# MCA Model for Simulating the Failure of Microinhomogeneous Materials

Authors: Yunliang Tan; Guirong Teng; Haitao Li
Journal: Journal of Nanomaterials
 (2008)

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## Abstract

A special cellular automaton (CA) model called mechanical cellular automaton (MCA) is built, in which inhomogeneity of material cell can be evaluated by Weibull's distribution function. In MCA, such material physical parameters as energy or strength and anisotropy defined by the different transferring energies along different directions are adopted. To simulate a crack or a weak zone, a cluster of weak energy cells are disposed. By the MCA, examples of failure evolving process of material containing one crack, two parallel cracks, and two cracks by bridge conjunction are simulated. The results show for inhomogeneous materials that their failure behaviors are much different from homogeneous ones, self-organization must be considered, and traditional homogeneous mechanics cannot be applied by copying word to word.


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## Body

## 1. Introduction

As  an  efficient method  for self-organization, cellular
automaton is proposed by von Neumann and Burks [1], and is gradually developed into a
dynamical evolving method in statistical mechanics [2].  It is considered as a
powerful tool to analyze evolutionary theory of life game [3]. Considering the complexity
of a system, Wolfram has made great contribution to computation of CA model [4, 5].Based on the rationale
of CA, the effects of spatial heterogeneities of environment and/or asymmetries
in the interactions among the individuals are analyzed [6] and applied in
material science. For example, particle-size distributions (PSDs) of soil
samples are analyzed by fractal methods and different scaling domains,
characterizing different ranges of particle sizes, are identified  based on CA. Modelling and controlling of
microstructure as well as grain size during thermal processing operations are
essential for tailoring the properties of polycrystalline materials; to investigate
them, a cellular automaton model, with advanced features for continuous
tracking of grain boundary and precise estimation of curvature, is developed [7],
in which namely parabolic growth behavior and the time invariant nature of
grain size distributions are
accurately described by Weibull's distribution function. It is shown that the predicted
topological features during the grain growth closely resemble experimental
observations in succinonitrile polycrystals.Also, because of
the parallel evolving function of CA, it is a useful technique for improving
the classical numerical method. For example, reversible computing is a paradigm,
in which it can reflect physical reversibility [8]. A new cellular automaton
(CA) model which reproduces isotropic time-evolution patterns observed in the Belousov-Zhabotinsky
reaction is proposed by Nishiyama et al. [9]. Lattice Boltzmann method, finite
element method, and cellular automata are coupled by Kwon and Hosoglu [10], and are tested
by wave propagation problems in 1D and 2D [11]. A three-dimensional cellular automaton
(CA) model is coupled with finite-element (FE) heat flow calculation in order to
predict the solidification grain structures in a Ni-base superalloy [12], in
which Gaussian distribution of nucleation sites is adopted and the Kurz-Giovanola-Trivedi
model is extended to multicomponent alloys to account for the growth kinetics
of dendrite tip. Much more, a cellular automata finite element (CAFE) model is developed
to simulate the strain localization phenomena that occur during various
plastometric tests, as
well as real industrial processes [13].Compared with
other materials, rock mass is one kind of most complicated materials in the
world, because of discontinuity, rheology, anisotropy, inhomogeneity, and so
on. Generally, rock mass has
obvious scale-feature behavior, and it can be divided into macro- and microscales
simply.In macroscale, a
commonly used model for fault-tolerant computation based on cellular automata is
built by McCann and Pippenger [14], in
which the degree (the number of neighboring cells
on which the state transition function depends) needed to achieve fault
tolerance is concerned when the fault rate is high (nearly 1/2), both some
traditional transient fault models (where faults occur independently in time
and space) and a recently introduced combined fault model including
manufacturing faults (which occur independently in space, but which affect
cells for all time) are considered.
Earthquake can be defined as a spatially extended dissipative dynamic system
that naturally  evolves into a critical state with no characteristic time or
length scale. A two-dimensional CA model capable of reproducing some prominent
features of earthquake data is presented [15]. Else, a proposed model with
continuous state and discrete time, comprising cell-charges and aiming at
simulating earthquake activity with the usage of potentials, is proposed by Georgoudas
et al. [16]. It is tested as well as calibrated using the recorded events of
the area in Northeastern Greece, centred at Xanthi, Thrace
as initial conditions. The simulation results are found in good quantitative
and qualitative agreement with the Gutenberge Richter (GR) scaling relations.In microscale, a
new theory of physical cellular automaton (PCA) for simulating the failure
evolving process of microheterogeneous materials is first set up by Tan et al. [17]
according to the basic law of transferring energy. PCA breaks through the
localization of traditional cellular automaton, and it is shown that the simulating
results by PCA can agree with the laboratory results perfectly. By using PCA
model, the anisotropic and discontinuous rock mass failure process and
corresponding acoustic emission (AE) are effectively simulated [18, 19]. Much
more, a PCA model considering elastoplastic property of rock has been developed
by Feng et al. [20] to simulate the 2D heterogeneous rock specimens under
uniaxial cyclic load compression test, and the influences of inhomogeneity,
softening coefficients, specimen sizes, and spatial distributions of material
parameters.In this paper,
the failure evolving processes of inhomogeneous rock containing single crack,
discontinuous cracks bridge, and double parallel cracks are simulated by
building a special CA model called mechanical cellular automaton (MCA).


## 2. Model Descriptions

### 2.1. Basic Model of CA

In a cellular automaton (CA), there are identical elements,
usually located in a regular array that can be one-dimensional,
two-dimensional, or even higher dimensional. The update rules are as follows: the
elements or cells of a CA are spread over the nodes of a rectangular grid, each
of them capable to be in one of a finite set of possible states. The neighborhood
of each cell can be defined as the set of the four nearest neighbors (North,
South, East and West), the set of the eight nearest neighbors (N, NE, E, SE, S,
SW, W, NW), or other bigger neighborhoods.Conventional  CA
models employ spatial lattices; each lattice element takes one of a set of
discrete values. The lattice evolves over discrete time steps, with element
values changing according to rules relating to the surrounding elements. CA
models are typically used to investigate “complex systems" [5], which exhibit
extreme sensitivity to initial conditions, boundary conditions, history and
element properties, and may have many different outcomes due to the existence
of multiple local equilibria. Thus, a cellular automaton is a discrete
dynamical system, and is composed of cells, cell lattices, neighbor rules, and
so on. The state of a cell is determined only by the states of itself and its neighbors
at previous step. CA has some advantages of time evolution, space
discretization, locality and parallelization, and so on. In order to build a
cellular automaton, the relationships between model components are needed to be clear (Figure 1).Figure 1
The relation between CA components.


### 2.2. MCA Model for Microheterogeneous Materials

Many kinds of material
(e.g., rock, concrete, etc.) are typical microheterogeneous ones; damages may
occur firstly in weak zones under certain load conditions. From their initial
damage to regional fracture, it is a typical self-organization process. As a
dynamical evolving technique, CA provides a base for simulating the fracture
process of microheterogeneous materials in methodology, because CA is a kind of
mathematical model to simulate the self-evolving process of a system caused by element interacts one
to another nonlinearly in it under a random initial condition [1], but
traditional CA  cannot simulate microscale failure evolving process effectively in
physical meaning because of the complexity of microheterogeneous properties of rock-like
materials. After the equivalent physical and mechanical properties for all cells
are given in a form of energy, the rock failure process as well as nonlinear
dynamic behavior scan can be
simulated by transformation of energy from one cell to the others, based
on the basic evolving rules, and thus it is called mechanical cellular automaton
(MCA).In MCA, a cell
is defined by mapping a real minor homogenous region inner material. As shown
in Figure2, a material specimen like rock can be divided into N×N elements or
grids in terms of homogeneity of material, and every element is considered as one
cell. According to the rationale of self-similarity of microstructure of rock,
enough cells can approach the real microstructure of rock equivalently.Cell grids.
(a)
Microstructure of rock(b)
Cell gridsFor a cell located at(xi,yi), if its deposited energy is regarded as strain energy, and
is denoted as Ge(xi,yi,ti) at time ti,
then it can be calculated by (1)Ge(xi,yi,ti)=12σiεi=12σi2Ei,  where σi is the maximum stress in
the ith cell at time step ti, and Ei is Young’s modulus of the ith cell.Given the threshold of
theith cell damage energy GeTh(xi,yi),
the corresponding failure strength [σi] of the ith cell is equivalent to (2)[σi]=(2Ei⋅Geth(xi,yi))1/2.If theith cell at time step
ti
satisfies (3)Ge(xi,yi,ti)≥Geth(xi,yi)orσi≥[σi], the cell is
damaged and the energy stored in it releases and transfers to other neighbor
cells. The total damaged cell number at step ti is counted by Nd(ti).To simulate rock damage gradually, a cell is
divided 4 or more subcells obeying the rule of multiplier4k, in order to satisfy the cell dividing symmetry for four quadrants
in a plane, where k=0,1,2,…. Obviously,
the larger the value k, the more the
subcell numbers. So a cell damage grade is considered as a gather of subcell
fracturing, as shown in Figure 3. In terms of the rationale of damage
mechanics, the damage variable for a cell can be  described as(4)D(xi,yi,t)=1−n(t)s*si, where
si
is the ith cell area,
s*
is
one subcell area, and n(t) is the count of damaged subcell at
time step t. If D(xi,yi,t)=0, it indicates that the cell i is in the undamage state; if D(xi,yi,t)=1, it
indicates that the cell i is in
complete damage state; if  0<D(xi,yi,t)<1, it indicates that
the cell i is in partial damage
state. By counting the energy releases of all damaged cells and subcells at time t, we can obtain the equivalent
energy releases diagram versus time step.Damage evolving process of a cell.
(a)
Failure process of a cell(b)
Energy changesThe damage
process of microheterogeneous material can be simulated by the following
procedures.Step 1.
According
a probability distribution to initialize the primary energyGe(xi,yi,ti),
the damage threshold energy GeTh(xi,yi), the
potential probability P(i) of
transferring energy from cell i to
neighbor cells after cell i being damaged, which indicates
the heterogeneous property of material. Also the weak zones are dominated by
decreasing the Ge(xi,yi,ti) value. After a cell fractures, the dissipative
energy is in the form of acoustic emission, heat dissipation as well as light, which
can be expressed by energy dissipative efficient NH(t) in order to
simulate a dissipative system.Step 2.
Input
energy from boundaries. In each evolving step, a cell is chosen randomly and is
endowed with certain energy. This can simulate the cell energy obtained in the
period of no any cell failure.Step 3.
Check  the energy of each cell in the cell space, if the
energy of a cell satisfies (3), then this cell is taken as in the failure
state; its energy is counted as zero and transfers to the neighborhood cells. If
all cells are checked no failure, that is, every cell energy is less its damage
threshold, then a complete evolving step runs out.Step 4.
Repeat
the process from Steps1 to 3, until simulation task arrives at the goal.


### 2.3. Inhomogeneity

For a very heterogeneous material like rock, it is not suitable to
describe the mechanical behavior and fracture process by using an
oversimplified model. Generally, a statistical approach is useful to
characterize the heterogeneity of rock. Previous work [18] indicates that
Weibull’s function can describe very well the distribution of material
parameters of rocks, and the homogeneous
efficient m can be adopted in Weibull’s
distribution function (5)φ(u)=mu0(uu0)m−1e−(u/u0)m, where u is the strength or the damage energy
threshold of material, φ(u) is the density of distribution, u0 is the average strength or damage energy
threshold of material for all cells. The efficient m indicates homogeneous extent of material, that is, the larger for
value m, the more homogeneous for
materials, as shown in Figure 4.Figure 4
φ(u) of different m.


### 2.4. Anisotropy

Such  material as
rock behaves not only inhomogeneity but also anisotropy in microscale, each
microelement in the interior
of such material like rock behaves different potential of transferring
energy along different direction. Weak element is inferior for transferring
energy, but strong element is superior for transferring energy. The potential
of transferring energy is a function of Young's modulus, which can be denoted
by the concept of connecting weight for transferring energy. Assume that the connection
weight of energy celli transferring
to the neighborhood cell j at time t is w(i,j), for Moorse's neighborhood cells (Figure 5), it
satisfies (6)∑j=18w(i,j)=1. If w(i,j)≠w(i,k)(j=1,2,…,8;k=1,2,…,8;j≠k), it is called typical anisotropy. After
cell i releases its total energy ΔGe
at time step t, the total energy of
cell j at time step t+1 is expressed as (7)Ge(xj,yj,t+1)=Ge(xj,yj,t)+w(i,j)⋅ΔGe. Obviously, w(i,j)
is another type of expression for anisotropy.Figure 5
Moorse's neighbor cells.


### 2.5. Crack

There exist many cracks in such materials as rock and
concrete, which cause
their strengths to decrease obviously. Also, a failure inner material originates from the crack
zone firstly. In MCA model, a crack is set up by a weakening zone in cell space
accordance with the crack length and direction, as shown in Figure6. The cells
in a crack obey the following evolving rules.Figure 6
A crack set up by MCA.(a)
The cells in crack zones are assigned in low damage
energy thresholdGeTh(xi,yi). Generally, the damage
energy threshold is considered as only 1/100 of average damage energy threshold
of the whole cells.(b)
All cells in crack zone have high absorbing energy
potential and less releasing energy potential. This can be done by given
absorbing energy weight in a large value of 0.8–1.0 and releasing energy weight
in a small value 0.01–0.1, which means the crack cell can absorb most energy
but release less energy like a “black body.”


## 2.1. Basic Model of CA

In a cellular automaton (CA), there are identical elements,
usually located in a regular array that can be one-dimensional,
two-dimensional, or even higher dimensional. The update rules are as follows: the
elements or cells of a CA are spread over the nodes of a rectangular grid, each
of them capable to be in one of a finite set of possible states. The neighborhood
of each cell can be defined as the set of the four nearest neighbors (North,
South, East and West), the set of the eight nearest neighbors (N, NE, E, SE, S,
SW, W, NW), or other bigger neighborhoods.Conventional  CA
models employ spatial lattices; each lattice element takes one of a set of
discrete values. The lattice evolves over discrete time steps, with element
values changing according to rules relating to the surrounding elements. CA
models are typically used to investigate “complex systems" [5], which exhibit
extreme sensitivity to initial conditions, boundary conditions, history and
element properties, and may have many different outcomes due to the existence
of multiple local equilibria. Thus, a cellular automaton is a discrete
dynamical system, and is composed of cells, cell lattices, neighbor rules, and
so on. The state of a cell is determined only by the states of itself and its neighbors
at previous step. CA has some advantages of time evolution, space
discretization, locality and parallelization, and so on. In order to build a
cellular automaton, the relationships between model components are needed to be clear (Figure 1).Figure 1
The relation between CA components.


## 2.2. MCA Model for Microheterogeneous Materials

Many kinds of material
(e.g., rock, concrete, etc.) are typical microheterogeneous ones; damages may
occur firstly in weak zones under certain load conditions. From their initial
damage to regional fracture, it is a typical self-organization process. As a
dynamical evolving technique, CA provides a base for simulating the fracture
process of microheterogeneous materials in methodology, because CA is a kind of
mathematical model to simulate the self-evolving process of a system caused by element interacts one
to another nonlinearly in it under a random initial condition [1], but
traditional CA  cannot simulate microscale failure evolving process effectively in
physical meaning because of the complexity of microheterogeneous properties of rock-like
materials. After the equivalent physical and mechanical properties for all cells
are given in a form of energy, the rock failure process as well as nonlinear
dynamic behavior scan can be
simulated by transformation of energy from one cell to the others, based
on the basic evolving rules, and thus it is called mechanical cellular automaton
(MCA).In MCA, a cell
is defined by mapping a real minor homogenous region inner material. As shown
in Figure2, a material specimen like rock can be divided into N×N elements or
grids in terms of homogeneity of material, and every element is considered as one
cell. According to the rationale of self-similarity of microstructure of rock,
enough cells can approach the real microstructure of rock equivalently.Cell grids.
(a)
Microstructure of rock(b)
Cell gridsFor a cell located at(xi,yi), if its deposited energy is regarded as strain energy, and
is denoted as Ge(xi,yi,ti) at time ti,
then it can be calculated by (1)Ge(xi,yi,ti)=12σiεi=12σi2Ei,  where σi is the maximum stress in
the ith cell at time step ti, and Ei is Young’s modulus of the ith cell.Given the threshold of
theith cell damage energy GeTh(xi,yi),
the corresponding failure strength [σi] of the ith cell is equivalent to (2)[σi]=(2Ei⋅Geth(xi,yi))1/2.If theith cell at time step
ti
satisfies (3)Ge(xi,yi,ti)≥Geth(xi,yi)orσi≥[σi], the cell is
damaged and the energy stored in it releases and transfers to other neighbor
cells. The total damaged cell number at step ti is counted by Nd(ti).To simulate rock damage gradually, a cell is
divided 4 or more subcells obeying the rule of multiplier4k, in order to satisfy the cell dividing symmetry for four quadrants
in a plane, where k=0,1,2,…. Obviously,
the larger the value k, the more the
subcell numbers. So a cell damage grade is considered as a gather of subcell
fracturing, as shown in Figure 3. In terms of the rationale of damage
mechanics, the damage variable for a cell can be  described as(4)D(xi,yi,t)=1−n(t)s*si, where
si
is the ith cell area,
s*
is
one subcell area, and n(t) is the count of damaged subcell at
time step t. If D(xi,yi,t)=0, it indicates that the cell i is in the undamage state; if D(xi,yi,t)=1, it
indicates that the cell i is in
complete damage state; if  0<D(xi,yi,t)<1, it indicates that
the cell i is in partial damage
state. By counting the energy releases of all damaged cells and subcells at time t, we can obtain the equivalent
energy releases diagram versus time step.Damage evolving process of a cell.
(a)
Failure process of a cell(b)
Energy changesThe damage
process of microheterogeneous material can be simulated by the following
procedures.Step 1.
According
a probability distribution to initialize the primary energyGe(xi,yi,ti),
the damage threshold energy GeTh(xi,yi), the
potential probability P(i) of
transferring energy from cell i to
neighbor cells after cell i being damaged, which indicates
the heterogeneous property of material. Also the weak zones are dominated by
decreasing the Ge(xi,yi,ti) value. After a cell fractures, the dissipative
energy is in the form of acoustic emission, heat dissipation as well as light, which
can be expressed by energy dissipative efficient NH(t) in order to
simulate a dissipative system.Step 2.
Input
energy from boundaries. In each evolving step, a cell is chosen randomly and is
endowed with certain energy. This can simulate the cell energy obtained in the
period of no any cell failure.Step 3.
Check  the energy of each cell in the cell space, if the
energy of a cell satisfies (3), then this cell is taken as in the failure
state; its energy is counted as zero and transfers to the neighborhood cells. If
all cells are checked no failure, that is, every cell energy is less its damage
threshold, then a complete evolving step runs out.Step 4.
Repeat
the process from Steps1 to 3, until simulation task arrives at the goal.


## 2.3. Inhomogeneity

For a very heterogeneous material like rock, it is not suitable to
describe the mechanical behavior and fracture process by using an
oversimplified model. Generally, a statistical approach is useful to
characterize the heterogeneity of rock. Previous work [18] indicates that
Weibull’s function can describe very well the distribution of material
parameters of rocks, and the homogeneous
efficient m can be adopted in Weibull’s
distribution function (5)φ(u)=mu0(uu0)m−1e−(u/u0)m, where u is the strength or the damage energy
threshold of material, φ(u) is the density of distribution, u0 is the average strength or damage energy
threshold of material for all cells. The efficient m indicates homogeneous extent of material, that is, the larger for
value m, the more homogeneous for
materials, as shown in Figure 4.Figure 4
φ(u) of different m.


## 2.4. Anisotropy

Such  material as
rock behaves not only inhomogeneity but also anisotropy in microscale, each
microelement in the interior
of such material like rock behaves different potential of transferring
energy along different direction. Weak element is inferior for transferring
energy, but strong element is superior for transferring energy. The potential
of transferring energy is a function of Young's modulus, which can be denoted
by the concept of connecting weight for transferring energy. Assume that the connection
weight of energy celli transferring
to the neighborhood cell j at time t is w(i,j), for Moorse's neighborhood cells (Figure 5), it
satisfies (6)∑j=18w(i,j)=1. If w(i,j)≠w(i,k)(j=1,2,…,8;k=1,2,…,8;j≠k), it is called typical anisotropy. After
cell i releases its total energy ΔGe
at time step t, the total energy of
cell j at time step t+1 is expressed as (7)Ge(xj,yj,t+1)=Ge(xj,yj,t)+w(i,j)⋅ΔGe. Obviously, w(i,j)
is another type of expression for anisotropy.Figure 5
Moorse's neighbor cells.


## 2.5. Crack

There exist many cracks in such materials as rock and
concrete, which cause
their strengths to decrease obviously. Also, a failure inner material originates from the crack
zone firstly. In MCA model, a crack is set up by a weakening zone in cell space
accordance with the crack length and direction, as shown in Figure6. The cells
in a crack obey the following evolving rules.Figure 6
A crack set up by MCA.(a)
The cells in crack zones are assigned in low damage
energy thresholdGeTh(xi,yi). Generally, the damage
energy threshold is considered as only 1/100 of average damage energy threshold
of the whole cells.(b)
All cells in crack zone have high absorbing energy
potential and less releasing energy potential. This can be done by given
absorbing energy weight in a large value of 0.8–1.0 and releasing energy weight
in a small value 0.01–0.1, which means the crack cell can absorb most energy
but release less energy like a “black body.”


## 3. Simulations for Specimens Containing Cracks

As a valid
approach, CA is used to simulate the force-epoch and AE-epoch of coal specimen
(Figure7) successfully by Tan et al. [21]. This supplies primary evidence that
MCA may be used to simulate the crack extending in rock.Figure 7
AE simulations of coal by CA [21].In this article,
a specimen of microheterogeneous material like rock of 10 cm wide and 10 cm high divided by50×50cells is taken. Weibull’s distribution is utilized for approaching
inhomogeneity. Let the average Young modulus E=1e06MPa and homogeneous
efficient m=8, the average failure
strength [σ]=200MPa and homogeneous
efficient m=4. Three cases are
considered: only one crack of length 2 cm,
two parallel cracks in 2 cm space and 2 cm long, two cracks in
2 cm long by bridge
conjunction.For only one
crack case as shown in Figure8, the initial damage occurs mainly in the region
near crack end although some damage occur closing the midregion of crack also.
As simulating step increases, the damage extends along the two ends in the
direction of near inclination 30°, which is in accordance with the results of limestone
rock compressive split test by Al-Shayea [22], as shown in Figure 9.Failure process of a specimen containing one
crack.
(a)
Step 0(b)
Step 1(c)
Step 5(d)
Step 10Figure 9
CSNBD specimens after failure (see
[22]
by Al-Shayea).For two parallel cracks, because of the interaction of
cracks, although some damages occur near the crack ends, the large-scale
damages occur in the regions
vertical to the cracks (i.e., in two sides of the two parallel cracks) as shown
in Figure10. The results indicate multinatural fissures or cracks take a lead
rule in the failure process of rock. This
result is similar to the wing crack pattern  II/III of crack coalescence which is a sandstone-like material containing two
parallel inclined frictional cracks under uniaxial compression experimented by Wong and Chau [23] (Figure 11), but much more disorder damage zones near
cracks occur because of microheterogeneous property of rock. If this idea has
been generalized to the macroscale, for an example in underground engineering,
a group of caves take controllable action for surrounding rock mass failure
zones, which is valuable for the reinforcement of underground engineering.Failure process of specimen containing two parallel cracks.
(a)
Step 0(b)
Step 2(c)
Step 8(d)
Step 15Nine different patterns
of rock crack coalescence (see
[23]
by Wong and Chau).
(a)(b)(c)(d)(e)(f)(g)(h)(i)For the case of two cracks by bridge conjunction, as shown
in Figure12, by analogy
to a single crack case, the damages occur in the regions of crack ends. But as
the failure evolves, the bridge zone is destroyed by high energy concentration
at last. The damage extends asymmetrically along two ends of cracks, and
failure zone forms irregularly because of inhomogeneity also. This case is
similar to shear crack
pattern of crack coalescence to a sandstone-like material containing two cracks by bridge
conjunction under
uniaxial compression generally (Figure 11),
which is experimented by Wong and Chau [23]. But some damage  zones also occur far away from
cracks, this also because of microheterogeneous property of rock. Thus, for
inhomogeneous materials, their failure behaviors are much different from the
homogeneous ones, so homogeneous mechanics cannot be adopted by copying word to word.Failure process of specimen containing two cracks by bridge conjunction.
(a)
Step 0(b)
Step 2(c)
Step 10(d)
Step 20(e)
Step 30(f)
Step 50


## 4. Conclusions

Some kinds of
materials (e.g., rock, concrete, etc.) have many complicated properties such as
inhomogeneity, anisotropy, discontinuity both in microscale and macroscale. Their
damage process is a typical self-organization evolving one. A special CA model
called MCA has been built, in which inhomogeneity of material cells is
evaluated by Weibull's distribution function, the failure index can be represented
by energy or strength, anisotropy is defined as the difference of transferring
energy along different direction, and the crack is constructed by a cluster of
weak energy cells.By MCA, three
cases of crack: only one crack, two parallel cracks, and two cracks by bridge
conjunction, are simulated. The results show, for only one crack, that the damage extends
along the directions of crack end near inclination30°
which is in accordance with the result
fracture mechanics. For two parallel cracks, because of the interaction of
cracks, although some damages occur near the crack ends, the large-scale
damages occur in the regions
vertical to the cracks. For the case of two cracks by bridge conjunction,
although the damage evolution is analogous to one single crack case, but the bridge is
destroyed by high energy concentration at last, the damage extends
asymmetrically along two ends of cracks, so an irregular failure zone forms, just
because of inhomogeneity. Thus, for inhomogeneous materials, their failure behaviors
are much different from homogeneous ones, so traditional homogeneous mechanics is
limited for application.



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*Source: 946038-2009-01-20.xml*