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Research

Article submitted to journal

Subject Areas:

mechanical engineering, materials

science

Keywords:

cellular automata, ﬁnite elements,

cleavage, polycrystals, grain

boundary, three-dimensional, Fortran,

coarrays

Author for correspondence:

A. Shterenlikht

e-mail: mexas@bris.ac.uk

Three-dimensional cellular

automata modelling of

cleavage propagation across

crystal boundaries in

polycrystalline

microstructures

A. Shterenlikht1, L. Margetts2

1Mech Eng Dept, The University of Bristol, Bristol BS8

1TR, UK

2Directorate of IT Services, The University of

Manchester, UK and Oxford e-Research Centre, The

University of Oxford, UK

A three-dimensional cellular automata with rectilinear

layout is used in this work to create and cleave

polycrystalline microstructures. Each crystal is deﬁned

by a unique randomly generated orientation tensor.

Separate states for grains, grain boundaries, crack

ﬂanks and crack fronts are created. Algorithms for

progressive cleavage propagation though crystals

and across grain boundaries are detailed. The mesh

independent cleavage criterion includes the critical

cleavage stress and the length scale. Resolution of

an arbitrary crystallographic plane within a 26-cell

Moore neighbourhood is considered. The model is

implemented in Fortran 2008 coarrays. The model

gives realistic predictions of grain size and mis-

orientation distributions, grain boundary topology

and crack geometry. Finally we show how the

proposed cellular automata model can be linked to a

ﬁnite element model to produce a multiscale fracture

framework.

c

The Author(s) Published by the Royal Society. All rights reserved.

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1. Introduction

Cellular automata (CA) modelling of physical systems is a well established ﬁeld [1–3]. CA is a

discrete time - discrete space framework. The model space is partitioned into identical cells with

a ﬁnite number of states. A state of a cell at the next time increment is determined by the state

of this cell and the states of some neighbourhood at the previous time increment. Fixed or self-

similar model boundaries can be used. This simple framework gives rise to a surprisingly rich

range of behaviours, some of which are suitable for simulating physical processes such as lattice

gas diffusion, phase transitions, wave propagation, multi-phase ﬂuids [2]. In solid mechanics it

is popular to superimpose continuum ﬁelds, such as temperature, stress or strain, over the CA

space. A combination of CA and ﬁnite elements, sometimes referred to as CAFE or CA-FE, has

been used successfully for predicting ductile to brittle transitional fracture [4], oxide cracking in

hot rolling [5], grain instability [6], solidiﬁcation [7,8], friction stir welding [9], recrystallisation

[10–13] and dynamic strain induced transformation [14].

The vast majority of CA models explored over the years are two-dimensional. However, there

are physical processes which cannot be accurately represented by 2D models. Polycrystalline

fracture is one example. Speciﬁcally, transgranular cleavage propagation across grain boundaries

cannot be modelled adequately by a 2D model, because grain boundary accommodation failure,

due to mis-orientation of preferred cleavage planes in the neighbouring grains, cannot be taken

into account [15–17].

It is important to highlight the major difference between the cellular automata approach and

Voronoi tessellation [18,19]. The crystals produced by the Voronoi method have easy geometrical

description: faces, edges, vertices. In contrast a CA produced crystal is just a collection of

connected cells. A CA grain has neither faces, nor edges or vertices. This might appear to be a

disadvantage. However, one must remember that Voronoi polyhedra is an idealisation of crystal

shapes in real polyscrystalline materials. Indeed, one might argue that ‘blobs’ of irregular shape,

produced by the CA approach, are closer to nature, as seen through the microscope, than nicely

deﬁned Voronoi polyhedra. However, the major advantage of the CA approach over the Voronoi

tessellation is in the ability of the CA framework to model grain competition, recrystallisation,

grain boundary migration and other phenomena resulting in the evolution of the microstructure.

This is easily achieved in the CA model, precisely because the crystals are not deﬁned by the

geometric means. Evolution of microstructure will require a lot more work if is to be implemented

via the Voronoi tessellation approach. However, Voronoi tessellation can be useful for setting the

initial cellular morphology [10].

This paper is concerned with the design of a 3D CA model, feature rich and ﬂexible enough to

represent a wide range of polycrystalline microstructures and transgranular cleavage propagation

in those.

The following notation is used in this work. Tensors of rank 2 are shown in bold: R. Vectors

and scalars are in the upright math type: x. Cell states are sans serif: c.

2. The cellular automata model

A 3D CA space with cubic cells and 26-cell Moore neighbourhood is created. The CA model is a

rectilinear brick with d1,d2and d3cells along dimensions 1,2 and 3 respectively. The total number

of cells in the model is D=d1×d2×d3.

First a polycrystalline grain microstructure is created by a simple solidiﬁcation process in the

following way. All cells are initially considered to be of liquid state, cL= 0.Nrandomly chosen

cells represent grain nuclei. These are assigned states cG∈[1 . . . N ]. Each grain (single crystal)

is assigned a randomly chosen orientation tensor, Rc. Each iteration of the solidiﬁcation process

a liquid cell can acquire the state of one of the 26 randomly chosen neighbours. This is shown

schematically in Fig. 1. This process is continuing until there are no liquid cells left in the model.

Both ﬁxed and self-similar boundary conditions can be used [6].

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0

0 0 0

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neighbourhood

cell

CA array grid

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0000

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(a) (b) (c)

Figure 1. (a) A 2D slice of the 3D 26-cell Moore neighbourhood, showing a central liquid cell, c= 0, acquiring the state of

one randomly chosen neighbour, indicated by an arrow, grain number 1, c= 1, in this example; (b) and (c) A 2D illustration

of a single iteration of 3D solidiﬁcation. Six liquid cells changed state to solid in this increment; three cells attached to

grain 1, and the other three attached to grain 24. The arrows show the copying of cells states.

The model described above will produce equiaxed microstructure. By changing the initial

distribution of grain nuclei, other popular distributions can be easily achieved, e.g. bimodal or

columnar [6]. Our prior work suggested that, in order to achieve results independent of grain

resolution, the CA model must be created with a sufﬁcient resolution, N/D < 10−5i.e. more than

105cells must be used, on average, to represent each grain [6].

A CA method itself has no concept of length or time scales. These scales are assigned to CA

by the user, based on the exact physical process of interest. In relation to this work, this means

that a polycrystal structure produced by the above algorithm can represent micro- or nano-crystal

structures. One major difference between nano- and micro-crystalline materials is that in nano-

materials the volume occupied by grain boundary regions becomes comparable to that occupied

by grain interiors. The fraction of grain boundary volume could be as high as 50% or even

more [20,21]. This result emerges naturally from this CA model. If one considers all cells having

neighbours belonging to other grains, as grain boundary cells, then their number (volume) will

increase dramatically with decreasing mean grain size.

3. The quasi-cleavage algorithm

We use tensor, index-free, notation. All sub- and super-scripts are not to be confused with lower

and upper indices of index notation.

Orientation of each crystal with respect to the spatial (cellular) coordinate system (CS) is given

by the rotation tensor, Rc, with the usual meaning: a vector in the crystal CS, xc, is transformed

into a vector in the spatial CS, xs, as xs=Rc·xc.

The stress tensor in the spatial CS, ts, is transformed into the stress tensor in the crystal CS as

tc=Rs·ts·R−s, where Rs≡R−c≡(Rc)−1≡(Rc)T.

It is assumed that cleavage is controlled by the normal stress on a crystallographic plane, tn.

For a plane {hkl}, with normal nhkl , the normal stress is thkl =nhkl ·tc·nhkl.

Each crystallographic plane is assumed to have a particular surface energy, γhkl. We postulate

that the work of cleavage is equal to the surface energy. It is further assumed that the work of

cleavage is thkl times the distance necessary to break the atomic bonds. Following Gilman [22],

this distance is taken equal to a0, the relaxation distance, which is the atom diameter in the

cleavage plane. The cleavage condition is thkl a0=γhkl , from which the stress required to cleave

the {hkl} plane can be calculated:

thkl =γhkl/a0(3.1)

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Example values for iron, from [22], are γ100 = 1440,γ110 = 1710 and γ111 = 5340 erg/cm2

(1erg/cm2= 10−3J/m2) and a0= 1.37 ×10−10m. This gives: t100 = 1.05 ×104MPa, t110 = 1.25 ×

104MPa, t111 = 4.90 ×104MPa.

In a CA model these values must be scaled down, because the CA model is not applicable at

the atomic scale but only at some intermediate, meso-scale, and because material imperfections

and micro-plasticity elevate the stresses locally around the crack fronts [23]. Micro-plasticity has

not yet been implemented in this model. Hence, in strict terms, the results are valid for some

simpliﬁed polycrystalline material. This is the reason we use the term quasi-cleavage instead of

simply cleavage.

When the cleavage condition of Eqn. (3.1) is satisﬁed, the model crack advances for a

characteristic length, which must be taken smaller than the corresponding characteristic length in

the model, e.g. the mean grain size. Together the cleavage criterion and the characteristic cleavage

length form a cleavage model independent of the CA resolution.

More complex cleavage criteria have been proposed in the literature. These are based on

detailed molecular dynamics and quantum mechanics analysis of inter-atomic bond potential,

and formulation of cohesive zone type bond breaking models [24–26].

In bcc crystals there are 24 symmetric rotation tensors, R1...24

sym , including the identity tensor. If

nhkl is a unit normal vector to some {hkl} plane, then n1...24

hkl =R1...24

sym ·nhkl are 24 normal vectors

describing planes of the same class. The weakest plane is that which maximises thkl:

tmax

hkl = max

hkl (thkl) = max

i=1...24(ni

hkl ·tc·ni

hkl)(3.2)

where the only planes under consideration are {100}, {110} and {111}, although the surface energy

of {111} planes is so high that it is practically impossible to cleave those. The normals to the

planes of the maximum normal stress are nmax

100 ,nmax

110 and nmax

111 . From Eqn. (3.1) cleavage will

occur when tmax

hkl ≥γhkl/a0, or when pmax

hkl =tmax

hkl /(γhkl/a0)≥1.

The ﬁrst step is to ﬁnd all pmax

hkl and the orientations of the corresponding planes, Alg. 1. The

second step is choosing the weakest plane and setting the cleavage cell state accordingly, Alg. 2.

The outputs are the unit vector, ns, normal to the active cleavage plane, in the spatial CS, and the

cleavage cell state, s. Vector ncis ﬁrst calculated in the crystal CS and then rotated to the spatial

CS, ns. Cell states c100 ,c110,c111 represent cleavage crack edges on {100}, {110}, {111} planes. The

ﬂag is true if cleavage condition is met, and false otherwise.

Algorithm 1: Cleavage algorithm, calculating maximum normal stresses and their planes

input :ts,Rc,γ100 ,γ110 ,γ111 ,a0

output:p100 ,p110 ,p111 ,pmax ,nmax

100 ,nmax

110 ,nmax

111

tc=R−c·ts·Rc

tmax

100 = maxi=1...24(ni

100 ·tc·ni

100)→nmax

100

tmax

110 = maxi=1...24(ni

110 ·tc·ni

110)→nmax

110

tmax

111 = maxi=1...24(ni

111 ·tc·ni

111)→nmax

111

p100 =tmax

100 /(γ100/a0);p110 =tmax

110 /(γ110/a0);p111 =tmax

111 /(γ111/a0)

pmax = max(p100, p110 , p111)

4. Cleavage representation in the cellular model

In the fracture cellular array, cells are initially intact. A number of crack nuclei, i.e. cells with

cleavage crack edge states, c100 or c110 , are positioned within the model. For example, the crack

nuclei can be scattered at random, representing pre-existing micro- or nano-cracks in the material.

In this manner, growth and/or interaction of a single or multiple cracks can be modelled.

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Algorithm 2: Cleavage algorithm, calculating the cleavage plane

input :p100 ,p110 ,p111 ,pmax ,nmax

100 ,nmax

110 ,nmax

111

output:ns,s, ﬂag

ns=0;s= 0 ; ﬂag = false

if pmax ≥1then

cleavage on {110} plane →ﬂag = true ;nc=nmax

100 ;s=c100

if p110 > p100 then

cleavage on {110} plane →nc=nmax

110 ;s=c110

if p111 > p100 and p111 > p110 then

cleavage on {111} plane →nc=nmax

111 ;s=c111

ns=Rc·nc

Next we scan over all intact cells. If an intact cell has a cleaved neighbour, such that the vector

connecting the cleaved and the intact cells, e, is on or near the cleavage plane, then the state of the

central cell is changed to the given cleavage state. Note that it is possible that the given cleavage

state and the neighbour cleavage state will differ. The key decision in this approach is choosing

a suitable threshold, t, for deciding when ebelongs to the cleavage plane, deﬁned by ns. This

problem is analysed in Appendix A. Assuming that such threshold can be chosen, the algorithm

is summarised in Alg. 3.

Algorithm 3: Cleavage algorithm, propagating cleavage crack through the cellular model.

input :ns,s,t

output: cell state change

for all cells do

pick cell i

for all 26 neighbours of ido

pick neighbour j

if cell jcleaved then

eis a unit vector connecting cells iand j

if e·ns< t then

cell istate is changed to s

exit

The cleavage criterion can be easily changed from a fully deterministic to a probabilistic, e.g.

if e·ns< t then there is a probability that the state of cell iis changing to c. This probability will

be inversely related to e·ns.

Alg. 3changes states only of the neighbouring cells. Thus the speed of cleavage propagation

in this algorithm is 1 cell/increment. With the use of the characteristic length scale, any crack

propagation speed is achievable.

Algs. 1,2and 3are combined to simulate cleavage propagation across the whole cellular grain

array G. The algorithm grows cracks in a similar way to grain growth algorithm: any intact cell

of the fracture array Fis allowed to join a cleavage crack if the following 3 conditions are met: (1)

it has a neighbouring crack front cell, (2) it lies on the cleavage plane and (3) the resolved stress

is high enough. If tcis changing very slowly, compared to the cleavage propagation speed, Algs.

1and 2need to be run only when the grain boundary is crossed, i.e. when the current cell, gi, in

the grain array, G, differs from the state of the neighbour, gold. The resulting algorithm is shown

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in Alg. 4. However, Alg. 4does not take into account the full complexity of cleavage propagation

across a grain boundary in 3D, see e.g. [15,27,28].

Algorithm 4: Complete cleavage algorithm, top level view.

input : grain cellular array G, fracture cellular array F,Rcarray , ts

output: possible cell state change to cleaved

gold = 0

for all cells in Fdo

pick cell i ; read its grain number gi; read its fracture state fi

if fi=intact then

if gi6=gold then

run Algs. 1,2→ns,s, ﬂag

gold =gi

if ﬂag = true then

run Alg. 3→change fito s

5. Crossing a grain boundary

The inﬂuence range in a CA model is one cell size. In this respect CA is closer to a molecular

dynamics approach, albeit on larger spatial and time scales, where the energy potentials are

invariably of a very close range, than to a weak formulation of continuum mechanics, where some

form of global equilibrium is usually maintained. Creating and maintaining global entities, such

as geometrical planes, edges or vertices, is very computationally expensive in CA formulation. In

CA a grain boundary is simply a cluster of cells of identical state gi, each of which has a neighbour

of a different type, gj. Although it would be possible, in principle, to ﬁt a plane over this cluster,

e.g. via a linear minimisation, this is not done in this work due to high computational costs. A

grain boundary edge is a cluster of cells of identical state, each of which has neighbours of at least

two different states.

Analysis of grain boundary fracture typically involves quantities such as crystallographic

types of grain boundaries and grain boundary plane orientation. These quantities are not

available in this CA formulation. Hence, simulating cleavage propagation across a grain

boundary involves some extra considerations, compared to approaches where grains are

modelled as polyhedra, see e.g. [15]. In such geometrical (global) model the process is simple:

as soon as a cleavage crack reaches a grain boundary at some spatial point, a cleavage plane in

the following, adjacent, grain is fully determined, thus allowing for the analysis of the fracture of

the boundary fragment deﬁned by the grain boundary plane and by the two cleavage planes in

both grains.

In contrast, in a cellular (local) model, there is no global cleavage plane deﬁned in a grain.

Hence each crack front cell at the grain boundary has a a chance of starting a new cleavage crack

in the adjacent grain. If left unchecked, this process quickly leads to a proliferation of cleavage

cracks on multiple parallel crystallographic planes in the next grain. This situation is shown

schematically in Fig. 2. Such a model is, of course, not physical. It must not be confused with

river patterns which are sometimes seen in cleavage fracture surfaces (see e.g. [29] and references

thereof).

Note that the geometrical model is not physical either. It simply looks at the ﬁnal result of the

cleavage propagation and tries to reproduce it. It is doubtful that the physical reality of cleavage

propagation across the grain boundary is close to the global geometrical view.

To prevent the nonphysical cleavage crack proliferation scenario illustrated in Fig. 2, only the

ﬁrst cleavage crack cell that touches the grain boundary is allowed to start a new cleavage crack

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crack with grain boundary

grain 1

grain 2 grain boundary

cleavage crack

in grain 1

cleavage cracks

proliferation of

in grain 2

intersection of grain 1 cleavage

Figure 2. Schematic of cleavage crack proliferation at grain boundary crossing due to using only the local neighbourhood

CA analysis. The arrows show the predominant cleavage propagation direction, from right (grain 1) to left (into grain 2).

The dashed line represents the intersection of the cleavage crack in grain 1 with the grain boundary. Note that a single

cleavage crack in grain 1 can initiate cleavage cracks in grain 2 at any point where it crosses the grain boundary.

grain 2

grain boundary

cleavage crack

in grain 1

intersection of grain 1 cleavage

crack with grain boundary

cleavage crack

in grain 2

intersection of grain 2 cleavage

crack with grain boundary

grain 1

Figure 3. Schematic of the accommodation failure of the grain boundary to allow a complete separation of the model

due to a running cleavage crack. The region of the boundary that has to fail to allow for a complete separation is shown

hatched. This is a conventional interpretation of grain boundary accommodations failure, see e.g. [16].

in the adjacent grain. The grain boundary is marked as failed immediately, and after that no

other cleavage crack is permitted to cross this grain boundary. To this end the grain boundary

connectivity array is created at the beginning of the cleavage simulation. The array contains an

entry for each grain boundary, which is inact initially, and is updated to failed when crossed by a

cleavage crack. A failed grain boundary cannot be crossed by another cleavage crack.

Another problem is that of simulating the accommodation failure of the grain boundary, to

allow for a complete separation of the parts of the model. The region of accommodation failure is

shown hatched in Fig. 3. Again, the local nature of the CA approach makes this hard, because the

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(a) (b)

Figure 4. Two examples of simulated equiaxed polycrystalline microstructures, showing a cubic volume populated with (a)

40 grains, and (b) 40,960 grains. The scale is arbitrary. Gray scale (colour version online) denotes a unique grain number

that is linked with a randomly assigned rotation tensor, Rc. Hence if the spatial volumes in (a) and (b) are considered

identical, then the mean grain size in (b) is smaller than in (a). Alternatively, if the mean grain size is considered identical

in (a) and (b), then example (b) represents a much bigger spatial volume of material than (a).

knowledge of global geometrical quantities is required, speciﬁcally the relative spatial orientation

of the regions of the grain boundary with respect to the cracks on both sides of the boundary.

In the absence of this information, when a grain boundary cell is analysed, the immediate

neighbourhood information is insufﬁcient to decide whether the cell should fail or not.

6. Results

The model was implemented in modern Fortran 2008 [30]. The code is available under 2-

clause BSD licence from http://sourceforge.net/projects/cgpack/. The code uses coarrays for

portability and performance. It is designed to be highly scalable so that it can be used on high

performance computers. The code has shown good scalling up to 32,000 processors on HECToR,

the UK national supercomputer [31].

Fig. 4shows 2 examples of simulated equiaxed microstructures. In both cases a cubic volume

was modelled. In the ﬁrst case it was populated with 40 grains, and in the second example 40,960

grains were grown.

Fig. 5shows two predicted grain size histograms, with 5,120 grains, Fig. 5(a) and with

40,960 grains, Fig. 5(b). The value of this data is that it allows for a direct comparison with

the experimental measurement. Another expected, but still very important, result is that the

maximum grain size in the model is increasing with the model size. For the model with 5,120

grains the biggest grain is about 3.5 times bigger than the mean, Fig. 5(a), while for the model with

40,960 grains, the biggest grain is 4 times the mean size, Fig. 5(b). This observation is important

because cleavage is often thought of as the weakest link model, and the largest grains will have

the lowest toughness [32]. Hence the bigger the model, the higher the chances of representing an

extremely low fracture energy event.

Fig. 6shows two important predictions obtained with a 640 grain model of an equiaxed

microstructure with periodic (self-similar) boundary conditions. Fig. 6(a) shows the predicted

grain boundary mis-orientation distribution. The model prediction closely matches the theoretical

results of [33]. The maximum calculated value is 62.5◦, whereas the theoretical maximum is 62.8◦.

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(a) (b)

Figure 5. Grain size histograms from simulated equiaxed microstructures, showing (a) data from the model with 5,120

grains, and (b) data from the model with 40,960 grains, shown in Fig. 4(b). Note the increase in the relative size of the

biggest grain with increasing the number of grains in the model.

(a) (b)

Figure 6. Useful predictions from a 640 grain equiaxed microstructure model, showing (a) the histogram of predicted grain

boundary mis-orientations in an equiaxed microstructure. The shape, the peak (45◦) and the maximum angle (62.45◦)

match the theoretical predictions given e.g. in [33]; and (b) the histogram of the number of neighbouring grains, which

roughly follows the normal distribution, with the mean around 15 neighbours. Note that this prediction is not easy to

validate experimentally because most, if not all experimental measurements of microstructures are done on 2D slices,

and 3D information can be extrapolated only using some additional assumptions.

Fig. 6(b) shows the predicted distribution of the number of neighbouring grains. The mean value

is around 15 neighbours, whereas in 3D geometric models, with the popular 14-hedra grain shape

(the Kelvin polyhedron), each grain has 14 neighbours sharing a face [15,34]. This similarity is all

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(a) (b)

Figure 7. A model with 2 randomly oriented grains and a single cleavage micro-crack at a randomly chosen location

in grain 1, showing (a) Two {100} cleavage cracks in both grains, with grain cells removed for clarity, and (b) the same

superimposed with grain 1. Note an irregular grain boundary and irregular intersections of both cracks with the grain

boundary.

the more striking given that a grain in a cellular model has no deﬁned faces, edges or vertices.

This comparison adds validity to the 3D cellular automata grain modelling results.

Fig. 7shows an example of a model with 2 grains, with ﬁxed boundary conditions, in which

a single cleavage micro-crack, i.e. a cleavage nuclei cell, was assigned a random location in grain

1. The results show the state of the model after 70 cleavage iterations. Both cracks are on {100}

planes. Discretisation of a randomly oriented plane within a 26 cell neighbourhood is very coarse,

see Appendix A. This means that two planes of a different orientation, passing through the same

cell, might be discretised identically, unless their orientations are substantially different. In some

cases the mis-orientation angle must be 45◦for the planes to be discretised differently in a cellular

model. Appendix Agives more details.

The grain boundary, and the intersections of both cracks with the grain boundary are irregular,

i.e. hard to describe by global geometrical parameters, see Fig. 7. We note that there is a

school of thought that insists that fractal geometry, rather than differential geometry, is the only

meaningful approach when dealing with polycrystalline fracture surfaces [35–39]. Also note that

the grain boundary area between the two intersections with the cleavage cracks in Fig. 7(b) would

ultimately fail by some other mechanism, possibly ductile shear [15,16,22,27–29]. This modelling

result is consistent with the theoretical framework, see Fig. 3and [15,16].

Fig. 8shows an example of non-physical 1D cracks. This is an artifact of allowing only 2

neighbouring cells to be associated with the cleavage plane passing through the central cell in

Alg. 3. As shown in Appendix A, if t= 0.1733, then for any nsthere will always be at least 2

neighbouring cells on the cleavage plane. As the orientation of these 2 cells with respect to the

the central cell are always the same, as long as nsis constant, the only possible resulting crack

shape is 1D straight lines. An intuitive solution to this problem is to increase t"a little bit", to make

sure there are always at least 4 neighbouring cells on the cleavage plane. However, in this case

there are such nsdirections, for which the neighbouring cells considered to lie on the cleavage

plane are not on the same geometrical plane. Alg. 3then results in 3D cracks, which are equally

non-physical, see Fig. 9.

Fig. 10 shows the the grain boundary accommodation fracture for the two {110} cleavage

cracks. As described in Sec. 4, at present, we have no good physically sound local criteria to

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(a) (b)

Figure 8. An example of 1D cracks produced, when only 2 neighbouring cells are recognised as lying on the cleavage

plane. This is a rare but possible modelling artifact when the threshold, t= 0.1733. See section 4and Appendix Afor

more details. This ﬁxed boundary model included 2 grains. (a) The {100} crack is planar in grain 1, but the cracks in

grain 2 are discretised as 1D linear cracks. The grains are not shown for clarity. (b) Grain 1 and the grain boundary are

superimposed on the cracks. Note the irregularity of the grain boundary.

(a) (b)

Figure 9. Two views of the same nonphysical 3D crack. This is a rare but possible artifact of the 3D cellular model if the

threshold, t > 0.1733.

decide which parts of the grain boundary should fail to achieve accommodation for the two

cleavage cracks on either side of the boundary. Therefore the area of accommodation fracture

simply grows outward from the crack intersections with the boundary. The end product of this

process is that the whole of the grain boundary is fractured, which is not physical.

Putting all the bits of the model together, Fig. 11 shows cleavage propagation through a model

with 20 grains, after 300 cleavage iterations. Note that a signiﬁcant fraction of grains exhibit

multiple cracks on parallel planes. Since the model explicitly forbids more than one crack to

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(a) (b)

(c)

Figure 10. A model with two grains, showing (a) the two {110} cleavage cracks in two grains. Neither the grains nor the

grain boundary are shown for clarity; (b) same view as in (a) but with the fractured part of the grain boundary also shown,

and (c) a different view of the fractured grain boundary. Note that the fractured grain boundary area simply grows away

from the intersection locations with each iteration. The end result of this process is that the whole of the grain boundary

is fractured, which is not physically sound.

propagate across any grain boundary, this means that cracks propagates into these grains from

different boundaries. For example, refer to Fig. 11(c). A grain in the front bottom corner of the

model has 3 cleavage cracks. It is likely that one of these cracks propagated into this grain from

the grain to its top left, and the other two cracks propagated from the two grains to its right. Note

that this example did not include the feedback from the microstructure to the structural model,

i.e. there is was no reduction of tsin cracked grains. In practice, a cracked grain would receive a

dramatically reduced ts, making further cracks on the same crystallographic planes very unlikely.

7. Combining with ﬁnite elements for a multi-scale model

A complete CAFE (cellular automata ﬁnite element) multi-scale model was constructed by linking

the CA model with the ParaFEM ﬁnite element library [40,41]. Conventional localisation and

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(a) (b)

(c) (d)

Figure 11. A cleavage propagation example with 20 grains, showing: (a) grains and grain boundaries on the surface

of the model, (b) 3D view of grain boundaries in the interior of the model, (c) cleavage cracks superimposed on grain

boundaries, and (d) only cleavage cracks, with all other structures removed for clarity.

homogenisation strategies [18,26,42] can then be used to exchange the information between the FE

and the CA parts of the model. In this example the FE stress tensor was redistributed over the CA

according to the proximity of the cells to crack ﬂanks, to replicate the stress intensity dominated

stress ﬁelds. Cells closest to the crack ﬂanks receive much higher stress tensors than the FE value.

Cells sufﬁciently removed from crack ﬂanks receive lower stress tensors than the FE value. The

mean value of the CA stress tensors is equal to the FE stress tensor. The homogenisation is done

via scaling of the FE Young’s modulus according to the size of the cleavage crack. A representative

example is given below.

A10 ×10 ×10mm3cube of material is considered. The material model is linear elastic, with

the Young’s modulus of 200GPa. The boundary conditions are: u1(x1= 0) = 0,u2(x2= 0) = 0,

u3(x3= 0) = 0. Distributed loading is applied along x3on a small element of surface x3= 10,

with the total applied force of 1kN. The mean grain diameter is taken as high as 1mm, primarily

to ease the visualisation, because large grains result in large trans-granular cracks. The complete

model thus implements 1,000 randomly oriented grains. A single micro-crack is introduced at

x1=x2= 0,x3= 5mm. The results in Fig. 12 show the deformed ﬁnite element mesh and the

crack patterns produced in three runs of the model.

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(a) (b)

(c) (d)

Figure 12. Three runs of a CAFE model in a steel-like material with large grains, showing: (a) the deformed ﬁnite element

model with the distortion artiﬁcially magniﬁed for clarity, (b), (c) and (d) clusters of cracks on 100 planes (light gray or

yellow online) and on 110 planes (dark gray or green online) forming a large main crack that is macroscopically normal to

the maximum principal stress direction in that region. Each particular crack scenario is unique. Analysis of a statistically

signiﬁcant number of runs of this CAFE model can lead to the prediction of scatter in cleavage crack propagation results,

such as the fracture energy.

8. Concluding remarks and future work

The 3D cellular automata framework, designed and implemented as described, has been shown to

be capable of simulating progressive quasi-cleavage propagation though a polycrystal. The model

is based on the fracture stress criterion and is made mesh independent by including a length scale.

The model takes into account random orientations of individual crystals. The immediate next step

is to add orientation dependent grain boundary energies [43,44]. Energies of the cleavage cracks

might at ﬁrst be taken equal to free surface energies, γhkl. Then the total energies of fracture can be

calculated simply by summing the number of cells of cleavage states, chkl , and of grain boundary

fracture states, cgb, multiplied by their energies. In this respect 1D and 3D crack artifacts present

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c

1

2

3

eee

e

ee

e

e

e

e

e

ff

f

e

cc

cc c

c

c

c

c

c

c

Figure 13. This 3×3×3cluster of 27 cubic cells represents the 26-cell nearest neighbourhood of the central cell. 6

neighbours share a face ("f"), 12 neighbours share an edge ("e") and 8 neighbours share a corner ("c"). A Cartesian CS

is a natural choice for this model.

a major obstacle. 1D line crack artifacts will result in artiﬁcially low energy, and 3D solid body

crack artifacts will dramatically increase the fracture energy. Hence, it is necessary to resolve the

issue on non-physical 1D and 3D cracks if realistic fracture energy prediction is aimed for.

Data accessibility. All data is accessible from http://sourceforge.net/projects/cgpack/.

Disclaimer. AS contributed to all sections. LM contributed to the creation of the CAFE model, Sec. 7. Both

authors gave ﬁnal approval for publication.

Acknowledgements. This work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk).

This work also used HECToR, the UK’s national high-performance computing service, which was provided

by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd. HECToR was funded by

the Ofﬁce of Science and Technology through EPSRC’s High End Computing Programme. This work

also used the computational facilities of the Advanced Computing Research Centre, the University of

Bristol (http://www.bris.ac.uk/acrc/) and the N8HPC Service, funded by the N8 consortium and EPSRC

EP/K000225/1, coordinated by the Universities of Leeds and Manchester (http://n8hpc.org.uk).

Conﬂict of interests. We have no competing interests.

A. Resolving arbitrary plane within a 3D CA cell neighbourhood

In 3D CA a cell has 26 nearest neighbours: 6 share a common face, 12 share a common edge and

8 share a common corner (vertex), see Fig. 13. This 3D space partitioning scheme is popular in

CA because it is particularly easy to visualise and implement in a computer program, simply by

using a three-dimensional array.

A Cartesian CS, with arbitrary origin, is aligned with cell axes. In single or poly-crystal models

a crystal is represented by a cluster of cells. A crystal can have arbitrary orientation with respect

to the global (cellular) CS. In contrast the cell neighbourhood can be discretised only into 26

directions, one for each neighbouring cell. Thus a problem arises when crystallographic directions

or planes have to be resolved within discrete cellular models.

A problem of particular importance is trans-granular cleavage, where material is separated

along a crystallographic plane, e.g. {100} or {110} plane in bcc crystals. To simulate cleavage in a

cellular model one must answer this question for every cell: which neighbouring cells are on the

cleavage plane?

Let’s illustrate the problem on a two-dimensional example, see Fig. 14. Let nbe the unit vector

normal to some cleavage plane, and ei, i = 1 ...8, is a unit vector connecting a cell with one of

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n

cp

F F

n

cp

F

F

i

cp

?

?n

?

?

e

ei

(a) (b) (c)

Figure 14. Two-dimensional illustration of the problem of deciding which neighbouring cells are on the cleavage plane

(cp, deﬁned by its normal, n). When nis aligned with one of the cell direction vectors, ei, the answer is clear because

the cleavage plane passes directly through the centres of two neighbouring cells ("F" for failed), as in cases (a) and (b).

In general, no eivectors are normal to n, and it is unclear which, if any, cells (marked "?") should be considered to lie on

the cleavage plane. Possible answers are zero, two or four.

its 8 neighbours. If n·ei= 0, for any ithen the question of which neighbouring cells are on the

cleavage plane is trivial, see Fig. 14(a) and (b). In general case, the answer to this question is not

clear. To help us answer it in general case, we pose this problem: ﬁnd nthat would maximise

the minimum angle between the cleavage plane and each of ei. This problem is solved by

ﬁnding maxn(mini=1...8|n·ei|), which we denote z. Looking at Fig. 14(c) this ‘MaxMin’ criterion

corresponds to the cleavage plane at equal angles to two adjacent cell direction vectors. Therefore

the angle is θ=π/8 = 22.5◦,n= (cos(θ+mπ

4),sin(θ+mπ

4)) for any integer mand z≈0.38.

One can then construct a cleavage criterion based on θor z. One possible criterion can be that

a neighbouring cell iis considered to lie on the cleavage plane if |n·ei|≤ z.

In three-dimensional case this analysis is more complex. If the centre of the central cell in

Fig. 13 is given coordinates (0,0,0), then the directions to centres of the neighbouring cells are

given by the following 26 unit vectors, ei, i = 1 ...26. There are six common face cells: (±1,0,0),

(0,±1,0),(0,0,±1); twelve common edge cells: (±1/√2,±1/√2,0),(0,±1/√2,±1/√2),

(±1/√2,0,±1/√2), and eight corner cells: (±1/√3,±1/√3,±1/√3). Consider an arbitrary unit

vector n. We formulate two geometric problems.

Problem 1: ﬁnd maxn(mini=1...26 |n·ei|)for all possible n.

Problem 2: ﬁnd minn(maxi=1...26 |n·ei|)for all possible n.

Although only problem 1 is related to the cleavage analysis, problem 2 is the opposite of

problem 1, so we solve it as well.

(a) Problem 1

Fig. 15 shows that the model has 3 planes of symmetry. It is therefore sufﬁcient to choose nonly

from one coordinate corner. For simplicity, we choose the corner with all coordinate axes positive.

If n= (n1, n2, n3)solves problem 1, then so do the eight vectors (±n1,±n2,±n3). Furthermore,

because the problem is symmetrical with any permutation of the coordinate axes, a further six

vectors are obtained from each nby permutation of its components. In total there are 48 vectors

nwhich solve problem 1.

Refer to Fig. 15. The origin is at the central cell and the unit vectors show locations of the

centres of some of the neighbouring cells. Due to three symmetry planes the space can be divided

as left/right, top/bottom and front/back. In this terminology nis in the right top front corner.

The problem is that of ﬁnding nthat makes biggest angles (but < π/2) with three eivectors. The

ﬁrst vector can be chosen arbitrarily. We choose e1= (0,0,1). To maximise the angles between

n, the other two vectors must lie in other corners. In the right bottom front corner, vector

making the biggest angle with nis either (0,1/√2,−1/√2) or (1/√2,−1/√2,0). We choose the

ﬁrst of these: e2= (0,1/√2,−1/√2). This leaves the third vector from left bottom front corner,

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n

0,

1

√2,−1

√2

(0,0,1)

1

√2,

1

√2,0

1

√3,−1

√3,−1

√3

1

√3,

1

√3,

1

√3

(0,1,0)

(1,0,0)

Figure 15. Schematic illustration of cell direction vectors, ei, i = 1 . . . 26. The central cell is at the origin. Unit vectors

connect the central cell with the centres of its neighbours. Only vectors used in the solution of the problems are shown.

e3= (1/√3,−1/√3,−1/√3). All other vectors in the front corners are either reﬂections of the

chosen vectors about the symmetry planes, or make smaller angles with n, or angles > π/2with

n. The problem is then reduced to solving the following equations.

n·e1=n·e2;n·e2=n·e3(A 1)

for n1and n2;n2

3= 1 −n2

1−n2

2. We omit the algebra and give the solution:

n1=ρn2;n2= 22 + √2

3 + 2√2+ρ2!−1/2

where ρ=2 + √3 + √2

1 + √2(A 2)

This gives n≈(0.8916,0.4183,0.1733) or any of its 47 equivalents constructed as discussed

above. Solution to problem 1 is then

max

n

( min

i=1...26 |n·ei|)≈0.1733.(A 3)

(b) Problem 2

Again refer to Fig. 15. We are now looking for nsuch that it makes the biggest possible angles

(but < π/2) with closest vectors e. We choose e1= (1,0,0). Its two closest vectors in the same

corner are: (1/√2,1/√2,0) or (1/√2,0,1/√2). We choose the ﬁrst of these: e2= (1/√2,1/√2,0).

The only choice for the third closest vector in the same corner is e3= (1/√3,1/√3,1/√3). As in

problem 1, we then solve Eqns. (A 1). The answer is:

n2

1= 9 −2(√6 + √2); n2=n1(√2−1) (A 4)

This gives n≈(0.8865,0.3672,0.2817) or any of its 47 equivalents constructed as discussed

above. Solution to problem 2 is then:

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min

n

( max

i=1...26 |n·ei|)≈0.8865.(A 5)

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