Journal of Nanomaterials

Volume 2008, Article ID 946038, 7 pages

http://dx.doi.org/10.1155/2008/946038

## MCA Model for Simulating the Failure of Microinhomogeneous Materials

Key Laboratory of Mine Disaster Prevention and Control, Shandong University of Science and Technology, Qingdao, Shandong 266510, China

Received 11 July 2008; Revised 19 October 2008; Accepted 25 October 2008

Academic Editor: Xuedong Bai

Copyright © 2008 Yunliang Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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).

##### 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 Figure 2, a material specimen like rock can be divided into 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.

For a cell located at , if its deposited energy is regarded as strain energy, and is denoted as at time , then it can be calculated by where is the maximum stress in the th cell at time step , and is Young’s modulus of the th cell.

Given the threshold of the th cell damage energy , the corresponding failure strength of the th cell is equivalent to

If the th cell at time step satisfies the cell is damaged and the energy stored in it releases and transfers to other neighbor cells. The total damaged cell number at step is counted by Nd().

To simulate rock damage gradually, a cell is
divided 4 or more subcells obeying the rule of multiplier , in order to satisfy the cell dividing symmetry for four quadrants
in a plane, where . 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 where
is the *i*th cell area,
is
one subcell area, and is the count of damaged subcell at
time step *t*. If , it indicates that the cell *i* is in the undamage state; if , it
indicates that the cell *i* is in
complete damage state; if , 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.

The damage process of microheterogeneous material can be simulated by the following procedures.

*Step 1. *According
a probability distribution to initialize the primary energy ,
the damage threshold energy , the
potential probability 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 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 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 Steps 1 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 where *u* is the strength or the damage energy
threshold of material, is the density of distribution, 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.

##### 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 cell *i* transferring
to the neighborhood cell *j* at time *t* is , for Moorse's neighborhood cells (Figure 5), it
satisfies If , 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 is expressed as Obviously,
is another type of expression for anisotropy.

##### 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 Figure 6. The cells in a crack obey the following evolving rules.

(a)The cells in crack zones are assigned in low damage energy threshold . 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 (Figure 7) successfully by Tan et al. [21]. This supplies primary evidence that MCA may be used to simulate the crack extending in rock.

In this article, a specimen of microheterogeneous material like rock of 10 cm wide and 10 cm high divided by cells is taken. Weibull’s distribution is utilized for approaching inhomogeneity. Let the average Young modulus MPa and homogeneous efficient , the average failure strength MPa and homogeneous efficient . 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 Figure 8, 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 , which is in accordance with the results of limestone rock compressive split test by Al-Shayea [22], as shown in Figure 9.

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 Figure 10. 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.

For the case of two cracks by bridge conjunction, as shown in Figure 12, 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.

#### 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 inclination 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.

#### Acknowledgments

The study was supported by National Natural Science Foundation of China (no. 50534080, no. 50674063), Natural Science Fund of Shandong Province (no. Y2004F11), Project no. J06N04 of Educational Department of Shandong Province, Key Laboratory of Mine Disaster Prevention and Control, Shandong University of Science and Technology, and Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.

#### References

- J. von Neumann and A. W. Burks,
*Theory of Self-Reproducing Automata*, University of Illinois Press, Urbana, Ill, USA, 1966. - L. S. Schulman and P. E. Seiden, “Statistical mechanics of a dynamical system based on Conway's game of Life,”
*Journal of Statistical Physics*, vol. 19, no. 3, pp. 293–314, 1978. View at Publisher · View at Google Scholar - J. M. Smith,
*Evolution and the Theory of Games*, Cambridge University Press, London, UK, 1982. - S. Wolfram, “CA as models of complexity,”
*Nature*, vol. 311, pp. 419–424, 1984. View at Google Scholar - S. Wolfram, “Computation theory of cellular automata,”
*Communications in Mathematical Physics*, vol. 96, no. 1, pp. 15–57, 1984. View at Publisher · View at Google Scholar - H. Fort, “On evolutionary spatial heterogeneous games,”
*Physica A*, vol. 387, no. 7, pp. 1613–1620, 2008. View at Publisher · View at Google Scholar - S. Raghavan and S. S. Sahay, “Modeling the grain growth kinetics by cellular automaton,”
*Materials Science and Engineering A*, vol. 445-446, pp. 203–209, 2007. View at Google Scholar - K. Morita, “Reversible computing and cellular automata—a survey,”
*Theoretical Computer Science*, vol. 395, no. 1, pp. 101–131, 2008. View at Publisher · View at Google Scholar - A. Nishiyama, H. Tanaka, and T. Tokihiro, “An isotropic cellular automaton for excitable media,”
*Physica A*, vol. 387, no. 13, pp. 3129–3136, 2008. View at Publisher · View at Google Scholar - Y. W. Kwon and S. Hosoglu, “Application of lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems,”
*Computers and Structures*, vol. 86, no. 7-8, pp. 663–670, 2008. View at Publisher · View at Google Scholar - M. J. Leamy, “Application of cellular automata modeling to seismic elastodynamics,”
*International Journal of Solids and Structures*, vol. 45, no. 17, pp. 4835–4849, 2008. View at Publisher · View at Google Scholar - S.-M. Seo, I.-S. Kim, C.-Y. Jo, and K. Ogi, “Grain structure prediction of Ni-base superalloy castings using the cellular automaton-finite element method,”
*Materials Science and Engineering A*, vol. 449–451, pp. 713–716, 2007. View at Publisher · View at Google Scholar - L. Madej, P. D. Hodgson, and M. Pietrzyk, “The validation of a multiscale rheological model of discontinuous phenomena during metal rolling,”
*Computational Materials Science*, vol. 41, no. 2, pp. 236–241, 2007. View at Publisher · View at Google Scholar - M. McCann and N. Pippenger, “Fault tolerance in cellular automata at high fault rates,”
*Journal of Computer and System Sciences*, vol. 74, no. 5, pp. 910–918, 2008. View at Publisher · View at Google Scholar - I. G. Georgoudas, G. Ch. Sirakoulis, and I. Andreadis, “Modelling earthquake activity features using cellular automata,”
*Mathematical and Computer Modelling*, vol. 46, no. 1-2, pp. 124–137, 2007. View at Publisher · View at Google Scholar - I. G. Georgoudas, G. Ch. Sirakoulis, E. M. Scordilis, and I. Andreadis, “A cellular automaton simulation tool for modelling seismicity in the region of Xanthi,”
*Environmental Modelling & Software*, vol. 22, no. 10, pp. 1455–1464, 2007. View at Publisher · View at Google Scholar - Y. L. Tan, H. Zhou, Y. J. Wang, and Z.-T. Ma, “Physical cellular automaton theory for simulating the failure process of micro-heterogeneous material,”
*Acta Physica Sinica*, vol. 50, no. 4, pp. 709–710, 2001. View at Google Scholar - H. Zhou, Y. J. Wang, Y. L. Tan, and X. Feng, “Study on physical cellular automata model of rock mass failure (I)—basic model,”
*Chinese Journal of Rock Mechanics and Engineering*, vol. 21, no. 4, pp. 475–478, 2002. View at Google Scholar - H. Zhou, Y. L. Tan, X. Feng, and Y. J. Wang, “Study on physical cellular automata model of rock mass failure (II)—examples,”
*Chinese Journal of Rock Mechanics and Engineering*, vol. 21, no. 6, pp. 782–786, 2002. View at Google Scholar - X. Feng, P.-Z. Pan, and H. Zhou, “Simulation of the rock microfracturing process under uniaxial compression using an elasto-plastic cellular automaton,”
*International Journal of Rock Mechanics and Mining Sciences*, vol. 43, no. 7, pp. 1091–1108, 2006. View at Publisher · View at Google Scholar - Y. L. Tan, H. Zhou, and Y. J. Wang, “PCA model for simulating AE and chaos in rock failure process,”
*The Chinese Journal of Nonferrous Metals*, vol. 12, no. 4, pp. 802–807, 2002. View at Google Scholar - N. A. Al-Shayea, “Crack propagation trajectories for rocks under mixed mode I-II fracture,”
*Engineering Geology*, vol. 81, no. 1, pp. 84–97, 2005. View at Publisher · View at Google Scholar - R. H. C. Wong and K. T. Chau, “Crack coalescence in a rock-like material containing two cracks,”
*International Journal of Rock Mechanics and Mining Sciences*, vol. 35, no. 2, pp. 147–164, 1998. View at Publisher · View at Google Scholar