: The turn-around for Gilsocarbon at varying temperatures for; dimensionalchange (left) and relative Youngs modulus (right) [87]

4.3.2 Radiolytic oxidation

Radiolytic oxidation is a concern for some gas cooled reactors, specifically those cooled

by a non-inert gas such as CO2 in the UK’s Magnox and AGR reactors. The CO2 within

the coolant can, when irradiated, split into CO and an oxidising species. This oxidising

species, as with the rest of the coolant can move into “open-porosity” where it can react with

carbon atoms on the graphitic pore surface, increasing the pore size (and hence porosity)

and producing CO [48, 88, 89]. This can result in significant weight-loss, particularly in the

40


matrix phase [12], to the graphite without any noticable change to the external component

dimensions.

It has been established that the change of mechanical properties, such as tensile or

compressive strength σ or Young’s modulus E, resulting from the fractional weight loss,

x, follows an exponential decay law according to the relationship proposed by Duckworth

[90], Ryshkevitch [91] and expanded by Knudsen [92] for porous materials:

σ = σ0 e−aθ (4.2)

E = E0 e−bθ (4.3)

where σ0 and E0 denote the pore-free values of tensile or compressive strength and Youngs

modulus respectively. θ is the pore volume fraction/porosity and a and b are dimensionless

constants. Alternatively, as outlined by Burchell et al. [93] and Berre et al. [94], the

relationship can be given in terms of fractional weight loss:

σ = σ0 e−Ax (4.4)

E = E0 e−Bx (4.5)

Kelly et al. [95] measured the constants A and B for Gilsocarbon as 4.0 and 3.6 respectively.

Buch [70] derived an expression relating the constants to the pore aspect ratio, Ar:

b = 1+0.594Ar (4.6)

41

Chapter 5

Size effect of quasi-brittle structures

Size effect in materials and structures has been evident since the times of Leonardo da Vinci

[96] who speculated that the strength of a rope is inversely proportional to its length. It is

of particular importance for quasi-brittle materials, where the typical engineering structure

size can vary significantly in length scale from those that can be appropriately tested [97].

The concept of size effect was developed by Mariotte in 1686 to form the basis of what

is now commonly known as the weakest link theory (WLT). Mariotte suggested that the

reason for this inverse proportionality was the increased probability of a failure-inducing

defect in a long rope. Without this defect the failure strength of both ropes would be

equal. Griffith’s work, which became the basis for LEFM, experimentally demonstrated

this strength increase on decreasing glass fibre diameters [22]. A historical overview of

such developments is given by Timoshenko [98].

5.1 Statistical Size Effect

5.1.1 Power laws

It is commonly known that physical systems involving no characteristic length will scale

through a power law [97]. The reasons behind this can be understood by considering the

response to loading, Y and Y ′ of two geometrically similar components of size D and D′

respectively:

Y = f (D), Y ′ = f (D′) (5.1)

Work by Bažant [99] showed that without a characteristic size the following relation must

be true:

42

CHAPTER 5. SIZE EFFECT OF QUASI-BRITTLE STRUCTURES

f (D′)f (D)

= f(

D′

D

)(5.2)

Which can be solved only by the power-law given in Equation 5.3 as only then will the con-

stant, c1, relating to some characteristic length, cancel out when substituted into Equation

5.2. The term s also represents a constant.

f (D) =

(Dc1

)−s

(5.3)

Figure 5.1: (a) Geometrically similar structures of different sizes; (b) power scaling laws[97]

Figure 5.1 illustrates the power-law scaling of nominal strength, σN , with compon-

ent size for elastic, elastic-plastic materials and LEFM. It has been shown that for both

strength/yield criteria for elastic/elastic-plastic materials respectively there is no size effect

with components of geometrically similar dimensions failing under the same value of σN in

the absence of a characteristic length [99]. This result corresponds to a value of s equal to 0.

Also shown in Figure 5.1 is the Weibull distribution, a distribution based upon WLT which

produces a power-law size effect [99]. The value of s shown, 16 [100], is considered typical

for concrete, deriving from an empirically fit Weibull modulus. WLT and specifically the

Weibull distribution will be described more in the following sections.

Geometric size effect in LEFM The geometric size effect predicted in LEFM can be

shown by equating the stress intensity factors of two geometrically similar cracks or flaws,

Figure 5.2 [24]:

KI = σ∗1√

πa f( a

W

)= σ

∗2

√πλa f

(λaλW

)(5.4)

43


Figure 5.2: Geometrically similar flaws in two components [24]

σ∗2 =

σ∗1√λ

(5.5)

Hence the value of s for LEFM, as shown in Figure 5.1, is 0.5. Experimental results do not

always verify this, even for materials which are considered to have a brittle response, with

the increase of strength with decreasing specimen size often exaggerated [24].

5.1.2 Weakest Link Theory

The WLT was originally constructed by Peirce [101] using extreme value statistics for a

series statistical system whereby a material is modeled as a chain with n links of distrib-

uted strengths. The failure strength of the component is essentially dictated by the failure

strength of its weakest link. This is more conventionally formulated in terms of component

survival or failure probabilities [37]. If each of the n links contains a failure probability of

P f then the survival probability of the ith link can be expressed as:

(Ps)i =[1−(Pf)

i

](5.6)

The component survival probability is the product of the survival probability of all n links1.

Ps =n

∏i=1

(Ps)i =n

∏i=1

[1−(Pf)

i

]∼= n

∏i=1

exp[−(Pf )i

]= exp

[−

n

∑i=1

(Pf )i

](5.7)

1the 3rd equality follows the Maclaurin series expansion for exp(−Pf

)taken to the linear order [102].

44


Equation 5.7 suggests that as the number of links, n, increases under a constant load the

probability of survival, Ps will decrease as the probability of a weak link increases, illus-

trating a clear size effect. This concept is shown in Figure 5.3.

Figure 5.3: (a) A chain with links of distributed strength; (b) failure probability of a givenelement; (c) a structure with a population of micro-cracks, each with a differing probabilityof becoming critical [97]

Extreme Value Statistics Statistically modelling brittle failure where fracture propagates

from one of many existing microscopic flaws falls into the bracket of extreme value statist-

ics. There are considered to be three types of extreme value distribution, whereby if several

sets of values are sampled from a distribution and the maxima (or minima in the case of

the Weibull distribution) from each set are collated into a new set, than this set will be rep-

resented by one of only three distributions; Gumbel, Fretchet and Weibull. For the sake

of brevity only Weibull will be considered here. For information on Gumbel and Fretchet

distributions please refer to Nemeth and Bratton [37].

Weibull distribution There are many variations on the weakest link theory with the

Weibull distribution being the most commonly referenced [103, 104]. The overview of

the Weibull derivation expressed here follows that presented by Nemeth and Bratton [37].

Weibulls distribution assumes that within a volume, V , of brittle material there exists a

critical stress, σ , which when present at a flaw of size l will lead to catastrophic crack

propagation. If there exists a distribution of flaw sizes, the critical strength, σc of a flaw of

length L can be generalised as follows:

L≥ l σc ≤ σ (5.8)

45


L < l σc > σ (5.9)

A crack density function can be defined, η(σ), describing the amount of flaws in a unit

volume which satisfy σc ≤ σ . For an incremental volume4Vi, the probability of failure of

the ith link, where the critical strength is σi becomes:

Pf (σi) = [η(σi)4Vi] (5.10)

Substituting this into Equation 5.7 gives the survival probabilty of the entire volume as a

function of the failure probabilities of the individual incremental volumes:

Ps(σi) = exp

[−

n

∑i=1

η(σi)4Vi

](5.11)

If a stress of σ is applied to the entire volume where σi = σ for all increments then the

global survival probability of V is:

Ps(σ) = exp [−η(σ)V ] (5.12)

Again using Equation 5.7, the entire component failure probability can be evaluated:

Pf (σ) = 1−Ps(σ) = 1− exp [−η(σ)V ] (5.13)

Or accounting for differing stresses throughout the volume:

Pf (σ) = 1−Ps(σ) = 1− exp[ˆ−η(σ)dV

](5.14)

The Weibull distribution is obtained if a power law is used to describe the crack density

function, η(σ). The Weibull three-parameter function is shown in Equation 5.15:

η(σ) =1

V0

(σ −σu

σ0

)m

=

(σ −σu

σ0V

)m

(5.15)

The three parameters in the model are defined as followed:

σu is the value of σ below which the probability of component failure is zero. Equat-

ing this parameter to zero will produce the two parameter Weibull distribution.

σ0V is the scale parameter which incorporates the characteristic volume, V0, and the

stress value at which 37% of samples under tensile load would not fail.

46


m is the Weibull modulus. This parameter is a dimensionless representation of the

variation of strength in the material. This is evaluated using the best fit to empirical

data.

From Equation 5.14, the two parameter Weibull equation can be decomposed to include an

effective volume, V e:

Pf = 1− exp[ˆ

V−(

σ

σ0V

)m

dV ] = 1− exp[−Ve

(σ f

σ0V

)m

] (5.16)

where Ve =´

V

(σaσ f

)mdV with σa denoting the level of stress at position a(x,y,z) and σ f is

the maximum stress within the effective volume. This implies that for two components (de-

noted with subscripts 1 and 2) of identical relative dimensions but different size, Equation

5.16 can be used to equate their failure probabilites to give a relation between component

volume and the maximum stress at a given point for a given failure probability [37]:

σ f 2

σ f 1=

(Ve1

Ve2

) 1m

(5.17)

Equation 5.17 shows that strength will reduce with increasing component size, with the

reduction related to the Weibull modulus, m.

Aside from its use in describing size effect, much work has been done on using the

Weibull distribution as a failure criterion for graphite to explain the spread of experimental

results, with an extensive review given in Nemeth and Bratton [37]. An example of an

analysis of graphite strength using the Weibull distribution is shown in Figure 5.4, where S

is the probability of fracture. This will be discussed more in Section 7.1.

5.2 Deterministic Size Effect

As mentioned, the power law distribution relating failure strength to size is strongly re-

liant on the absence of a characteristic length. As discussed in Sections 1.4 and 4.2.4,

a characteristic length exists in quasi-brittle materials in the form of the significant FPZ

size. Nemeth and Bratton [37] present evidence from several works which suggest that the

Weibull approach, as a method of modelling size effect purely statistically, is insufficient

for modelling graphite, with similar works drawing the same conclusions for cement-based

materials [31, 100].

At the length scale of a single micro-crack the behaviour can be described as LEFM,

however at the scale that flaws interact the component will exhibit what appears to be plastic

47


Figure 5.4: An example of a Weibull plot for graphite; (a) tensile specimens (b) bendspecimens [105]

behaviour. As the component size increases to such an extent that the microstructure of the

material becomes insignificant the material behavior in the presence of a flaw will tend

towards LEFM. In this way the size effect response of quasi-brittle structures will need to

tend from that of a plastic material for small structures to that of LEFM for larger structures,

Figure 5.5. This is termed the deterministic size effect, initially experimentally observed by

Leicester [106] and Walsh [107] for concrete with Leicester attempting to fit these results

to the aformentioned power-law.

Nonlinear

fracture

mechanics

Elasticity or

Plasticity

Linear fracture

mechanics

Log(size h)

Nom

inal str

ess a

t fa

ilure

, N

2

1

Figure 5.5: Size effect relations produced from elastic and plastic yield criteria, LEFM andNLFM (reproduced from [19])

To accurately model the deterministic size effect the characteristic length (FPZ length)

must be taken into account in order to bridge the transition between the two powerlaws

48


which describe the distinct statistical size effect behaviour of quasi-brittle materials above

and below the characteristic length [16]. One way of doing this is through the application

of Non-Linear Fracture Mechanics (NLFM), Figure 5.5, which will be discussed further in

the following chapter.

49

Chapter 6

Non-linear fracture mechanics

NLFM aims to introduce the non-linear energy dissipation processes found in the non-

negligible process zone ahead of a crack in a continuum-based failure model. These fracture

models have mainly been developed for concrete and extensive reviews can be found in the

following references [14, 16, 25]. Such models can be generally classified into two types;

• Models which implicitly represent the non-linear zone, mainly for numerical imple-

mentation in Finite Element analysis methods [21]. These include the cohesive or

smeared crack models.

• Models which modify LEFM, replacing the flaw with one of “effective” properties

which represent the non-linearity. These include the two-parameter fracture model,

size-effect model and the effective crack model.

For brevity only the first type, which is of more direct relevance to the presented work,

is considered here. The reader is referred to the aforementioned texts for an overview of

modified LEFM methods. The derivations provided are based around those found in the

cited papers and other standard texts [16, 17].

The first attempt to model the behaviour in an FPZ was the Dugdale-Barenblatt (cohesive-

zone) approach to correct for small-scale yielding in a linear-elastic material. The concept

behind such models originates from Dugdale’s Strip-Yield Model [108] and Barenblatt’s

Cohesive force model [109], which were originally proposed as alternatives to Irwin’s

modification of Griffith’s energy balance criterion (i.e. a correction for plastic flow in

LEFM), Appendix A.3.2. This concept has been extended and developed as the basis for

two of the most significant FPZ models, mainly in application to concrete and cement:

• Fictitious crack model (FCM)

50

CHAPTER 6. NON-LINEAR FRACTURE MECHANICS

• Crack band model (CBM)

which are examples of Cohesive zone/Discrete crack and Smeared Crack models respect-

ively.

6.1 Cohesive zone/Discrete crack models

In cohesive zone models, of which the Fictitious Crack Model (FCM) is the most com-

monly referenced, closure stresses are modelled as acting on the faces of a smoothly clos-

ing fictitious crack, which extends the actual crack across the FPZ. The location of the

effective crack tip and hence FPZ size is found at the point at which the stress intensity

factor becomes zero, Equation A.35 [16]. With micro-cracking and other softening mech-

anisms modelled along a discrete line through the FPZ rather than continuously distributed

throughout, these models are also sometimes called Discrete Crack Models [27].

Fictitious Crack Model (FCM) The FCM, proposed in 1976 by Hillerborg et al. [78],

extended this concept, with application to concrete. Unlike Dugdale’s strip-yield model,

where the closure stresses are constant at the material yield strength, Hillerborg et al. as-

sumed that the stresses along the fictitious crack length could be expressed as a function

of the crack-opening displacement, σ(δ ). This relationship was hypothesised as a unique

material property which dictates softening within the FPZ. This is the same approach first

proposed in the Barenblatt model, however in the Barenblatt model the size of the FPZ was

assumed to be small in comparison to the crack so that brittle fracture could be modelled.

This assumption is not the case in the FCM and hence the stress distribution in the FPZ is

required as a material parameter [16]. The crack is deemed to be able to propagate when

the closure stress at the tip of the fictitious crack reaches the materials tensile strength, σt

as shown in Figure 6.1(a). At this moment the crack-opening displacement (COD) at the

actual crack-tip is defined as the critical value δc, Figure 6.1(b). The closure stress at this

point is zero.

Energy is absorbed in opening the crack. Under the assumption of a unique σ − δ

relationship, the true fracture energy is given by:

G f =

ˆ 0

σt

σdσ =

ˆδc

0σdδ (6.1)

This energy is equal to the area underneath the σ − δ graph, Figure 6.1(a). By fitting the

curve so that the true fracture energy, G f , equals the critical energy release rate for crack

51


Figure 6.1: The fictitious crack model for quasi-brittle materials; (a) the stress-displacementresponse (left); (b) the crack opening displacement at the ficticious crack tip (right) [17]

propagation, Gc the LEFM energy balance approach is modelled. The choice of a suitable

softening curve to match the energy rates is critical in replicating material behavior. Many

softening curves have been proposed, such as linear, bilinear, power-law and exponential,

Figure 6.2. In (incorrectly) assuming that all the material along the length of FPZ has

reached the tensile strength of the material, σt with a toughness of G f , the same approach as

Irwin in his plastic zone correction (Appendix A.3.1 - Equations A.23 and A.27), Hillerborg

et al. were able to derive a characteristic length expression which roughly approximates the

FPZ length [16]:

lp 'E ′G f

σ2t

(6.2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

σσt

δ δ

linear

bilinear

exponential

Figure 6.2: Example softening curves

Cohesive zone models require two material parameters to describe fracture behaviour.

In the FCM these parameters are the fitted softening relationship, σ(w), in the FPZ and

specific fracture energy G f . In reality, any two independent parameters can be used, includ-

ing lp [16]. There are limitations to the FCM, including the assumption of a unique σ −δ

52


material relationship, which is not always correct as the FPZ in quasi-brittle materials can

be substantial enough that edge effects may need to be accounted for. This results in a size

effect, where fracture parameters depend on the size of the FPZ as eluded to in Section

4.2.4. Furthermore this method relies on a curve fitting approach to find the most suitable

softening curve and as a result the correct macro-scale response is not always achievable.

6.2 Smeared crack models

The FCM fundamentally operates by assuming a discrete crack, with the FPZ approximated

to a discrete line in a fictitious crack, wherein tension softening occurs according to the

function, σ(δ ). Smeared crack models differ in that an approach akin to damage-mechanics

is followed, with the damage smeared over a region, e.g. a finite element, rather than a

discrete line. Bažant’s Crack Band Model (CBM) is the most widely used model of this

type [110].

Crack Band Model (CBM) Bažant [110] proposed that if the fracture energy, G f , of

the discrete crack were to be “smeared” over a fixed FPZ band, of width, h, then a strain-

softening relationship σ(ε) would too be an adequate approximation, with the strain related

to the crack opening displacement and fracture energy, Figure 6.3. Research based on

experimental data has shown that the width of this band can be optimally approximated

at a value equal to three times the maximum aggregate size [19]. The critical strain εc is

then provided via a fracture criterion. In this way the stress would decrease according to

increasing elastic strain as opposed to increasing crack opening displacment. This crack

band model has been further developed by Oh and Bažant [19], Bažant and Cedolin [111]

and Cedolin and Bažant [112] amongst others. The total strain can be split into elastic and

inelastic components εc and ε respectively.

εt = εc + ε (6.3)

The constitutive relation for a 2D isotropic material including both elastic and inelastic

components of strain can be expressed as:

εxx

εyy

εxy

=1E ′

1 ν ′ 0

ν ′ 1 0

0 0 E ′G

σxx

σyy

σxy

+

0

ε

0

(6.4)

53


h

f't

(a) (b)

f't

c

t

E

Figure 6.3: (a) Smeared micro-cracking in a band of width h; (b) The inelasitc deformationin the FPZ represented by an equivalent inelastic strain (reproduced from [16])

where E ′ = E1−ν2 and ν ′ = v

1−νfor plane strain and E ′ = E and ν ′ = ν for plane stress. It is

initially assumed that the micro-cracks within the crack band zone all begin parallel to the

stress applied normal to the crack faces, σyy, as shown in Figure 6.3. In this way the inelastic

strain ε only increases strain in the direction normal to crack faces. Work by Phillips and

Zienkiewicz [113] introduced a shear-retention, or aggregate interlocking factors factor, β :

σxy

εxy= βG (6.5)

where 0< β ≤ 1. This factor was introduced to allow for the physically realistic restrictions

shearing movement due to crack surface roughness and aggregate interlocking [16]. With

this factor the constitutive relations become:

εxx

εyy

εxy

=1E ′

1 ν ′ 0

ν ′ 1 0

0 0 E ′βG

σxx

σyy

σxy

+

0

ε

0

(6.6)

A further improvement to increase physical realism can be introduced by including a scalar

damage parameter, ω , ranging from 0 to 1 for the undamaged and complete failure states

respectively [114]. The reduced stiffness, E ′s = E ′(1−ω), of the crack band zone with

progressive damage in the direction of loading can then be accounted for in the constitutive

relations in a coupled approach to damage using a secant stiffness matrix:

εxx

εyy

εxy

=1E ′

1 ν ′ 0

ν ′ 11−ω

0

0 0 E ′G

σxx

σyy

σxy

+

0

ε

0

(6.7)

54


This progressive damage will produce the tension softening response of concrete, however

this response will be dictated by the choice of softening curve used to evolve the damage

ω = f (εyy). This is a similar scenario to the FCM method with different proposed softening

curve approximations. The fracture energy can be expressed as:

G f =

ˆ h

0

ˆεc

0σyy (ε)dεdx (6.8)

which, assuming the strains do not change over the crack band, equals:

G f = hˆ

εc

0σyy (ε)dε (6.9)

where εc =δch corresponds to the critical crack opening displacement δc in the FCM [19].

Equation 6.9 reduces to Equation 6.1 as h tends to 0.

55

Chapter 7

Local approach to material failure

The limitations to global material failure modelling and fracture mechanics and the limited

applicability of NLFM, as discussed in Chapters 2 and 6, led to the development of more

locally based approaches to failure. Local approaches aim to predict micro failure initiation

and development through into the initiation of a macro sized flaw in areas where global

approaches are unreliable, i.e. in the FPZ. Such models have been previously developed

for fracture of ductile [115–117], brittle [30] and quasi-brittle materials. Introducing local

behavior can essentially be done in two ways (which can be used in combination):

• Definition of a failure criterion that accounts for local fracture mechanisms.

• Incorporation of local behaviour into the global constitutive relations.

A brief review of literature will be undertaken for local approaches to fracture. For further

details, the reader is referred to Pineau [32].

7.1 Failure criterion

7.1.1 Global failure criterion

Criterion for global material failure are classically based around the global material stress

state for an elastic continuum. The simplest failure criterion involves material failure occur-

ring when a critical value of stress or strain is reached. This is shown for a brittle specimen

under uniaxial tension in Figure 7.1, where failure occurs when the material reaches its

tensile strength, σt at a critical value of strain, εt .

This is more formally extended to two (and three) dimensions as the Rankine, or max-

imum stress, failure criteria, Figure 7.2(a), where failure occurs when the maximum prin-

56

CHAPTER 7. LOCAL APPROACH TO MATERIAL FAILURE

σ

ε

σ

ε

Figure 7.1: A simple failure criteria where the bond fails at the tensile stress/strain condi-tions, σt , εt

cipal stress reaches the uniaxial tensile or compressive strength, σc, such that:

−σc < {σ1,σ2}< σt (7.1)

σ2

σ1

σt

σt

-σc

-σc

σ2

σ1

σy

σy

-σy

-σy

Von Mises

Maximum Shear

Figure 7.2: (a) The Rankine failure envelope (left); (b) The Von-Mises and Maximum Shearfailure envelopes (right).

For ductile materials, where the yield point is considered the point of failure, other fail-

ure criteria are considered more accurate. The Tresca, or maximum shear stress, criterion,

associated the yield point with the maximum shear stress in the material, derived from a

uniaxial tensile test, Figure 7.2(b). For a 2D stress state [118]:

τmax =σ1−σ2

2(7.2)

57


For uniaxial tension at the yield point σ1 = σY and σ2 = 0. This implies at the yield point:

σ1−σ2 = σY (7.3)

Another failure criterion for materials at yield is the Von-Mises, or shear-strain energy,

criterion, Figure 7.2(b). This is also known as the octahedral shear stress, or distortion

energy criterion. This was derived by splitting the principal stresses into volumetric, σV ,

and deviatoric components, σD:

σi = σV +σ

Di (7.4)

where i = 1,2,3 and σV =σ1+σ2+σ33 .

Using these components the strain energy stored in the material upon deformation can

be decomposed into strain energy due to a volume change, i.e. from σV , and due to distor-

tion or shear, i.e. from σDi . The strain energy due to change in shape can be used to derive

the Von-Mises yield criterion (full derivation can be found in Benham et al. [118]):

(σ1−σ2)2 +(σ2−σ3)

2 +(σ3−σ1)2 = 2σ

2Y (7.5)

σ2

σ1

σt

σt

-σc

-σc

Mohr-Coulomb

Max stress

Figure 7.3: The Mohr-Coulomb failure envelope.

A further failure criterion, the Mohr-Coulomb criterion, for brittle materials can be de-

scribed by combining the Rankine and Tresca failure criterion, Figure 7.3(a). This criteria,

derived from the Mohrs circle concept, allowed the material to fail differently under purely

tensile and compressive loads, and incorporated the affect of shear stress from the Tresca

58


criteria.

Much work has been developed for finding a suitable failure criterion for quasi-brittle

materials in a triaxial stress state, in particular concrete. This has included the use of all the

aforementioned brittle and yield criteria. Some more significant works include [119–121]

as well as the use of the Drucker-Prager yield criterion, an extension of Von-Mises [122].

However most of these are phenomenologically based, with parameters found from curve

fitting procedures from experimental results [123] and also fail to replicate variations of

strength over repeated tests [37], a phenomena dictated largely by distributions of micro-

structural features. For graphite it has been shown that modelling failure behaviour accord-

ing to a strength of material approach, with failure occurring at a critical value of stress,

strain or elastic strain energy density, is insufficient for tensile loading as a result of ignor-

ing the underlying microstructure [54].

7.1.2 Statistical failure criterion

Attempts have been made to model the implications of micro-structural defects and the

corresponding mechanisms of graphite, described in Section 4.2.3, using statistical failure

criterion, such as Weibull analysis [37, 54], as discussed with relation to the size effect in

Chapter 5.1.2. In this approach the micro-structural model is implied through the weakest

link assumption, rather than modelled. The use of the Weibull distribution as a local failure

criterion was introduced by Beremin for the application of cleavage fracture [30]. This

statistical model has been questioned in cleavage fracture [124] and quasi-brittle fracture

[31] due to the failure to explicitly account for actual mechanistic failure at the micro-scale.

Other early methods attempted to base graphite’s failure behaviour on the Griffith failure

model. Such an approach is termed the “fracture mechanics model” by Tucker et al. [54],

where the defected microstructure of the graphite, Figure 7.4(b), is represented by a crack

which fails according to Griffith criteria, Figure 7.4(a) located at the area of maximum

stress. Although such an approach aims to represent the microstructure, again it cannot

represent the actual failure mechanisms occurring at the micro-scale, especially using the

simplified crack conditions and geometries assumed in LEFM.

Further improvements, beginning with the work of Buch [70], developed a micro-

structural based failure criterion to define the conditions at which a cleavage flaw will be

generated large enough to fail according to LEFM, using input parameters such as distribu-

tions of particle size and porosity. The failure probability of each individual grain is used

to evaluate the probability that a flaw of critical size in accordance to the Griffith criterion

59


Figure 7.4: An LEFM approximation (a) of a material with microstructure (b) [54]

will develop.

Rose and Tucker extended Buch’s work [71] to generate the Rose-Tucker model, where

the material is discretized into cubes, the size of which is representative of the particle size

within the graphite. Each particle, and hence each cube, is deemed to have a random orient-

ation which will cleave at a prescribed tensile stress. The presence of porosity is included

through the designation of cleavage strength of 0. The failure of sufficient adjacent cubes to

create a critically sized Griffith flaw represents specimen failure, Figure 7.5. One downfall

of this model is the uncoupled affect of the creation of a microflaw with the surrounding

stress field. Essentially as a new pore is created the stress around this pore will remain in

the same state, despite the clear stress raising affects.

Figure 7.5: The Rose-Tucker graphite failure model [54].

A further development of the Rose-Tucker model was developed by Burchell over sev-

eral works [7, 11, 67]. This followed the same discretisation procedure as the Rose-Tucker

model but the onset of cleavage was only considered in areas of concentrated stress, i.e.

around porosity. Initial pores within the model are treated as micro-cracks with cleavage

planes perpendicular to the applied tensile stress on the cube by which they are represented.

60


In this way the stress in surrounding cubes is concentrated. The total failure probability of

a row of particles is dictated by the probability of the concentrated stress from the initial

pore being sufficient to fail each successive particle once the stress is resolved onto the

corresponding cleavage planes, Figure 7.6.

Figure 7.6: The crack front in the Burchell model can extend through an successive particlesif the stress resolved onto the cleavage plane is large enough to cause cleavage [37].

McLachlan and Tucker [69, 125] extended the Burchell model, introducing a reliance

on the stress concentrating capacity of pores on their shape. In this manner pores were

deemed as “active”, or essentially micro-cracks, where the stress around is concentrated, or

“passive”, where the pore shape incurs little stress concentration and hence is less likely to

intiate cleavage.

A review of these statistical failure criterion models was undertaken in Tucker et al.

[54], Tucker and McLachlan [69], and Nemeth and Bratton [37]. These papers concluded

that the models which aim to explicitly model the affects of microstructure, Rose-Tucker,

Burchell, McLachlan-Tucker, performed better than those which implicitly suggest the

presence of microstructure, Weibull, and those which involve microstructure without con-

sideration of the interactions at this scale, fracture mechanics model. However all these

methods are based around a weakest link assumption so are inherently unable to fully model

the interaction processes at the micro-scale.

7.2 Local constitutive equations

A local continuum approach to material failure can rely on both Fracture Mechanics and

Continuum Damage Mechanics (CDM). Introducing damage mechanics allows the damage

to be evolved to the point of macro-crack initiation, after which the critical conditions for

component failure are defined by a failure criterion [20] dictated by fracture mechanics or

61


a critical value of strength or damage (as covered in the previous section). With the initial

conditions and loading known the strain consitutive equations can be used to evaluate the

corresponding stress and strain fields, through numerical methods such as Finite Element

Method [21].

The development of damage can be undertaken in an uncoupled approach, as shown in

Figure 7.7, where the stress and strain fields are recalculated for an incrementally increasing

load or time step until the damage parameter, which has been evaluated from these fields,

has reached a critical level. In this approach, outlined in Appendix B.2, the stress and strain

are evaluated without any additional input from the damage parameter, with any interactions

ignored.

Strain constitutive

equations

Method of

calculation

Stress and

strain field

histories

Critical

conditionsStructure

Loading

Initial conditions

Crack

initiation

Crack

propagation

Damage evolution

laws

Crack propagation

laws

Damage

mechanics

Fracture

mechanics

Figure 7.7: An uncoupled local approach to fracture (adapted from [20])

The CBM, previously discussed in Section 6.2, adapted this methodology to include a

continuous representation of the initiated crack. This uncoupled approach again has separ-

ate and distinct constitutive and damage evolution equations, but the progressive damage is

fed back into the constitutive laws after each time or load step (Equation 6.7) until a critical

value is reached, Figure 7.8. In this way the fracture mechanics methodology is avoided.

Further advances include coupling the damage and strain into the constitutive equa-

tions in order to represent the softening-effect damage has on the material stiffness through

stress redistribution, Figure 7.9. Coupling the damage into the constitutive equations us-

ing the effective stress concept (Appendix B) increases failure analysis complexity as the

stress-strain-damage field has to be evaluated at each step, with crack propagation repres-

ented by the progression of spatial points which have reached the critical value of damage

corresponding to zero stiffness (and hence the presence of a macro-crack) [126].

62


Strain constitutive

equations

Method of

calculation

Stress and

strain field

histories

Critical

conditionsStructure

Loading

Initial conditions

Crack

initiation

Crack

propagation

Damage evolution

laws

Damage

mechanics

Figure 7.8: An uncoupled local approach to fracture with a continuous representation of acrack (adapted from [20])

There are many proposed methods for derivation of constitutive laws for damage-coupled

local approaches in application to quasi-brittle materials. These can be phenomenologically

based, where the parameters for the relations are obtained by curve fitting to macroscopic

experimental data [27, 28]. As discussed previously with relation to the FCM and CBM,

this is not ideal as curve fitting is required separately for different loading cases and com-

ponent geometries. More recent works have tended to focus on building the constitutive

models inclusive of the actual failure mechanisms at the micro-structural level. Bažant and

Prat [127] provide an extensive reference list relating to both macroscopic phenomological

models (including plasticity models and damage-plasticity models) and micromechanics

constitutive damage laws.

7.3 Modelling approaches for graphite

The primary method of structural analysis of graphite components throughout industry and

academia is the Finite Element (FE) method [21]. As such much of the research work

regarding graphite and indeed other quasi-brittle materials has involved developing more

accurate FE models, both from continuum and local approach perspectives.

FE models have been used for validation of experimentally obtained linear-elastic frac-

ture parameters [26] and calculation of continuum (microstructure-free) strain fields for

comparison with strain-fields due to microstructure [57]. Hall et al. modelled an idealised

filler particle and polycrystalline graphite microstructure to highlight dominant mechanisms

63


Strain-damage

constitutive equations

Method of coupled

calculation

Stress and strain

and damage

field histories

Critical

conditionsStructure

Loading

Initial conditions

Crack initiation

and propagation

Figure 7.9: A coupled local approach to fracture (adapted from [20])

of fast neutron irradiation damage [87]. Moreover researchers have explored image-based

modelling, constructing a finite-element mesh directly onto an image of the graphite mi-

crostructure [94, 128, 129]. Such analysis can mimic a specific microstructure effectively,

however it raises issues in terms of at what length scale the microstructure model becomes

representative of the material at a larger scale. Significant research has been directed to-

wards defining a constitutive law which accounts for the change in material properties due

to different radiation damage mechanisms [130–135] and structural damage [136–139] us-

ing various brittle or plastic damage models (briefly described in Appenidx B.2) in ap-

proaches more akin to a local approach. Local constitutive relations can also be found

phenomologically through inverse methods, using FE models to iteratively calculate mater-

ial properties which produce an experimentally measured response [140]. Such laws can be

easily incorporated into commercial FE software packages such as ABAQUS [141] through

user defined material subroutine (UMAT) for more effective stress analysis or coupled with

fracture mechanics methodologies for modelling crack growth, such as the eXtended Finite

Element Method (XFEM) [142]. Moreover, researchers have attempted to model the local

heterogeneity of graphite through stochastic means, namely the Stochastic/Random Finite

Element Method [143–145], whereby random fields are incorporated into the determin-

istic FE. Recent works by Saucedo and Marrow have implemented the mesh-free cellular

automata method into an FE model (Finite Element Microstructure MEshfree, FEMME)

in an attempt to explicitly model the graphite microstructure in regions of interest, without

dependence on mesh density [146, 147].

64

Chapter 8

Discrete local approaches - lattice models

Discrete methods show promise in allowing the development of micromechanical based

constitutive equations. Cundall [148] originally developed the Discrete Element Method

(DEM) as a numerical framework to simulate discontinuous problems found in geological

and rock mechanics. This approach was further developed by Cundall and Strack [33] in

application to the flow and interaction of granular media. Such Discrete Element Methods

consider the physical system as a composition of discrete particles, with a node at the centre

of each particle. These particles are free to move according to Newton’s second law upon an

applied force. Upon contact with another particle, the affect of the frictional and interaction

forces on the particle’s motion is calculated. As a result, the important variables for this

method, once the initial position of the particles are specified, are the applied force and

displacement, as opposed to the stress and strain as used in the Finite Element method

[28, 149]. Cundall’s method was extended to study fracture at the micro-structural scale of

rocks [150] and aggregate based composites [151].

More modern developments of particle methods include peridynamics [152] and Smoot-

hed Particle Applied Mechanics (SPAM) [153], the latter of which is equivalent to Smoothed

Particle Hydrodynamics (SPH) [154] but for solids applications. These have been applied

to the study of quasi-brittle structures [155, 156]. These approaches offer a promising

alternative to standard continuum models, where extra constitutive relations are required

for defect interaction and coalescence. Although these particle methods can model damage

progression, there remains no method (to the author’s knowledge) to distribute a initial state

of damage such as porosity and hence explicitly represent the actual mechanisms involved

in failure at the micro-structural scales.

A second branch of discrete method is the lattice models. Lattice models treat the ma-

terial as a parallel system (unlike the weakest link methods, which are essentially statistical

65

CHAPTER 8. DISCRETE LOCAL APPROACHES - LATTICE MODELS

systems based in series), with nodes at cell centres formed into a network through bond

connections to neighbouring nodes, or nodes within a region of influence. These bond con-

nections, represented by discrete elements, in the form of elastic beams [157], springs [34],

fuses [158] or custom elements, are capable of transfering elastic tensile or compressive

forces. Connecting bonds can break due to overextension, removing the ability to transfer

forces between particles. Failure of a bond results in the redistribution of the load, rather

than instantaneously failing the whole system. This is considered to provide a closer repres-

entation of micro-crack interaction and accumulation of damage in quasi-brittle materials,

with the local heterogeneity represented by the failure properties assigned to the discrete

elements.

The parallel basis of lattice models can prove computationally intensive [37] suggesting

engineering scale component models are impractical. However such models can be benefi-

cial in areas where it is necessary to model local behaviour, e.g. in the FPZ. Furthermore,

lattice models have the same advantages over continuum modelling as particle methods,

in that no additional assumptions for crack development or propagation are necessary, but

offer a further advantage in that an initial state of damage can be introduced. The literature

review presented in this chapter will focus on models of a lattice framework although there

is significant overlap in theory and naming conventions between lattice and particle models.

8.1 Background

Development of lattice models, actually predates that of the discrete element method, hav-

ing been used in the field of elasticity since the 1940s [159]. Burt and Dougill [160] are

considered the first to have applied this to model materials of a heterogeneous nature, when

they constructed a random truss joint system with distributed truss strength and stiffness.

The main application of such models, as is generally the trend with all quasi-brittle stud-

ies, has been directed at cementitious and aggregate based materials, stemming from initial

models by Bažant and Tabbara [40] and Schlangen and Van Mier [157].

Bažant et al. [40] proposed a Random Particle Model, generalising previous work from

Zubelewicz and Bažant [151], based on Cundall’s DEM formulation. In his model Bažant

considered only axial forces between particles. In such a way this is considered similar

to the truss network developed by Burt and Dougill [160] although based on a network

of particles, hence bridging the concepts of discrete particle and lattice models. Until this

model, development of discrete element models and lattice models had been in parallel.

66


Schlangen and Van Mier [157] proposed their 2D Lattice Fracture Model using beam

elements to connect nodes into a triangular lattice (where connecting the midpoints of 6 tri-

angles form a repeating hexagonal cell), as shown in Figure 8.1. The beam properties were

dictated by the material phase within which it was situated when the lattice was projected

onto a synthetic microstructure. Both of these early works provided ample demonstration of

the dependence of the macroscopic non-linearity on the initiation and development of mech-

anisitic microstructutral failure events, through recreation of a realistic force-displacement

relation.

Figure 8.1: Different lattice arrangements [157, 161]

8.2 Variations of lattice model

In a general sense, there are many possible methods of constructing a lattice in order to

model a quasi-brittle material, including the calibration, choice of element, network ar-

rangement, method for incorporating material heterogeniety and element failure criterion

[162].

8.2.1 Calibration

The calibration of element properties is generally based on the energy equivalence method-

ology, proposed by Morikawa et al. [163] and further developed by Mustoe and Griffiths

[164, 165]. In this methodology the strain energy, U in a typical lattice unit or repeated unit

is equated to the strain energy in the equivalent continuum:

Ucell =Ucontinuum (8.1)

Ucontinuum =12

ˆV

σεdV (8.2)

67


For the simplest case of a spring bond with a single DOF the strain energy is given by the

standard relationship:

Ubond =12

ku2 (8.3)

where k is the spring stiffness and u is relative displacement between the ends of the spring.

The strain energy within the cell is given by the total strain energy of each bond within it:

Ucell = ∑bond

Ubond (8.4)

The equivalent procedure for a triangular and square lattice of beam elements (of both

Euler-Bernoulli and Timoshenko formulation) can be found in [44].

The methodology of energy equivalence is analagous to the Cauchy-Born rule, whereby

continuum constitutive relations are derived from the atomistic scale [166]. This procedure

is by no means the exclusive method of calibration, with differing methods discussed with

regards to specific lattice models in Section 8.3.

8.2.2 Lattice network arrangement and choice of element

Many arrangements have been used for lattice models as shown in Figure 8.1. These can

be regularly arranged in 2D; e.g. triangular [161], square [167] or rhombic configuration,

or 3D; cubic [168] or more complex FCC or HCC arrangements [169]; using a variety

of different elements including fuses [158], truss [40] or beams [157] and springs [34].

It can be argued that simple configurations, such as 2D and cubic lattice, do not provide

a physical representation of the actual arrangement and coordination of grains and will

inherently fail to model materials with the correct elastic properties. There is evidence that

the regularity in the aformentioned lattices introduces bias in the propagation direction of

a macro-crack [35, 170]. As a result irregular/random configurations have been proposed,

including configurations obtained through Voronoi tessellation and Delaunay triangulation

procedure [34, 171–173]. However, these random arrangements only directly represent the

unique configuration being modeled.

One of the constraints of lattice models is the ability to reproduce a range of Poisson’s

ratio values suitable for an isotropic linear elastic material. This has been shown to result

from both the network arrangement and choice of element within the model. Studies using

truss or spring elements which transmit only normal forces were shown to be limited to a

fixed value for the Poisson’s ratio, ν = 0.33 for the 2D Random Particle Model by Bažant

68


and Tabbara [40] and ν = 0.25 for the FCC arrangement by Donze and Magnier [174].

To overcome this restriction many researchers attempted to model non-axial degrees

of freedom between neighbouring particles or sites. This can be done either by using

beam elements, although such structural elements are deemed to be physically unrealistic in

modelling the actual microstructure interactions [175], or by introducing additional shear

springs. The latter was used by Griffiths and Mustoe, who showed that a triangular lat-

tice can be used to represent a material with a Poisson’s ratio, ν < 0.25 (plane strain) and

ν < 0.33 (plane stress) [165]. 3D models still exhibit problems; the cubic and FCC and

HCC arrangements are only representative of a linear elastic isotropic material when ν = 0

[168, 169]. The Distinct Lattice Spring Model (DLSM), described in Section 8.3 overcame

this problem by introducing negative spring stiffness, although the physical meaning of this

is in question [176]. Furthermore there remains the issue with such models that they fail to

remain rotationally invariant when introducing extra degrees of freedom between particles.

8.2.3 Method for generating and incorporating heterogeneity

There are considered to be 2 ways of generating representative heteregeneous material prop-

erties [157]:

• Random distribution of bond properties, according to a given distribution [177, 178].

The distribution can reflect the degree of heterogeneity present in the material in

comparison to the structure size being modelled.

• Generation of a synthetic microstructure, either directly from imaging techniques or

as a statistical representation of the random nature of the material microstructure.

If a synthetic microstructure is generated then there are considered to be 3 general meth-

odolgies for projecting/incorporating the material heterogeneity onto the lattice [179]; the

particle overlay lattice, centre particle lattice and long range interaction lattice.

Lattice with particle overlay This approach involves superimposing a lattice onto the

synthetic microstructure [157, 178], Figure 8.2. The properties of the lattice bonds are

dictated by the micro-structural phase onto which it is projected. In the case of concrete the

are 3 different bonds properties - matrix, aggregate and interface.

Centre particle lattice The centre particle lattice, such as in [169] is constructed by con-

necting the centre points of neighbouring features, e.g. particles or aggregates, with bonds,

69


Figure 8.2: A lattice superimposed onto a synthetic concrete microstructure (adapted from[161]).

Figure 8.3: A centre particle lattice configuration (adapted from [179]).

Figure 8.3. In this way the interactions between features is included through the calibra-

tion of bond properties. This method provides an advantage that only nodes that represent

particles are generated; a possible computational benefit. However this method provides

calibration issues as each bond represents a multiphase system and provided this issue can

be overcome, each calibration is phenomenlogical and specific to the material composition

for which it was undertaken [179]. This model approach stems from the generation of a

lattice framework from particle methods as mentioned previously [40, 151].

Long range interaction Both of these main types of lattice exist with only short range

interaction, i.e. to the nearest nodal neighbours, however longer range interaction may be

introduced, as in the work of Burt and Dougill [160] and Bažant and Tabbara [40].

70


8.2.4 Constitutive law and failure criterion

As mentioned in Section 7.1, there have been many failure criteria proposed to model the

global behaviour of brittle, ductile and quasi-brittle materials. The failure criterion for

individual elements in a lattice need to represent either the material phase within which it is

situated for a particle overlay lattice, or the interaction between particles in a centre particle

lattice. Generally the failure criteria is integrated within the constitutive behaviour of the

element itself.

For a spring or truss model, where each spring supports a single degree of freedom

corresponding to a simple uniaxial stress-strain relationship, the simplest criteria is the

elasto-brittle failure seen previously in Figure 7.1, wherein a critical value of stress, strain

or strain energy results in failure. Beam elements capable of representing more complic-

ated stress fields can carry failure criterion derived from the Rankine, Tresca or Von-Mises

failure/yield criterion [161, 167].

It has been suggested that using such failure criterion, where after a critical value of

load or displacement the element is failed and the load redistributed around the remaining

bonds, can give an overly brittle response and without the inclusion of tension softening

at the element level a reliance on element size is introduced [178]. Although Bažant et

al. [40] proposed a softening relationship for the truss bonds between particles, this is

only a representation of the multiphase system in a centre particle lattice configuration as

opposed to the actual behaviour of the matrix phase. It was in fact Arslan et al. [161]

and subsequently Karihaloo et al. [44] who implemented the tension softening relation in

beam elements exclusively for the matrix phase only in a particle overlay lattice, keeping

the elements representative of aggregate and interface elasto-brittle. Both models showed

an improved representation of the tension softening response over early lattice models [40,

157]. A simple brittle tension softening criteria is shown in Figure 8.4. Both Arslan et

al. and Karihaloo et al. implemented the tension softening relation using a secant elastic

modulus, Es to represent the unloading of the bond without residual damage after the peak

load.

Van Mier discussed the use of an energy potential to dictate the 1D force-displacement

between nodes [180]. This is akin to the pair-potential used to represent the relationship

between two atoms in Molecular Dynamics studies whereby energy associated with a pair

of atoms derives exclusively from the separation of the atoms. Such potentials could be

used directly as a lower-scale foundation from which to derive the constitutive relationship

at the length-scale of interest [181]. Conversely the potentials could be used as a framework

71


σbond)

ε(bond)

σ

ε εf

E

Es

Figure 8.4: A failure criterion accounting for tension softening with modified secant elasticmodulus for unloading

from which to semi-empirically fit a relationship, in essence creating an analogue between

the interaction between atoms and that of lattice nodes.

8.3 Lattice models of note

In addition to the historical perspective in the preceding sections, it is worth briefly men-

tioning here the different lattice models (and closely related lattice-particle models) which

have undergone significant development over recent years and their features.

Lattice Discrete Particle Model (LDPM) Cusatis et al. [175, 182] originally developed

a 3D model they named the Confinement Shear Lattice (CSL) for simulation of concrete, ex-

tending the principles of random spatial distributions of discrete physical particles laid out

by Bažant and Tabbara [40]. Particles, representing aggregates, were joined by bonds cap-

able of transferring axial and shear stresses which simulated deformation and subsequent

failure of the matrix phase. Rather than the energetic calibration described in Section 8.2.2

the CSL relates strains and tractions on a series of microplanes to the stresses in a con-

tinuum cell in a methodology based on that described in the microplane models [183, 184].

The CSL was succesfully coupled to a Finite Element model and demonstrated good pre-

diction of crack paths [173] with other extensions involving random material property fields

[185, 186].

Later works by Cusatis et al. [187–189] introduced a new model, entitled the Lattice

72


Discrete Particle Model (LDPM) which closely followed the CSL framework but combined

principles and numerical solving methods from the Discrete Particle Method [190]. Both

the LDPM and its predecessors require phenomological calibration from experimental data.

Virtual Internal Bond Model (VIB) Gao and Klein [191] and Klein and Gao [192] de-

veloped the Virtual Internal Bond Model (VIB) utilizing the Cauchy-Born rule of crystal

elasticity. The Cauchy-Born rule is a homogenization technique generally used to provide

continuum constitutive laws from atomistic properties using an energetic calibration. In-

stead of homogenizing at the atomistic scale, the VIB generated a random-lattice of “ma-

terial particles” which were phenomonologically calibrated according to the material of

interest.

Extensions to the VIB include the Virtual Internal Pair-Bond Model (VIPB) [193] and

the Virtual Multi-Dimensional Internal Bond Model (VMIB) [194]. In the VIPB each bond

is replaced by two separate bonds. In this manner the fracture energy can be split into

a specimen size-independent initial fracture energy and a specimen size-dependent total

fracture energy, used with short and long range potentials respectively in order to recreate

the size-effect. In the VMIB the axial bond present in the VIB is complemented by shear

bonds in order to overcome restrictions on Poisson’s ratio (in VIB, maximum ν = 0.25

for plane strain and ν = 0.33 for plane stress). The random nature of the “particles” and

resulting bonds within the VIB model make it inherently non-physical.

Distinct Lattice Spring Model (DLSM) In the thesis of Zhao [176] the virtual non-

physical structure of the VMIB was further extended in an attempt to better approximate a

real microstructure. Spherical particles were distributed randomly with connectivity set by

joining overlapping particles by normal and shear bonds through their centres. In this man-

ner the new model was named the Real Multi-Dimensional Internal Bond Model (RMIB). In

Zhao’s thesis and the published works taken from it [195–198] the numerical implement-

ation of the RMIB with springs for bonds was referred to as the Distinct Lattice Spring

Model (DLSM). The developed solver was based on the Discrete Element Method.

The DLSM (inclusive of the RMIB), which can operate on regular or irregular geometry,

allows the calibration of spring properties from macroscopic elastic properties as opposed

to the reverse for VIB. As mentioned previously, the negative shear stiffness, which oc-

curs when ν > 0.25, allows a full range of Poissons ratio of elastic solids to be simulated.

Moreover the deformation of the shear springs are evaluated from local strain using a least

squares method as opposed to particle displacements, which Zhao argues allows the model

73


to remain rotationally invariant unlike other lattice models with shear bonds [176].

The model was intended for application on dynamic fracture of rocks [195, 196] with

additional theoretical and numerical works including wave propagation validation [199],

coupling with Numerical Manifold Method [200] and code parallelization [197, 198]. Rate

dependency has also been introduced into bond properties for PMMA [201] and sandstone

[202] with the latter work constructing the model directly from tomography data.

Lattice Solid Model (LSM) The Lattice Solid Model (LSM), also called LSMearth and

ESyS_Particle, has been in development since the late 1990s. Based on molecular dynam-

ics principles, and treated as a lattice of discrete physical particles, it was originally de-

veloped to simulate dynamic response of rocks during earthquakes [203, 204]. More recent

works have highlighted the need to include 6 degrees of freedom between particles (axial,

2× shear, twist and 2×bending) an extension on the axial and shear degrees of freedom

allowed in many other models. Spring constants were initially approximated [205, 206]

and then derived for different lattice arrangements [169]. Further applications of the LSM

have included a parallel implementation of the model for simulating fault gorge [207] and

coupling with Lattice-Boltzmann method to link fluid flow in fractured systems [208].

Rigid Body Spring Network (RBSN) The Rigid-Body Spring Network (RBSN) was

originally proposed by Kawai in 1978 as a more computationally-efficient alternative to

the conventional elements used in the Finite Element Method [209]. The model consisted

of rigid sections which interacted with one another via a system of springs that controlled

behaviour over mutual surfaces. This concept was extended by Bolander and Saito [34] for

modelling crack paths in concrete beams, whereby the geometry was discretized using a

Voronoi tesselation of randomly distributed points, with spring stiffness uniquely defined

by the resultant geometry. The model was extended to 3D and given a fracture criterion

which reflects a cohesive law over cell facet areas by Bolander and Burton [210]. In this

case 6 degrees of freedom were represented by springs between boundaries.

Other researchers who have implemented the RBSN methodology in various forms in-

clude Nagai et al. [211], Vorechovský and Eliáš [212], Eliáš and Vorechovský [213], Grassl

and Jirásek [214], Grassl et al. [39] and Xenos et al. [215].

Volume Compensated Particle Model (VCPM) The Volume Compensated Particle Model

is a relatively new model. Although current studies are limited to 2D, with planned work for

an extension into 3D, the model builds upon concepts from molecular dynamics [216, 217]

74


in introducing a volumetric energy potential in addition to the pair potential mentioned in

Section 8.2.4 [218]. This volumetric term accounts for the strain energy resulting from

the cells change of volume. This particular notion of a multi-body potential will be dis-

cussed further in Section 12.1.1. Further studies extended the model to include interactions

between particles and non-nearest neighbours to allow for more realistic crack patterns

[219, 220].

8.4 Site-Bond lattice model

The lattice model used within this research is the Site-Bond model derived by Jivkov and

Yates [41]. A very brief overview of the model is given here with the reader referred to

the published work in Appendices C-I and the corresponding descriptions in Chapter 10 for

further details.

The model was proposed in an attempt to overcome the restrictions of current lattice

models which, as mentioned, can only model materials with a Poisson’s ratio within a

certain range or require non-physical processes to do so. The Site-Bond lattice is construc-

ted from a regular tesselation of truncated octahedral units cells, Figure 8.5(a). The trun-

cated octahedron was shown to be the regularly tessellating unit cell that was geometrically

closest to the average cell found in a large scale Voronoi tessellations of randomly distrib-

uted points in space [221]. In this way it is considered representative of an actual generic

microstructure. A site is located at the centre of each cell and interacts with neighbouring

sites via 14 deformable bonds, Figure 8.5(b): six bonds of length 2L (2L is the cell size)

in principal directions (through square faces), and eight bonds of length√

3L in octahedral

directions (through hexagonal faces). This formation of lattice allows for a meso-scale rep-

resentation of material microstructure, with the site at the centre of each cell representative

of a micro-structural feature, e.g. the particle or aggregate phase of graphite and concrete

respectively. The cell size can be allocated according to actual feature size, e.g. average

or distributed grain or particle size and initiators of microscopic failure for quasi-brittle

materials can be represented in the bond failure criteria, e.g. micro-cracks and porosity.

The initial state of distributed porosity can be accommodated at the boundary interfaces,

with further porosity introduced due to interface failure as a result of mechanical loading.

In this way the initiating damage mechanism is representated such that the interaction and

coalscence can be subsequentially captured.

The truncated octahedron was initially proposed as a unit cell for use in the study of

75


Figure 8.5: (a) Cellular representation of material; (b) the skeletal bond structure

stress corrosion cracks between grains [222, 223]. Not until later was the capability of such

a model to recreate macroscopic elastic behaviour of any isotropic material explored [41].

In this work, beam elements were used as bonds between sites in a numerical study. The

model was demonstrated to be capable of reproducing elastic behaviour upto a Poisson’s

ratio of 0.5. It was shown by Jivkov et al. [224] that it was possible to use such a model

for the study of progressive damage evolution, without further constitutive assumptions

regarding crack path direction and flaw coalescence. In this work it was shown that such a

framework could represent the accumulative damage process from initiation through to the

interaction and coalescence of flaws in cement under tensile and compressive loading. This

work was applied to more complex compressive loading conditions for concrete, using a

porosity distribution obtained from X-ray tomography, in [225] and extended in [226].

76

Chapter 9

Generalised Continuum

Generalised continuum theories (sometimes called enriched continuum theories [227]) have

been developed to deal with materials for which the continuum approximation is insuffi-

cient due to the presence of non-negligible micro-structural features, such as quasi-brittle

materials. There are considered to be 3 types of generalised continuum theory [228]:

• “Weak” non-local/strain-gradient theories, whereby the 3 degrees of freedom in the

form of displacements are enriched by including the gradient of strain into the con-

stitutive laws [229].

• “Strong” non-local/non-local integral theories, whereby the stress at any point is re-

lated to the state of the whole body [229].

• Couple stress or micropolar theories, where the rotations of continuum points are

considered as extra degrees of freedom. Such theories are of interest in areas of this

research.

The common ground in such theories is the introduction of a length scale. In couple stress

theory, unlike the non-local theories, it is possible to directly link this length scale to the

micro-structural features of the material to provide a physical basis [42]. As referred to in

Chapter 2, the naturally introduced length scale and rotational invariance in couple stress

theories may prove useful in discrete models [44–47]. As such an outline of a consistent

couple stress theory is given here.

9.1 Couple stress theory

In couple stress theory the continuum is modelled as a set of rigid points within the con-

tinuous mass. Each point has 3 additional degrees of freedom in comparison to the classical

77

CHAPTER 9. GENERALISED CONTINUUM

continuum theory, in the form of rotations. These rotations are associated with the couple

stresses in the same way that the force stresses (defined solely as stresses in a classical the-

ory) are associated with the point displacements, Figure 9.1. Furthermore the loads related

to these rotations are described completely by a body couple. By deriving the equilibrium

equations inclusive of these couple stresses it is found that the stress tensor is now in fact

non-symmetric, a characteristic of polar-continua [43]. Couple stress theory is by no means

a new development. It was first used to describe material behaviour early in the 20th cen-

tury by the Cosserat brothers [230] after the idea of couple stress was presented by Voigt

[231]. This theory was on the whole ignored until the 1960s when a rejuvenation of couple

stress research occurred due to a desire to understand the mechanisms behind micro-crack

growth for more accurate crack assessment and characterisation. Hadjesfandiari and Dar-

gush [43] provide an overview of the historical developments of couple stress theories and

these developments will be briefly summarised here.

Figure 9.1: The stresses present on a 2D couple stress element under static load [232]

In the original Cosserat theory the rotations associated with the couple stresses were

defined as the independent micro-rotations of the points within the continuum. These are

considered unrelated to the rotation of the material at the macro-scale. Developments in

the 1960s took a different approach by constraining the micro-rotation to be equal to the

macro-rotation. The theory produced from such assumptions was not without problems,

despite allowing the use of the gradient of the rotation vector as the curvature. More spe-

cifically, the indeterminacy of the spherical part of the couple stress tensor proved problem-

atic and for this reason the theory is referred to as the indeterminate couple stress theory

[233] or Cosserat model with constrained rotations [234]. More recent advances [235, 236]

reinstated the independence of the micro-rotation from the macro-rotation in a branch of

78

CHAPTER 9. GENERALISED CONTINUUM

couple stress theory called micropolar theories (sometimes referred to as Cosserat model

with free rotations [237]). This effectively models the material as a collection of discrete

particles, raising concerns about whether this can actually represent a continuum at all. The

most recent development was developed by Hadjesfandiari and Dargush [43], wherein a

consistent couple stress theory is derived using only true kinematical quantities, effectively

introducing natural size dependence into the continuum theory. An outline of the consistent

couple stress theory developed by Hadjesfandiari and Dargush is included in the published

work presented in Appendix C.

Cosserat theory and the developing micropolar elasticity have established themselves

with many applications. For example, the structure of bone [238, 239], cellular and granu-

lar materials [240] provide varying length scales associated with micro-structural features.

Furthermore, discrete materials such as jointed rocks have found that the discrete nature of

the continuum models in micropolar theory provide accurate and useful insight into their be-

haviour [241]. Despite the promise of micropolar elasticity, there are problems which limit

its acceptance as an additional improvement on classical continuum mechanics. Boundary

conditions and material moduli have proved difficult to prescribe and measure respectively,

providing difficulty in model characterization [42].

79

Part III

Contribution to the field

80

Chapter 10

Modelling and published work

This thesis is presented in the form of published or submitted work. The portfolio of work

described by published works can be found in Appendices C-I1. In this Chapter each pub-

lished work will be outlined with a brief overview including context and discussion. Works

during the first 12 months of the project focused on model calibration, Section 10.1, and

initial studies on early iterations of the Site-Bond model, Figure 10.1. Later works, from

12-24 months and 24-36 months as shown in Figures 10.2 and 10.3 respectively, focused

on more complete validated studies of the microstructure-informed Site-Bond model, Sec-

tion 10.2. It should be mentioned that the author has featured as a named author on other

publications although the relevence of the work or the magnitude of contribution from the

author is not deemed significant enough to warrant discussion [242–244].

1Please note that although the content of published works remains unchanged from the published form,the paper itself may be presented in its unpublished format or with changes made to formatting (namely pagemargins) to provide continuity with the rest of this thesis.

81

CHAPTER 10. MODELLING AND PUBLISHED WORK

Figure 10.1: Completed works during 0-12 months of the project

Figure 10.2: Completed works during 12-24 months of the project

82


Figure 10.3: Completed works during months 24-36 months of the project

10.1 Model calibration

Meso-scale features and couple stresses in fracture process zone Calibration of the

micro-structurally informed lattice model which forms the initial project objective may

require the inclusion of micropolarity and couple stress theory to obtain the required six

degrees of freedom between each lattice node. Prior to calibration it must be determined

whether the inclusion of couple stresses is beneficial to the calibration procedure and can in-

deed naturally introduce a length scale into local material behaviour modelling. Hence, this

work was a pre-requisite to calibration of the Site-Bond model with spring-bonds. This was

done by constructing a simple finite element model consisting of rigid particles within a de-

formable matrix. The relative rotations and displacements between particles were measured

for 3 different loading cases, with varying degrees of couple stress; hydrostatic compres-

sion, pure twist and pure bending, and were related to the energy within the unit cell. The

energy within the unit cell was used as an indication of size effect. The magnitude of rota-

tions between particles was used to suggest a reduction of springs within each bond prior

to calibration.

This work was presented at the 13th International Conference of Fracture in Beijing,

China. The corresponding conference paper can be found in Appendix C. Further work

towards verifying the physical realism of couple stress theory was planned in the form of

experimental validation, although it was never undertaken due to project time-constraints.

83


A meso-scale site-bond model for elasticity: Theory and calibration This paper, for

which Dr Mingzhong Zhang is the main author, derives analytically the spring constants

for the Site-Bond model in terms of the macroscopic elastic constants of the material. The

version of the Site-Bond model around which this publication focuses, involves the repres-

entation of bonds with six independent elastic springs resisting three relative displacements

and three relative rotations between sites, Figure 10.4. This yields four spring types [169]

with axial, kn, shear, ks, twisting, kt , and bending, kb, stiffness, which could, in general, be

different for principal and octahedral directions.

Figure 10.4: The 6 degrees of freedom represented by springs in the Site-Bond model

The model is calibrated by equating the strain energy in the unit cell and corresponding

continuum for typical global elastic properties of graphite under a homogeneous displace-

ment field. The lattice arrangement has been shown to generate macroscopic cubic elasti-

city. In this branch of elasticity 3 independent elastic constants are required to describe the

material behaviour as demonstrated in Equation 10.1 where W is strain energy density and

E, ν and µ are Youngs modulus, Poission’s ratio and shear modulus respectively.

Wcontinuum = f (E,ν ,µ) (10.1)

Under the initial assumption of a classical continuum (i.e. without micropolarity), relative

angular motion is ignored and the Site-Bond model will be composed of only axial and

shear spring types, the stiffnesses of which are different for the principal and octahedral

directions. The strain energy density within such a unit cell is shown in Equation 10.2.

Wdiscretecell = f((kn,ks)

principal ,(kn,ks)octahedral

)(10.2)

84


This creates an indeterminate problem when calibrating the stiffness values [169], with an

infinite number of stiffness constant combinations that will generate the cubic elasticity

constants. As such, the calibration is over determined. Without micropolarity (or another

suitable method) to make the calibration procedure determinant, the shear stiffnesses in the

principal and octahedral directions are considered to be related, i.e. there are three spring

constants to determine from the three cubic elasticitity constants.

This work was presented at the 2nd Global Conference on Materials Science and En-

gineering, Xianning, China with the corresponding paper published in Materials Research

Innovations, Appendix D.

10.2 Microstructure-informed model

Lattice-spring modelling of graphite accounting for pore size distribution This con-

ference paper presents an initial iteration of the completed first objective - a micro-structura-

lly informed lattice model. The model is developed using an initial numerical calibration

assumption (which was later improved on analytically in the previously described paper).

In the same manner as the analytical calibration, strain energy in the continuum was equated

to the strain energy in the discrete cell, the latter given in terms of stiffnesses and displace-

ments. Rather than a more rigorous derivation the spring constants were calculated by

equating the continuum and discrete energies for several loading cases and then solving for

the spring constants simultaneously. A population of porosity, produced statistically from

experimental data, was introduced onto the model with the constraint of a single pore per

cell face used to scale the cell size. Progressive damage was modeled for several loading

cases representative of the region ahead of a crack tip. This work provides an initial step

towards deriving a load independent damage evolution law which could be implemented in

a continuum model and utilised as part of a structural integrity assessment.

This work was presented at the 7th International Conference on Materials Structure

and Micromechanics of Fracture in Brno, Czech Republic. The corresponding paper was

published in Key Engineering Materials2, Appendix E.

2Two minor issues regarding the Introduction were brought to the attention of the author following pub-lication. Firstly, it is stated that “Synthetic graphite is manufactured from petroleum cokes”. Although this istrue of some graphites. Secondly, it is stated that Gilsocarbon is “a relatively fine-grained graphite”. Gilso-carbon is in fact a medium grained graphite. This information is presented correctly throughout the rest of thethesis.

85


Discrete lattice model of quasi-brittle fracture in porous graphite An extension of the

work applying the Site-Bond model to graphite was published in a special edition of Mater-

ials Performance and Characterisation, Appendix F. The analytically derived calibration

constants presented in Appendix D were implemented and an improved representation of

length scale was included by allocating a cell size according to the average distance between

particles. Distributions of micro-structural features were obtained from the literature. The

model was used to explore the effect of radiolytic oxidation on two different grades of nuc-

lear graphite by using increasing porosity as analogue to the effects of radiolytic oxidation.

In this manner the model provides the foundations for an integrity assessment methodology

for ageing plants, although more information on the change of pore size distribution with

increasing oxidation is required.

Fracture energy of graphite from microstructure-informed lattice model A further

extension of the previous microstructure-informed model was presented at the 20th European

Conference of Fracture in Trondheim, Norway, with the corresponding paper subsequently

published in Procedia Materials Science, Appendix G. The main extension from the pre-

vious works was the increased computational program, which included more detailed cov-

erage of porosities. This allowed the focus of the work to shift towards exploration of the

relationship between tensile strength and porosity, with comparison to literature.

Site-bond lattice modelling of damage process in nuclear graphite under bending It

was shown in a preliminary study that the Site-Bond model when using three linear springs

(of type Spring 2 in commerical software ABAQUS [141]) was unable to model the elastic

line of a cantilever specimen due to the non-linear displacement field. The overall response

was overly stiff, Figure 10.5. As a result the basis of the bond was changed from spring

bundles to a single spring-like connector bond whereby the stiffness is related to a change

of bond length only. This provides compatiblity with the geometric discretization theory

[245], whereby balance of angular momentum is dictated a single force between displaced

positions of sites. The new model was used to reproduce the force-displacement response of

a Gilsocarbon micro-cantilever specimen using pore size distributions and volume fractions

obtained from pycnometry and mercury porosimetry. Cell size scaling methods from the

previous models were inappropriate for reproducing actual specimen geometry (and incid-

entally for modelling at the length-scale of the cantilever specimen, which is significantly

lower than that of a single filler Gilsocarbon particle), an aspect of the model which may

prove useful for industrial structural integrity assessement. As such a cell sensitivity study

86


was undertaken with the size of each cell scaled to match the specimen dimensions.

Figure 10.5: The displacement of a cantilever beam in the y-direction along the beam length(z-direction)

This work was presented at the 23rd International Conference on Structural Mechanics

in Reactor Technology in Manchester, UK. The conference paper was published in the

corresponding conference proceedings, Appendix H.

Multi-scale modelling of nuclear graphite tensile strength using the Site-Bond lattice

model The final piece of work for this project was published in Carbon3, Appendix I. The

work aimed to develop a multi-scale modelling methodology for the Site-Bond, with global

response derived from the responses of the independent phases. Microstructure-informed

Site-Bond models were produced separately for the filler particles and matrix phase from

experimental data. The obtained behaviour was used to inform a multi-scale Site-Bond

model. The simulated values of Young’s modulus and tensile strength proved a good match

with values from literature.

3It was brought to the attention of the author following publication that the definition of the graphite used inthe Material and microstructure section should read “IM1-24 Gilsocarbon polygranular nuclear graphite”.

87

Chapter 11

Conclusions

• Lattice models are a branch of discrete, local approach models which allow the con-

struction of a micro-mechanically based material constitutive law capable of gener-

ating the expected non-linear quasi-brittle response of nuclear graphite.

• A methodology for informing a lattice model with microstructure information has

been developed over several published works with a view to increasing the under-

standing of deformation and fracture behaviour of nuclear graphite. Studies have

explored choice of bond element, appropriate meso length-scale, calibration of bond

stiffness constants and microstructure mapping.

• Models produced using this methodology have been validated against experimental

data in the form of elastic constants, material properties and general quasi-brittle

behaviour to good effect.

• The model has been used to explore damage evolution in an initial attempt to develop

a load-independent damage law for possible uses in the continuum models used for

structural integrity assessment.

• The effect of porosity, as an analogue to radiolytic oxidation, on mechanical response

has been explored, providing a platform for the assessment of ageing AGR plants.

• More advanced studies have included reproducing the force-displacement response

of a micro-cantilever specimen and a multi-scale model, whereby a global model is

built up from the response of models of the individual phases.

88

Chapter 12

Further work

The results from this research have been generally positive, justifying further research into

the Site-Bond model. In this chapter several areas of potential future work are discussed.

12.1 Model calibration

The analytical calibration procedure used throughout this research can be improved upon,

refining the model’s capability to discretely represent a continuum volume. The inclusion

of micropolarity described in Appendix C was an attempt to do so, incorporating the effect

of local behaviour at an appropriate length scale into the calibration. Although the conclu-

sions drawn from the work were useful in justifiying assumptions made in the analytical

calibration, the failure to isolate a couple stress constant restricted the impact. As such,

further work may include the exploration of other theories neccessary for derivation of an

improved constitutive relationship between bond deformation and energy within a cell. Two

possible routes will be described here.

12.1.1 Improved relationship between bond deformation and energy

A calibration procedure may draw inspiration from Molecular Dynamics (MD), the origin

of most discrete models for solid mechanics applications, in particular the multi-body po-

tential theories of Finnis and Sinclair [216] and Daw and Baskes [217]. Discrete models

generally work on the same principle whereby energy is derived from a change in spatial

position of a set of discrete points given a constitutive link between force and displace-

ment (in a static sense). In the Site-Bond model discrete points represent micro-structural

features (filler particles in the case of graphite) whereas in MD they represent atoms.

89

CHAPTER 12. FURTHER WORK

In MD the interaction of atoms is approximated by a potential function [246]. Many

potential functions are defined as two-body or pair potentials where the energy between two

atoms is linked exclusively to the relative positions between the atoms. For a system of two

atoms, the resulting potential takes the form Ui j = f(ri j)

where ri j is the distance between

the atoms. Examples of such empirical potentials include the Lennard-Jones potential,

Equation 12.1 and Morse potential, Equation 12.2. In these potentials ε and α are constants

used for curve fitting. Potentials are selected depending on their suitability for a specific

application.

Ui j = 4ε

[(σ

ri j

)12

−(

σ

ri j

)6]

(12.1)

Ui j = ε

[e−2α(ri j−r0)−2e−α(ri j−r0)

](12.2)

Using only pair-potentials to describe atom interactions has significant disadvantages, such

as an inability to fully reproduce elastic constants. Independent works in the 1980s by

Finnis and Sinclair [216] and Daw and Baskes [217] explored the use of a semi-empirical

multi-body potential where an additional term is introduced to account for the local dens-

ity of atomic sites in BCC and FCC crystals respectively. The Finnis-Sinclair multi-body

potential is especially relevant to the site-bond model, which is also constructed in a BCC

arrangement. In multi-body potentials the energy per atom can be split into the pair poten-

tial energy, un and the multi-body term, or cohesive potential, up:

utot = un +uP (12.3)

un and up are given by Equations 12.4 and 12.5 respectively where A is a constant, ri is the

distance from an atom at the BCC crystal origin to the ith atom.

uP =12 ∑

i 6=0V (ri) (12.4)

un =−A f (ρ) (12.5)

The Finnis-Sinclair model introduced the function f (ρ) =√

ρ , where the density function,

ρ , defined in Equation 12.6 is based on a tight-bonding technique and density functional

theory.

90


ρ = ∑i 6=0

φ (ri) (12.6)

Put simply, the pair potential is dependent solely on the distance between two atomic sites.

For each pair-potential between sites the cohesive potential introduces an energy which is

summed over all the sites within a specified radius. In this sense, the force between two

sites then depends on the density of sites around them as well as their separation. From

these definitions it may be seen that the pair-potential is an analogue to the constitutive law

currently used in the Site-Bond model. By using this energy in isolation the relative spatial

changes of sites are considered but it may be that the deformation of the local cell around

each site is unaccounted for. In the Site-Bond model the local density of sites around

a central site defines the local volume around that particular site. As such the cohesive

potential which incorporates this local density can then be interpreted as a volumetric term

whereby it accounts for an energetic change resulting from a deformation of a local volume,

or unit cell, around a central site. The use of such volumetric terms is not entirely new in

lattice models having recently been introduced in the Volume Compensated Particle Model

discussed in Section 8.3 and introduced to an extent in the multi-scale study described in

Appendix I.

The Finnis-Sinclair model and similar models are semi-empirical with curve fit exper-

imental constants used to reproduce experimental observations for different elements. For

the purposes of the Site-Bond model it is appropriate to find a more general link between

changes in geometry and energy. The pair potential used in such a relationship may follow

the form:

up =12 ∑CA f

(4AA0

)(12.7)

where CA is a constant. It may be noted that Equation 12.7 uses a function of the relative

change of area rather than displacement (as is currently used). This is for two reasons. The

first is to conserve angular momentum, which must be followed as a result of a discretization

of elasticity [245]. The second is to be able to include a true representation of stress,

whereby the reduction in area due to bond extension is explicitly incorporated into the

constitutive law. At this stage it is unclear as to whether there is a definitive link between

a relative increase in bond length and the change in the corresponding change in area of

the cell face. If a link is found then it is deemed suitable to use either the change of area

explicitly or implicitly through the change of bond length within the constitutive law. In a

91


similar manner the volumetric, cohesive potential term may follow the form:

un =CV f(4VV0

)(12.8)

where CV is a constant.

12.1.2 Couple with dual graph

The Site-Bond model, and indeed any lattice based model, can be formulated as a mathem-

atical graph, where sites (or lattice nodes) and connecting bonds are represented as vertices

and edges respectively in a cell complex. Such formulations can be expressed as matrices

and implement the field of discrete exterior calculus as an analysis tool. Following the

terminology outlined in [242] where a more complete graph-based formulation of the Site-

Bond model is presented, the geometry of the model can be defined as a 3−complex. As

such the vertices, edges, faces and cell volumes in a graph are referred to as 0−, 1−, 2−and 3−cells respectively. When described in this fashion each graph has an associated dual

space/dual graph, whereby each geometric entity within a graph has a corresponding entity

in the associated dual. Taking the example of a graph A and its dual graph B, each vertice

in graph A corresponds to a cell volume in graph B. In the same manner each edge in graph

A corresponds to a face in graph B and so on.

The dual-graph concept may be used to describe a discretised volume more rigorously.

At present, the Site-Bond model calculates material response according to two variables,

namely lengths and areas or lengths and volumes (the discussion regarding multibody po-

tentials essentially proposes using areas and volumes). This is a means of modelling the

initial graph with little consideration of the influence of the dual graph. By simulating the

graph and its dual simulatenously, calculating equilibrium for both at each increment, al-

lows the material response to be calculated using lengths (and the relative change thereof)

exclusively. This is a topic for further consideration.

12.2 Explore the effect of porosity on graphite failure energy

at grain level

Exploring the effect of porosity on graphite failure energy at the grain level using lower

scale modelling would allow the development of a physically representative failure criteria

when accounting for porosity, which could be implemented within the Site-Bond model.

92


The failure criteria used in the model presented in this thesis (with the exception of the

multi-scale model presented in Appendix I) consists of a linear softening constitutive law

with an energetic failure criteria whereby the failure energy is calculated as the product

of the corresponding face area and the separation energy of graphite. Pores are represen-

ted on bonds through the reduction of the bond failure energy. This approach has given

promising results in reproducing the “graceful” pre-peak softening of quasi-brittle mater-

ials and matching experimentally measured tensile strength values. However, this failure

criteria and the changes under the influence of porosity is somewhat arbritary and it may be

beneficial to derive the affect of porosity on bond energy directly from lower scale models.

12.3 Inclusions and validation of physical phenomena

Although attempts have been made to validate the model at every opportunity, further proof

of concept for different grades of graphite in different component geometries and under

different loading conditions or environments will be beneficial. The hinderance in many

cases of validation is finding consistent data, either microstructure information from which

to populate the model or global response data for comparison. Taking the data used for the

paper outlined Section 10.2 as an example, although all data was obtained from reputable

sources, the values of elastic moduli are dynamic measurements which may differ from

static values and the porosity and filler volume fraction disclosed are lower than some val-

ues used in industry. Such discrepencies may introduce discrepencies between simulation

results and the physical response of graphite specimens. Extensions to the Site-Bond model

may include:

• Directional bias in certain graphite grades. As discussed in Chapter 4 many grades of

graphite have acicular particles with preferential alignment. This could be introduced

into the model either by stretching the cell dimensions along the axis of maximum

particle length or introducing a cleavage direction of particle to simulate the cleavage

directional bias along the particle aligned axis.

• Irradiation in nuclear graphite. Modelling irradiation effectively in the Site-Bond

model is dependent on suitable characterisation of progressive changes to the micro-

structure due to irradiation [11].

• Creep. Although not covered in particular detail in the review of literature, irradiation

creep and the fundamental understanding of the underlying mechanisms, is a signific-

93


ant challenge when considering nuclear graphite. Inclusion of viscoelastic behaviour

into the constitutive relationships of bonds may have potential with relations to such

an application [247].

• Multi-physics phenomena. A long-term goal of the Site-Bond model would be to

couple different physical phenomena to produce a multiphysics model in a similar

manner to other lattice models [208]. Of particular interest would be the coupling

of the site-bond model developed for solid mechanics applications here and that de-

veloped for diffusive reactive transport in porous media [248]. This would be of

particular interest for geological materials.

12.4 Structural integrity assessment

Although the research in this thesis provides a solid platform, further work is needed to

help the Site-Bond model meet the needs of industry with regards to structual integrity

assessment. The damage evolution study presented in Appendix E could be extended to

allow for direct input into an industry standard continuum model. Moreover the studies

related to radiolytic oxidation and ageing plant assessment described in Appendices F and

G may be built upon, incorporating the change of both porosity and pore size distribution

over time which results from radiolytic oxidation. Finally, once further proof of concept is

achieved the numerical stability and computational efficiency of the model maybe refined

to produce a user-friendly tool.

94

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117

Part IV

Appendices

118

Appendix A

Linear Elastic Fracture Mechanics

(LEFM)

This appendix gives a brief historical overview of linear elastic fracture mechanics (LEFM),

to compliment the presented work. The information in this appendix is based on derivations

from standard fracture mechanics and solid mechanics texts [16, 17, 118, 249] with some

additional background reading [18].

A.1 Energy approach

LEFM was originally developed by Griffith [22] in the 1920s as a means to explain the frac-

ture of brittle materials at stresses below the expected value (the value at which the atomic

bonds would be broken), a phenomena that Griffith believed was due to unobserved flaws

within the material which magnified the local stress producing a reduced global strength.

His experimental work provided a link between fracture stress and flaw size. He used pre-

vious work by Inglis [250] which had introduced the stress concentration factor.

A.1.1 Stress concentration factor (Inglis)

The work of Inglis [250] demonstrated evidence that stress was concentrated at flaws, spe-

cifically, elliptical holes, lowering the global strength of the infinitely large flat plates in

which they were situated. For the elliptical flaw shown in Figure A.1 the stress at the flaw

tip (point A) can be expressed as :

σA = σ

(1 +

2ab

)= σ

(1 + 2

√aρ

)(A.1)

119

APPENDIX A. LINEAR ELASTIC FRACTURE MECHANICS (LEFM)

Figure A.1: An elliptical hole in a flat plate [17]

The stress concentration factor, K, was defined as the ratio between σA and σ . The termaρ

indicates a significant reliance on flaw shape. K varies between 3 for a circular flaw

(a = b) to infinitely large when the flaw develops into a crack (radius of curvature ρ =

0, b→ 0). This is suggestive that any sharp crack should concentrate stress to an infinite

level that guarantees failure upon any non-zero load. This is clearly a physically unrealistic

prediction. In all materials the stress levels ahead of a defect are reduced to various extents

by processes dissipating strain energy, e.g. plasticity, micro-cracking, void formation and

growth, crazing etc.

A.1.2 Griffith approach

Griffith employed the first law of thermodynamics to perform an energy balance, rather

than focusing on the apparently infinite stress at the crack tip. He proposed that crack

propagation will only become energetically favourable if the elastic strain energy released

is in excess of the energy required to produce a new crack surface (surface energy). He

defined the total energy within a cracked plate, U , as:

U =U0 +Ue +Us−F (A.2)

where:

U0 is the elastic energy of the loaded uncracked plate, which is a constant,

Ue is the change in energy due to the introduction of a crack into the plate,

U s is the change in energy due to the creation of two new crack surfaces,

120


F is the external work on the plate, which is equal to U0 under displacement control

conditions, i.e. the derivative of F with respect to crack extension equals 0.

For an incrementally increasing crack area dA, the equilibrium condition for crack growth

is achieved when:

d(Us +Ue)

dA= 0 (A.3)

The elastic strain energy released by crack formation can be evaluated from:

Ue =12

ˆa

σ(x)4(x,a)dx (A.4)

where σ(x) is the stress distribution around the crack and 4(x,a) is the vertical crack

opening.

Figure A.2: The strain energy released around a crack of length 2a (reproduced from [118])

Griffith used the work of Inglis [250] to show that for a through thickness crack of

length 2a in an infinite plate, where B is the plate thickness, Figure A.2:

Ue =πσ2a2B

E(A.5)

where E is the Young’s modulus of the material. This can be shown to be:

Ue =πσ2a2k

E(A.6)

where k = 1− v2 for plane strain and k = 1 for plane stress. ν is Poissons ratio. For the

above crack, the surface energy U s is given by:

121


Us = (2aB)2γ (A.7)

where γ is the surface energy per unit area. Substituting A.6 and A.7 into A.3 gives:

d2(4acBγ)

dadB=

d2(πσ2

f a2cB

E )

dadB(A.8)

2γ =πσ2

f ac

E(A.9)

Resulting in a fracture stress of:

σ f =

(2Eγ

πac

) 12

(A.10)

For crack propagation to be energetically favourable the elastic strain energy released upon

crack formation must exceed that required to create the new crack surfaces, Figure A.3.

This same analysis can be applied to other crack shapes.

Crack Energy

Us

Ue

Us + Ue

ac

A - equilibrium

Figure A.3: The prediction of the Griffith energy balance for energetically favourable frac-ture (reproduced from [118])

A.1.3 Modified Griffith approach

The Griffith model accurately predicted the failure stress of glass but predictions for metals

were found to be conservative. In 1948 Irwin [23] modified the Griffith model in a bid to

better describe the behaviour of metals.

Griffith’s model included only the materials surface energy with relation to the work of

fracture. This surface energy is a measure of the total energy of broken atomic bonds in a

122


unit area and as such represents only purely brittle materials, in which fracture occurs upon

breaking of these bonds. Irwin’s modification proposed that the plastic strain, as a result

of dislocation motion around the crack tip, observed in studies by Orowan [251] could be

used to approximate the dissipated energy in such plastic flow. Orowan [252] later went on

to independently propose a similar modification. The modified equation is given by:

σ f =

(2E(γs + γp)

πa

) 12

(A.11)

where γp is plastic work (per unit area of surface created) and γs is the material surface

energy.

The work of fracture, or fracture energy, which for the original Griffith theory consists

solely of the surface energy is redefined as the combined expression for surface and plastic

energy:

w f = γs + γp (A.12)

The plastic work is typically considerably larger in metals than the surface energy and hence

improves on the conservative estimate given from the original model. The same concept is

equally valid for introducing other energy dissipation mechanisms.

A.1.4 Energy release rate

In 1956 Irwin [253] derived the concept of the energy release rate from the modified Griffith

theory to avoid having to use surface energy and plastic contributions in calculations. The

energy release rate, or crack driving force, G is defined as the derivative of potential energy

(i.e. the available energy for crack growth) with respect to crack area. At the moment of

fracture equilibrium Equation A.3 is satisfied and the energy release rate reaches a critical

value Gc, a material property representing the materials fracture toughness or resistance to

fracture.

Gc =dUs

dA(A.13)

For a perfectly brittle material:

Gc = 2γs (A.14)

or accounting for other energy dissipation mechanisms:

123


Gc = 2w f (A.15)

So using the previous cracked example, the energy release rate and hence critical energy

release rate can be expressed as:

G =πσ2a

E(A.16)

Gc =πσ2

f ac

E(A.17)

which is clearly the same as Equation A.9. The crack energy release rate can be shown to

be equal for both load controlled and displacement controlled loading, another advantage

over the modified Griffith approach where load is assumed constant during crack extension.

A.1.5 The R curve

The R curve is a graphical representation of the stability of the crack growth for a given

crack energy release rate. The value of G at which crack extension occurs, Gc, is deemed

equal to a term R, a material property defining its resistance to crack extension.

To determine the state of crack stability, both G and R are plotted against crack size,

a. These are simply called the driving force and resistance curves respectively. When the

crack energy release rate increases with crack length at a lower rate than the material’s

resistance, the crack is deemed stable:

dGda

<dRda

(A.18)

When the crack energy release rate increases with crack length at the same rate as the

material’s resistance, the crack is deemed stable, the crack is on the point of instability:

dGda

=dRda

(A.19)

When the crack energy release rate increases with crack length at a greater rate than the

material’s resistance, the crack is deemed unstable:

dGda

>dRda

(A.20)

For a flat R curve, Figure A.4(a), the material’s resistance to crack extension remains con-

stant so there is a unique value for Gc. This is not so simple for a rising R curve, Figure

124


A.4(b), where materials are characterised by the value of G at which Equation A.19 is

generally used.

Figure A.4: (a) Flat R curve (b) rising R curve [17]

A.2 Stress intensity approach

Figure A.5: Coordinate and element definition ahead of a crack tip [17]

In 1957 Irwin [254], using past work by Westegaard [255], demonstrated that the stress

field around a crack tip in an elastic material, as shown in Figure A.5, takes the form (neg-

lecting higher order terms) of:

σi j =KI√2πr

fi j (θ) (A.21)

where r is the distance from the crack tip and fi j is a geometrical function of the angle θ .

The stress is shown to be proportional to 1√r , such that a stress singularity is reached at the

crack tip, and also a constant KI . Irwin defined this constant as the stress intensity factor.

If this constant is known then the entire crack-tip stress field can be evaluated. KI is the

constant for mode I loading, Figure A.6, although the concept remains for mode II and III

125


loading. Derivations of standard results for these modes can found in the cited texts for

these Appendices. In mixed mode loading the total stress tensor is found from summing

the stress tensors from individual loading modes.

Figure A.6: The 3 modes of loading for a crack [17]

Irwin noted that the stresses appear proportional to√

πa and as such a general expres-

sion for the stress intensity factor for different geometries can be expressed as:

KI = σ√

πa f( a

W

)(A.22)

where W is the width of the cracked plate and as before 2a is the crack width.

The value of the stress intensity factor at fracture, KIc, may be used as an alternative to

the energy release rate as a material property which measures fracture toughness. Despite

the stress intensity factor providing a locally characterising material description through

stress analysis as opposed to the global energy change which quantifies the Griffith method,

both provide an identical approach to fracture mechanics. Although both approaches are

equivalent, the use of the stress intensity factor provides consistency across geometries

through f( a

W

). By combining Equation A.16 and Equation A.22, considering fracture

mode I, G can be expressed as a function of KI for linear elastic materials, such that:

G =K2

IE ′

(A.23)

where E ′ = E for plane stress and E ′ = E1−ν2 . This can be proved for all crack geometries

under each mode of fracture (for full derivation see [17]).

126


A.3 Crack tip yielding

LEFM represents only fully elastic materials which may undergo plastic work only at the

scale of dislocations in metals or similar. Even with this plastic work, stresses in the crack

domain reach a singularity at the infinitely sharp crack tip. As mentioned this is a non-

physical problem and in reality the crack tip cannot be infinitely sharp, with crack-tip yield-

ing creating a plastic zone around the tip. As the size of this zone grows and is no longer

small in comparison to the appropriate geometry, the validity of LEFM is reduced. Two

methods were proposed in the early 1960′s to introduce a plastic-zone correction for small-

scale yielding within LEFM, where the yield or plastic zone size can be estimated; the Irwin

approach and the Dugdale-Barenblatt cohesive zone approach. If yielding above this scale

occurs then this non-linear behaviour must be accounted for with elastic-plastic fracture

mechanics (EPFM).

A.3.1 Irwin’s plastic zone correction

Irwin’s plastic zone correction [256] assumed the plastic zone at the crack tip was circular.

From Equation A.21 the normal stress σyy along the x-axis (crack plane) at θ = 0, is given

by:

σyy =KI√2πr

(A.24)

Initially the size of the plastic zone can be estimated by finding the distance away from the

crack at which the stress reaches the value at which yield occurs, i.e. the distance is when

σyy = σY S at a distance ry. The solution for plane stress is given by Equation A.25.

ry =1

2π

(KI

σY S

)2

(A.25)

Assuming the stress cannot exceed the yield value (i.e the material is elastic - perfectly

plastic so any effects from strain hardening are ignored), the stress along the crack plane

in the plastic zone remains at the yield value. This initial assumption, as shown in Figure

A.7, essentially cuts off the elastic stress distribution at the yield stress value. Equilibrium

is left unsatisfied as the stresses above the “cut-off” point (the cross-hatched area on Figure

A.7) are ignored rather than redistributed due to the constraint of a elastic-perfectly plastic

material.

Redistributing the additional stress into a larger plastic zone of size rp can be done by

127


Figure A.7: Estimates of the plastic zone size for small-scale yielding. The distances ry andrp represent the first-order and second order estimates respectively [17].

balancing the forces. The larger estimate of the plastic zone size rp is shown to be twice

that of the initial estimate, ry from Equation A.25:

σY Srp =

ˆ ry

0σyydr =

ˆ ry

0

KI√2πr

(A.26)

rp =1π

(KI

σY S

)2

(A.27)

Irwin represented the local reduction of stiffness in the plastic zone with an increased ef-

fective crack length, where the effective crack ends in the centre of the plastic zone at a

distance of ry:

ae f f = a+ ry (A.28)

This effective crack size can be used to evaluate an effective stress intensity factor, Ke f f .

A.3.2 Dugdale-Barenblatt cohesive zone/strip concept

The cohesive zone concept was independently developed by Dugdale [108] and Barenblatt

[109] based on a strip of material ahead of the crack tip which has yielded. Although

they work on the same principle, superimposing cohesive forces onto a crack under tension

causing a smooth crack closure, both models were developed for different applications. In

128


essence only Dugdale’s strip yield model was developed to correct for small-scale yielding

in LEFM. Barenblatt’s “cohesive force” model was developed for brittle materials by mod-

elling forces between atoms at the crack tip. The conceptual similarities in the models has

led to both being referred to as Dugdale-Barenblatt or cohesive zone/strip models.

A.3.2.1 Dugdale strip yield model

In Dugdale’s strip yield model, Figure A.8, the plastic work is situated along a strip ahead

of the crack. This model is similar to Irwin’s approach in that the crack is given an effective

length, which exceeds its actual length. However in this model the crack faces within the

plastic zone of length ρ , ahead of a crack of half length a, where ρ � a, are deemed to be

subject to a constant closure stress the magnitude of which equals the yield stress, σY S. For

this model the strip yield is treated as the cohesive zone.

Figure A.8: The strip-yield model [17].

Figure A.9: The crack-opening force, P, acting at a distance x from the crack’s centre-line[17].

For the actual crack distance as shown in Figure A.9, the stress due to the tensile load,

P applied to the crack face is:

σ (x) = σ 0≤ |x|< a (A.29)

For the plastic zone, the distance between the effective and actual crack length, this tensile

load is counteracted by the closure stress:

σ (x) = σ −σY S a≤ |x|< a+ρ (A.30)

129


So the stress intensity factor at the effective crack tip due to the tensile load which is applied

over the entire crack length is given by (using Equation A.22):

Kσ = σ√

π(a+ρ) (A.31)

The stress intensity factor at the effective crack tip due to the closure stress over the plastic

zone can be expressed as:

Kclosure =σY S√

π(a+ρ)

ˆ a+ρ

a

√(a+ρ)+ x(a+ρ)− x

dx (A.32)

Kclosure =σY S√

π(a+ρ)

ˆ a+ρ

a

√(a+ρ)− x(a+ρ)+ x

dx (A.33)

for the crack tips in the positive and negative x direction respectively. Solving the sum of

these integrals gives:

Kclosure =−2σY S

√a+ρ

πarccos

(a

a+ρ

)(A.34)

The plastic zone length ρ is evaluated by equating the stress intensity factors from the

remote tension and closure stress. This gives:

aa+ρ

= cos(

πσ

2σY S

)(A.35)

Ke f f can then be estimated using the sum of a and ρ as the effective crack length. This

estimate was improved by Burdekin and Stone [257] who used a Westergaard complex

stress function [255] to derive an expression for the crack-opening displacement from the

strip-yield model. This was used to used calculate the J-integral and hence Ke f f . Both the

concepts of crack-opening displacement (COD) and the J-integral are used more extens-

ively for yielding which exceeds small scale (Elastic-Plastic Fracture Mechanics) and is not

explained here.

A.3.2.2 Barenblatt cohesive force model

Barenblatts cohesive force model worked on the same concept but stresses varied with

deformation rather than remaining constant at the yield stress, Figure A.10. As a result

Equations A.29 and A.30 become:

130


q(x)

Figure A.10: The cohesive force model (adapted from [17])

σ (x) = σ 0≤ |x|< a (A.36)

σ (x) = σ −q(x) a≤ |x|< a+ρ (A.37)

A simple comparison between pure LEFM and its plastic zone corrections can be found in

[17].

131

Appendix B

Continuum Damage Mechanics (CDM)

This section gives a brief overview of the background and theory of Continuum Damage

Mechanics. The information presented is sourced from the cited papers, standard text books

[29] and review papers [27, 258].

Continuum damage mechanics (CDM), initially developed by Kachanov [114] to model

creep in metals, provides a complimenting approach to fracture mechanics framework in

modelling material failure. In a purely fracture mechanics approach the effect of the dis-

continuity created by a single crack within a continuum is modelled analytically. For the

FPZ of quasi-brittle materials, the high-volume of cracks and discontinuities and their dy-

namic interaction processes, creates difficulties with this approach.

For quasi-brittle materials, CDM indirectly quantifies the accumulated damage by con-

sidering the degrading effect of the micro-crack population on the macroscopic material

properties such as strength, stiffness and toughness. In this way the high volume of cracks

is essentially “smeared out” within the FPZ continuously. As CDM maintains the classical

continuum assumption, numerical implementation in commercial finite element software

proves relatively simple, in comparison to fracture mechanics where the discontinuities in

the mesh neccesitate remeshing unless mesh-less methods can be sufficiently developed.

B.1 Damage parameters

The CDM framework begins by representing a materials state of damage with one or more

damage variables, which can be of scalar or tensorial form. The continuum damage concept

dictates that the final stage of damage will be when the continuum assumption is broken,

i.e. the development of a discontinuity or macro-crack. A scalar damage variable, D can

be defined with a value of 0 corresponding to an undamaged state and a value of 1 cor-

132

APPENDIX B. CONTINUUM DAMAGE MECHANICS (CDM)

responding to final fracture. A scalar variable is suitable for isotropic loading, i.e. when

the microflaws are randomly distributed or are small enough to assume so. If a scalar vari-

able is unsuitable, i.e for a combination of damage mechanisms or material anisotropy, then

multiple scalar parameters or a tensor may be used.

Damage can be directly measured, through micro-structural observations and a change

in net cross sectional area, or indirectly through the change in physical material macro-

scopic properties as a result of the micro-structural changes [259]. Such properties include

a change in density due to void fraction increase, acoustic emission and most commonly a

change in the mechanical behaviour, characterised by the effective stress. This is an exten-

sion of the effective area concept, Figure B.1, proposed by Kachanov [114]. In this concept

the effective area of the fictitious undamaged state, dA, of a bar under tensile load, dF can

be expressed as a function of the scalar damage parameter D and the area of the damaged

state, dA:

dA = (1−D)dA (B.1)

This can be used to evaluate the effective stress:

σ =dFdA

=1

(1−D)

dFdA

=σ

1−D(B.2)

and hence:

D = 1− σ

σ(B.3)

Figure B.1: The concept of a fictitious undamaged state, on which the effective stress prin-ciple is based [29].

The effective stress concept (and similarly the effective strain concept) equates the re-

133


sponse of a damaged material under an stress, σ , to the response of a fictitious undamaged

volume of the same material under an effective stress, σ . This assumption forms the hypo-

thesis of strain equivalence [29]. In this way:

ε =σ

E (D)(B.4)

ε =σ

E0(B.5)

where E0 is the Young’s modulus of the undamaged and fictitious undamaged volumes of

material and E (D) is the Young’s modulus of the damaged material (either secant or tangent

modulus). Introducing these expressions into Equation B.3 yields:

D = 1− E (D)

E0(B.6)

Without the assumption of isotropic damage, a scalar parameter becomes insufficient to

describe the damage behaviour. In such cases tensorial parameters are required [260].

B.2 Damage evolution and constitutive laws

Once the initial state of damage is known (from the effective stress concept), a descriptive

law mapping the progression of the damage variable as time or load increases is necessary.

This can be derived on a thermodynamic basis, the details of which are out of the scope

of this brief review. For more details consult one of the given references for this appendix.

Once the damage parameter and evolution law are known, the damage can be progressed

until the initiation of failure in terms of a macro-crack. There are two basic approaches to

the damage analysis:

• Uncoupled approach. The uncoupled analysis assumes that the stress and strain

fields are not reliant on the damage within the material. Under initial loading condi-

tions the constitutive laws allow evaluation of the stress and strain field. If the initial

damage state is also known (or assumed to be 0), then the damage evolution law can

be used to evaluate the progression of damage in a given time or load step and hence

determine the resulting damage field. This damage field is used as a failure prediction

for the structure, but has no affect on the subsequent failure analysis for increasing

time or load. This neglect of the interaction between damage and constitutive laws

introduces error but also provides simplicity.

134


• Coupled approach. Coupled analysis recognises the interaction between stress,

strain and damage. In this way the material constitutive laws are derived to take into

account damage and the effect on the material deformation as time or load progresses.

As a result, unlike in the uncoupled approach, the stress, strain and damage can be

evaluated simultaneously. However such procedures require the full field history of

damage and hence increase complexity.

Examples of both types of model can be found in the literature [29, 126, 261, 262]. Moreover

the formulation of different damage models differs depending on the type of material, with

a general distinction evident between elastic and plastic materials:

• Elastic/Elastic-Brittle damage models. In such models damage applies to the elastic

response of an elastic material [139].

• Damage-Plasticity/Elastic-Plastic damage models. In such models damage applies

to elastic-plastic materials. This can incorporate the elastic response exclusively,

leaving the plastic response “undamaged” (uncoupled), or both elastic and plastic

regimes (coupled) [263].

135

Appendix C

Meso-scale features and couple stresses

in fracture process zone

136

13th International Conference on Fracture June 16–21, 2013, Beijing, China

-1-

Meso-scale features and couple stresses in fracture process zone

Craig N Morrison1, 3,*, Andrey P Jivkov2, 3, John R Yates3

1 Nuclear FiRST Doctoral Training Centre

2 Research Centre for Radwaste and Decommissioning 3 Modelling and Simulation Centre

Dalton Nuclear Institute, The University of Manchester, Manchester, M13 9PL, UK

* Corresponding author: [email protected]

Abstract Generalized continuum theories such as couple stress theory have the potential to improve

our understanding of material deformation and fracture behaviour in areas where classical continuum

theory breaks down at, for example, the length scale of meso-scale features within the fracture process

zone. The couple stress theory considers not only relative displacements between these features but

also relative rotations, introducing a natural length scale. A model has been developed of a low

stiffness matrix containing suitably situated high stiffness particles to simulate the presence of defects

at the meso-scale. This has been used to assess the descriptive potential of a novel consistent couple

stress theory. The model has been subjected to a set of displacement fields selected to produce strain

energies with varying contributions from the coupled stresses. The results demonstrate the effect of

particle size to spacing ratio on the elastic energies. These can be used to evaluate the couple stress

constant as well as validate the constant experimentally for specific materials.

Keywords: generalized continuum; meso-scale defects; FE analysis; strain-curvature energy; size

effect

1. Introduction

Analysis of materials at engineering length scales is based upon assumptions of classical

continuum behaviour. This is adequate for most macro-scale analyses but, when considering

smaller length scales where cracks, notches and defects introduce stress concentrations, the

material microstructure is known to have a significant impact on material behaviour [1]. Local

approaches, which incorporate mechanistic understanding of material failure behaviour at the

length scale of their relevant features, are beneficial for linking microstructures to

macroscopic responses [2]. However, the widely used weakest link (WL) assumption has been

challenged as a realistic method of modelling size effects in cleavage [3] and quasi-brittle

fracture [4] as a result of failing to account for the interaction processes during failure.

Discrete methods have shown promise for modelling materials undergoing such fracture.

Lattice models consist of nodes connected into a lattice via springs [5], beams [6] or other

discrete elements with the properties of these connections allowing a micro structurally

informed response. Lattice modelling differs in principle from previous local approach

models by using a statistically parallel system, where loads are redistributed upon the

breaking of a single bond, rather than the ultimate failure seen in WL systems. This is

considered to be a closer representation of the interaction and coalescence of micro-cracks

and flaws, which characterize quasi-brittle materials such as graphite [7] and cement-based

materials [8]. The work presented here explores aspects of the site-bond model developed by

Jivkov and Yates [9] and used for studies of damage evolution from distributed porosity in

cements [10]. Work on this model has shown that evaluating the stiffness coefficients of the

bonds using strain energy equivalence between the discrete model and a classical continuum

creates an indeterminate problem. Use of a generalized continuum theory, such as couple

stress theory, offers a possible solution to this indeterminacy.


-2-

In couple stress theory (CST) each point within the continuum has three additional degrees of

freedom, point rotations. These are associated with couple stresses as the classical (force)

stresses are associated with strains. A general CST was proposed early in the 20th century by

the Cosserat brothers [11]. It went largely unnoticed until the 1960s when a desire to

understand the mechanisms behind micro-crack growth for more accurate crack assessment

rejuvenated interest. One branch of CSTs considers point micro-rotations to be independent of

the macro-rotations; the rotations derived from the displacement gradient [12, 13]. These are

known as micropolar theories or Cosserat models with free rotations [14]. While such a view

appears to be well suited for use with discrete lattice methods, it is difficult to establish a link

between a continuum and a discrete representation of a material containing features that are

following the deformations of the bulk. For such situations it is more plausible to assume that

the micro-rotations are equal to the macro-rotation. This assumption led to a branch of CSTs

known as Cosserat models with constrained rotations [14, 15]. Initially, these were based on

couple stresses work-conjugate to the macro-rotation gradient. As a consequence, the

spherical part of the couple stress tensor remained undetermined. Recently, Hadjesfandiari

and Dargush [16] proposed a consistent CST using true kinematic quantities to remove the

indeterminacy of the couple stress tensor.

The consistent CST [16] naturally introduces a length parameter. This is of key importance for

the local material behaviour. But the calibration of the CST requires that the length parameter

is physically related to the material microstructure; the sizes and distances between

characteristic features that disturb the symmetry of the stresses. We report on work in progress

investigating whether a medium with features can be used to calculate bond responses to

bending and torsion in the discrete model [9] and if this can be used to calibrate the consistent

CST.

2. Theory and model

2.1 Generalised continuum

The kinematics of a material point under small deformation is given by the displacement

gradient, Eq. (1), where comma denotes differentiation, rounded parenthesis denotes

symmetric part and square parenthesis denotes skew symmetric part of the tensor. The

symmetric (strain tensor eij) and the skew symmetric (rotation tensor ωij) parts are given by

Eq. (2) and Eq. (3), respectively. The right hand side of Eq. (3) gives the rotation tensor as a

vector using the permutation tensor.

𝑢𝑖,𝑗 = 𝑢(𝑖,𝑗) + 𝑢[𝑖,𝑗] (1)

𝑢(𝑖,𝑗) = 𝑒𝑖𝑗 = 1

2(𝑢𝑖,𝑗 + 𝑢𝑗,𝑖) (2)

𝑢[𝑖,𝑗] = 𝜔𝑖𝑗 = 1

2 (𝑢𝑖,𝑗 − 𝑢𝑗,𝑖) = 𝜖𝑗𝑖𝑘𝜔𝑘 (3)

In classical continuum mechanics, the elastic potential depends solely on the strain tensor. In

generalised continuum, additional potential is carried by the gradient of the rotation vector, Eq.

(4). The symmetric part of this gradient, χij in Eq. (5), represent “pure” twists, and the skew

symmetric part, κij in Eq. (6), represent “pure” curvatures, which can be given by a vector as

shown with the right-hand size.

𝜔𝑖,𝑗 = 𝜔(𝑖,𝑗) + 𝜔[𝑖,𝑗] (4)

𝜔(𝑖,𝑗) = 𝜒𝑖𝑗 = 1

2 (𝜔𝑖,𝑗 + 𝜔𝑗,𝑖) (5)


-3-

𝜔[𝑖,𝑗] = 𝜅𝑖𝑗 = 1

2 (𝜔𝑖,𝑗 − 𝜔𝑗,𝑖) = 𝜖𝑗𝑖𝑘𝜅𝑘 (6)

Generally, the energy potential of the rotation gradient leads to a non-symmetric force stress

tensor, σji, Eq. (7), and the introduction of a couple stress tensor, μji, Eq. (8). When the force

stress is taken as work conjugate to the strain tensor, and the couple stress is taken as work

conjugate to the gradient of the rotation, the symmetric part of the couple stress tensor

becomes indeterminate.

𝜎𝑗𝑖 = 𝜎(𝑗𝑖) + 𝜎[𝑗𝑖] (7)

𝜇𝑗𝑖 = 𝜇(𝑗𝑖) + 𝜇[𝑗𝑖] (8)

Hadjesfandiari and Dargush [16] suggested a solution to this problem by demonstrating that

the entire rotation gradient does not have energy potential, only the curvature tensor, Eq. (6).

Thus the deformation energy consists of the work done by the force stress on the strain and

the work done by the couple stress on the pure curvature. The symmetric part of the rotation

gradient, Eq. (5), has no forces associated with it, which solves the problem of the

indeterminate spherical part of the rotation gradient. For an isotropic material, the elastic

potential is given by [16]:

𝑊(𝜀, 𝜅) = 1

2 𝜆 (𝜀𝑘𝑘)2 + 𝜇 𝜀𝑖𝑗𝜀𝑖𝑗 + 8𝜂 𝜅𝑖𝜅𝑖 (9)

where λ and μ are Lamé parameters and η is a material couple stress constant.

According to this theory, a homogeneous displacement field, such as hydrostatic compression

Eq. (10), does not introduce rotations and hence curvatures.

𝑢1 = 𝑥1 𝑢2 = 𝑥2 𝑢3 = 𝑥3 (10)

A displacement field producing pure twist, Eq. (11), introduces rotations, Eq. (12), but no true

curvatures[16]:

𝑢1 = −𝜃𝑥2𝑥3 𝑢2 = 𝜃𝑥1𝑥3 𝑢3 = 0 (11)

𝜔1 = −1

2𝜃𝑥1 𝜔2 = −

1

2𝜃𝑥2 𝜔3 = 𝜃𝑥3 (12)

A displacement field corresponding to pure bending of a beam, Eq. (13), introduces non-zero

rotations, Eq. (14), that result in a single non-zero curvature, Eq. (15), [16]:

𝑢1 = −1

𝑅𝑥1𝑥3 𝑢2 = −

𝜈

𝑅𝑥2𝑥3 𝑢3 =

𝜈

2𝑅(𝑥2

2 − 𝑥32) −

1

2𝑅𝑥1

2 (13)

𝜔1 = 𝜈𝑥2

𝑅 𝜔2 =

𝑥1

𝑅 (14)

𝜅3 = 1−𝜈

2𝑅 (15)

where is Poisson’s ratio and R is the radius of curvature of the beam central axis.

2.2 Discrete site-bond model

The site-bond model [9] uses a discrete lattice, based on a regular tessellation of material

space into truncated octahedral cells, Fig. 1(a). The lattice derives from the cellular structure

when material particles, attached to cell centres, interact via deformable bonds. The bond

properties relate to their ability to transfer shear and axial forces as well as torsion and

bending moments to satisfy the six degrees of freedom of each site. A site requires 14 bonds

to connect it to its neighbours, Fig. 1(b): six bonds of length 2L (2L is the cell size) in

principal directions (through square faces), and eight bonds of length √3L in octahedral


-4-

directions (through hexagonal faces). Development of this model involves bond

representations with six independent elastic springs resisting three relative displacements and

three relative rotations between sites. This yields four spring types with axial, Kn, shear, Ks,

twisting, Kt, and bending, Kb, stiffness [17], which could, in general, be different for

principal and octahedral directions.

Figure 1. Cellular representation of material (a); and unit cell with bonds (b).

2.3 FE model of elastic continuum with rigid particles

For our investigation, we use a finite element model of a cube of an elastic material

surrounding a truncated octahedral cell of size 2L. The material has a unit modulus of

elasticity and Poisson’s ratio = 0.375. Rigid cubic particles are introduced in the cube, so

that one particle, P0, is positioned in the centre of the cell, while others are positioned outside

the cell in the principal and octahedral directions as shown in Fig. 2. Three different loading

conditions are used: (H) hydrostatic compression; (T) pure twist; and (B) pure bending. These

are applied via displacement fields on the cube surfaces; examples for pure twist and pure

bending are given in Fig. 3. In all cases we calculate elastic energies, , within the unit cell

surrounding the central particle. With no particles present, the cell elastic energy is given by

the classical continuum solution, since no features exist to disturb stress symmetry. The

displacement magnitudes for the three loading cases are selected so that without particles is

the same, 0.

Figure 2. Particle additions in the principal (left) and octahedral (right) directions.

Figure 3. Displacement maps for the loading cases of pure twist (left) and pure bending (right).


-5-

To link the cell energies to kinematic quantities we calculate particle translations and rotations

using three unit vectors, n1, n2 and n3 normal to three orthogonal faces of a particle, see Fig. 4.

After deformation these vectors remain orthogonal as the particles are rigid, given with t1, t2

and t3 in Fig. 4. The coordinates of these, arranged in columns, form the transformation

matrix, T, for the particle. The particle motion can be represented by a single rotation, θ given

by Eq. (16), around a normalized axis, α given by Eq. (17). The components of the rotation

vector for a particle are calculated by Eq. (18). The relative rotations between central and any

other particle are expressed in the coordinate system defined by the particular pair using the

corresponding transformation.

𝜃 = 𝑐𝑜𝑠−1 (𝑡𝑟𝑎𝑐𝑒 (𝐓)−1

2) (16)

𝛼 =1

2 sin (𝜃) [

𝑇32 − 𝑇23

𝑇13 − 𝑇31

𝑇21 − 𝑇12

] (17)

𝜔𝑖 = 𝜃 × 𝛼𝑖 , 𝑖 = 1, 2, 3 (18)

Figure 4. Orthogonal vectors, (n1, n2, n3) and (t1, t2, t3) describing the orientation of P0 before and

after deformation respectively.

3. Results and Discussion

For the particle arrangement used (a central particle and all 14 particles of the site-bond model)

we have used four different particle sizes relative to the cell size in order to investigate the

effect of cell to particle size ratio, . The ratios are 3 (large particles), 4, 6, and 12 (small

particles).

In the case of hydrostatic loading no rotations of particles were observed and the relative

displacements between P0 and any principal or octahedral particle were only axial,

conforming to Eq. (10). In the site-bond model these relative displacements should be resisted

by axial springs with stiffness coefficients Knp and Kn

o, respectively. The relative axial

displacements were found independent of , scaling with the applied displacements. The cell

energies were found to be dependent on , as shown in Figure 5. For small particles, = 12,

the cell energy approached 0; the presence of particles has negligible effect. The cell energy


-6-

and displacements from this case can be used to calibrate a linear combination of Knp and Kn

o.

Note, that in general this case is not sufficient for calibrating stiffness coefficients separately.

However, the particle size effect appears to be very small, suggesting that the two stiffness

coefficients could be assumed equal and given from the continuum solution, following [17]

for example.

Figure 5. Normalised cell energy vs size ratio under hydrostatic compression.

In the case of pure twist the cell energies were found to be dependent on , Fig. 6. While this

is similar to the case of hydrostatic compression the effect of particle size is substantially

larger. All relative displacements between P0 and the principal particles were zero,

conforming to Eq. (11). The relative displacements between P0 and octahedral particles were

zero axial and non-zero transversal, again conforming to Eq. (11). This means that the only

activated linear springs in the site-bond model are the shear springs in octahedral direction

with stiffness Kso.

Figure 6. Normalised cell energy vs size ratio under pure twist

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

0 5 10 15

No

rmal

ise

d e

ner

gy, Ω

/Ω0

Cell size / Particle size

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0 5 10 15

No

rmal

ise

d e

ner

gy, Ω

/Ω0



-7-

The calculated relative rotations between P0 and the particles in the two directions are shown

in Fig. 7. The non-zero relative rotations between P0 and principal particles were twists as

expected from Eq. (12) and the magnitude shown in Fig. 7 is the twist of the particle in

direction X1. It is seen that this is 3-4 orders of magnitude smaller that the relative rotations

between P0 and octahedral particles. The latter were found to be bend-type rotations only,

conforming to Eq. (12), and suggesting that the only moment springs activated are the

bending springs in the octahedral direction with stiffness Kbo. The magnitudes of the

bend-rotations of octahedral particles were found nearly independent of particle size. These

results suggest that for the case of pure twist, the elastic strain energy is accumulated in the

shear and bending springs in the octahedral direction only, and one can calibrate a linear

combination of Kso and Kb

o, which should depend on the cell to particle size ratio. From this

perspective the results support the analytical derivations in [16], where pure twists do not

contribute to the elastic energy. From here, it can be speculated that the torsion springs in the

principal directions could be omitted from the site-bond model, i.e. Ktp = 0. This should be

supported by considering other cases with curvature-free displacement fields which introduce

twists of principal particles according to the theory.

Figure 7. Relative rotation vs size ratio under pure twist

The cell energy dependence on for the case of pure bending is shown in Fig. 8. The effect of

particle size is smaller than in the case of pure twist, but not negligible. The relative

displacements between P0 and the octahedral particles were non-zero axial and tangential,

conforming to Eq. (13). The relative displacements between P0 and the principal particles also

conform to theory with non-zero axial for particles in X2, non-zero tangential for particles in

X1 and zero displacements of particles in X3. This means that the activated linear springs in

the site-bond model would be axial and shear springs in all octahedral directions with stiffness

coefficients Kno and Ks

o, and axial and shear springs in two principal directions with stiffness

coefficients Knp and Ks

p.

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0 5 10 15

Mag

nit

ud

e o

f re

lati

ve r

ota

tio

ns


Principal

Octahedral


-8-

The calculated relative rotations of the particles are shown in Fig. 9. The only non-zero

relative rotation of principal particles was found to be the bend-type rotation of the particles in

X1, conforming to Eq. (14). However, this relative rotation was found at least an order of

magnitude smaller than the magnitude of the relative rotation between P0 and octahedral

particles. The latter contains twist and bend components, suggesting an activation of both the

torsion and bending springs in the octahedral direction with stiffness coefficients Kto and Kb

o.

The much smaller rotation of the principal particles can be used to approximate the

kinematics and assume zero rotation of principal particles. Thus the elastic energy is taken by

the linear springs in all directions and the torsion and bending spring in the octahedral

direction, allowing for calibration of a linear combination of Kno, Ks

p, Kno, Ks

o, Kto and Kb

o,

which should depend on the cell to particle size ratio. As in the case of pure twist it can be

speculated that the bending springs in the principal direction can be omitted, i.e. Kbp = 0, but

this needs to be supported by considering other displacement fields introducing bend-type

rotations of principal particles. It can be further speculated, that the torsion stiffness in the

octahedral direction should be zero, i.e. Kto = 0. This could be deduced from the theoretical

requirement that pure twists have no energy potential, but requires further investigation. If

these were shown to be true, the bending case would provide a calibration for the linear

combination of Kno, Ks

p, Kno, Ks

o, Kbo.

Figure 8 . Normalised cell energy vs size ratio under pure bending

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

0 5 10 15

No

rmal

ised

en

erg

y, Ω

/Ω0



-9-

Figure 9. Relative rotation vs size ratio under pure bending

From the three cases considered in this work, it is clear that a separation between all required

stiffness coefficients is not possible. However, there is a good indication that the site-bond

model should not contain torsion springs and possibly bending springs in the principal

directions. One way to check this is pure twist along the octahedral axis, which theoretically

should provide twist and bend rotations of all particles. If this proves to be the case, the elastic

energy of the continuum with features should be taken by the deformations of the remaining

springs. Additional loading cases are necessary to determine the stiffness coefficients of these

springs. If these are determined from curvature-free loading cases alone, a good strategy for

the calculation of the coupled-stress constant can be proposed. The site-bond model with

constants calibrated from curvature-free loading cases can be subjected to a case introducing

curvature energy according to theory, for example pure bending, and any excess of energy

between the site-bond model and classical continuum should be attributed to curvature energy.

4. Conclusions

We have proposed a methodology for calibrating the spring constants of a special lattice

model using a micromechanical model of a material containing features.

The comparison of the results with the consistent couple-stress theory suggests that

some of the possible moment springs in the lattice could be omitted, reducing the

complexity and increasing the correspondence between continuum couple-stress theory

and discrete representation.

We have demonstrated that in all loading cases considered there is an effect of the

distance to size ratio of the features, which must be taken into account when calibrating

the constants. This suggests that actual microstructure data needs to be used for

calibrating the site-bond model.

The loading cases considered were not sufficient for complete determination of the

spring constants of the discrete model. Further work is necessary with loading cases that

provide different linear combinations of activated springs’ kinematics.

There is the potential that the lattice model, if fully calibrated with curvature-free

loading cases, can provide a means of determining the couple-stress constant for a

material with given microstructure properties such as average particle size and distance.

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0 5 10 15

Mag

nit

ud

e o

f re

lati

ve r

ota

tio

ns


Principal

Octahedral


-10-

Acknowledgements

The support from EPSRC, via Nuclear FiRST Doctoral Training Centre, to Morrison, from

EPSRC via grant EP/J019763/1 and BNFL to Jivkov, and from EDF R+D to Yates is

gratefully acknowledged.

References

[1] G.B. Sinclair, A.E. Chambers, Strength size effects and fracture mechanics: What does

the physical evidence say? Eng Fract Mech 26 (1987) 279-310.

[2] J. Lemaitre, Local approach of fracture. Eng Fract Mech 23 (1986) 523–537.

[3] A.P. Jivkov, D.P.G. Lidbury, P. James, Assessment of Local Approach Methods for

Predicting End-of-Life Toughness of RPV Steels. In Proc. PVP2011 (2011) paper 57546,

Baltimore, Maryland.

[4] Z.P. Bažant, S.-D. Pang, Activation energy based extreme value statistics and size effect

in brittle and quasibrittle fracture. J Mech Phys Solids 55 (2007) 91–131.

[5] A. Pazdniakou, P.M. Adler, Lattice Spring Models. Transp Porous Med 93 (2012)

243–262.

[6] E. Schlangen, E. Garboczi, Fracture simulations of concrete using lattice models:

computational aspects. Eng Fract Mech 57 (1997) 319–332.

[7] N.N. Nemeth, R.L. Bratton, Overview of statistical models of fracture for nonirradiated

nuclear-graphite components. Nucl Eng Design 240 (2010) 1–29.

[8] P. Grassl, D. Grégoire, L. Rojas Solano, G. Pijaudier-Cabot, Meso-scale modelling of the

size effect on the fracture process zone of concrete. Int J Solids Struct 49 (2012) 1818–1827.

[9] A.P. Jivkov, J.R. Yates, Elastic behaviour of a regular lattice for meso-scale modelling of

solids. Int J Solids Struct 49 (2012) 3089–3099.

[10] A.P. Jivkov, M. Gunther, K.P. Travis. Site-bond modelling of porous quasi-brittle media.

Mineral Mag 76 (2012) 94-99.

[11] F. Cosserat, E. Cosserat, Theory of Deformable Bodies. A. Hermann et Fils, Paris, 1909.

[12] R.D. Mindlin, Micro-structure in linear elasticity. Arch Ration Mech An 16 (1964) 51–78.

[13] W. Nowacki, Theory of Asymmetric Elasticity. Pergamon Press, Oxford, 1986.

[14] M. Garajeu, E. Soos, Cosserat Models Versus Crack Propagation. Math Mech Solids 8

(2003) 189–218.

[15] R.A. Toupin, Theories of elasticity with couple-stress. Arch Ration Mech An 17 (1964)

85-112.

[16] A.R. Hadjesfandiari, G.F. Dargush, Couple stress theory for solids. Int J Solids Struct 48

(2011) 2496–2510.

[17] Y. Wang, P. Mora, Macroscopic elastic properties of regular lattices. J Mech Phys Solids

56 (2008) 3459–3474.

Appendix D

A meso-scale site-bond model for

elasticity: Theory and calibration

147

Meso-scale site-bond model for elasticity:theory and calibration

M. Zhang*1, C. N. Morrison1,2 and A. P. Jivkov1

A meso-scale site-bond model is proposed to simulate the macroscopic elastic properties of

isotropic materials. The microstructure of solids is represented by an assembly of truncated

octahedral cells with sites at the cell centres and bonds linking the nearest neighbouring sites.

Based on the equivalence of strain energy stored in a unit cell to strain energy stored in a

continuum of identical volume, the normal and shear stiffness coefficients of bonds are derived

from the given macroscopic elastic constants: Young’s modulus and Poisson’s ratio. To validate

the obtained spring constants, benchmark tests including uniaxial tension and plane strain are

performed. The simulated macroscopic elastic constants are in excellent agreement with the

theoretical values. As a result, the proposed site-bond model can be used to simulate the

macroscopic elastic behaviour of solids with Poisson’s ratios in the range from 21 up to 1/2.

Keywords: Site-bond model, Elasticity, Lattice spring, Discrete2continuum equivalence, Isotropy

IntroductionThe classical homogeneous elasticity is widely used todescribe the macroscopic linear mechanical behaviour ofmost materials, even though they are actually hetero-geneous from a microscopic point of view. However, themechanical response of heterogeneous quasi-brittlematerials, such as concrete, rock, graphite or ceramics,cannot be modelled realistically without explicit con-siderations of their underlying microstructures. Thisrequires numerical approaches which are able to accountfor not only the elastic stage, but also the initiation,growth, interaction and coalescence of micro-cracks. Thediscrete lattice approach, usually called the meso-scaleapproach, shows potential to meet this requirement.1 Inlattice models,1–10 the microstructures of materials arerepresented by an assemblage of unit cells or particles.The lattice sites are placed at the centres of the cells. Thedeformation of the represented continuum arises from theinteractions between the lattice sites. The neighbouringcells are linked through interface bonds, which can berepresented by lattice beam elements or lattice springs.Compared to the continuum finite element modelling,lattice models have been shown to be more suitable forfracture simulation because of their discrete nature.

The lattice models have been successfully applied to themodelling of quasi-brittle materials. The macroscopicstress–strain curve for concrete is obtained by using alattice beam model based on a two-dimensional regularlattice with hexagonal unit cells.2,3 However, this lattice

cannot be used for isotropic elastic materials withPoisson’s ratio larger than 1/3 in plane stress and 1/4 inplane strain.4 Based on the simplest regular lattice withcubic cells, a three-dimensional (3D) lattice beam modelhas been proposed by Schlangen5 to simulate the crackdevelopment in concrete. It has been shown that thislattice is only suitable for materials with zero Poisson’sratio.6 With respect to lattice spring models, Wang andMora7 developed two 3D lattices using face-centred cubicand hexagonal closely-packed arrangements. Each pair ofsites in the lattice network is connected by spring. It wasfound that only isotropic elastic material with Poisson’sratio of zero can be represented by these lattices, which isthe same as cubic lattices. To overcome these limitations,a site-bond based on a bi-regular lattice of truncatedoctahedron cells has been recently proposed by Jivkovand Yates1 for meso-scale modelling of solids. The bondsof the site-bond assembly are modelled with structuralbeam elements. It has been demonstrated that this site-bond model is able to represent isotropic elastic materialswith Poisson’s ratios up to 1/2.

The main purpose of this work is to reformulate thesite-bond assembly presented in Ref. 1 by modelling thebonds with two types of elastic springs instead ofstructural beam elements to further study the capabilityof this lattice arrangement for the macroscopic elasticbehaviour of practical interest. The stiffness coefficientsof springs are analytically determined by equating thestrain energy stored in the discrete and continuum cell.The derived spring constants are validated throughnumerical analyses.

Site-bond modelIn the site-bond model, the microstructure of a realmaterial is represented by tessellating the space intotruncated octahedral cells, as shown in Fig. 1a. The

1Modelling & Simulation Centre and Research Centre for Radwaste &Decommissioning, The University of Manchester, Manchester M13 9PL,UK2Nuclear FiRST Doctoral Training Centre, The University of Manchester,Manchester M13 9PL, UK

*Corresponding author, email [email protected]

S2-982

� W. S. Maney & Son Ltd. 2014Received 15 September 2013; accepted 12 December 2013DOI 10.1179/1432891714Z.000000000537 Materials Research Innovations 2014 VOL 18 SUPPL 2

truncated octahedron was found to be the best choicefor a regular representation of real materials comparedto the cube, the regular hexagonal prism or the rhombicdodecahedron.1,6 Each cell has six equal square facesand eight equal hexagonal faces. The cell centre isconsidered as a site, which is connected with itsneighbouring sites by 14 bonds, six bonds B1 in principaldirections through square faces and eight bonds B2 inoctahedral directions through hexagonal faces, asillustrated in Fig. 1b.

In this study, the bonds are modelled with elasticsprings. The sites have six independent degrees offreedom: three translational and three rotational. Inprinciple, each bond should contain six springs: onenormal, two shear, one twisting, and two bending springsin order to resist the relative displacement and relativerotations between the two adjacent cells. However, it waspresented that the twisting stiffness kt and bendingstiffness kb are related to the shear stiffness ks andnormal stiffness kn, respectively, with the contribution ofkt and kb to the macroscopic elasticity effectivelynegligible in comparison to ks and kn.7 Therefore, onlythe normal and shear springs in principal and octahedraldirections are considered herein, as shown in Fig. 1c.

Derivation of spring constants for site-bond modelIn this section, the spring constants are derived from themacroscopic elastic parameters by equating the strainenergy stored in a unit cell Ucell to the associated strainenergy in the equivalent continuum system Ucont

Ucell~Ucont (1)

The strain energy of the continuum system is given by

Ucont~1

2

ðV

sedV~1

2CijkleijeklV (2)

where C represents the stiffness tensor of the material, eis the strain field and V is the system volume. The strainenergy stored in a unit cell can be expressed as a sum ofthe strain energies stored in each internal bond Ub

Ucell~X

b

Ub~1

4

XNb

b

k(b)n u(b)

n u(b)n zk(b)

s u(b)s u(b)

s

� �(3)

in which u(b)n and u(b)

s stand for the relative displacements

in the normal direction and transverse direction of the

bond, respectively. Let us assume that the bond b linkstwo sites A and B, then the relative normal and sheardisplacements in the 3D global system X1X2X3 can bewritten as

u(b)n ~Du

(b)i j

(b)i (4)

u(b)s ~Du

(b)i {u(b)

n j(b)i (5)

in which

Du(b)i ~eijDxj~eij(xjB{xjA) (6)

The direction vector of the bond j(b)i is given as

j(b)i ~

(xiB{xiA)

LAB

(7)

where xiA and xiB are the positions of the sites, LAB isthe length of the bond. By substituting equations (4)–(7)into equation (3) and performing tensor and vectormanipulations, equation (3) can be expressed as

Ucell~

1

4

XNb

b

L(b)� �2

k(b)n j(b)

i eijj(b)j j(b)

k eklj(b)l zk(b)

s eklj(b)l {j(b)

i eijj(b)j j(b)

k

� �ekmj(b)

m {j(b)n enmj(b)

m j(b)k

� �h i (8)

in which L(b) is the length of a bond b. Then by equatingthe total strain energy stored per unit volume V, thestrain energy density rcont, to that in the unit cellrcell~Ucell=V and by using Cauchy’s formula,8–10 thestress tensor of the continuum system can be obtained as

sij~Lrcont

Leij

~1

2V

XNb

b

L(b)� �2

k(b)s eilj

(b)l j(b)

j z k(b)n {k(b)

s

� �eklj

(b)i j(b)

j j(b)k j(b)

l

h i

(9)

Finally, the elastic stiffness tensor can be given as

Cijkl~Lsij

Lekl

~1

2V

XNb

b

L(b)� �2

k(b)s dikj(b)

j j(b)l z k(b)

n {k(b)s

� �j(b)

i j(b)j j(b)

k j(b)l

h i

(10)

where dik is the Kroecker’s delta.Considering the site-bond assembly shown in Fig. 1

and assuming that the unit cell size in the principaldirections is L, the lengths of bonds B1 and B2 are L and31/2L/2, respectively. The volume of the unit cell V is L3/2. For each bond, k(b)

n and k(b)s are the normal and shear

spring constants for bonds B1 and B2, which are denotedas kp

n and kps , ko

n and kos , respectively in the following

1 Cellular lattice: a site-bond assembly; b unit cell with bonds; c normal and shear springs

(8)

(9)

(10)

Zhang et al. Meso-scale site-bond model for elasticity: theory and calibration

Materials Research Innovations 2014 VOL 18 SUPPL 2 S2-983

sections. The direction vectors of bonds B1 and B2 aregiven in Table 1.

Hence, by using equation (10) and assuming the twoshearing spring stiffness coefficients are equal within eachbond type but different between the two bond types, B1

and B2, we get the components of the stiffness tensor

C1111~2

3L3kp

nzkonz2ko

s

� �~C2222~C3333 (11)

C1122~2

3Lko

n{kos

� �~C1133~

C2233~C2211~C3311~C3322 (12)

C1212~2

3L3kp

s zkonz2ko

s

� �~

C1313~C2323~C2121~C3131~C3232 (13)

Other Cijkl~0 (14)

It can be seen from equations (11)–(14) that there are onlythree independent elastic constants. This indicates that thesite-bond assembly generates macroscopic cubic elasticity.With the Voigt notation, which is the standard mapping fortensor indices, the spring constants can be expressed as

kpn~{ko

nzL

2C11z2C12ð Þ (15)

kps ~{ko

nzL

22C12zC44ð Þ (16)

kos ~ko

n{3L

2C12 (17)

Meanwhile, from equations (11)–(14), we find thatonly when 3kp

n{6kps {2ko

n{kos ~0, the site-bond assem-

bly is able to yield macroscopic isotropic elasticity. Forisotropic materials, Hooke’s law in terms of matrix formcan be written as

s1

s2

s3

s4

s5

s6

2666666664

3777777775~

C11 C12 C12 0 0 0

C21 C11 C12 0 0 0

C21 C21 C11 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C66

2666666664

3777777775

e1

e2

e3

2e4

2e5

2e6

2666666664

3777777775

~E

(1zn)(1{2n)

1{n n n 0 0 0

n 1{n n 0 0 0

n n 1{n 0 0 0

0 0 0 (1{2n)=2 0 0

0 0 0 0 (1{2n)=2 0

0 0 0 0 0 (1{2n)=2

2666666664

3777777775

e1

e2

e3

2e4

2e5

2e6

2666666664

3777777775

(18)

where E and n are Young’s modulus and Poisson’s ratio,respectively.

Combining equation (18) and equations (15)–(17), therelationship between the linear stiffness coefficients ofthe bonds and the macroscopic material constants canbe established. However, the four spring stiffness cannotbe uniquely determined because there are only threeequations of equilibrium. To solve this over-determinedproblem, the shear stiffness kp

s of bond B1 in principaldirections is assumed to be zero, since the shear stiffnessko

s has components in principal directions and thecontribution of kp

s to macroscopic elasticity can berepresented in terms of ko

s , as seen in equation (13).Thus, the other spring constants kp

n, kon and ko

s can bedetermined as follows

kpn~

EL

4(1zn)(1{2n)

kon~

(1z2n)EL

4(1zn)(1{2n)

kos ~

(1{4n)EL

4(1zn)(1{2n)

(19)

It can be seen that the shear stiffness kos will become

negative when the Poisson’s ratio n exceeds 1/4. Thisindicates that the physical Poisson’s ratio range resultingfrom the site-bond model is{1vnƒ1=4, since thenegative spring constant seems non-physical. However,it is proved by molecular dynamics simulations that thenegative stiffness still has a physical explanation at themolecular level.10 Therefore, the shear spring withnegative stiffness can be added in order to model amaterial with a Poisson’s ratio higher than 1/4 but lowerthan 1/2.

Benchmark testsNumerical benchmark tests are carried to validate thederived spring stiffness coefficients. The macroscopicYoung’s modulus and Poisson’s ratio are selected asE511 000 MPa and n50?2. The stiffness coefficients ofsprings are calculated from equation (19). A cubic site-bond arrangement with size of 10L by 10L by 10L isused for simulations. This means that there are 10 unitcells in each principal direction; illustration given inFig. 2. The assembly is subjected to various loadingconditions, i.e. uniaxial tension and plane strain, toestimate the macroscopic Young’s modulus, Poisson’sratio and modulus of rigidity independently.

For uniaxial tension, the sites X150, X250 and X350are fixed in the X1, X2 and X3 direction, respectively. Adisplacement of L in the X3 direction is applied at sitesX3510L and other sites are free, which induces amacroscopic tensile strain et~e3~L=10L~0:1. Themacroscopic Poisson’s ratio n for tension and compres-sion is calculated according to n~{e1=e3 or n~{e2=e3,in which e1~u1=10L and e2~u2=10L are identical. Here,

Table 1 Direction vectors of bonds B1 and B2 in the site-bond assembly

Bond type Bond no. b j(b)i

B1 1 (1, 0, 0)2 (0, 1, 0)3 (0, 0, 1)4 (21, 0, 0)5 (0, 21, 0)6 (0, 0, 21)

B2 7 (1/31/2, 21/31/2, 21/31/2)8 (1/31/2, 1/31/2, 21/31/2)9 (1/31/2, 21/31/2, 1/31/2)10 (1/31/2, 1/31/2, 1/31/2)11 (21/31/2, 1/31/2, 1/31/2)12 (21/31/2, 21/31/2, 1/31/2)13 (21/31/2, 1/31/2, 21/31/2)14 (21/31/2, 21/31/2, 21/31/2)

(18)


S2-984 Materials Research Innovations 2014 VOL 18 SUPPL 2

u1 and u2 stand for the average displacements in the X1

and X2 directions of sites on plane X1510L andX2510L, respectively. The macroscopic modulus ofelasticity is estimated using E~st=et, where the macro-scopic stress in the X3-direction st is computed byst~f3=(10L:10L) and f3 is the reaction force at sites onplane X350. Figure 3 demonstrates the contour plot ofthe simulated von Mises stress by site-bond model underuniaxial tensile loading. The calculated macroscopicelastic modulus E is 11 225 and Poisson’s ratio n is0?1973, which have a relative error of about 2?05 and1?35%, respectively, compared to the imposed values.This shows that the proposed site-bond model is capableof simulating the elasticity of solids under uniaxialtension with a very good accuracy.

With respect to plane strain tension test, the sites onboth X150 and X1510L are fixed in the X1 direction.

The sites on X250 are fixed in the X2 direction. The siteson X350 are fixed in the X3 direction. The displacementsof L in the X2 and X3 directions are imposed at sitesX2510L and X3510L, respectively. The other sitesare free. Thus, two macroscopic tensile strainse2~L=10L~0:1 and e3~L=10L~0:1 are imposed onthe lattice and the strain in the X1 direction is zero. Themacroscopic modulus of rigidity can be computedaccording to G~ s3{l(e1ze2ze3)½ �=2e3, in which themacroscopic stress in the X3 direction s3 is calculatedusing s3~f3=(10L:10L) and f3 is the measured reactionforce at sites on plane X350. The Lame’s first parameteris calculated by l~En= (1zn)(1{2n)½ �. The Poisson’sratio and Young’s modulus are obtained asn~s1=(s2zs3) and E~ s3{n(s1zs2)ð Þ=e3. It is foundthat the Young’s modulus and Poisson’s ratio obtainedfrom plane strain tension test are the same as thosederived from uniaxial tension test. The calculated shearmodulus according to the method as introduced before-hand is 4691, which has a 2?35% difference relative tothe theoretical value of G~11000= 2(1z0:2)½ �~4583:33MPa. This means that the site-bond model with thederived normal and shear spring constants is able tosimulate the elastic behaviour of an isotropic materialunder shear loading.

To investigate the influence of the number of unit cellson the estimated macroscopic elastic constants, a set ofcubic cellular lattices with various sizes from L3 to10 648L3 is generated. Simulations are performed with

3 Simulated von Mises stress under uniaxial tension

4 Predicted Young’s modulus against number of cells in

principal direction

5 Predicted Poisson’s ratio against number of cells in

principal direction

2 Generated lattice 10L610L610L


Materials Research Innovations 2014 VOL 18 SUPPL 2 S2-985

the plane strain boundary conditions. The predictedmacroscopic Young’s modulus, Poisson’s ratio andshear modulus against the number of unit cells in eachprincipal direction of a cubic region are plotted inFigs. 4–6.

It can be seen that the simulated macroscopic Young’smodulus and shear modulus decrease with the increasein the number of cells in principal direction. On thecontrary, the estimated Poisson’s ratio increases withthe increasing size of cellular lattice. Eventually, thesimulated elastic constants tend to their correspondingtheoretical values with the increase in the size of cellularlattice. This is attributed to the smaller boundary effectwhen the size of region is larger. When the number ofcells in principal direction is higher than 20, theestimated results are close enough to the theoreticalvalues.

The generated site-bond model can be considered as avalid representation of isotropic elastic material. Basedon the definition of criterion for bond failure andrelevant implementation as presented in a previous studyby Jivkov et al.,6 the fracture process and damageevolution in quasi-brittle materials can be simulated byusing the proposed site-bond model. This is a subject ofongoing work. In addition, the effects of microstructureparameters, such as porosity, pore size distribution andconnectivity of solid phase, on the macroscopic beha-viour, stress–strain response of quasi-brittle materialswill be investigated. The results of these aspects will bereported in future publications.

ConclusionThis work presents a meso-scale model for macroscopicelasticity of solids. The model is based on a cellularlattice of truncated octahedrons, filling the spacecompactly. The cellular architecture is transformed intodiscrete site-bond lattice with bonds containing normaland shear springs. The spring stiffness coefficients areobtained as functions of macroscopic elastic constants.

From the findings of the present study, the followingconclusions can be drawn.

1. The site-bond assembly represents generally amacroscopic cubic elasticity and is able to deliver anymacroscopic isotropic elasticity.

2. The physical Poisson’s ratio range results from thesite-bond model is {1vnv1=2.

3. For uniaxial tension test, the estimated macro-scopic Young’s modulus and Poisson’s ratio show a verygood agreement with the theoretical ones.

4. For plane strain test, the measured macroscopicmodulus of rigidity fit very well with the theoreticalvalue.

5. The size of cellular lattice plays an important rolein the accuracy of simulation due to boundary effect.When the number of cells in principal direction is higherthan 20, the obtained simulation results are close enoughto the theoretical data.

6. The proposed site-bond model is regarded as a verygood representation of isotropic elastic materials, andwill be applied to simulate the fracture process anddamage evolution in quasi-brittle materials.

Acknowledgements

M. Zhang and A. P. Jivkov acknowledge the supportfrom EPSRC via grant no. EP/J019763/1, ‘QUBE:Quasi-Brittle fracture: a 3D experimentally-validatedapproach’, and from BNFL for the Research Centre forRadwaste & Decommissioning. C. N. Morrison greatlyappreciates the support from EPSRC via NuclearFiRST Doctoral Training Centre.

References1. A. P. Jivkov and J. R. Yates: ‘Elastic behaviour of a regular lattice

for meso-scale modelling of solids’, Int. J. Solids. Struct., 2012, 49,

3089–3099.

2. E. Schlangen and J. G. M. van Mier: ‘Experimental and numerical

analysis of micromechanisms of fracture of cement-based compo-

sites’, Cem. Concr. Compos., 1992, 14, 105–118.

3. C. S. Chang, T. K. Wang, L. J. Sluys and J. G. M. van Mier:

‘Fracture modelling using a micro structural mechanics approach.

I: Theory and formation’, Eng. Fract. Mech., 2002, 69, 1941–1958.

4. D. V. Griffiths and G. G. W. Mustoe: ‘Modelling of elastic

continua using a grillage of structural elements based on discrete

element concepts’, Int. J. Numer. Meth. Eng., 2001, 50, 1759–1775.

5. E. Schlangen: ‘Crack development in concrete. Part 2: Modelling of

fracture process’, Key Eng. Mater., 2008, 73–76, 385–387.

6. A. P. Jivkov, D. L. Engelberg, R. Stein and M. Petkovski: ‘Pore

space and brittle damage evolution in concrete’, Eng. Fract. Mech.,

2013, 110, 378–395.

7. Y. Wang and P. Mora: ‘Macroscopic elastic properties of regular

lattices’, J. Mech. Phys. Solids, 2008, 56, 3459–3474.

8. M. Ostoja-Starzewski: ‘Lattice models in micromechanics’, Appl.

Mech. Rev., 2002, 55, 35–60.

9. G. Wang, A. Al-Ostaz, A. H.-D. Cheng and P. R. Mantena:

‘Hybrid lattice particle modelling: Theoretical considerations for a

2D elastic spring network for dynamic fracture simulations’,

Comput. Mater. Sci., 2009, 44, 1126–1134.

10. S. Zhao and G. Zhao: ‘Implementation of a high order lattice

spring model for elasticity’, Int. J. Solids. Struct., 2012, 49, 2568–

2581.

6 Predicted shear modulus against number of cells in

principal direction


S2-986 Materials Research Innovations 2014 VOL 18 SUPPL 2

Appendix E

Lattice-spring model of graphite

accounting for pore size distribution

153

Lattice-spring modeling of graphite accounting for pore size distribution

Craig N Morrison1,a, Andrey P Jivkov1,b, Gillian Smith2,c and John R Yates1,d 1School of MACE, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

2Interface Analysis Centre, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK

[email protected], [email protected], [email protected], [email protected]

Keywords: Nuclear graphite; Porosity; Meso-scale deformation; Micro-cracks; Macroscopic behaviour

Abstract. Lattice models allow length scale dependent micro-structural features and damage

mechanisms to be incorporated into analyses of mechanical behaviour. They are particularly

suitable for modelling the fracture of nuclear graphite, where porosity generates local failures upon

mechanical and thermal loading. Our recent 3D site-bond model is extended here by representing

bonds with spring groups. Experimentally measured distributions of pore sizes in graphite are used

to generate models with pores assigned to the bonds. Microscopic damage is represented by failure

of normal and shear springs with different criteria based on force and pore size. Macroscopic

damage is analysed for several loading cases. It is shown that, apart from uniaxial loading, the

development of micro-failures yields damage-induced anisotropy in the material. This needs to be

accounted for in constitutive laws for graphite behaviour in FEA of cracked reactor structures.

Introduction

Nuclear-grade graphite has featured in over 100 reactors [1] with its main functions being a fast

neutron moderator and structural material. It also forms an integral part of a potential Generation

IV Very High Temperature Reactor (VHTR). The integrity of graphite, as with all structural reactor

components, is critical for their fitness for purpose. Understanding graphite’s fracture behaviour is

essential for approving plant life extensions and predicting in-service performance.

Synthetic graphite is manufactured from petroleum cokes and a binder material; usually coal-tar

pitch. The resulting micro-structural features – grain size, pore size/density, are strongly influenced

by the manufacturing process and the structure of the coke and binder particles used. The work in

this paper focuses on Gilsocarbon, a relatively fine-grained graphite, used in the UK Advanced

Gas-cooled Reactors (AGRs). Graphite microstructure consists of 3 phases; a matrix of graphitized

binder particles, relatively large filler particles (derived from coke) and porosity [2]. In Gilsocarbon

the spherical filler particles, ranging from 0.3-1.5mm in size, result in near isotropic mechanical

properties [3]. The 3 main porosity populations [4], ranging from nm to mm in size, total

approximately 20% of virgin graphite volume. Gas evolution cracking occurs within the matrix as

gas bubbles form when liquid pitch, impregnated to increase density, boils during baking [3].

Calcination and Mrozowski cracks form within filler particles due to uneven thermal expansion and

shrinkage as the graphite heats and cools during calcination and graphitization respectively [2,5].

Graphite, alongside rock, concrete and cement, is considered a quasi-brittle material [6], with

failure occurring when distributed micro-cracks coalesce into a critically sized flaw. As a result,

graphite exhibits a reduced stiffness upon loading (similar to plasticity) prior to failure [2] despite

remaining brittle. Using local approaches to model graphite failure can capture the dependence of

macro-scale integrity on meso-scale features by incorporating mechanistic understanding of failure

at the length scale of features [7]. However current methods rely on weakest link (WL)

assumptions and fail to account for the micro-crack interactions apparent in quasi-brittle materials

[8]. Lattice models are a branch of discrete, local approach models, consisting of nodes connected

into a lattice through discrete elements including springs [9] and beams [10]. Element properties

allow a material response according to actual mechanistic failure data. Unlike WL methods, lattice

models are based around a parallel statistical system, with load redistribution amongst remaining

bonds once a bond is broken. Such models have been developed for graphite [4,10,11] and cement

[12]. The work presented here furthers the site-bond lattice model developed by Jivkov and Yates

[13]. The model has been calibrated using a generalized continuum theory, with bond stiffness

constants evaluated through strain energy equivalence of a discrete unit cell and classical

continuum [14].

Theory and model

The site-bond model is based upon a tessellation of space into truncated octahedral cells, Fig 1a.

Particles, attached to cell centres, interact via deformable bonds, with 14 bonds connecting a

particle to its neighbours, Fig 1b. The 6 bonds in the principal directions and 8 bonds in the

octahedral directions can have different mechanical responses. The energetic calibration for this

model [14] suggested that as a first assumption particle rotations could be ignored, with bonds

represented only by normal and shear springs. Spring elastic constants Kaxprinc

, Kaxoct

, Kshprinc

and

Kshoct

were calibrated for plane stress and plane strain for a graphite of typical properties E=11Gpa,

v=0.2 [15] with values 4.041 x104, 4.939 x10

4, 2.245 x10

3 and 4.49 x10

3 respectively.

The behaviour of shear springs is illustrated in Fig 2a, where the shaded area represents energy

dissipated in spring failure. For axial springs, failure in compression is not permitted and the

behaviour is shown in Fig 2b.

Gilsocarbon microstructure was modeled, assuming porosity occurs exclusively within the

matrix phase, with calcination and Mrozowski cracks ignored. Filler particles with normally

distributed sizes were randomly located with total volume fraction of 20%. Pores with normally

distributed sizes were located in the matrix with a constraint preventing coalescence and 5%

volume fraction. This porosity was distributed to the faces of the cellular structure. The ratio

between pore and face areas determined the failure energy of the corresponding springs, so that Gc

varied between zero for very large pores and one for very small pores. The model was subject to

uniaxial tension; u1 = u, (x2 and x3 directions remain unconstrained), as well as to fields experienced

ahead a crack: high-constraint plane strain (u1 = u2 = u, u3 = 0) and lower-constraint plane strain (u1

= 0.5u2 = u, u3 = 0). For uniaxial tension the damage can be characterized by the parameter DE, Eq.

(1), as principal stresses σ2 = σ3 = 0.

. (1)

For the plane strain cases, 4 damage parameters are required, Eq. (4-5), evaluated from a

decomposition of nominal stress/strain into deviatoric and volumetric components of sα, eα and σh,

εv respectively, Eq. (2-3), where σh is hydrostatic stress and εv is volumetric strain.

Fig 1. Cellular representation of material

(a); and unit cell with bonds (b).

Fig 2. Spring

failure criteria

for shear

springs(a);

and normal

springs (b).

Displacement -2(uc)sh

Load

(a) (b) (Fc)

sh

(uc)sh

Displacement -10(uc)ax

Load

(Fc)ax

2(uc)sh

-(uc)sh

-(Fc)sh

(uc)ax 2(uc)

ax

Shaded area

= (Gc)sh

Shaded area

= (Gc)ax

-10(Fc)ax

. (2)

where . (3)

. (4)

. (5)

Where K and μ are the bulk and shear moduli respectively.

Results

Uniaxial tension exhibits quasi-brittle behaviour with softening before failure as expected, Fig 3.

The similarity of damage parameters, DE and Dμ suggests that the material remains isotropic, Fig 4.

Upon loading in plane strain under both high and low constraint the damage parameter Dμ

becomes dependent on direction suggesting anisotropy is introduced, Fig 5b and Fig 6b.

Fig 4. The

damage

parameters for

uniaxial tension,

DE (a); Dμ (b).

Fig 5. The

damage

parameters for

plane strain

high constraint

DK (a); Dμ (b).

Fig 3. The stress/strain

response of the model

under uniaxial tension.

(a) (b)

(a) (b)

Discussion and conclusions

The results show that the development of micro-damage in porous graphite is strongly dependent

on the loading. Uniaxial loading seems to be the only case where the material remains isotropic

with damage evolution. This suggests that a macroscopic damage evolution law based on a single

damage parameter is appropriate only for uniaxial states. For material ahead of a macroscopic

crack, high-constraint, Fig 5, and low constraint, Fig 6, damage evolves differently in different

loading directions. Negative damage parameter observed in Fig 5, is not surprising; it merely

shows that the transverse shear modulus in y-direction increased above the initial isotropic value as

the transverse shear modulus in x-direction decreased. It is also clear that damage of shear moduli

is larger than the damage of bulk modulus. The results suggest that damage evolution laws for

complex loading cases need to be based on three independent invariant of the stress tensor.

Presently, it is not clear whether a load-independent evolution law can be developed for materials

with given pore space structures with the strategy presented here. This is a subject of ongoing

work.

References

[1] IAEA report (2006).

[2] A. Hodgkins, T. J. Marrow, M. R. Wootton, R. Moskovic, and P. E. J. Flewitt: Mater. Sci.

Tech. Vol. 26 (2010), p. 899.

[3] M. R. Joyce, T. J. Marrow, P. Mummery, and B. J. Marsden: Eng. Fract. Mech. Vol. 75

(2008), p. 3633.

[4] M. R. Bradford and A. G. Steer: J. Nucl. Mater. Vol. 381 (2008), p. 137.

[5] E. Schlangen, P. E. J. Flewitt, G. E. Smith, a. G. Crocker, and A. Hodgkins: Key Eng.

Mater. Vol. 452–453 (2010), p. 729.

[6] M. Mostafavi and T. J. Marrow: Fatigue Fract. Eng. Mater. Struct. Vol. 35 (2012), p. 695.

[7] J. Lemaitre: Eng. Fract. Mech. Vol. 23 (1986), p. 523.

[8] Z. P. Bažant and S.D. Pang: J. Mech. Phys. Solids Vol. 55 (2007), p. 91.

[9] A. Pazdniakou and P. M. Adler: Tran. Porous Med. Vol. 93 (2012), p. 243.

[10] E. Schlangen and E. Garboczi: Eng. Fract. Mech. Vol. 57 (1997), p. 319.

[11] N. N. Nemeth and R. L. Bratton: Nucl. Eng. Design Vol. 240 (2010), p. 1.

[12] P. Grassl, D. Grégoire, L. Rojas Solano, and G. Pijaudier-Cabot: Int. J. Solids Struct. Vol. 49

(2012), p. 1818.

[13] A.P. Jivkov and J.R. Yates: Int. J. Solids Struct. Vol. 49 (2012), p. 3089.

[14] C. N. Morrison, A. P. Jivkov, and J. R. Yates: Proc. ICF13 (Beijing, China, 2013), p. S31-

016.

[15] M. Holt: Issues of scale in nuclear graphite components (PhD Thesis, University of Hull,

2008).

Fig 6. The

damage

parameters for

plane strain

low constraint,

DK (a); Dμ (b).

(a) (b)

Appendix F

A discrete lattice model of quasi-brittle

fracture in porous graphite

158

Materials Performance andCharacterization

Craig N. Morrison,1 Mingzhong Zhang,2 Andrey P. Jivkov,2 and John

R. Yates2

DOI: 10.1520/MPC20130077

Discrete Lattice Model ofQuasi-Brittle Fracture inPorous Graphite

VOL. 3 / NO. 3 / 2014

Craig N. Morrison,1 Mingzhong Zhang,2 Andrey P. Jivkov,2 and John R. Yates2

Discrete Lattice Model of Quasi-BrittleFracture in Porous Graphite

Reference

Morrison, Craig N., Zhang, Mingzhong, Jivkov, Andrey P., and Yates, John R., “Discrete

Lattice Model of Quasi-Brittle Fracture in Porous Graphite,” Materials Performance and

Characterization, Vol. 3, No. 3, 2014, pp. 414–428, doi:10.1520/MPC20130077.

ISSN 2165-39923

ABSTRACT

Lattice models allow the incorporation of length-scale-dependent

microstructural features and damage mechanisms into analyses of the

mechanical behavior of materials. We describe our 3D lattice implementation

and its use in fracture simulations. The method is particularly suitable for

modeling fractures of nuclear graphite. This is a quasi-brittle material in which

there is considerable non-linearity prior to final fracture caused by the inherent

porosity, which triggers a field of local distributed failures upon mechanical

and thermal loading. Microstructure representative models are generated with

experimentally measured particle and pore size distributions and volume

densities in two graphite grades. The results illustrate the effect of distributed

porosity on the emerging stress–strain response and damage evolution. It is

shown how the failure mode shifts from graceful, plastic-like behavior

associated with substantial energy dissipation via distributed damage at lower

porosities, to glass-like behavior with negligible energy dissipation at higher

porosities. Thus, the work proposes a microstructure-informed methodology

for integrity assessment of aging structures, where porosity increase is driven

by environmental factors, such as radiation of nuclear graphite components.

Keywords

nuclear graphite, porosity, brittle-ligament lattice, damage evolution, quasi-brittle behavior

Manuscript received October 17,

2013; accepted for publication

January 20, 2014; published online

June 18, 2014.

1

Mechanics and Physics of Solids

Research Group, Modelling and

Simulation Centre, The Univ. of

Manchester, Oxford Rd.,

Manchester M13 9PL, United

Kingdom; and Nuclear FiRST

Doctoral Training Centre,

Manchester M13 9PL,

United Kingdom, e-mail:

[email protected]

2

Mechanics and Physics of Solids

Research Group, Modelling and

Simulation Centre, The Univ. of

Manchester, Oxford Rd.,

Manchester M13 9PL,

United Kingdom.

3

This paper is a contribution to a

Special Issue of Materials

Performance and Characterization

on “Fracture Toughness,” Guest

Editors, Bojan Podgornik and

Votjeh Leskovsek, Institute of

Metals and Technology, Ljubljana,

Slovenia.

Copyright VC 2014 by ASTM International, 100 Barr Harbor Drive, P.O. Box C700, West Conshohocken, PA 19428-2959 414

Materials Performance and Characterization

doi:10.1520/MPC20130077 / Vol. 3 / No. 3 / 2014 / available online at www.astm.org

Introduction

Nuclear-grade graphite has been used as a fast neutron moderator in over 100

reactors throughout the world [1], with planned use in the generation IV very-high-

temperature reactor. A main advantage of graphite over other moderator materials,

such as light and heavy water, is its suitability for use as a structural material. As a

result, developing a means of predicting the structural integrity of graphite under

complex loading and in demanding environments, such as a nuclear reactor core, is

critical for predicting plant lifetime and in-service performance.

Nuclear graphite has a feature-rich multiphase microstructure consisting of

petroleum or pitch coke filler particles distributed within a matrix of binder material,

usually coal-tar pitch. Distributed within both the matrix and filler phases is a popu-

lation of porosity. This population can be broadly grouped into three main catego-

ries [2]:

• Gas evolution cracking—During manufacture, liquid pitch is impregnated intothe graphite to increase density. In the subsequent baking of the green article,the pitch boils resulting in the evolution of gas through the matrix phase [3].

• Calcination cracks—During the heating and cooling procedures involved incalcinations, the variation of the thermal expansion/shrinkage of the two solidphases of graphite’s microstructure induces cracks [2,4].

• Mrozowski cracks—These form for similar reasons to calcination cracks dur-ing the graphitization process [2,4].

These flaws can vary in length scale, ranging from the nm scale to mm scale

depending on the grade of graphite in question. The variation of input materials and

manufacturing processes strongly influences the resulting microstructure so that dif-

ferent grades of graphite can have widely different properties.

The work in this paper focuses on two grades of graphite for potential use in

future generations of high-temperature reactors. IG110 is a fine-grained graphite of

isotropic macroscopic properties currently in use in the Japanese high-temperature

test reactor (HTTR), which first reached criticality in November 1998 [5]. It also

remains a possibility for generation IV high-temperature reactors. Another possibil-

ity for high-temperature reactors is PGX, a medium-grained semi-isotropic graphite.

The use of these two grades allows an effective comparison of the proposed method-

ology for grades of varying pore and particle sizes.

Component-scale failure of graphite has been shown to be dependent on its

discrete multiphase microstructure, with the primary failure mechanism being the

coalescence of micro-cracks into a critically sized flaw [6]. Such a response is consid-

ered to be a property of the class of quasi-brittle materials, allowing graphite to be

considered alongside rock, concrete, and cement in its failure behavior. Distinctive

properties of quasi-brittle materials include a reduced stiffness upon loading prior to

failure and a residual load-carrying capacity beyond the peak load [7]. The initial

softening appears similar to plasticity although the individual phases remain brittle.

Failure modeling of quasi-brittle materials has led to the development of a num-

ber of local fracture models, whereby mechanistic understanding at the length scale

of microstructural features—referred to as the meso-scale in this context—can be

incorporated into component scale models [8]. In this way, the macro-scale depend-

ence on these microstructural failure mechanisms can be recreated in a physically

MORRISON ET AL. ON DISCRETE LATTICE MODEL 415


realistic manner. Most current local approaches, such as weakest link (WL) assump-

tions, are phenomenological, with parameters calculated by curve fitting to macro-

scopic experimental data [9]. Such approaches do not account for the chain of

events leading to macroscopic failure, from micro-crack nucleation, through micro-

crack interaction, and, to coalescence, observable at the microscale of quasi-brittle

materials [10].

A continuum-based approach, intended to account for micro-crack nucleation

and growth, is the cohesive zone modeling [9,11], where the failure initiation and

local softening in the quasi-brittle materials are represented by special cohesive

elements. The advantage of this approach is that it is easily incorporated into the

existing finite element formulations, and indeed cohesive elements are offered as

standard in most finite element packages. However, the cohesive zone modeling

relies on the introduction of cohesive elements; hence, the modeling predefines the

potential crack paths. Further, the formulation of the constitutive behavior of the

cohesive elements is phenomenological, involving calibration against observed

macroscopic behavior. Microstructure-informed models, such as the one offered in

this work, have the potential to assist in deriving mechanistic constitutive laws for

cohesive zone modeling.

Discrete approaches offer a promising method of developing material constitu-

tive equations based on the actual micromechanical processes occurring. Lattice

models form a branch of local discrete models, wherein a lattice is formed through

the connection of nodes via discrete elements including springs [12] and beams [13].

The designation of element properties as representative of the micromechanical

failure mechanisms can allow a realistic macro-mechanical response. Unlike WL

methods, where bond failure signals total sample failure, lattice models form a statis-

tically parallel system. This allows redistribution of load once a bond is broken

among the remaining bonds.

Lattice models have been successfully developed for graphite [4,14] and concrete

[13,15]. Previous models have been limited in their ability to reproduce a full range

of Poisson’s ratio for elastic materials. Regular two-dimensional (2D) beam-lattices

using hexagonal unit cells, such as [10], can only be used for an isotropic elastic

material with Poisson’s ratio below 1/3 and 1/4 in plane stress and plane strain,

respectively [15]. This is sufficient for most of the materials classified as quasi-

brittle. However, the evolution of damage via micro-crack formation and growth is

intrinsically a 3D phenomenon, and physically realistic studies require 3D lattices.

Regular 3D lattices, essential for realistic fracture modeling, suffer the same prob-

lems. Simple cubic lattices [16] and more complex face-centered cubic (FCC) and

hexagonal closely-packed (HCP) lattice arrangements [17] have been demonstrated

to be suitable only for materials with a Poisson’s ratio of 0 [18]. The 3D lattice used

in this work offers a significant improvement, as it can represent any isotropic mate-

rial with a Poisson’s ratio of up to 1/4, similar to 2D lattices in plane strain, as shown

in the next section.

The work extends the site-bond 3D lattice model developed by Jivkov and Yates

[19]. Sites at the center of unit cells are connected to neighboring sites by bonds.

Previous work on this model using beam elements as bonds have demonstrated its

ability to represent isotropic elastic materials with Poisson’s ratio between �1 and

0.5 [19]. One of the issues with using beam elements is that no unique



correspondence between the discrete and the continuum response of a cell can be

established without the use of generalized continuum theory [19]. Therefore, in the

current work, bundles of linear springs are used to replace the beam elements as

bonds.

This paper presents a methodology for construction of a meso-scale site-bond

3D lattice accounting for experimentally measured particle and pore size distribu-

tions as an extension on previous work [20]. A parametric study between a surface-

based damage parameter and a standard mechanical damage parameter under

uni-axial tension is used to establish a relationship between the volume fraction of

distributed pores and progressive mechanical damage. It is demonstrated how the

increase of porosity because, for example, of radiation-induced microstructure

changes, embrittles the material, which can provide a scientific underpinning for

integrity assessment of aging graphite components.

Theory and Model

The site-bond model uses a regular tessellation of space into truncated octahedral

cells as a representation of the material microstructure (Fig. 1(a)). This choice is dic-

tated by the observation that the truncated octahedron is the regular shape closest to

the average cell in Voronoi tessellations of real microstructures, in terms of faces,

edges, and vertices [21]. Hence, our cellular representation of the microstructure can

be considered as an initial topological homogenization. The truncated octahedron

comprises of six square faces, normal to principal axes, and eight hexagonal faces,

normal to the octahedral axes. The cellular lattice is used to maintain a link to the

microstructure features to be accounted for in the model. In addition, a discrete

lattice of sites located at cell centers, and bonds connecting neighboring sites, is

generated for computational purposes. As a result, each site is connected to its

neighbors by 14 bonds, six in the principal directions, B1, with lengths equal to the

cell size in principal direction, L, and eight in the octahedral directions, B2, with

lengths equal to H3/2 L (Fig. 1(b)).

In this site-bond model, the bonds between sites are modeled as bundles of

elastic springs. Generally, a bundle should contain six springs to represent the six

possible degrees of freedom between sites, three translational/linear springs (one

normal/axial and two shear), and three rotational/angular springs (one twist/axial

FIG. 1 (a) The cellular site-bond lattice with a single highlighted unit cell; (b) unit cell with bonds; and (c) normal and shear

springs.



and two bending). From a mechanical perspective, the linear and the angular springs

should resist the symmetric (strains in continuum) and the skew-symmetric

(rotations in continuum) parts of the local displacement gradient, respectively.

Mathematically, these two tensors are fully uncoupled, meaning that no relation

between linear and angular spring behavior can be established. The linear springs

can be calibrated using the correspondence between classical continuum strain

energy in the cell and in the springs under identical displacement fields; this is

explained below. However, the angular springs cannot be resolved without recourse

to a generalized continuum theory, such as [22], and the corresponding material

length-scale required in such a theory. Experimental work is planned to establish

the validity of the generalized continuum description and the relation between the

material length-scale and real microstructure length-scales.

A recent paper considered the bonds to be represented by bundles of linear

springs only, with one normal and two shear springs per bond [23] as shown in

Fig. 1(c). In such a case, and assuming the two shear springs in a bundle to have

equal stiffness, there are four spring constants to be calibrated using energy equiva-

lence between the discrete and continuum cells: kpn and kps , normal and shear stiff-

ness in principal direction, and kon and kos , normal and shear stiffness in principal

direction. It has been shown analytically that such a configuration represents a case

of macroscopic cubic elasticity, with three elastic constants describing the material

behavior [23]. The resulting indeterminacy, with three elastic constants available for

calibration of four spring stiffness constants, could be solved in a number of ways. It

has been suggested that the shear stiffness in the principal direction kps be set to zero

[23]. The local behavior is not affected by this decision, because shear on planes par-

allel to the principal directions is also resisted by the shear and normal springs in

the octahedral direction. Furthermore, it has been shown that for an isotropic mate-

rial [23], with the selection kps ¼ 0, the remaining spring stiffnesses in this model are

related to the elastic modulus, E, the Poisson’s ratio, �, and the cell size, L, by:

kpn ¼EL

4 1þ �ð Þ 1� 2�ð Þ ; kon ¼1þ 2�ð ÞEL

4 1þ �ð Þ 1� 2�ð Þ ; kos ¼1� 4�ð ÞEL

4 1þ �ð Þ 1� 2�ð Þ(1)

It can be seen that the shear stiffness kos will become negative when Poisson’s ratio �

exceeds 1/4. This indicates that the site-bond model with springs can represent

isotropic materials with Poisson’s ratio in the range �1 < � � 1=4. The result differs

from the analysis using beams [19] and the reason is that the angular springs have

been neglected in [23] and in the current work, whereas the beams introduce specific

coupling between bending and shear.

The spring stiffnesses for the models of the two graphite grades, IG110 and

PGX, were evaluated using Eq 1, and the macroscopic material properties shown in

the cell size calculation is described in the next section.

Table 1 contains data obtained from Refs 24 and 25. The cell size calculation is

described in the next section.

Distribution of Material Features

Statistical distributions of filler particle and pore sizes for the two grades of nuclear

graphite in the virgin state were generated according to experimental data from



recent works by Kane et al. [26]. A lognormal distribution was used to reproduce the

particle distribution parameters. The pore distributions were produced using third-

order polynomial parameters specified in [26]. Basic parameters for the particle and

pore size distributions and volume fractions for the two grades in their virgin state

are given in Table 2.

The procedure to construct the site-bond lattice with a suitable meso-length

scale is as follows:

• A cellular structure contained in a region is specified in terms of numberof cells in the three principal directions. Thus, for Nx, Ny, and Nz cellsin the three directions, the cellular structure will contain C¼Nx�Ny�Nzþ (Nx�1)� (Ny�1)� (Nz�1) cells.

• Particles with assumed spherical shape are assigned to all cells, with one parti-cle per cell with random size from the experimental particle size distribution.

• A representative cell length, L, is calculated using Eq 2, using the volume ofthe cellular structure as a function of cell length (Eq 3), the total volume of thedistributed particles, and the particle volume fraction h.

L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2XCi¼1

viparticle

Ch

3

vuuut(2)

Vlattice ¼ CL3

2(3)

Such an assignment can be used to assess how representative a cellular structure

with C cells is for prescribed particle size distribution and volume density. If the

scatter in calculated cell length for different random assignments of particles in C

cells is smaller than some value, then the cellular structures with C cells can be

accepted as sufficiently representative. Figure 2 illustrates a sensitivity study of the

length-scale dependence on the number of cells. With this result, and accepting a

maximum standard deviation of 6.02 and 27.9 (4 % and 5 % of the mean cell length)

TABLE 1

Material properties for PGX and IG110 graphite.

IG110 PGX

E (MPa) 9800 8300

� 0.14 0.11

Note: Data for IG110 from Ref 23; data for PGX from Ref 24.

TABLE 2

Particle and pore size parameters reproduced from experimental data.

IG110 PGX

Mean particle diameter (lm) 27 92

Mean pore area (lm2) 98 197.9

Particle volume fraction 20 % 20 %

Porosity in virgin state 14.73 % 21.49 %



for IG110 and PGX, respectively, we justify the use of a 15� 15� 15 cellular

structure in the subsequent simulations as a compromise between accuracy and

computational expense. This structure contains 6119 cells and 39|046 bonds.

In the paper by Kane et al. [26], it was observed that, at the obtained image reso-

lution, less than 1 % of the pores present in both grades of graphite occurred within

the filler particles. Hence, it is reasonable to assume in our porosity modeling proce-

dure that all porosity occurs in the matrix phase of graphite, and we can ignore the

calcination and Mrozowski cracks. Furthermore, as with the particles, the pores are

assumed to be spherical.

Following lattice construction and length-scale calculation, pores are distributed

to the faces between cells. Multiple pores can be allocated to each cell face to reach

the desired pore volume fraction (porosity). This is done in a succession of passes

over the cell faces in a random manner. An illustration of distributed porosity on

cell faces, equivalently bond centers, is given in Fig. 3. Multiple pores per face are

not shown for clarity of the figure.

Multiple lattices were generated for the two grades of graphite for varying pore

volume fractions from the virgin state up to a maximum of 60 %, with the increase

in volume fraction giving an insight into the effects of increased porosity over a

nuclear reactor lifetime on damage. It is assumed that as the porosity increases both

the particle and porosity statistical size distributions remain the same.

Mechanical damage upon loading is represented by progressive failure of the

bonds in the site-bond lattice. The behavior of the two types of springs is shown in

Fig. 4. The shaded areas represent the energy dissipated upon failure of the spring,

Gc, which is different for different bonds, depending on the assigned pores. The

energy is calculated with Gc¼ c Ac, where c is the energy lost in creation of two

surfaces, and Ac is the intact area of the face, i.e., the original face area minus

the cross-section area of all pores assigned to the face. The material parameter

c¼ 9.7 J/m2 is taken from the literature as the enthalpy for creation of two surface in

graphite [27]. Notably, this is the energy of separation of ideal atomic lattice of

FIG. 2 The sensitivity of the produced cell size from a sample from a given population for varying number of cells for (a) IG110

graphite; and (b) PGX graphite. Error bars indicate one standard deviation for 1000 different samples of particle size

distributions.



graphite and should not be compared to fracture energy directly. Therefore, this

value is identical for the two graphite grades analyzed here. The fracture energy of a

particular grade is an emergent property from the underlying energy of separation

and its microstructure features and distribution. It can be represented by the area

under the simulated stress–strain curve and will be different for different failure

modes, e.g., normal or shear failure. The pore areas allocated to each face are the

maximum pore cross-sectional areas. This is justified to a large extent in the context

of the discrete computational model where the pore can be placed anywhere between

the sites and have the same failure effect on the spring bundle. Thus, Gc tends

toward the maximum spring failure energy (material surface energy) for pore size

approaching zero, and tends toward zero for pore size approaching face area. Bonds

are removed prior to simulations if the initial pore area exceeds the area of the face

on which it is situated.

The assumption used for these criteria is that if the spring damage initiates at a

critical relative displacement, then the damage evolution is linear (softening branch)

FIG. 4 Failure criteria for (a) shear springs; and (b) axial springs.

FIG. 3

A visual representation of the

site bond model with pores

distributed on faces between

cells.



and terminates with spring failure at two times the critical displacement. This is a

reasonable assumption for local brittle failure and allows for direct calculation of the

critical displacement and force from knowledge of the failure energy and initial

spring stiffness. Notably, the axial springs are not allowed to fail in compression,

and the behaviors of the two types of springs are uncoupled in the current model.

Equal failure energies for the axial and shear springs in a bond have been used.

Results and Discussion

Sample site-bond lattices with distributed porosity, with a range of volume fractions,

for each grade of graphite were loaded under displacement control in uni-axial ten-

sion. The simulations were performed in steps of increasing boundary displacements

until the final rupture of the samples. This was an automated solution, such that

multiple spring failures could potentially occur within a loading increment. Whereas

this differs from previous works [17,27], where the load increment was driven by

individual failures, it represents the behavior that would be observed when testing

real samples. At each increment end, the macroscopic stress was determined from

the reaction forces on the boundary with prescribed displacements and the physical

area of this boundary; the macroscopic strain was determined from the current dis-

placement level and the length of the sample. As the strains and stresses are small

within the loading range to failure for this material, the obtained engineering strain

and stress are approximately equal to the true values.

Data for the models analyzed is given in Table 3. This includes the model length

scale derived from each specific particle distribution to cells, the fraction of initially

removed bonds because of excessive pore size on corresponding faces, and the initial

elastic moduli determined prior to damage initiation in the lattice. Notably, the elas-

tic moduli, given in Table 1 and used for the spring stiffness calibration, were derived

with specimens containing virgin state porosities. This is the reason for the differ-

ence between the virgin state elastic moduli in Tables 1 and 3. In principle, the model

can be calibrated without difficulties to account for given porosity (missing bonds)

and provide required macroscopic elastic modulus. However, the initial calibration

of the full lattice is preferable here to facilitate comparison between cases with

increasing porosity. It also helps to note that the porosity effect on the elastic modu-

lus is not linear and arises from the specific combination of a particle size

TABLE 3

Basic parameters of the analyzed models.

Pore Volume Fraction (porosity) Cell Size (lm) Fraction of Bonds Removed Initial Young’s Modulus (MPa)

IG110 0.1473 (virgin) 148.4 0.026 9474.6

0.3 159.1 0.086 8417.6

0.6 148.8 0.319 5083.0

PGX 0.2149 (virgin) 607.0 0.002 8377.4

0.4 547.0 0.268 4938.7

0.6 628.1 0.352 4226.5

Note: Cell size derived from particle distribution to cells; fraction of removed bonds because of large pores on faces; elastic modulus ofvoided model prior to damage.



distribution (via the length scale) and a pore size distribution. For example, compare

the difference between the virgin states of the two grades where the higher porosity

grade, PGX, is stiffer than the lower porosity grade, IG110, relative to the benchmark

values in Table 1, because of the different relation between the cell size (particle sizes)

and the pore sizes.

Figure 5 illustrates the emergent stress–strain curves for both grades of graphite

with different porosities. The initial linear behavior in all cases is governed by corre-

sponding macroscopic elastic modulus, listed in Table 3. Beyond the linear response,

clear quasi-brittle non-linearity prior to the fracture point is observed in Fig. 5 for

cases with low porosities (below 30 %). For higher pore volume fractions, very

little non-linearity is exhibited; the samples failed in a strictly brittle manner at mar-

ginally higher strains than for the cases with lower porosity. The last points of each

stress–strain curve correspond to final failure, i.e., the separation of the modeled vol-

ume into disjoint regions. Further insight into these behaviors is gained by consider-

ing the evolution of the mechanical damage.

FIG. 6 The variation of the mechanical damage parameter DE with applied strain.

FIG. 5 The stress–strain response under of uni-axial tension of (a) IG110 graphite; and (b) PGX graphite.



Figure 6 shows the development of the mechanical damage parameter DE with

increasing applied strain. The damage parameter DE is defined as a relative change

of the material’s elastic modulus:

DE ¼ 1� EE0

(4)

It can be seen that for IG110 graphite (Fig. 6(a)), at relatively low values of applied

strain (<0.03), the DE for all porosity volume fractions is increasing, with a greater

rate of increase for higher porosity volume fractions. This is an expected behavior

and suggests non-linearity in the corresponding stress–strain curves (Fig. 6(a)),

which is not clearly identifiable for all curves because of the scales used. The dam-

age evolution in IG110 continues with increasing rate at the virgin state, with

increasing but smaller rate at 30 % porosity, but with decreasing rate at 60 %

porosity prior to failure. Similar damage evolution behavior is observed in PGX

(Fig. 6(b)), where, however, the 40 % porosity exhibits decreasing damage evolu-

tion rate before failure, and the 60 % porosity exhibits a negligible increase of DE

before failure.

The observed behavior of decreasing damage evolution rates at higher porosities

suggest that the graphite grades studied exhibit “avalanche” behavior, typical of

purely brittle failure. The bonds remaining in the corresponding models continue to

carry the load until the sudden simultaneous rupture of all bonds required for sam-

ple disintegration. This is particularly pronounced in PGX at 60 % porosity where

practically no mechanical damage developed prior to final failure. The avalanche

behavior is potentially the result of the large number of pores required on each bond

to reach the high pore volume fractions. Consequently, the accumulated pore size

and from there the failure strength of all bonds is approximately equal, specifically

in 60 % porosity PGX. This results in a simultaneous rapture of a bond population

critical for the sample integrity. Although the result that higher porosity embrittles

the material may seem counterintuitive, it correlates with the observations that irra-

diation causes porosity increase and reduction of fracture toughness [28].

These observations require further analysis of the developing micro-crack popu-

lation following the procedure outlined in [29]. Specifically, it is necessary to develop

understanding and quantify the relation between failed area and its topology (sur-

face-based damage measure), and the emerging mechanical damage. Nevertheless,

with this pilot study we have established a methodology for microstructure-

informed prediction of the fracture energy of graphite. This can be calculated as the

area under the stress–strain curve for a particular microstructure, which includes all

energy loss via distributed micro-cracking and the eventual avalanche-like rupture,

divided by the sample area. A more precise quantification of the surface-based

damage would improve the fracture energy estimates. Current results show that the

increase of porosity reduces the “gracefulness” of the graphite behavior with corre-

sponding reduction of the fracture energy, which can be used for integrity assess-

ment of aging components, where environmental factors cause porosity increase.

The predicted failure stress in the virgin states can be compared to the limited

data available for these grades [30]. The comparison shows that our predictions are

half of the mean experimental values reported. The reason for this discrepancy could



be the assumption for the spring behavior shown in Fig. 4, where a large softening

branch has been introduced. This was done by considering that the graphite matrix

contains a system of nano-pores, which could act as a lower-scale softening mecha-

nism. On the other side the selected separation energy of the matrix does not include

the presence of nano-porosity. These options require further consideration. With

the current separation energy the results suggest that the spring behavior should be

assumed with negligible softening, in which case the stress values shown in Fig. 5

would roughly double.

One point to be noted is that in the present work the increase of porosity has

not been accompanied by any change in the statistical pore size distribution. In

some cases, this could be unrealistic. However, the model provides the means to

calculate the failure energy should changes in size distribution and volume density

of pores be measured experimentally. In addition, the results presented here are spe-

cific to uniaxial loading and hence useful for prediction of fracture energy. The

methodology, however, can be used to derive damage evolution laws for incorpora-

tion in continuum-base finite element analysis of cracked components. It has been

previously shown that under more complex stress states, such plane stress existing

ahead of a crack front, the evolution of damage results in an elastic anisotropy [29].

Derivation of a damage tensor for such loading cases under the proposed methodol-

ogy will be a subject of further publications.

The presented model for elastic-brittle local behavior can be extended to

elastic–plastic or visco-elastic behavior prior to spring failure initiation with the use

of non-linear springs and dampers. This has been done for other lattice models [31]

and is a subject of ongoing work for the lattice proposed here.

The approach developed here is aimed at linking microstructure features

to macroscopic response. It is demonstrated how a macroscopic property, e.g.,

stress–strain curve or damage, emerges from the underlying microstructure and the

separation energy of defect-free graphite atomic lattice. The presented results are

specific to the uniaxial tension of a volume element of a limited size. These allow for

derivation of cohesive zone behavior and fracture toughness associated with the

particular loading considered. One important extension of the work will consider

different deformation conditions, specifically those present ahead of a macroscopic

crack. This way, the model could be used for the analysis of the fracture process

zone size. The parameters controlling this size are still not clearly understood.

According to Aliha and Ayatollahi [32], Awaji et al. [33], and Claussen et al. [34],

the size of the fracture process zone scales with the squared ration of the fracture

toughness and tensile strength. However, Ayatollahi and Aliha [35] have pointed

out that the size is related to the average grain size alone. The modeling of the entire

macro-crack tip region with microstructure-informed lattice will require a signifi-

cant computational effort. As a first step into understanding the different damage

evolutions within the macro-crack tip region, we have planned simulations with rep-

resentative volume elements subjected to the deformation conditions at a number of

spatial positions. The results of such simulations will be analyzed to estimate the

need for large-size lattice covering the entire macro-crack tip region and reported in

future communications. Before completing the planned program, it is difficult to

estimate the effects of grain, pore, and cell sizes on the macroscopic fracture behav-

ior of graphite.



Conclusions

A procedure to construct a meso-scale site-bond lattice model is described that

accounts for experimentally measured particle and pore size distributions and vol-

ume fractions. The procedure is particularly suitable for the analysis of the collective

behavior of micro-cracks, which is the precursor for the non-linear behavior in

quasi-brittle materials, such as graphite.

For the graphite grades investigated, the predicted stress–strain behavior at low

porosities, in virgin state and up to 30 %, is typical of a quasi-brittle material. The

failure is preceded by a significant dissipation of energy by distributed micro-

cracking. This is reflected in the growth of the macroscopic mechanical damage.

At higher porosities, the predicted stress–strain behavior shifts to a typical brit-

tle, characterized by sudden failure with reduced or negligible energy dissipation

from distributed micro-cracking. The failure is an avalanche-like event, whereas a

population of bonds critical to the component integrity fail simultaneously.

The procedure can therefore be used to estimate the changes in material fracture

energy as a function of the changes in the microstructure with aging. Three areas

requiring further investigation are identified: (1) provision of experimental evidence

for the behavior of graphite as a generalized continuum at the particle length scale

and the corresponding calibration of angular springs in bond bundles; (2) quantita-

tive analysis of micro-crack population to relate surface-based damage to mechanical

damage for prediction of fracture energy; and (3) analysis of complex stress states to

derive damage evolution laws for continuum finite element analysis.

ACKNOWLEDGMENTS

C.N.M. greatly appreciates support from EPSRC via Nuclear FiRST Doctoral

Training Centre. M.Z. and A.P.J. acknowledge support from EPSRC via grant

EP/J019763/1, “QUBE: Quasi-Brittle Fracture: A 3D Experimentally Validated

Approach,” and from BNFL for the Research Centre for Radwaste & Decommission-

ing. J.R.Y. acknowledges support from EDF R&D.

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Appendix G

Fracture energy of graphite from

microstructure-informed lattice model

175

Available online at www.sciencedirect.com

ScienceDirect

Procedia Materials Science 3 (2014) 1848-1853

www.elsevier.com/locate/procedia

2211-8128 © 2014 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU),

Department of Structural Engineering.

20th European Conference on Fracture (ECF20)

Fracture energy of graphite from microstructure-informed

lattice model

Craig N Morrison*, Mingzhong Zhang, Andrey P Jivkov

Mechanics and Physics of Solids Research Team, School of Mechanical, Aerospace and Civil Engineering, The University

of Manchester, Manchester, M13 9PL, UK

Abstract

Graphite remains a key structural material in the nuclear industry, the integrity assessment of which in

demanding reactor environments is critical for safe operation of plant. Fracture of graphite is preceded by

growth and coalescence of distributed micro-cracks within a process zone, classifying it as a quasi-brittle

material alongside cement-based and ceramic materials. The evolution of a micro-crack population to failure is

well represented by discrete lattice models, e.g. (Wang and Mora 2008). Here, a recently developed 3D lattice

(Jivkov and Yates 2012), with elastic spring elements and brittle-damage behaviour is used to generate

microstructure representative models of two graphite grades at a representative meso length scale. Micro-

cracks are represented by spring failures and the macroscopic damage results from their collective behaviour.

Presented results capture a transition from graceful, plastic-like failure at lower porosities, with energy

dissipation via micro-cracking, to glass-like behaviour with negligible energy dissipation at higher porosities.

The results are in good agreement with experimental data. Thus, the proposed methodology can calculate

fracture energy from the stress-strain curve, or formulate cohesive and damage evolution laws for continuum

models, based exclusively on microstructural features.

© 2014 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Norwegian University of Science and Technology

(NTNU), Department of Structural Engineering.

Keywords: Nuclear graphite; Porosity; Meso-scale model; Quasi-brittle behaviour; Damage evolution

* Corresponding author.

E-mail address: [email protected]

2

1. Introduction

Nuclear-grade graphite has been used as both a fast neutron moderator and structural material in

over 100 reactors worldwide. The ability to fulfill both purposes gives it a distinct advantage over

water-based reactors and has lead to its inclusion in the design of the Generation IV Very High

Temperature Reactor. Consequently both future reactor core designs and life extensions of existing

plants require a suitable means of predicting the structural integrity of graphite in the demanding

environment and complex loading states found in a reactor core.

Graphite has a feature-rich microstructure consisting of multiple phases, the size and structure of

which are dependent upon the raw materials and manufacturing process used. This has allowed for

the development of a broad range of graphite grades, with properties tailored to a specific purpose.

The microstructure consists of filler particles, derived from petroleum or pitch cokes distributed

within a matrix of graphitized binder material; usually coal-tar pitch. Distributed throughout both

of these phases are 3 main populations of porosity formed at different stages of the manufacturing

process; gas evolution cracks, calcinations cracks and Mrozowski cracks (Bradford and Steer 2008;

Joyce et al. 2008), the length scale of which ranges from nm to mm scale depending on the grade in

question.

Current preferred modeling procedures used to investigate component scale integrity are based

on a classical continuum, such as the finite element method. These inherently fail to model the

effect of microstructure on macroscopic behavior, assuming all points in an element behave

homogeneously. This is problematic when modeling graphite components where failure has been

shown to be dependent on its discrete multiphase microstructure (Mostafavi and Marrow 2012),

primarily the result of micro-crack coalescence into a critically sized flaw. Such a response suggests

graphite falls into the class of quasi-brittle materials alongside rock, concrete and ceramics. Upon

loading quasi-brittle materials typically exhibit a reduction of stiffness prior to failure, similar to

plasticity. Moreover, a residual load carrying capacity beyond the peak load is demonstrated

(Hodgkins et al. 2010).

Attempts have been made at developing continuum-based approaches which account implicitly

for microstructural affects, notably in the fracture process zone (FPZ) ahead of an existing crack.

One approach of note, easily implemented in the finite element method is cohesive zone modeling

(Borst 2002; Elices et al. 2002). This method represents the local softening found in quasi-brittle

materials ahead of a crack in the properties of cohesive elements. However problems arise from the

pre-definition of a crack path and the phenomenological calibration and curve fitting of parameters

required from observed macroscopic behavior for different loading cases.

Such failings of continuum approaches have led to the development of a number of local fracture

models for quasi-brittle materials. These ‘meso-scale’ models aim to capture the dependence of

macro- or component scale failure on microstructural features by representing the corresponding

failure mechanisms at the feature length scale in a physically realistic manner (Lemaitre 1986).

However, most current local approaches are again phenomenological (Borst 2002) and as such

inherently ignore the actual failure mechanisms observed at the microscale of quasi-brittle materials

(Bažant and Pang 2007). Microstructure-informed models, such as the one offered in this work,

have the potential to assist in deriving mechanistic constitutive laws suitable for use in the

aforementioned continuum approaches such as cohesive zone modeling.

Lattice models, such as the model described in this paper, are a category of local discrete models,

where discrete elements, including springs (Pazdniakou and Adler 2012) and beams (Schlangen and

Garboczi 1997) are used as connections between nodes to form a parallel network. The response

and failure of elements can incorporate actual mechanisms at the length scale of microstructure

features, allowing a physically realistic macro-mechanical response. Global failure is the result of

cumulative micro-failures into a macro-sized flaw, with load redistribution occurring upon failure

3

of single bonds. Previous lattice models for graphite (Nemeth and Bratton 2010) and concrete

(Schlangen and Garboczi 1997) have either been 2D models, questionable in their ability to

simulate the intrinsically 3D phenomenon of micro-crack damage evolution, or limited in their

ability to reproduce a full range of Poisson’s ratio for elastic materials (Wang and Mora 2008).

This work extends the site-bond 3D lattice model developed by Jivkov and Yates (Jivkov and

Yates 2012) with application to nuclear graphite using bundles of springs as discrete bonds

connecting sites at the centre of unit cells. Previous construction methodology accounting for

experimentally measured particle and pore size distributions is improved (Morrison et al. 2014b;

Morrison et al. 2014a) and used to investigate the effects of increasing porosity on tensile strength

and damage evolution with a view to understanding graphite’s response to radiation damage.

2. Theory and Model

The theory behind the site-bond model is briefly presented here. Details are given in another

contribution to this volume (Jivkov et al. 2014) and in previous works (Jivkov and Yates 2012;

Morrison et al. 2014a; Morrison et al. 2014b). The model consists of a regular tessellation of

truncated octahedral cells – the regular shape considered the closest representation of a generic

microstructure (Kumar et al. 1992). The truncated octahedron has six square faces, normal to the

principle axes, and eight hexagonal faces, normal to the octahedral axes. Sites at cell centers are

connected to neighboring sites by bonds, with bond lengths varying for principal and octahedral

directions.

Fig. 1. Cellular lattice: (a) site-bond assembly; (b) unit cell with bonds; (c) normal and shear springs.

In this work each bond is represented as a bundle of elastic-brittle normal and shear spring. The

need for further angular springs to represent all degrees of freedom between sites is a subject of

ongoing work. Spring constants, given in Eq.1, are calibrated by equating the strain energy in a

single discrete cell to the energy in the equivalent continuum (Zhang et al. 2013).

(1 2 ) (1 4 ); 0; ;

4(1 )(1 2 ) 4(1 )(1 2 ) 4(1 )(1 2 )

p p o o

n s n s

EL v EL v ELk k k k

v v v v v v

(1)

where E and are Young’s modulus and Poisson’s ratio of the material respectively,

and

represent the stiffness of normal and shear springs in principal directions and and

represent

the stiffness of normal and shear springs in octahedral directions. L is the cell size, i.e. distance

between sites in the principal directions.

4

3. Distribution of Material Features

Experimentally measured particle and pore size distributions, reproduced from (Kane et al.

2011) for two graphite grades, IG110 and PGX, were used to microstructurally inform

corresponding site-bond models at increasing pore volume fractions (porosity) in order to simulate

the effect of radiation damage. A meso-length scale was naturally introduced by randomly

assigning particles to sites of a lattice of specified size, in this case 15 x 15 x 15 cells. The cell size

was calculated by equating the specified particle volume fraction of each grade with the known

volume of the distributed pores and lattice size (Morrison et al. 2014b). A consistent cell size was

used for each grade of graphite to allow direct comparison between cases of varying porosity. Pores

were randomly distributed to faces between cells over a succession of passes, with multiple pores

allocated to each face to ensure the desired porosity was reached. Each pore is assumed spherical

and bisected by cell faces such that the pore cross-sectional area on each face is maximum. In a

previous work (Morrison et al. 2014b) springs without a pore present fail at a prescribed energy,

namely the product of the cell face area through which the bond passes and the graphite energy of

separation (Abrahamson 1973). This produced tensile strengths of half the experimentally

measured values. Here, each spring was allowed double the peak force of previous works,

corresponding to a factor of four increase in failure energy when springs constants are maintained.

The presence of pores on a cell face is incorporated into the corresponding spring failure criteria.

Failure energy decreases from the pore-free value according to the ratio of the face area which is

pore-free and the total face area. The material properties used for calibration of each grade and key

statistical parameters are given in Table 1.

Table 1. Material properties and microstructure data for IG110 and PGX graphite.

Grade IG110 PGX Reference

Elastic modulus E (MPa) 9800 8300 (Products n.d.), (Kaji et al. 2001)

Poisson’s ratio 0.14 0.11 (Products n.d.), (Kaji et al. 2001)

Mean particle diameter (μm) 27 92 (Kane et al. 2011)

Mean pore area (μm2) 98 197.9 (Kane et al. 2011)

Particle volume fraction 20% 20% (Kane et al. 2011)

Porosity in virgin state 14.73% 21.49% (Kane et al. 2011)

4. Results and Discussion

Fig. 2 shows the simulated stress-strain curves for specimens with different porosities. At lower

porosity values both grades exhibit a typically ‘graceful’ quasi-brittle response, with pre-peak

softening similar to plasticity. As porosity increases this shifts to a more brittle response with

minimal energy dissipation corresponding to avalanche-failure of multiple bonds. Simulations at

each porosity were repeated with different spatial distributions of pores to check the validity of a

lattice of 15 cell length as a representative volume element (RVE). The stress strain response

remained consistent for both grades at every porosity value, although there was variation in the

tensile strength of the IG110 grade at a given porosity. This suggests that the model size for IG110

is not an RVE with respect to damage evolution. The failure to capture potential post-peak

softening in the stress-strain curves is a subject of ongoing work, including the consideration of

more representative failure criteria derived from lower scale simulations. Inclusion of a softening

tail to the response will allow derivation of mechanistic constitutive laws for continuum damage

modelling of fracture.

5

Fig. 2. Stress-strain response to failure for: (a) IG110; (b) PGX at varying levels of porosity.

The introduction of an improved failure criterion has resulted in a good match between obtained

tensile strength values at the virgin state and experimental data (Ishihara et al. 2004). As porosity θ

is increased the change in tensile strength has been shown to exponentially decay approaching zero

(Berre et al. 2008; Ishihara et al. 2004) according to the Knudsen relationship (Knudsen 1959),

Eqn. 2. The results obtained from our model, Fig. 3, demonstrate a similar exponential decay,

however the end value is non-zero such that the approximation in Eqn. 3. is followed, where m

equals 5 and 9 for IG110 and PGX respectively. The reason for this is that the model possesses a

resistance to instantaneous failure due to a number of bonds without pores even at high porosities.

Even if the failure is of avalanche type, the system requires external work to overcome the failure

energies assigned to these bonds. By increasing the porosity an improved estimate can be obtained.

m

ff e 0 (2)

010

mm

ff ee (3)

Fig. 3. Relationship between tensile strength and porosity for: (a) IG110; (b) PGX.

Fig. 4 illustrates the evolution of damage for both grades at each value of porosity, where

damage is defined at the relative degradation of the corresponding Young’s modulus. At low

porosity in IG110 the damage evolves at an ever increasing rate until failure. As porosity increases

damage evolution again starts at any increasing rate, however prior to failure the evolution rate

begins to decrease. As porosity progressively increases this declining rate of increase occurs faster

as the material’s response becomes more brittle. The response of PGX appears similar with a

6

quicker decline to a brittle response. The results suggest this model can be used to analyse and

further understand damage evolution ahead of a macroscopic crack for a material of known

microstructure with a possibility to aid in the prediction of FPZ size – a phenomenon not yet fully

understood in quasi-brittle materials (Awaji et al. 2006).

Fig. 4. Damage evolution for: (a) IG110; (b) PGX; at various porosities

5. Conclusions

This work describes a method of constructing a site-bond model to account for experimentally

measured pore and particle size distributions and volume fractions, such that a suitable meso-scale

length scale is achieved. We have demonstrated its potential to derive material constitutive

behavior emerging from micro-crack growth and coalescence – the prominent microstructural

failure mechanism for quasi-brittle materials, such as graphite. Two graphite grades have been

investigated, with ‘graceful’ quasi-brittle found at lower porosities whereby energy is dissipated

with distributed micro-failures and hence an increasing rate of damage evolution prior to global

failure. Increase of porosity leads to a more brittle response where final failure is sudden with

avalanche-like failure of numerous bonds simultaneously. In this case, rate of damage evolution is

constantly decreasing. Tensile strength is shown to exponentially decay with increased porosity,

although the decay relationship differs from literature data.

The method has potential for use in exploring phenomena which are not currently well

understood, e.g. the FPZ size, and deriving constitutive and damage evolution behavior for a

microstructure under aging processes for use in continuum methods. Further work includes;

additional consideration of angular spring inclusion on receipt of experimental evidence exhibiting

graphite’s characteristics as a generalized continuum and exploration of the correct surface energy

of graphite required for deriving spring failure criteria.

Acknowledgements

Morrison appreciates the support from EPSRC via Nuclear FiRST Doctoral Training Centre.

Zhang and Jivkov acknowledge the support from EPSRC via grant EP/J019763/1, “QUBE: Quasi-

Brittle fracture: a 3D experimentally-validated approach”, and from BNFL for the Research Centre

for Radwaste & Decommissioning.

7

References

Abrahamson, J., 1973. The surface energies of graphite. Carbon, 11(4).

Awaji, H., Matsunaga, T., Choi, S.-M., 2006. Relation between Strength, Fracture Toughness, and Critical Frontal Process Zone Size in Ceramics. Materials Transactions, 47(6), pp.1532–1539.

Bažant, Z.P., Pang, S., 2007. Activation energy based extreme value statistics and size effect in brittle and quasibrittle

fracture. Journal of the Mechanics and Physics of Solids, 55(1), pp.91–131. Berre, C., Fok, S.L., Mummery, P.M., Ali, J., Marsden, B.J., Marrow, T.J., Neighbour, G.B., 2008. Failure analysis of the

effects of porosity in thermally oxidised nuclear graphite using finite element modelling. Journal of Nuclear

Materials, 381(1-2), pp.1–8. Borst, R. de, 2002. Fracture in quasi-brittle materials: a review of continuum damage-based approaches. Engineering

fracture mechanics, 69, pp.95–112.

Bradford, M.R., Steer, A.G., 2008. A structurally-based model of irradiated graphite properties. Journal of Nuclear Materials, 381(1-2), pp.137–144.

Elices, M., Guinea, G.V., Gómez, J., Planas, J., 2002. The cohesive zone model: advantages, limitations and challenges.

Engineering Fracture Mechanics, 69(2), pp.137–163. Hodgkins, A., Marrow, T.J., Wootton, M.R., Moskovic, R., Flewitt, P.E.J., 2010. Fracture behaviour of radiolytically

oxidised reactor core graphites: a view. Materials Science and Technology, 26(8), pp.899–907.

Ishihara, M., Sumita, J., Shibata, T., Iyoku, T., Oku, T., 2004. Principle design and data of graphite components. Nuclear Engineering and Design, 233(1-3), pp.251–260.

Jivkov, A.P., Morrison, C.N., Zhang, M., 2014. Site-bond modelling of structure-failure relations in quasi-brittle media. In

20th European Conference on Fracture, Trondheim, Norway. p. Paper #689. Jivkov, A.P., Yates, J.R., 2012. Elastic behaviour of a regular lattice for meso-scale modelling of solids. International

Journal of Solids and Structures, 49(22), pp.3089–3099.

Joyce, M.R., Marrow, T.J., Mummery, P., Marsden, B.J., 2008. Observation of microstructure deformation and damage in nuclear graphite. Engineering Fracture Mechanics, 75(12), pp.3633–3645.

Kaji, Y., Gu, W., Ishihara, M., Arai, T., Nakamura, H., 2001. Development of structural analysis program for non-linear

elasticity by continuum damage mechanics. Nuclear Engineering and Design, 206(1), pp.1–12. Kane, J., Karthik, C., Butt, D.P., Windes, W.E., Ubic, R., 2011. Microstructural characterization and pore structure analysis

of nuclear graphite. Journal of Nuclear Materials, 415(2), pp.189–197. Knudsen, F.P., 1959. Dependence of Mechanical Strength of Brittle Polycrystalline Specimens on Porosity and Grain Size.

Journal of the American Ceramic Society, 42(8), pp.376–387.

Kumar, S., Kurtz, S.K., Banavar, J.R., Sharma, M.G., 1992. Properties of a three-dimensional Poisson-Voronoi tesselation: A Monte Carlo study. Journal of Statistical Physics, 67(3-4), pp.523–551.

Lemaitre, J., 1986. Local approach of fracture. Engineering Fracture Mechanics, 23, pp.523–537.

Morrison, C.N., Jivkov, A.P., Smith, G., Yates, J.R., 2014a. Lattice-spring modeling of graphite accounting for pore size distribution. Key Engineering Materials, 592-593, pp.92–95.

Morrison, C.N., Zhang, M., Jivkov, A.P., Yates, J.R., 2014b. A Discrete Lattice Model of Quasi-brittle Fracture in Porous

Graphite. Materials Performance and Characterisation, 3(3), p.In press. Mostafavi, M., Marrow, T.J., 2012. Quantitative in situ study of short crack propagation in polygranular graphite by digital

image correlation. Fatigue & Fracture of Engineering Materials & Structures, 35(8), pp.695–707.

Nemeth, N.N., Bratton, R.L., 2010. Overview of statistical models of fracture for nonirradiated nuclear-graphite components. Nuclear Engineering and Design, 240(1), pp.1–29.

Pazdniakou, A., Adler, P.M., 2012. Lattice Spring Models. Transport in Porous Media, 93(2), pp.243–262.

Products, Toyo Tanso Carbon., Special Graphite and Compound Material Products. Schlangen, E., Garboczi, E., 1997. Fracture simulations of concrete using lattice models: computational aspects.

Engineering Fracture Mechanics, 57(2), pp.319–332.

Wang, Y., Mora, P., 2008. Macroscopic elastic properties of regular lattices. Journal of the Mechanics and Physics of Solids, 56(12), pp.3459–3474.

Zhang, M., Morrison, C.N., Jivkov, A.P., 2013. A meso-scale site-bond model for elasticity: Theory and calibration.

Material Research Innovations, In press.

Appendix H

Site-bond lattice modelling of damage

process in nuclear graphite under

bending

183

Transactions, SMiRT-23

Manchester, United Kingdom - August 10-14, 2015

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SITE-BOND LATTICE MODELLING OF DAMAGE PROCESS IN

NUCLEAR GRAPHITE UNDER BENDING

Craig N Morrison1, Mingzhong Zhang2, Dong Liu3 and Andrey P Jivkov1

1 Modelling and Simulation Centre, School of Mechanical, Aerospace and Civil Engineering, The

University of Manchester, Manchester, UK 2 Advanced & Innovative Materials Group, Department of Civil, Environmental and Geomatic

Engineering, University College London, London, UK 3 Interface Analysis Centre, School of Physics, University of Bristol, Bristol, UK

ABSTRACT

Graphite is used as neutron moderator and structural material in the core of the UK's fleet of Magnox and

Advanced Gas-cooled Reactors (AGRs). The graphite cores are non-replaceable in these two designs and therefore potentially life-limiting. Graphite is a multi-phase, aggregated and porous material which could

have a non-linear stress-strain response because of distributed damage accumulation within the material

prior to rupture: quasi-brittle characteristics. Lattice models provide a way of capturing the resulting non-

linear behaviour by incorporating microstructural features and damage mechanisms within the discrete

system. Here, the 3D site-bond model (Jivkov and Yates 2012) is used to simulate a near-isotropic nuclear

reactor core Gilsocarbon graphite under bending in a micro-cantilever test (Liu et al 2014).

Experimentally measured pore-size distributions and volume densities are used for model construction. Previous work on graphite site-bond modelling (Morrison et al 2014b) is further developed to consider

pore effect on the deformation and failure behaviour of the bonds. Damage evolution and accumulation

with increasing load is simulated by the consecutive removal of bonds subject to failure criterion. The simulated mechanical properties and force-deflection relationship were validated by experimental results.

Keywords: Site-bond model; Nuclear graphite; Micro-cantilever bending; Local damage; Microstructure

INTRODUCTION

Graphite has been used in the nuclear industry as a fast neutron moderator since the first demonstration of a chain nuclear reaction in the 1940s. Moreover, its high strength at elevated temperatures within reactors

has allowed it to be coupled as a structural component in some reactor designs (Mantell 1968). The three-

phase microstructure of graphite varies depending on the raw materials and the manufacturing process

used. Filler particles, deriving from petroleum or pitch coke, are dispersed within a binder material

matrix, usually consisting of graphitised coal-tar pitch. Both of these solid phases host various

populations of pores covering different length scales (Jenkins 1962). Graphite is believed to belong to the

quasi-brittle class of materials (Hodgkins et al 2010), with structural integrity considered to be dependent on material failures at the length scales of its prominent microstructure features. The initiation of brittle

micro-cracks, considered to be at pores, dissipates energy resulting in a reduction in the global stiffness

prior to peak stress in a similar manner to plasticity; and this is followed by a post peak softening. Ultimate failure is considered to be the result of micro-crack growth and coalescence into a flaw of

critical size.

Global modelling strategies, such as the finite element method (Turner et al 1956), assume scale-

independent homogeneous behaviour within model elements and as such fail to account for the effects of

microstructure failure mechanisms in the material response. This is considered to be inappropriate in

certain cases particularly when modelling situations at length scales where the material response is

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dictated by phenomena at the micro-scale, such as the region ahead of a crack tip. In such cases additional

local information is required.

Various modelling approaches have been proposed to account for local effects either statistically, using

weakest link statistics (Nemeth and Bratton 2010), or within continuum-based framework such as

cohesive zone models (de Borst 2002). However, these approaches are limited by the reliance on

phenomenological calibration against macro-scale data. Lattice models are a branch of discrete models whereby nodes are connected by elements into a statistically parallel network. Such models have been

developed for quasi-brittle materials including graphite (Schlangen et al 2010; Nemeth and Bratton 2010)

and concrete (Schlangen and Garboczi 1997, Schlangen 2008) with incorporation of microstructural information into element properties allowing the production of macro-scale behaviour inclusive of micro-

failure mechanisms. Many of these are 2D, as 3D configurations investigated could not reproduce desired

Poisson’s ratio values. The model used in this work is a 3D site-bond model proposed by Jivkov and

Yates (Jivkov and Yates 2012), which has been shown to be able to reproduce the range of Poisson’s ratio

required for quasi-brittle materials.

The work described here focuses on Gilsocarbon graphite, named as such due to the Gilsonite coke from

which it is manufactured, which is used as moderator and life-limiting core structural material in the 14 operating Advanced Gas Cooled Reactors (AGR) in the UK. Specifically the site-bond model will be used

to simulate the load-displacement response achieved experimentally in a micro-cantilever experiment

using statistical microstructure data for Gilsocarbon graphite. Results presented include experimentally

validated force-displacement relations using a 2-phase model, with relatively good agreement considering

the large spread of experimental results. The accumulation of damage and global failure is shown to have

been captured.

EXPERIMENTAL PROCEDURE

Mechanical testing over a range of length scales is important for materials, such as graphite, exhibiting a significant size-effect as a result of prominent heterogeneous microstructure. One particular novel

technique for testing at the micro-scale uses micro-cantilever specimens milled into a bulk sample by

focused ion beam. The technique is outlined in more detail by Liu et al (Liu et al 2014) with the results

from that particular work forming the experimental validation used here. The new procedure adopted

produced no taper at the root of the micro-cantilevers and the setup allows the observation of the

cantilever beam throughout the loading process, Fig. 1a. Several graphite beams were milled and tested.

The experimental sample used for modelling was chosen as 2×2×10 μm. The results for this particular

specimen were not published by Liu et al (Liu et al 2014) but have been provided courtesy of the authors,

Fig. 1b.

Figure 1. (a) A micro-cantilever under load (Liu et al 2014) (b) failure point of the modelled specimen

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THEORY AND METHOD

The site-bond model represents material volume through the use of a regular lattice of truncated

octahedral cells, Figure 2a. This shape is considered the regularly tessellating shape most capable of

representing a generic microstructure (Kumar et al 1992), and as such the lattice can be considered a

topological homogenization of a microstructure. Sites are located at the centres of cells with each site connected to its 14 nearest neighbours by bonds. In this manner, the sites and bonds are considered the

dual complex of the cell volumes and face areas respectively (Jivkov 2014a; Jivkov 2014b). Of the 14

bonds extending from a site, 6 bonds will pass through square faces in the principal directions and 8 bonds will pass through hexagonal bonds in the octahedral directions, yielding two geometrically distinct

bond types, labelled B1 and B2 in Figure 2b. Principal bonds, B1, and octahedral bonds, B2, have lengths

of L and √3L/2 respectively, where L is the cell length. The proportion of the cell volume directly

associated with each bond is termed the support volume, Figure 2c.

Figure 2. (a) The cellular structure of the site-bond model; (b) unit cell with bonds; (c) support volume,

highlighted in red, for a bond through a principal face

The topology of the site-bond model represents the aggregate microstructure of quasi-brittle materials by

allocating a filler particle/aggregate, the size of which is chosen from a distribution, to each site. As such

the bonds between sites represent the brittle micro-failures which occur both within and between the

particles. First models for concrete (Jivkov et al, 2012; Jivkov, 2014) used beam elements for bonds which introduced micro-polarity not amenable to analytic calibration. Later models for graphite

(Morrison et al 2014a, Morrison et al 2014b) and cement-paste (Zhang et al 2014a) represented the bonds

with spring bundles, where each spring resisted separate translational degree of freedom. Spring

behaviour was calibrated by equating energy within a cell to that of a continuum (Zhang et al 2014b).

This methodology enabled a good representation of isotropic elastic materials with Poisson’s ratio

ranging from -1 to 0.5. However, it was not compatible with the geometric discretization theory (Yavari

2008), whereby balance of angular momentum dictated a single force between displaced positions of sites. As a result the model failed to capture deformations in non-linear displacement fields, such as

bending. To overcome this, bonds are represented in this work by connector elements, which come as

standard in the commercial finite element software ABAQUS (Simulia 2013), and the behaviour is calculated as geometrically non-linear. Connectors allow damage, failure and subsequent energy

dissipation to be readily introduced. The 1D connector element which represents each bond has a single

stiffness value representing the resistance to a length change. The stiffness values of the axial springs

from the previous calibration methodology (Zhang et al 2014b) are given in Equations 1a and 1b. These

are used as a first approximation in the present work; a calibration procedure more suitable for this model

is a subject of ongoing work.

(c)

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�� ; �

� � �� (1a,b)

The full mechanical response of a bond is shown in Figure 3. The peak force, Fd and displacement at

which damage initiates, Ud are calculated from the bond stiffness and failure energy, Gc, which in turn is

the sum of the elastic GE and damage GD energies. The value for Gc is taken as the product of the face

area through which the bond passes, A0, and the enthalpy for creation of two surfaces in graphite, γ. The

value of γ is taken as 9.7 J/m2 (Abrahamson 1973). Once the relative displacement of the bond reaches the

displacement Uf it is deemed to have failed and is removed from the simulation. Initially Uf is set at 2Ud.

Figure 3. Bond failure criteria, relative force, F against relative displacement, U

Due to the models highly nonlinear behaviour, several controls were utilized and default settings relaxed

in order to overcome the difficulties in obtaining a converged solution. Viscous regularization was

implemented to damp the dissipation of energy released to the surrounding elements upon damage and

subsequent failure of an element. A damping coefficient of 1×10-5 was used, a value considered to help

the solution converge without introducing significant artificial energy into the system. Moreover the

extrapolation method was suppresed to prevent excess iterations when finding the equilibrium position of

nodes connected to failed elements and the quasi-newton method for quasi-static analyses was utilized.

MICROSTRUCTURE MAPPING

In previous works for graphite using the site-bond model (Morrison et al 2014a, Morrison et al 2014b) the

cell size was calculated by equating the model volume as a function of the cell size, the volume of the

filler particles distributed to the model and the known volume fraction. As such the model was scaled at a

suitable meso-length scale. The work presented here differs in that the model dimensions are determined

from the described experimental specimen as opposed to scaling cells at a suitable meso-length scale.

This is a useful alteration in that it allows direct experimental validation but also a necessity as the filler

particles in Gilsocarbon graphite are typically on the mm scale, significantly larger than the actual specimen being modelled. As a result a mesh refinement study was required, determining the best suited

dimensions of the model in terms of number of cells by evaluating the ability to reproduce the input value

of Young’s modulus, E. The value of E used to calibrate the model corresponds to the ‘pore-free’ value of

15 GPa, calculated using nano-indentation at the Manchester School of Materials (Berre et al 2006).

Beams were first modelled without any porosity, each with the 2×2×10 μm specimen dimensions but with

an increasing number of cells. Boundary conditions corresponded to the experimental setup, with the face

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at the base of the cantilever fixed and a displacement applied to the nodes along the top edge of the

opposite surface. Figure 4 shows the results of the study, whereby the number of cells along the beams

cross-section are plotted against E, calculated according to simple beam bending theory. The limiting

assumptions in this theory are most probably the reason that the value of E appears to be tending to a

value slightly above the input value. A cross section of 12 cells was chosen as a compromise between

accuracy and the considerable computational expense. Figure 4b shows the the lattice size with the applied boundary conditions.

Figure 4. Cell sensitivity study

The pore distributions and volume fractions used to populate the model followed those obtained by

Laudone et al (Laudone et al 2014) using pycnometry and mercury porosimetry. The pore size

distribution from one of the samples presented by Laudone is shown in Figure 5. The distribution covered

the porosity expected within the cantilever beam, ranging from 4 nm diameter to 15 μm, although

realistically due to the decreasing number density with increasing pore diameter, the largest pores seen in

the model typically did not exceed 0.9 μm in diameter.

Figure 5. The pore size distribution used to populate the model, recreated from (Laudone et al 2014)

0

20

40

60

80

100

120

1 10 100 1000 10000 100000

Cu

mu

lati

ve

co

ntr

ibu

tio

n t

o

acc

ess

ible

po

rosi

ty

Pore diameter (nm)

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The distributions presented by Laudone accounted only for pore volume accessible to the mercury

intrusion, with the closed pore volume calculated separately from the density of the samples. The pore volume fraction used as input to the model was the average minimum total porosity (i.e. both open and

closed) across all the samples under the assumption that the closed porosity would follow the same size

distribution as the open porosity. This value was 19.14%.

Pores were selected randomly from the distribution before being assigned to a randomly selected bond.

This process was repeated until the desired pore volume fraction was achieved. This process was found to

be time-consuming and largely unnecessary when considering the smaller pores which made up the

majority of those selected. The mismatch between the cell size and the size of the smaller pores meant that many pores were assigned to each face when populating the model. This essentially smeared out the

effect of the smaller pores with the minority of larger pores selected from the tail-end of the distribution

dictating when the model reached the target porosity. In response to this an arbitrary cut-off point

(significantly lower than the critical pore size for a cell face) was chosen, whereby the cumulative volume

fraction of the pores beneath this point and thence the cumulative volume could be divided by the total

number of bonds and assigned to every face as a single effective-pore. A volume bias was implemented

so octahedral bonds received a greater share of this porosity than principal bonds in accordance with its greater associated support volume. This allowed a simple treatment of approximately 75% of the bonds

and resulted in each bond being 15% failed (i.e. 15% of its volume was occupied by pores) before larger

pores were assigned. Micro-cracks are considered to initiate at pores and as such pores are represented in the model by a

change in bond mechanical response. The pore-free response, Figure 3, was scaled according to the size

of the pore present on the particular face, as shown in Figure 6. The change of peak force from the pore-

free value, Fd, to the new value, F’d, is related to the change of an effective area, Equation 2:

��

� ��

(2)

where V is the support volume of the face and V’ is the support volume remaining after the corresponding pore volume is removed, Equation 3:

�� !�"# − ��# (3)

In the same manner, the displacement at which damage initiates, Ud, scales to the new value, U’d

according to the change an effective length, Equation 4:

%��%�

� ��

(4)

The stored elastic energy, GE, thus scales with the change of support volume, Equation 5:

%��%��

� �� (5)

Differently, the damage energy, GD, is considered to scale with the real face area, Equation 6:

&%�'�%��(��&%'�%�(��

� )�) (6)

where A’ is the face area remaining after the corresponding pore area is removed, Equation 7:

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*� � *�+,#+�#+ − *��# (7)

In this manner the failure displacement, Uf’ reduces proportionally less from the pore-free value than the

damage displacement Ud’ so response exhibits less softening as the pore size is increased and essentially

becomes more brittle.

Figure 6. Bond failure criteria accounting for the affect of porostiy

Bonds were removed prior to simulations if the assigned pores were larger than their support volumes. To

accomodate pores at the tail-end of the distribution, i.e. pores with volume significantly larger than the

support volume, an algorithm was implemented whereby if the volume of a pore was enough to fail a

bond, the excess pore volume was then designated to neighbouring bonds until there was no excess

volume. In this manner, the largest pores could fail over 200 bonds of the 94578 bonds in the model

depending on the predesessing conditions of the bonds.

RESULTS AND DISCUSSION

Figure 7a gives the force-displacement results for the cantilever. It can be seen that there is good

correlation between the initial stiffness of the experimental results and the two exhibited simulations.

There are however discrepancies between the peak load values and the corresponding displacement with

the simulation largely over-predicting these. This may be due to the use of the porosity value of 19.14%

given by Laudone et al (Laudone et al 2014) which was a lower bound. Furthermore it may be noted that

the spread of results from the simulation was small - different populations of porosity producing similar

initial stiffness. The only difference is in the accumulation of damage and failure point. However, the

experimental results have shown a much larger specimen-to-specimen variation in terms of the load-

displacement curves. For example, Figure 7a provides an example of this with a much lower stiffness

from a 2.3×2.3×13 μm cantilever beam when compared with that of a 2×2×10 μm specimen. This is

considered to be a consequence of the genuine microstructural heterogeneity of the Gilsocarbon graphite.

The experimental tests consisted of 4 loading-unloading cycles but after initial simulations it was deemed

unnecessary to repeat the process directly with minimal damage-induced hysteresis occurring until later in

the simulation as shown by the cumulative damage energy within the model from Figure 7b. The results

appear to be noisy with regular decreases in the damage energy. These dips in energy are in fact a result

of the viscous regularization imposed to maintain model stability.

23rd


Division II

Figure 7. Experimental and computational (a) force-displacement; and (b) damage energy results for the

2×2×10 μm cantilever beam.

The open-source software ParaView was used to visualise damage within the model. Figure 8 shows the initial porosity for simulation 2, where the clusters of failed bonds representing large pores are clearly

visible. Figures 9a and 9b show the development of failed bonds within the model - at an intermediate

load increment corresponding to a displacement of 2.2 μm and the final increment before final failure,

respectively. These represents the distributed damage within the deformed volume prior to the fracture at

peak load, which is consistent with the observation of other workers (Hodgkins 2010) It can be seen from

Figures 9a and 9b that damage occurs at the root of the beam leading to shearing at the wall. It appears

that failed bonds are primarily principal bonds and the damage progresses from the bonds loaded in tension along the top surface, vertically down through the sample as the load is increased and the tension

bearing bonds fail. This crack development from top to bottom, as seen in Figure 9c, is largely to be

expected, although it differs slightly from that seen experimentally, Figure 1b, where the crack begins on

the top surface but away from the root before progressing towards the root. This discrepancy suggests

there may be additional complexity of the local microstructure of that particular beam has not been

experimentally determined, hence not considered in the modelling.

Figure 8. Initial porosity for simulation 2

(a) (b)

23rd


Division II

Figure 9. Failed bonds after: (a) 3000 increments; (b) final increment. (c) Relative displacement in the

failed model

CONCLUSION

The site-bond technology has been adapted to consider a two-phase (matrix and pores) graphite micro-

cantilever beam. Results are promising and physically realistic, but slight discrepancy between

experimental and simulated force-displacement results has been observed. This can be attributed to the

large spread seen from sample to sample in the physical tests. The failure of the beam occurs after accumulation of damage at its root with the distributed porosity throughout the rest of the beam having

apparently negligible effect to the point of global failure. Further work will involve implementation of the

site-bond methodology to graphite under different loading conditions with a view to visualisation of damage accumulation and derivation of a damage evolution law. Furthermore, it is necessary to explore

new avenues for model calibration that are consistent with discretized elasticity theories (Yavari 2008).

ACKNOWLEDGEMENTS

Craig N Morrison acknowledges the support from EPSRC via Nuclear FiRST Doctoral Training Centre.

Mingzhong Zhang, Dong Liu and Andrey P Jivkov acknowledge the support from EPSRC via grants EP/J019763/1 and EP/J019801/1, “QUBE: Quasi-Brittle fracture: a 3D experimentally-validated

approach”. Moreover Mingzhong Zhang and Andrey P Jivkov acknowledge BNFL through the Research

Centre for Radwaste & Decommissioning.

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“Numerical modelling of the effects of porosity changes on the mechanical properties of nuclear

graphite”. Journal of Nuclear Materials, 352, 1-5.

(a) (b)

(c)

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de Borst, R. (2002). “Fracture in quasi-brittle materials: a review of continuum damage-based

approaches”. Engineering Fracture Mechanics, 69, 95-112. Hodgkins, A., Marrow, T.J., Wootton, M.R., Moskovic, R., Flewitt, P.E.J. (2010). “Fracture behaviour of

radiolytically oxidised reactor core graphites: a view”. Materials Science and Technology, 26(8),

899-907.

Jenkins, G. (1962). “Fracture in reactor graphite”. Journal of Nuclear Materials, 3(3), 280-286.

Jivkov, A.P. (2014). “Structure of micro-crack population and damage evolution in quasi-brittle media”.

Theoretical and Applied Fracture Mechanics, 70, 1-9.

Jivkov, A.P., Gunther M, Travis KP (2012). “Site-bond modelling of porous quasi-brittle media”. Mineralogical Magazine, 76(8), 2969-2974.

Jivkov, A.P., Morrison, C.N., Zhang, M. (2014a) “Site-bond modelling of structure-failure relations in

quasi-brittle media”, Procedia Materials Science, 3, 1872-1877.

Jivkov, A.P., Todorov, T., Morrison, C.N., Zhang, M. (2014b) “Application of analysis on graphs to site-

bond models for damage evolution in heterogeneous materials”, Proc. 11th World Congress on

Computational Mechanics and 5th European Conference on Computational Mechanics.

Jivkov, A.P. and Yates, J.R. (2012). “Elastic behaviour of a regular lattice for meso-scale modelling of solids”. International Journal of Solids and Structures, 49(22), 3089-3099.

Kumar, S., Kurtz, S.K., Banavar, J.R., Sharma, M.G. (1992), “Properties of a three-dimensional Poisson-

Voronoi tessellation: A Monte Carlo study”, Journal of Statistical Physics, 67(3-4), 523-551. Liu, D., Heard, P.J., Nakhodchi, S., Flewitt, P.E.J. (2014), “Small-Scale Approaches to Evaluate the

Mechanical Properties of Quasi-Brittle Reactor Core Graphite”. ASTM symposium, Graphite

Testing for Nuclear Applications: The Significance of Test Specimen Volume and Geometry and the

Statistical Significance of Test Specimen Population, ASTM International, 1-21.

Laudone, G.M., Gribble, C. M., Matthews, G. P. (2014). “Characterisation of the porous structure of

Gilsocarbon graphite using pycnometry, cyclic porosimetry and void-network modeling”. Carbon,

73, 61-70. Mantell, C.L. (1968). Carbon and Graphite Handbook, Interscience Publishers

Morrison, C.N., Zhang, M. Jivkov, A.P. (2014a). “Discrete lattice model of quasi-brittle fracture in

porous graphite”. Materials Performance and Characterization, 3, 414-429, ASTM International Morrison, C.N., Zhang, M., Jivkov, A.P. (2014b). “Fracture energy of graphite from microstructure-

informed lattice model”. Procedia Materials Science, 3, 1848-1853.

Nemeth, N. Bratton, R. (2010). “Overview of statistical models of fracture for nonirradiated nuclear-

graphite”. Nuclear Engineering and Design, 240, 1-29.

Schlangen, E. (2008). “Crack development in Concrete, Part 2: Modelling of Fracture Process”. Key

Engineering Materials, 385-387, 73-76.

Schlangen, E., Flewitt, P.E.J., Smith, G.E., Crocker, A.G., Hodgkins, A. (2010). “Computer Modelling of Crack Propagation in Porous Reactor Core Graphite”. Key Engineering Materials, 452-453, 729-

732.

Schlangen, E. and Garboczi, E. (1997). “Fracture simulations of concrete using lattice models: computational aspects”. Engineering Fracture Mechanics, 57(2), 319-332.

Simulia, Abaqus Version 6.13 Documentation, Dassault systemes, 2013.

Turner, M.J., Clough, R.W., Martin, H.C., Topp, L.J. (1956). “Stiffness and Deflection Analysis of

Complex Structures”, Journal of Aeronautical Sciences, 23(9), 805-824.

Yavari, A. (2008). “On geometric discretization of elasticity”, Journal of Mathematical Physics, 49(2),

022901.

Zhang, M., Morrison, C.N., Jivkov, A.P. (2014a). “A lattice-spring model for damage evolution in cement paste”, Procedia Materials Science, 3, 1854-1859.

Zhang, M., Morrison, C.N., Jivkov, A.P. (2014b). “A meso-scale site-bond model for elasticity: Theory

and calibration”. Materials Research Innovations, 18, 982-986.

Appendix I

Multi-scale modelling of nuclear

graphite tensile strength using the

Site-Bond lattice model

194

Multi-scale modelling of nuclear graphite tensile strength using thesite-bond lattice model

C.N. Morrison a, *, A.P. Jivkov a, Ye. Vertyagina b, T.J. Marrow b

a Modelling and Simulation Centre, School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, UKb Department of Materials, The University of Oxford, UK

a r t i c l e i n f o

Article history:Received 22 September 2015Received in revised form8 December 2015Accepted 30 December 2015Available online 2 January 2016

a b s t r a c t

Failure behaviour of graphite is non-linear with global failure occurring when local micro-failures,initiated at stress-raising pores, coalesce into a critically sized crack. This behaviour can be reproducedby discrete lattices that simulate larger scale constitutive responses, derived from knowledge ofmicrostructure features and failure mechanisms. A multi-scale modelling methodology is presentedusing a 3D Site-Bond lattice model. Microstructure-informed lattices of both filler and matrix constitu-ents or ‘phases’ in Gilsocarbon nuclear graphite are used to derive their individual responses. These arebased on common elastic modulus of “pore-free” graphite, with individual responses emerging frompore distributions in the two phases. The obtained strains compare well with experimentally obtaineddata and the stress-strain behaviour give insight into the deformation and damage behaviour of eachphase. The responses of the filler and matrix are used as inputs to a larger scale composite lattice modelof the macroscopic graphite. The calculated stress-strain composite behaviour, including modulus ofelasticity and tensile strength, is in acceptable agreement with experimental data reported in theliterature, considering the limited microstructure data used for model's construction. The outcomesupports the applicability of the proposed deductive approach to the derivation of macroscopicproperties.© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Synthetic graphite has been used in the nuclear industry as afast neutron moderator since the first demonstration of a chainnuclear reaction in the 1940s. Its retention of strength at elevatedtemperatures has allowed it to be used as a structural component inhigh temperature gasecooled reactor designs [1]. Nuclear graphitecan be regarded as having a three-phase microstructure, whichdepends on the rawmaterials and the manufacturing process used.Filler particles, which derive from calcined petroleum or pitch coke,are dispersed within a matrix of binder material, usually consistingof graphitised coal-tar pitch mixed with finely ground filler parti-cles. Both of these solid phases host populations of pores, with sizesthat cover the length scale from a few nm upwards to mm [2].Graphite belongs to the class of quasi-brittle materials, whichexhibit limited non-linear stress/strain response prior to maximumor peak stress, with a macroscopic effect akin to plasticity [3]. This

non-linearity is partly attributed to the generation and growth ofmicro-cracks, which occur at length scales dictated by the promi-nent microstructure features. As tensile strain is applied, micro-cracks initiate around the larger pores due to local stress amplifi-cation, and their effect leads to a reduction in graphite's stiffness[4]. Ultimate tensile failure results from crack growth and coales-cence into a flaw of critical size [4]. The continued evolution ofthese processes determines the behaviour beyond the peak stress;graphite may exhibit a limited post-peak softening or fail at peak-stress, depending on its microstructure and the loading condi-tions [3]. Therefore, the structural integrity of nuclear graphite iscontrolled by the organisation of the three phases (filler, matrix andpores) and the component's service conditions.

Conventional modelling strategies, such as the finite elementmethod [5], assume the behaviour within model elements is ho-mogeneous and scale-independent, which inherently fails to ac-count for the effects of microstructure failure mechanisms in thematerial response. This might be inappropriate in certain cases,particularly when modelling the graphite responses at lengthscales close to its microstructure features. If fracturemechanics is to* Corresponding author.

E-mail address: [email protected] (C.N. Morrison).

Contents lists available at ScienceDirect

Carbon

journal homepage: www.elsevier .com/locate/carbon

http://dx.doi.org/10.1016/j.carbon.2015.12.1000008-6223/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Carbon 100 (2016) 273e282

be used in the structural integrity assessment of nuclear graphite,the fracture process zone ahead of a macroscopic crack is of specificinterest, as this is influenced by microstructure features and spec-imen size [3], [6]. Integrity assessment requires local information insuch cases.

Local effects have been incorporated into various modellingapproaches. This can be done either statistically, using the weakestlink assumption [7], or within a continuum framework extended bycohesive zonemodels [8]. However, these approaches are restrictedby a dependence on phenomenological calibration against macro-scale data. Lattice models are a branch of discrete modelswhereby nodes are connected by elements into a statistically par-allel network. Such models have been developed for quasi-brittlematerials including graphite [7] [9], and concrete [10e11], incor-porating microstructure information into the element properties toallow the simulation of macro-scale behaviour as a consequence ofmicro-failure mechanisms. Lattice models can be constructed asirregular, representing specific, usually imaged, microstructures, orregular. Irregular models present a substantial problem, as thecalibration of lattice element properties with measured continuumproperties becomes a trial and error process. Regular models can becalibrated in many cases analytically and allow for up-scaling topotentially representative volume elements, although one issuewith most 3D regular lattices has been that they could not be tunedto reproduce desired Poisson's ratio values [12]. The regular 3D site-bond model proposed by Jivkov and Yates [13] is capable ofreproducing the range of Poisson's ratio required for quasi-brittlematerials, and this model has been further extended in thisresearch.

The original model, using beams as lattice elements, was firstlyapplied to study damage evolution in concrete in tension [14] andunder complex loads [15]. One problem with that formulation isthat the structural beams between nodes introduce local micro-polar behaviour in the lattice, i.e. rotation-dependent energy con-servation. That behaviour has not been confirmed experimentallyand consequently the calibration of beam properties remainsincomplete. To avoid this, the site-bond model has been reformu-lated with spring bundles, the stiffness coefficients of which havebeen calibrated analytically [16]. The work presented here makes asubstantial step in the development of the site-bond model with anew approach to represent lattice elements and their behaviour.Experimentally-derived distributions of microstructure features[17] are used to inform separate site-bond lattice models for fillerand matrix constituents (i.e. phases) in the microstructure of near-isotropic Gilsocarbon graphite. The results are compared withexperimental data of the tensile deformation of the individualconstituents, obtained recently by strain imaging of the micro-structure during a mechanical test of the same graphite [18]. Briefdetails of that experiment and the data obtained are provided here.The experimentally measured response of the separate micro-structure constituents is then used to inform a multi-scale site-bond methodology. Such a model may complement and scientifi-cally underpin the conservatism of structural integrity assessmentmethodologies for graphite, providing size estimates for areas ofsignificant local damage ahead of macroscopic cracks, or damageevolution laws for use in continuum scale models.

2. Theory and method

Within the site-bond methodology [13], material volume isrepresented with a discrete assembly of truncated octahedral cellsillustrated in Fig. 1(a). This choice of shape is in accordance withstatistical studies that demonstrate its suitability for representing atopologically averaged microstructure [19]. The computationalcounterpart of the regular assembly is a 3D lattice, or mathematical

graph, consisting of sites at cell centres, connected by bonds to 14neighbouring sites. This yields two distinct bond types, B1 and B2,which emanate from sites in the 6 principal (normal to squarefaces) and 8 octahedral (normal to hexagonal faces) directions,respectively, as illustrated in Fig. 1(b). The bond types have lengthsL and

ffiffiffi3

pL/2, with L representing the cell extension in the principal

directions. The bonds' behaviour is associated with an inter-cellvolume, called the support volume, formed by the two pyramidswith common base at the face normal to the bond; Fig. 1(c) showsthe support volume of a principal bond. An example of a site-bondmodel is given in Fig. 1(d).

Previously the site-bond model has been applied to the three-phase graphite microstructure, i.e. matrix, filler particles andpores, with filler particles considered to be located at sites and tooccupy fractions of cell volumes, pores considered to be located insome support volumes, and the remaining volume occupied bymatrix [20]. In this manner, the network of bonds, modelled asbundles of independent springs, one axial and two transversal,represent the potential micro-failures both within and betweenparticles. With the mapping of particles to sites, the model lengthscale, L, has been determined from experimentally measured par-ticle size distribution and volume fraction. With known L, thesprings' stiffness coefficients have been calculated from the energyequivalence between discrete and continuum cells under homo-geneous strain fields according to the procedure outlined by Zhanget al. [16]. This methodology, derived specifically for the site-bondgeometry, allows for accurate representation of isotropic elasticmaterials with Poisson's ratio ranging from �1 to 0.5, animprovement on previous lattice arrangements where only zeroPoisson's ratio has been allowed [12].

However, when the observable microstructure cannot beconsidered as a three-phase composite, but rather a two-phasecomposite with pores dispersed in a solid, the model length scalecannot be calculated in the same manner. In such a case (e.g.Ref. [21]), the model length scale is arbitrary, and similarly to thefinite element analysis, improved accuracy is achieved by reducing

Fig. 1. The site-bond lattice model; (a) cellular tessellation; (b) sites and bonds in atruncated octahedral cell; (c) bond support volume; (d) site-bond model representedas a network of sites and bonds. (A color version of this figure can be viewed online.)

C.N. Morrison et al. / Carbon 100 (2016) 273e282274

the scale, i.e. increasing the number of cells in the assembly rep-resenting the specimen. Further in Ref. [21] the bonds have beenrepresented by 1D connector elements in ABAQUS [22], rather thanspring bundles. The combination of connector elements with onlyaxial non-linear and dissipative physical response, and a geomet-rically non-linear formulation, i.e. finite deformation analysis,makes the representation more physically realistic. Firstly, springsrepresent only conservative behaviour, similar to inter-atomic po-tentials in molecular dynamics, while connectors allow for energydissipation. Secondly, the finite deformation analysis ensures con-servation of angular momentum at sites in accordance with recentadvances in geometric theory of solids [23].

The model presented in this paper combines and extends thedevelopments presented in Refs. [20] and [21]. The graphite isconsidered as a three-phase composite and the approach used inRef. [20] is applied at the composite level. However, the compositelevel properties are derived from separate models of filler particlesand matrix at the constituent level, where the approach used inRef. [21] is applied. The procedure for bond calibration follows [16],with the exception that only the axial stiffness coefficients are usedfor the connector elements. The axial stiffness coefficients ofprincipal and octahedral bonds are given by Equations (1) and (2)respectively, where E and n are macroscopic elastic modulus andPoisson's ratio.

Kpn ¼ EL

4ð1þ nÞð1� 2nÞ (1)

Kon ¼ ð1þ 2nÞEL

4ð1þ nÞð1� 2nÞ (2)

In the absence of transversal springs, the analytical results of[16] dictate an initial macroscopic Poisson's ratio of 0.25 must beused in order to maintain energetic equivalence in the calibrationprocedure. However, the finite deformation analysis reduces theemergent ratio to the prescribed value used in Eqns. (1) and (2)such that the actual value for graphite of 0.2 may be used.Further, local heterogeneity of graphite solid phases due to ar-rangements of crystals and the presence of unresolved porosity isrepresented by variable E for different bonds, following uniformlyrandom distribution within ±10% of a nominal value. This is inaddition to the stiffness changes resulting from resolved poreswithin each phase, the introduction of which is described in Section2.3.

In summary, the present work introduces a two-scale approachfor graphite modelling, where filler particles and matrix aremodelled separately as two-phase materials (pores dispersed insolids) and their responses are used to inform a larger scale two-phase model of graphite (filler particles and matrix).

2.1. Pore-free bond behaviour

The bond response follows a linear relationship in compressionand a linear-softening relationship in tension as shown in Fig. 2. Thetensile behaviour encapsulates both the deformation of the bondsupport volume, V, by storing elastic energy, and the failure of theface between the two cells with area A, by dissipating energy insurface generation. The energy released upon bond failure, GC, isthe sum of the elastic energy stored at face failure initiation, GE, andthe dissipated energy in full face separation, GD. The force, Fd, andthe displacement, Ud, at face failure initiation can be calculatedfrom known GE and the bond stiffness. The failure displacement, Uf,can be calculated from known GD and the failure initiation point.

In a previous work on graphite grades IG110 and PGX (bothnearly isotropic) [20], the total released energy was equated

exclusively to the face separation energy, gA, where g is theenthalpy for creation of two surfaces in graphite, derived by atomicscale calculations to be 9.7 J/m2 [24]. This did not allow for deter-mination of Fd, Ud, and Uf, from bond stiffness and separation en-ergy alone and required an assumption for the ratio GD/GE,alternatively Uf/Ud. Irrespective of the selected ratio, the model wasnot able to predict correctly the relative tensile strengths acrossdifferent graphite grades without grade-dependent factors (i.e.microstructure dependent factors) to increase the released energy.The average filler particle size provides a significant three-folddifference between the two grades [25]. Therefore, it can bededuced that the grade-dependent factor should be related to avolumetric term, whereby the additional released energy is asso-ciated with the stored energy in support volumes.

Hence, here a separate scaling for GE and GD, by the supportvolume and face area, respectively, is proposed via volumetric andsurface constants, U and g:

GC ¼ GE þ GD ¼ UV þ gA (3)

While the value of g is the same as before, the value of thevolumetric constant, U, is not as easily derived from experimentalor atomic scale calculations. It accounts physically for the volu-metric deformation of the bond support volume. This arises notonly from the change in bond length but also from the necessity tomaintain solid unbroken cells, i.e. support volumes from the samecell remain in contact. The procedure for calibrating this constant isdescribed in Section 2.4.

2.2. Material and microstructure

The material considered is moulded IM1-24 Gilsocarbon (GCMBgrade) polygranular nuclear graphite, manufactured by Graftech(formally UCAR). The bulk material has weakly-anisotropic prop-erties; depending on orientation, the elastic Young's modulus isbetween approximately 11.6 and 11.9 GPa, with a Poisson ratio of0.2 and a tensile strength between 19 and 20 MPa [26]. It is one ofthe graphite grades used in the nuclear cores of the UK AdvancedGasecooled Reactor fleet. The same grade, from different billets,has been studied in previous work by some of the authors [27e29].

High resolution computed X-ray tomography data were ob-tained with a voxel size of 1.8 mm in experiment EE9036 at theDiamond Light Source (I12 beamline). Full details of the experi-mental conditions and standard back-projection tomographicreconstruction from radiographs are reported elsewhere [18]. Theimaged volume discussed here (4.32 � 4.32 � 4.81 mm) contains

Fig. 2. Bond failure criteria, relative force, F against relative displacement U

C.N. Morrison et al. / Carbon 100 (2016) 273e282 275

filler, matrix and pores. The pores have lower X-ray attenuation;there is also some incidental phase contrast due to the imagingconditions, which aids the detection of pores. The regions of fillerandmatrix can then be identified by themorphology of their pores;filler particles exhibit a characteristic onion-skin structure oflenticular pores, and the matrix has a less organised structure,Fig. 3.

A total of 55 filler particles and 25 matrix sub-volumes ofdifferent sizes were extracted from the dataset. The smallest vol-ume of a selected filler particle is 0.05 mm3, and the largest volumeis 4.4 mm3. The matrix volumes vary from 0.16 to 1.16 mm3. Theresults for both phases are shown in Fig. 4.

The microstructures may be segmented using an image in-tensity threshold to define the pores and solid graphite. All X-raytomography images were converted to 8-bit datasets before thesegmentation. It was not possible to apply a single threshold for thegrey-scale dataset, so the segmentation procedure was performedin ImageJ software using a multi-step thresholding with the cor-responding smoothing and binarisation steps for pore boundarydetermination and large pore filling. The thresholds were verifiedvisually by comparisonwith the original grey-scale image. The fillerparticle boundaries have been manually identified by using thevisible matrix pores, which surrounded a particle. These pores arequite large and have a well distinguishable structure that isdifferent from the lenticular pores of the particle. Parts of theparticle boundary connected with solid matrix were restored thenassuming the ellipsoidal shape of the particle. The shape of theunbroken Gilsocarbon filler particles is typically ellipsoidal, oftenclose to a spherical shape. Analyses of tomographic data from thesame graphite billet found that the fraction of filler particles varieswithin 14e29%; the fraction in the volume from Ref. [18] is 29%.

Within both phases the smallest pore volume that could beresolved was restricted by the resolution of the tomography data.The mean filler pore volume throughout the imaged volume is15080 mm3 (standard deviation 6320 mm3). The pores in the matrixhave very different shapes and cover a wide range of volumes from6 to 105 mm3; the most frequent pore volume is approximately100 mm3. The largest individual pores observed occupy a volume ofabout 106 mm3 in the filler and 105 mm3 in the matrix; more than90% of the total pore volume in a subset may be spatially combinedinto one large pore. It is important to note that the selection of

matrix volumes excluded regions that contained larger pores(>100 mm), which occur due to gas porosity. These pores have beenquantified using laboratory tomography data of lower resolution(Skyscan 1272) (10 mm/voxel), their fraction has been estimated as6.3% of the total volume of the sample.

The porosity fraction in the different phases was extracted for arandomly selected subset of 20 filler particles and 7matrix volumesfrom tomographed volumes of the same graphite billet [17]. Theobserved porosity fraction in the filler is typically lower than in thematrix; the mean porosity of the filler subsets is 12.2% with astandard deviation of 3.6%, while the matrix has average porosity of16% and a standard deviation of 3.1%. The cumulative probability forporosity observed in both filler and matrix phases are shown inFig. 5(a). A region was chosen at random for each phase, so that itspore size distribution could be used for the subsequent simulations.The pore size distribution from the selected region for both phasesis shown in Fig. 5(b) with the filler and matrix samples containing1204 and 24394 pores respectively.

The deformation of filler and matrix, up to an applied tensilestress of 7.5 MPa, has been studied by digital volume correlation(DVC) of X-ray computed tomography images, obtained during atensile test. Full details of test and the image correlation analysisare reported elsewhere [18]. Briefly, the DVC analysis of sub-volumes that contained filler and matrix was used to calculatethe axial strains in each xy-plane as the gradient of the averagevertical displacements in the z-direction, which corresponded tothe tensile axis. Only those displacements contained inside theellipsoidal volume that defined the filler particle have beenconsidered for the filler. In the studied volumes, the axial strain at250 MPa applied stress varies from 0.0003 to 0.001 in the filler andfrom 0.0005 to 0.0013 in the matrix; the average axial strain islarger in the matrix (774 mε ± 178 mε) than for the filler particles(667 mε ± 197 mε), which suggests the elastic modulus of the matrixmay be lower. The mean strain of the tomographed volume of thetensile sample at 250 MPa was measured to be 730 mε.

2.3. Pore-affected bond behaviour

The mapping of microstructure to the model follows the pro-cedure outlined in a previous work [21] whereby micro-cracks areconsidered to initiate at pores. Pores, with sizes selected at randomfrom an experimental pore size distribution, are assigned to faces of

Fig. 3. A reconstructed X-ray computed tomographic image of Gilsocarbon micro-structure. Coarse filler particles can be seen dispersed within a matrix of graphitizedpitch and finer (ground) filler particles. Pores can be seen as dark regions as a result oflow X-ray attenuation. Blue inserts show segmented structures of the pores within thematrix (top) and filler (bottom) sub-volumes. (A color version of this figure can beviewed online.)

Fig. 4. Filler particle and matrix sample volumes. (A color version of this figure can beviewed online.)


cells until the desired porosity is reached. The presence of porosityis reflected in changes of the tensile response of correspondingbonds, Fig. 6. The peak force changes from the pore-free value, Fd, toa new value, F'd, according to

F 0dFd

¼�V 0

V

�2=3

(4)

where V is the support volume of the bond and V0 is the supportvolume remaining after the corresponding pore volume isremoved, i.e.

V0 ¼ VSupport volume � Vpore (5)

In the same manner, the displacement at face failure initiation,Ud, changes to a new value, U'd, according to

U0d

Ud¼

�V 0

V

�1=3

(6)

Equations (4) and (6) represent pore-corrected force anddisplacement parameters via pore-corrected (or effective) areasand lengths, respectively.

The stored elastic energy at face failure initiation, GE, scales withthe change of support volume:

U0dF

0d

UdFd¼ V 0

V(7)

Differently, the damage energy, GD, scales with the face area:

�U0f � U0

d

�F 0d�

Uf � Ud

�Fd

¼ A0

A(8)

where A0 is the face area remaining after the corresponding porearea is removed:

A0 ¼ AFace area � Apore (9)

In this manner, the failure displacement, Uf’, reduces propor-tionally less from the pore-free value than the initiation displace-ment, Ud’, so the amount of softening is reduced as the pore size isincreased, resulting in an increasingly brittle response. The bondstiffness is also reduced.

Similarly to tensile behaviour, the presence of porosity also al-ters the compressive response of the bond. Compressive stiffness isreduced by the same factor as tensile stiffness according to poresize, although this decrease is only maintained for a relativedisplacement equivalent to the diameter of the pore present, afterwhich time the stiffness increases back to the original value.

In the process of random pore allocation to faces, some porevolumes may exceed the corresponding bond support volumes.Such bonds are removed from the model and the excess pore vol-ume, i.e. the difference between the allocated pore volume and theremoved support volume, is distributed to neighbouring bonds inthe same manner until all the volume is allocated. In this mannerthe size distribution of pores distributed to faces will be a repre-sentative sample of the experimental distribution, although thespatial distribution of pores will be entirely random.

2.4. Calibration of the volumetric constant

Preliminary studies using the pore representation outlined inSub-section 2.3 were undertaken to calibrate the volumetric con-stant. Four different grades of nuclear graphite, IG110, NBG-18, PGXand Gilsocarbon, were simulated with models scaled according tothe size and volume fraction of the corresponding filler particlesfollowing [20]. The microstructure information used for gradesIG110, NBG-18 and PGX, including pore size distributions, porosityand filler particle sizes was taken from the microscopy studies byKane et al. [25]. The filler particle size distribution used for the IM1-24 Gilsocarbon was that shown in Fig. 4. The microstructure dataused for all grades and the corresponding references are

Fig. 5. (a) Porosity distributions for the filler and matrix sub-volumes [17]; (b) poresize distributions for the randomly selected filler and matrix regions [17]. (A colorversion of this figure can be viewed online.)

Fig. 6. Bond failure criteria accounting for the affect of porosity


summarised in Table 1. A filler particle volume fraction may varywithin graphite grades. The value used was 0.2, an average of thetwo values obtained of 0.144 and 0.252 presented in Ref. [17].Although these values were specific to Gilsocarbon, the same fillerparticle volume fraction of 0.2 was used for all grades. The poredistributions taken from Ref. [25] do not differentiate betweenpores found in the matrix and filler phases. Hence, for simplicityonly the pore size distribution for the matrix phase, shown in Fig. 5,was used to calibrate the volumetric constant of the IM1-24 Gil-socarbon. It is not necessary to repeat the process for the fillerphase, because for a prescribed porosity its larger pores can beconsidered as represented in the model by the coalescence ofnumerous smaller pores assigned to one and the same lattice bond.

Measured Young's moduli, tensile strength and typical poros-ities for each grade in its virgin (i.e. as supplied, without any effectsof fast neutron irradiation or radiolytic oxidation) are shown inTable 1. To calculate the pore-free, axial stiffness coefficients ofbonds, pore-free values of the Young's moduli are required. Thesewere calculated with a series of normalised simulations withYoung's modulus equal to one, without porosity and with virginstate porosity. Several realizations with the latter were analysed,differing in the spatial distribution of pores but identical size dis-tributions. The simulations were performed without failure ofbonds, i.e. in the elastic regime of bond behaviour. From thesesimulations the ratios between the (unit) pore-free and the simu-lated average virgin-porosity moduli were calculated. The ratioswere used to scale the experimental virgin-porosity moduli topore-free moduli, reported in row 3, from where pore-free axialstiffness coefficients were calculated by Eqns. (1) and (2). The closeproximity of the calibrated pore-free modulus of Gilsocarbon,14955 MPa, to a pore-free value of 15 GPa, derived from nano-indentation experiments [30], gives confidence in the calibrationprocedure. In the absence of equivalent data (to the knowledge ofthe authors') for the other grades this is considered an adequatevalidation.

Models with calibrated coefficients were subject to displace-ment controlled uniaxial tension until failure, using the assumptionthat energy released at bond failure equals the energy of faceseparation, gA, specifically assuming as previously [20] [21].

GprelimE ¼ Gprelim

D ¼ 12Ag

�Uf ¼ 2Ud

�(10)

Failure was considered to be the point at which the simulationfailed to find equilibrium using a time increment size less than athreshold value (1 � 10�25 was deemed suitably small). Severalcontrol parameters were adapted to improve convergence in thehighly non-linear model. Viscous regularization was utilized toimprove the dissipation of energy from damaged bonds to thesurrounding bonds. A damping coefficient of 1 � 10�5 was chosenafter initial studies showed it gave the best compromise betweenaiding convergence and producing a consistent peak stress. Thequasi-Newton method was used for the analysis and the method ofextrapolation was suppressed, preventing excess iterations.

The difference in simulated and experimental values of tensile

strength, sSimT and sExpT , as a function of model cell size (volume), for

a model that does not include the proposed volumetric correction,is shown in Fig. 7 with red marks. Here, the cell size, shown inTable 1, reflects the different structures of the four graphite gradesin terms of particle size distribution. It is apparent that, when usingthe cell size as a representation of the average filler volume, therelationship between strength discrepancy and size is approxi-mately linear, i.e. the larger the average filler particle size the largerthe difference between simulated and expected tensile strength.

This relation between strength discrepancy and cell size occursfrom using the energy of face separation as a sole measure of en-ergy released upon bond failure, failing to account for the energeticseparation into area, GD ¼ gA, and volumetric terms, GE ¼ UV, asproposed in Section 2.1. As such the volumetric constant can becalibrated from the linear trend.

The peak bond force in the preliminary studies (without volu-metric term) and the proposed model can be expressed as:

FprelimD ¼ffiffiffiffiffiffiffiffiffiffiAgK

p(11)

FD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2UVK

p(12)

with stiffness coefficient, K, being the same for both models. Thelinear trend shown in Fig. 7 can be expressed as:

sExpT

sSimT

� 1 ¼ mS (13)

where m is the gradient of the linear trend and S is the cell size.From comparison between simulations and experiment, one canwrite:

Table 1Comparisons between grades for model inputs (Young's modulus, virgin porosity), the resulting model cell size and literature values of tensile strength and mean particle size.

Graphite grade Gilsocarbon (IM1-24) IG110 NBG-18 PGX

Typical young's modulus (MPa) 11600 [26] 9800 [31] 11500 [32] 8300 [33]Calibrated pore-free young's modulus (MPa) 14955 11327 12446 9717Virgin porosity (%) 19.14 [34] 14.73 [25] 13.97 [25] 21.49 [25]Tensile strength (MPa) 19e20 [26] 25.3 [35] 20 [32] 8.1 [35]Mean particle size (major axis length) (mm) 914 [17] 27 ± 2 [25] 360 ± 25 [25] 92 ± 7 [25]Cell size (mm) 1591 79.7 854 297

Fig. 7. Calibration of the volumetric term. (A color version of this figure can be viewedonline.)


sExpT

sSimT

¼ FDFprelimD

(14)

As such, by substituting Equations (11) and (12) into Equation (13)the volumetric term can be expressed as:

U ¼ ðmSþ 1Þ2Ag2V

(15)

The volumetric term is therefore a function of cell size (volume)and differs for bonds B1 and B2, according to their support volumes,V, and face areas, A. The gradient m, was calculated from Fig. 7 as0.0022. The units of the volumetric term are J/m3. Rerunning thesame simulations with the calibrated volumetric term producedresults with considerably less discrepancy from experimentalvalues, as shown in Fig. 7 with blue marks.

3. Single phase modelling and results

In this work a two-scale methodology is introduced in order tobuild up the composite response of graphite directly from themechanical response of the individual phases. In this section thesingle phase procedure used for both filler and matrix phases isoutlined; specifically, the modelling of filler particles and matrixincorporates the experimentally measured pore size distributionsof the individual phases.

Five site-bond models for each phase were generated with po-rosities randomly selected from the measured porosity distribu-tions shown in Fig. 5(a). Within each model, pores were randomlyassigned to faces with sizes from the corresponding measured poresize distributions, Fig. 5(b), until the required porosity for theparticular model was achieved. All models were constructed aslattices occupying cubic regions of 10-cell sides for computationalefficiency. Fig. 8 and Fig. 8(b) show the stress-strain response ob-tained from the filler models and matrix models, respectively. Theresponse is visibly different for both phases with significantly moreenergy dissipation and nonlinearity exhibited by the filler phasesimulations as a result of bonds entering the softening region of theconstitutive behaviour. The matrix phase shows less pre-peak non-linearity with sudden “avalanche” failure shortly after peak stress.One of the five matrix simulations failed to run (results not shown),presumably as a result of multiple failures occurring in the initialsolution increment. The stress-strain curves illustrate the effect ofvariable porosity on the responses of different phases, which willbe used as input to the composite level model in Section 4. Thereare significant variations of elastic modulus within the models foreach phase, which do not relate simply to the total porosity. Thissuggests that the response results from both the porosity value andthe different spatial distributions of pores across samples. It ap-pears that porosity increase generally leads to elastic modulusreduction with this trend more prominent within the filler phase.However, this is not a comprehensive trend with samples of com-parable porosities showing different moduli, which is attributed tothe spatial arrangements of the pores. Furthermore, it has beenshown that pore shape affects modulus [36] [37] although thisphenomenon is not yet represented in the current model.

The behaviour of each phase can be understood when the initialmodel states are considered. The initial porosity present on a fillerand matrix model are shown in Fig. 9. The brittle response of thematrix phase results from high proportion of bonds that are bothremoved pre-simulation due to porosity, on average 10.5%, anddamaged but not yet failed, on average 59.9%. The high number ofdamaged bonds explains the catastrophic failure, with a largenumber of damaged bonds reaching a critical load at the same

simultaneously. The more “graceful” failure of the filler phaseemerges from lower proportions of the same values, 7.1% and 30.9%respectively, which allow damage to evolve.

Table 2 lists the Young's modulus calculated in each simulationfor the initial load increment, which was sufficiently small so as nofailures occurred. It should be noted that some values of E for thefiller phase are higher than the pore-free value used to calibrate themodels. This is because of the introduced random distribution of Eto different bondswith 10% standard deviation. The average value iswithin 1.5% of the pore-free value of 14995 MPa, suggesting littleeffect of the filler pores on its stiffness. In contrast, the matrixporosity has a substantial effect on its stiffness, reducing the pore-free value by more than 20%.

Total strains are composed of an elastic part and permanent partarising from the generation and growth of micro-cracks. The stress-strain curves in Fig. 8 were used to extract the total strain for eachsimulation at a stress value of 7.5 MPa, which can be compared tothe experimentally measured strains in filler particles and matrix.The experimental values were measured at global tensile stress of7.5 MPa. Fig. 10 shows the cumulative probability of measured axialstrain in both filler and matrix samples, together with the cumu-lative probability of strains obtained by simulations. The cumula-tive probability of simulated strains arises from the five modelrealizations with different porosities per phase. Both experimental

Fig. 8. The stress-strain response of the; (a) filler simulations; (b) matrix simulations.Samples are ordered with increasing porosity, q. (A color version of this figure can beviewed online.)


and simulated results for the two phases indicate that for the sameglobal stress, matrix strains are higher than filler strains. The dif-ference in the pore systems of the matrix and the filler, a conse-quence of the distributions of porosity and pore volumes shown inFig. 5, results in lower stiffness and lower strength of the matrix.The lower stiffness can be attributed primarily to the higher matrixpore volume fraction, while the lower strength e to the higherpropensity to micro-crack generation.

Care should be taken with direct comparison of the

experimental and simulated results, since the simulations resultfrom a local stress of 7.5MPa as opposed to the global 7.5MPa in theexperiment; stiffer regions within a heterogeneous bulk specimenattract an increased amount of the load, leading to stress parti-tioning (similar to that observed in particulate composites) ac-cording to the phase properties and position within the specimen.The comparison does however yield some interesting discussionregarding the stress-state of the samples.

Both simulations of filler and matrix phases predict lower axialstrain at 50% probability than the experimental data; the fillerphase model has a larger discrepancy (25% difference from exper-iment as opposed to 15% for matrix phase). However, the lowercompliance of the filler phase would suggest that the filler particlesexperience a larger stress than the matrix under the same appliedglobal stress in experiments. The variability in porosity and poresize distribution in the microstructure is larger than that used toconstruct the models analysed, which may account for the smallervariation seen in the simulations. Themodelling approach is judgedto be promising; both phases were calibrated using the same pore-free value of E, leaving the resulting responses of the two phases toemerge from the porosity of the microstructures alone.

4. Composite modelling and results

The lattice for the matrix-filler composite was based on a cubiccellular structure of length 10 cells, giving C ¼ 1729 cells. Particlesizes from the experimental distribution, Fig. 4, were assigned atrandom to each site. The cell size was calculated from the model

Fig. 9. Examples of pore distribution in the model of: (a) filler phase sample 1F (modelsize ¼ 820 mm); and (b) matrix phase sample 1M (model size ¼ 980 mm). Pore di-ameters, depicted in microns, reflect those designated to each bond, where large poresfrom the experimental distribution are assigned as smaller pores over several bonds.Examples of such large pores are circled in blue. (A color version of this figure can beviewed online.)

Table 2The initial Young's modulus calculated from each simulation.

Sample Filler Matrix

Porosity, q (%) E (MPa) Average E (MPa) Standard deviation (MPa) Porosity, q (%) E (MPa) Average E (MPa) Standard deviation (MPa)

1 6.22 14644 14788 2387 13.37 12549 11454 10152 6.61 17239 14.67 108093 7.74 17024 18.53 104014 13.15 13369 17.75 120585 14.58 11665 18.65 n/a

Fig. 10. The distributions of axial strains in the; (a) filler phase and (b) matrix phaseobtained experimentally at a global load of 7.5 MPa and simulated at local load of7.5 MPa. (A color version of this figure can be viewed online.)


volume as a function of cell length L, Equation (16), the cumulativeassigned filler particle volume and the desired filler volume frac-tion, qF, according to Equation (17). The value of qF was taken as 0.2,an average of the two values obtained of 0.144 and 0.252 presentedin Ref. [17].

Vlattice ¼ CL3

2(16)

L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PC

i¼1viparticle

CqF

3

vuut(17)

Bonds, chosen at random, were assigned random properties ofthe filler samples, derived in Section 4. This process continued untilthe cumulative total of the bond support volumes matched thedesired particle volume fraction. The remaining bonds wereassigned random properties of the matrix samples, also derived inSection 4. Specifically, the stress-strain behaviour of the differentfiller and matrix samples (i.e. Fig. 8) were used to inform bondbehaviour. The local gradient of the non-linear stress-strainbehaviour (i.e. from Fig. 8) was used as input to the stiffness cali-bration used for the single phasemodels, Equations (1) and (2). Thisstiffness and the corresponding strain from the sample behaviourwere used to calculate the force-displacement relationship of eachbond.

The results of the composite simulations are shown in Fig. 11.The initial Young's modulus value of 12900 MPa is higher thantypical literature values for virgin Gilsocarbon. This may be a resultof failing to simulate the largest pores in the matrix phase (withsize >100 mm); these occupy 6.3% of the total volume and wereexcluded in the distributions of porosity in filler and matrix, re-ported in Vertyagina and Marrow [17], that were used in the sim-ulations. This extra porosity, which could be included in futuremodels either in the matrix phase models themselves, or as addi-tional porosity applied to the composite model, would slightlydecrease the elastic modulus. The tensile strength value obtained of13.6 MPa is lower than the value of 19e20 MPa quoted in theliterature [26]. The simulated tensile strength would be expected todecrease with inclusion of the larger pores, so this suggests that themodel does need further refinement, particularly with regards tothe peak stress values. Nonetheless, the comparison betweensimulation and experiment is encouraging at this stage of themodel development.

5. General discussion

It was found that the multi-scale model was more numericallystable and less computationally expensive than the single phasemodels. This is because the multi-scale model does not require thepores to be modelled explicitly. Instead the effects of pores arehomogenised within the models of the individual phases, whichare represented by a small number of possible stress-strain be-haviours that are reasonably consistent with experimental behav-iour (i.e. Fig. 10). In doing so there is no need to remove bonds priorto simulation as a result of porosity. This makes finding an initialequilibrium less expensive and hence improves the numericalstability of the model.

Despite these promising results there are still limitations withthe model, mainly linked to limited data and numerical controls.Firstly, in addition to the large pores that are currently neglected,there is finer scale porosity in the graphite microstructure [38] thatis unresolved using the experimental techniques described inSection 2.2. Its effect should be included in the pore-free modulus,however. Furthermore no consideration has been taken of residualstresses, which exist following manufacturing [39] and so affect onthe stress-state of each phase. With respect to numerical stability,the controls required to obtain convergence, including viscousregularization, and their impact on the model response need to befurther understood to increase confidence in the results and henceallow for reduced conservatism in safety assessments of graphitecomponents.

Further work includes an improved calibration procedurecompatible with theories of discrete elasticity [23]. Following this,more rigorous studies may be undertaken using the Site-Bondmethodology, deriving damage evolution laws and characterisingthe size of the fracture process zone for use in continuum scaleanalyses for structural integrity assessment of graphite.

6. Conclusions

A multi-scale modelling methodology is presented, wherebymicrostructure-informed Site-Bond lattice models of both fillerand matrix phases are used to construct a larger scale compositeSite-Bond model of the nuclear grade graphite Gilsocarbon. A keyfeature of the proposed model is that it requires a single cali-bration of the elastic properties of “pore-free” graphite, fromwhere it can predict the elastic properties of real graphite fromthe knowledge of microstructure characteristics, such as particleand pore density and size distribution. The single-phase modelresults suggest that the evolution of damage is more prevalent inthe filler phase than the matrix phase with filler modelsdemonstrating “graceful” stress-strain behaviour resulting frommicro-failures as opposed to the brittle failure seen in the matrixphase. Filler particles are shown to be stiffer than the matrixphase.

� The modulus and tensile strength value calculated from a multi-scale composite model, 12.9 GPa and 13.6 MPa respectively,informed with the responses of the single-phase models, areencouraging when compared to the values found in the litera-ture, 11.6 GPa and 19e20 MPa respectively. The proposed semi-empirical approach, using a calibrated pore-free stiffness andderiving longer scale behaviour from microstructure informa-tion, has the potential to develop into a deductive methodologyfor calculating emergent behaviour, when combined with richermicrostructure information and validated by damage charac-terisation experiments.Fig. 11. The stress-strain response of the multi-scale simulation


Acknowledgements

C.N.M acknowledges the support from EPSRC via Nuclear FiRSTDoctoral Training Centre. A.P.J acknowledges the support fromEPSRC via grant EP/J019801, “QUBE: Quasi-Brittle fracture: a 3Dexperimentally-validated approach”. Moreover A.P.J acknowledgesBNFL through the Research Centre for Radwaste & Decom-missioning. T.J.M. gratefully acknowledges the support of OxfordMartin School, and Ye.V. acknowledges the support of EDF EnergyGeneration.

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Lattice-Modelling of Nuclear Graphite for Improved