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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
00 0
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neighbourhood
cell
CA array grid
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000 00 0
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0000
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0
(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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