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Three-dimensional cellular automata modelling of cleavage propagation across crystal boundaries in polycrystalline microstructures


Abstract and Figures

A three-dimensional cellular automata (CA) 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 through 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 CA model can be linked to a finite-element model to produce a multi-scale fracture framework.
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Subject Areas:
mechanical engineering, materials
cellular automata, finite elements,
cleavage, polycrystals, grain
boundary, three-dimensional, Fortran,
Author for correspondence:
A. Shterenlikht
Three-dimensional cellular
automata modelling of
cleavage propagation across
crystal boundaries in
A. Shterenlikht1, L. Margetts2
1Mech Eng Dept, The University of Bristol, Bristol BS8
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
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 [13]. 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
[1013] 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 [1517].
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
00 0
CA array grid
0 0 0 0
000 00 00
00 0
0 0
0 0
1 1 1 1
1 1 1
1 1
1 1
240 2424
2424 24 24 24
24 24
24 24
24 24
24 24
24 24
1 1 1 1
1 1 1
1 1
1 1
240 2424
2424 24 24 24
24 24
24 24
24 24
24 24
24 24
0 0 0
000 00 0
00 0
0 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 < 105i.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·Rs, where RsRc(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= 103J/m2) and a0= 1.37 ×1010m. 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 [2426].
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:
hkl = max
hkl (thkl) = max
hkl ·tc·ni
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
100 = maxi=1...24(ni
100 ·tc·ni
110 = maxi=1...24(ni
110 ·tc·ni
111 = maxi=1...24(ni
111 ·tc·ni
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
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
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
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
6 Proc R Soc A 0000000
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. 1,2ns,s, flag
gold =gi
if flag = true then
run Alg. 3change 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
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
7 Proc R Soc A 0000000
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
8 Proc R Soc A 0000000
(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 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
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 45for 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 [3539]. 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 [15,16,22,2729]. 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
11 Proc R Soc A 0000000
(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)
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
13 Proc R Soc A 0000000
(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.
14 Proc R Soc A 0000000
(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
15 Proc R Soc A 0000000
cc 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
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 (
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 ( and the N8HPC Service, funded by the N8 consortium and EPSRC
EP/K000225/1, coordinated by the Universities of Leeds and Manchester (
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
16 Proc R Soc A 0000000
(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)) for any integer mand z0.38.
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,
17 Proc R Soc A 0000000
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
2. We omit the algebra and give the solution:
n1=ρn2;n2= 22 + 2
3 + 22+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
( 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:
1= 9 2(6 + 2); n2=n1(21) (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:
18 Proc R Soc A 0000000
( max
i=1...26 |n·ei|)0.8865.(A 5)
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... According to some recent studies, 3D-HEX CA can be used to simulate various phenomena, such as cleavage propagation across crystal boundaries [16], a coupled hydrogen porosity and microstructure during the solidification of ternary aluminum alloys [17,18], and grain refinement during the severe plastic deformation of micro-alloyed steel. These simulations can help understand the mechanisms and effects of different factors on the microstructure evolution and properties of materials [1][2][3][4]. ...
... Furthermore, some advantages of 3D-HEX CA over other methods, such as phase-field models, front tracking methods, and vertex models, are that they are simpler, faster, more flexible, and more scalable [19][20][21]. They can also capture complex features, such as grain boundaries, crack fronts, dislocation structures, and phase transformations, with a high accuracy [16,17]. For example, 3D-HEX CA can simulate the evolution of grain boundaries in polycrystalline materials under different driving forces and boundary conditions [16]. ...
... They can also capture complex features, such as grain boundaries, crack fronts, dislocation structures, and phase transformations, with a high accuracy [16,17]. For example, 3D-HEX CA can simulate the evolution of grain boundaries in polycrystalline materials under different driving forces and boundary conditions [16]. 3D-HEX CA can model the propagation of crack fronts in brittle materials with arbitrary shapes and orientations [22]. ...
Full-text available
Cellular automata (CA) modeling is a powerful and efficient tool for simulating the dynamic evolution of polycrystalline microstructures in modern materials and metallurgy studies, such as solidification, plastic deformation and recrystallization. We propose a novel model to calculate the shape factor of grains in three-dimensional hexagonal grid (3D-HEX) CA, which overcomes the disadvantages of 3D-HEX CA, such as complex algorithms and a long computation time. The shape factor is a quantitative measure of grain morphology based on the ratio of the surface area of the grain to its volume-equivalent-sphere and volume-equivalent-chain. It indicates how the shape of a grain or phase affects its mechanical properties, such as stiffness, deformation and fracture. Our model can easily calculate the shape factor for any grain by counting its surface cells and volume cells. We test our model on 1000 grains with different shapes (equiaxed, irregular and chain-like) by Monte Carlo (MC) methods. MC methods evaluate the validity of a calculation model by comparing the simulated outcomes with the observed or expected outcomes. The results show that our model can accurately describe the grain morphology and has a good comparability and generality.
... However, three-dimensional grain growth [6][7][8][9] can reflect the internal structure of materials more intuitively than two-dimensional grain growth. At present, some researchers are gradually exploring threedimensional space [10][11][12][13] on the basis of two-dimensional research. Wang et al. [14] considered the growth mechanism driven by curvature and thermodynamics and used 3D-CA model to simulate the normal austenite grain growth process. ...
... 11: e metallographic structure of the longitudinal section at different temperatures.Advances in Materials Science and Engineering ...
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Based on the thermodynamic conversion mechanism and energy transition principle, a three-dimensional cellular automata model of grain growth is established from the aspects of grain orientation, grain size distribution, grain growth kinetics, and grain topology. Also, the effect of temperature on the three-dimensional grain growth process of AZ31 magnesium alloy is analyzed. The results show that the normal growth of three-dimensional grains satisfies the Aboav-weaire equation, the average number of grain planes is between 12 and 14 at 420°C and 2000 CAS, and the maximum number of grain planes is more than 40. Grains of different sizes are distributed normally at different times, most of which are grains with the ratio of grain diameter to average grain diameter R/Rm ≈ 1.0, which meets the minimum energy criterion of grain evolution. The grain of AZ31 magnesium alloy increases in size with the increase of temperature, and the number of grains decreases with the increase in time. The angle between the two-dimensional slices of three-dimensional grains is approximately 120°, which is consistent with that of the traditional two-dimensional cellular automata. The relative error of grain size before and after heat preservation is in the range of 0.1–0.6 μm, which indicates that the 3D cellular automata can accurately simulate the heat preservation process of AZ31 magnesium alloy.
... Pioneering work on CAFE modelling for fracture was performed by [15] where they modelled fracture at the ductile-brittle transition regime in a thermomechanically controlled rolled steel. Innovations in high performance computing such as Co-array Fortran [16]coupled with the message passing interface [17] have since been used to reduce runtimes for the modelling of cleavage fracture using CAFE, enabling computationally intensive processes such as dynamic loading to be tackled [18]. Several other researchers have focused their efforts not only on fracture related topics, but also on various other subjects like oxide cracking [19], dynamic re-crystallization [20] and friction stir welding [21].A multi-scale modelling technique allows us to combine two or more different length and time scales which are often dissimilar in terms of their characterization due to the change in the scale [22]. ...
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Fracture is an inherently statistical phenomenon as it is a function of micro-structural heterogeneities such as distributed defects and inclusions. This is evidenced by scatter in the toughness of seemingly identical specimens. Therefore, deterministic approaches do not give full picture of scatter in fracture behaviour. More suitable probabilistic methods have been devised to describe the scatter associated with fracture. While the probabilistic approaches provide a sound scientific basis for capturing the scatter in the fracture data through assuming a probability for the presence of fracture initiators, their microstructurally agnostic assumptions can limit their predictive capability. This is because there is no information on the microstructure such as grain size and morphology, texture, and other important features considered in them. An alternative class of models which take into account the distribution of toughness is cellular automata finite element models (CAFE). CAFE models are stronger in simulating the scatter in the fracture data through their ability to represent the microstructure although so far, they have been limited to fully brittle or quasi-brittle materials. In addition, the CAFE models they are computationally expensive, and their running time can be prohibitive for their application to large scale engineering components thus reducing their appeal. In this study, a CAFE model was developed to take advantage of the microstructural fidelity of CAFE but presented within the context of a probabilistic fracture approach. The CAFE based model calculates the macroscopic strain from the continuum FE model. The strain is then used to load a model which is defined in the cellular automata space. The CAFE model then simulates the initiation and propagation of fracture in the microstructure to fully capture the heterogeneity of the material at the lower length scale. The critical stress acting normal to the cleavage plane of each grain is calculated in the CAFE model and used to decide the onset of cracking in a probabilistic manner; the stress depends on the orientation of the grain in which microcrack initiates as well as depending on the orientation of the surrounding grains. To evaluate its performance, the model was calibrated using a set of experimental fracture toughness data and the results of its prediction were compared with an intendent and separate set of warm prestress experiments of the same material. Good agreement between the prediction and the experiment of the second set was observed giving confidence in the model.
... These CA models, initially developed in 2D, was later extended to 3D and coupled with finite element (FE) heat flow calculations resulting in the so-called Cellular Automaton-Finite Element (CAFE) models [43,44,91,[141][142][143][144]. They are widely applied in investment casting [142], directional solidification [91] and an extensive range of microstructure evolution phenomenon including dendrites, micro-segregation, defects in different alloy systems [22,[145][146][147][148][149][150][151][152]. Also, different kinds of defect formation in casting production process during the last stage of solidification are controlled by CA simulations to achieve the desired microstructure [145]. ...
... These CA models, initially developed in 2D, was later extended to 3D and coupled with finite element (FE) heat flow calculations resulting in the socalled Cellular Automaton-Finite Element (CAFE) models [41,87,91,[138][139][140][141]. They are widely applied in investment casting [139], directional solidification [87] and an extensive range of microstructure evolution phenomenon including dendrites, micro-segregation, defects in different alloy systems [22,[142][143][144][145][146][147][148][149]. Also, different kinds of defect formation in casting production process during the last stage of solidification are controlled by CA simulations to achieve the desired microstructure [142]. ...
Metal additive manufacturing (MAM) technology is now changing the pattern of the high-end manufacturing industry, among which microstructure simulation gradually shows its importance and attracts many research interests. As the simulation targets, this paper summarizes the unique microstructure characteristics in MAM fabricated parts, Ti-6Al-4V as an example. Further discussions are focused on the development of solidification microstructure simulation methods as well as their capacity and applicability on MAM. Finally, the difficulties and suggested future research topics in MAM microstructure simulation are addressed.
... Then, the strain/stress concentration will occur on the closing neighborhood of 'dead' cell since it loses the load-bearing capacity. A framework (Shterenlikht, 2003;Shterenlikht and Margetts, 2015) has been established to locate such a closing neighborhood around the 'dead' cell. The local concentration factors, ‫ܥ‬ for ductile CA array 140 ...
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The effect of residual stress on fracture of materials or structures has been widely studied. However, its influence on ductile-to-brittle transition (DBT), a crucial phenomenon of structural materials, has rarely been investigated so far. In the present study, employing the eigenstrain method residual stresses are introduced into a bi-material specimen, where two configurations of crack and interface, e.g., one with interface perpendicular and one parallel to the crack extension, are designed to study the influence of residual stress. The DBT of the bi-material specimen in the presence of residual stresses is numerically studied by using the CAFE method where temperature dependent surface energy is implemented to calculate absorbed energy of Charpy impact testing specimen. It is found that residual stress generated in the two configurations affect the DBT in a similar manner. The DBT curves generally shift to higher temperature due to the decrease of absorbed energy with the increase of residual stress. It is found that the decrease of absorbed energy in both configurations is caused by the additional constraint on the notch root, which is induced by the residual stress and can facilitate the fracture.
This work presents Rousselier´s damage model parameters effect and their physical meaning on Charpy fracture propagation curves. Therefore, instrumented Charpy tests were performed at room test temperature to measure the load-displacement curve. The parameters were measured from a Grade A ship plate steel, employed for the construction of merchant ships. The effect of Rousselier´s model parameter was done by performing cellular automata finite element (CAFE) modelling, where Rousselier’s damage model was coded, and therefore Rousselier´s model parameters were incorporated using random number generators in the ductile arrays of cells, using Weibull distributions. Consequently, in each CAFE simulation, the model evaluates random values of Rousselier´s damage model parameters performing a more physically based modelling. The results showed that the present CAFE modelling was able to reproduce the hardening and fracture propagation regions of instrumented Charpy data. Furthermore, the present work showed a suitable Rousselier´s damage model parameters calibration procedure with Charpy data, and how each Rousselier´s model parameter can affect the hardening and fracture propagations regions when they are not properly calibrated, producing unrealistic results. Additionally, it can be observed that the present results can be used as a template for a better calibration of Rousselier´s damage model parameters in CAFE modelling.
With the increasing use of metal semi-solid forming technology in the production of high-quality copper alloy parts, foundational research on the robust preparation of copper alloy semi-solid slurry as key components and foundation of copper alloy semi-solid parts is critical. Therefore, based on the explosive nucleation theory and the 3D-CAFE method, the effects of the length of the melt-constrained flow-induced nucleation channel (185, 250, 315, and 365 mm), which is the core component of self-designed and developed fully enclosed melt-constrained flow-induced nucleation metal semi-solid slurry preparation device, on the temperature field distribution and microstructure of CuSn10P1 alloy during the of semi-solid slurry preparation was studied. The research results show that the numerical simulation results agreed with the experimental results. As the melt-constrained flow-induced nucleation channel length was increased from 185 to 365 mm, the chilling effect of the channel on the CuSn10P1 melt initially increased and then decreased. When the length of the melt-constrained flow-induced channel reached 315 mm, the most uniform temperature distribution, most homogeneous microstructure, and finest microstructure were obtained. Moreover, the grains had an average equivalent diameter of only 30.43 μm and with the distribution of 633/mm2.
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The conventional micromechanical approaches today are still not able to properly predict the ductile-to-brittle transition (DBT) of steels because of their inability to consider the co-operating ductile fracture and cleavage mechanisms in the transition region, and simultaneously to incorporate the inherent complexity of microstructures. In this study, a complete methodology with coupled cellular automata finite element method (CAFE) and multi-barrier microcrack propagation models is presented to advance the prediction of DBT. The methodology contains three key elements: (i) a multiscale CAFE modelling approach to realize the competition between ductile damage and cleavage fracture and embrace the probabilistic nature of microstructures, (ii) a continuum approach to estimate the effective surface energy for a microcrack to penetrate over particle/matrix interface, and (iii) a method to calculate the effective surface energy for the microcrack to propagate across grain boundaries. The prediction of DBT therefore needs only (1) the stress-strain curves tested at different temperatures, (2) the activation energy for DBT, (3) the ratio between the size of cleavage facets and cleavage-initiating defects, and (4) key statistical distributions of the given microstructures. The proposed methodology can accurately reproduce the experimental DBT curve of a low-carbon ultrahigh-strength steel.
This paper presents a novel computational framework for modelling multiscale fracture that can be used to solve engineering problems subject to dynamic loading. The framework simulates, mechanistically, at the mesoscale, the physical processes that lead to brittle fracture. A homogenisation step is used to translate the accumulation of damage from the mesoscale to the macroscale (as a reduced stiffness in the corresponding region of the structure). In order to achieve this, the multiscale framework couples together two open source Fortran packages; the macroscale ParaFEM with the mesoscale CASUP. ParaFEM is a highly parallel finite element analysis library used to model structures at the continuum scale. CASUP is a package that uses cellular automata to simulate brittle fracture in polycrystalline materials. A simple test problem involving a vibrating cantilever beam is used to demonstrate the simulation of dynamic cyclic loading, leading to brittle cracking. In the cellular automata software, there are a range of parameters that can be adjusted, such as the fracture energy and grain size. These are explored to demonstrate how they might affect the predicted structural integrity of the cantilever beam. Parallel performance is investigated using a Cray XC30 supercomputer, showing that the software can make efficient use of tens of thousands of cores. This paper highlights that modelling the physical mechanisms that lead to damage and plasticity could be an attractive alternative to phenomenological constitutive models. This work will be of interest to researchers and practitioners needing more precise predictions or a better understanding of damage propagation under cyclic or impact loading. With further development, this type of framework will enable the insilico design and evaluation of new material microstructures; leading to improved performance of components and devices subject to extreme operating conditions.
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In this paper a new method for the generation and meshing of arbitrarily shaped three-dimensional polycrys-talline models is presented. The discretization is based on Voronoi tessellation, which is shown to be statistically repre-sentative of the microstructure of polycrystalline materials. An original approach is introduced to define any possible (concave or convex) shape of the final domain, independently from the initial configuration of the aggregate. Firstly the Voronoi cells are cropped along arbitrarily oriented planes to generate a convex domain, and then an arbitrary num-ber of cuts are performed along planar surfaces to generate the final concave domain. Finally the grains are discretised separately and assembled together to create a finite element model. Several examples are presented to show the capability of generated virtual samples to simulate the behaviour of real polycrystalline materials. The macroscopic elastic properties of polycrystals consisting of anisotropic (trigonal) grains and the stress intensity factor at the tip of a sharp notch are eval-uated and compared both with analytical calculations and experimental evidences, showing excellent agreement.
This book provides a self-contained introduction to cellular automata and lattice Boltzmann techniques. Beginning with a chapter introducing the basic concepts of this developing field, a second chapter describes methods used in cellular automata modeling. Following chapters discuss the statistical mechanics of lattice gases, diffusion phenomena, reaction-diffusion processes and non-equilibrium phase transitions. A final chapter looks at other models and applications, such as wave propagation and multiparticle fluids. With a pedagogic approach, the volume focuses on the use of cellular automata in the framework of equilibrium and non-equilibrium statistical physics. It also emphasises application-oriented problems such as fluid dynamics and pattern formation. The book contains many examples and problems. A glossary and a detailed bibliography are also included. This will be a valuable book for graduate students and researchers working in statistical physics, solid state physics, chemical physics and computer science.
Three-dimensional models with irregular grain geometries and appropriate physical properties are needed to investigate fracture in polycrystalline metals and alloys. Creating such models is challenging but achievable using a two-stage process, suitable for any polycrystal. The processes described in this paper are illustrated by examples of brittle fracture in ferritic steel, zinc and nickel. The predicted crack path in a model is compared with the grain boundary fracture seen in three point bend specimens of nickel embrittled by sulphur.
Volume 1: Structural integrity assessment - examples and case studies, (I. Milne et al). Volume 2: Fundamental theories and mechanisms of failure, (B. Karihaloo, W.G. Knauss). Volume 3: Numerical and computational methods, (R. de Borst, H.A Mang). Volume 4: Cyclic loading and fatigue, (R.O. Ritchie, Y. Murakami). Volume 5: Creep and high-temperature failure, (A. Saxena, H.K.D.H. Bhadeshia). Volume 6: Environmentally-assisted fracture, (J. Petit, P. Scott). Volume 7: Practical failure assessment methods, (R.A. Ainsworth, K-H Schwalbe). Volume 8: Interfacial and nanoscale failure, (W. Gerberich, W. Yang). Volume 9: Bioengineering, (Y-W Mai, S-H Teoh). Volume 10: Index volume.
Theoretical density functions are obtained for the angle of disorientation (the least angle of rotation required to rotate a cube into a standard orientation) and for MinΛ100〉 (the least of the nine acute angles between the edges of a cube and the edges of a fixed reference cube). These density functions and their cumulative distribution functions have been evaluated numerically.