- Email: [email protected]
Discrete model for fragmentation with random stopping
Physica A 300 (2001) 13–24
www.elsevier.com/locate/physa
Discrete model for fragmentation with random stopping Gonzalo Hern$andez School of Engineering, Andres Bello University, Sazie 2320, Santiago, Chile Received 27 October 2000; received in revised form 15 June 2001
Abstract In this work, we present the numerical results obtained from large scale parallel and distributed simulations of a model for two- and three-dimensional discrete fragmentation. Its main features are: (1) uniform and independent random distribution of the forces that generate the fracture; (2) deterministic criteria for the fracture process at each step of the fragmentation, based on these forces and a random stopping criteria. By large scale parallel and distributed simulations, implemented over a heterogeneous network of high performance computers, di3erent behaviors were obtained for the fragment size distribution, which includes power law behavior with positive exponents for a wide range of the main parameter of the model: the stopping probability. Also, by a sensitive analysis we prove that the value of the main parameter of the model does not a3ect these results. The power law distribution is a non-trivial result which reproduces empirical c 2001 Published by Elsevier Science B.V. results of some highly energetic fracture processes. Keywords: Discrete fragmentation; Maximum force fracture; Random stopping
1. Introduction Fracture and fragmentation processes are common phenomena in nature and technology. In Refs. [1–3] Turcotte, Lawn and Wilshaw gave a long enumeration of natural fragment size distributions (fragments from weathering, asteroids, coal heaps, rock fragments from nuclear and chemical explosions, projectile collisions, etc.) for which power laws were measured with exponents ranging from 1.9 to 2.6 concentrating around 2.4. There are careful experiments in one dimension, see Ref. [4], in which long thin glass rods are broken by vertically dropping them. Depending on the height from which the glass rods are dropped, the fragment size distribution varies from a log-normal shape, for smaller heights, to a power law for increasing heights. E-mail address: [email protected] (G. Hern$andez). c 2001 Published by Elsevier Science B.V. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 3 4 3 - 0
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G. Hernandez / Physica A 300 (2001) 13–24
Continuous stochastic models for one dimensional fragmentation can be found in Ref. [5], where the fracture points are chosen randomly. Power law, exponential and log-normal fragment size distributions can be obtained depending on the a priori fracture points distribution. By the introduction of a minimal fragment size (a piece that cannot be broken further) and a Poisson distribution for the number of fragments in which pieces are broken, a power law behavior is obtained at some stages of the fragmentation process. Stochastic processes which are discrete in space and time have also been studied as models for fragmentation phenomena using cellular automata. In Ref. [6] two and three dimensional cellular automata are proposed to model the power law distribution resulting from shear experiments of a layer of uniformly sized fragments. The fracture probability of a fragment is determined by the relative size of its neighbors. A larger probability is obtained for fragments with a larger number of equal size neighbors. The process continues until no blocks of equal size are neighbors. In all cases a power law fragment size distribution is obtained, with an exponent that depends on the parameters of the model. In Ref. [7], a model for the fragmentation of gas clouds via gravitational condensation is developed, under the assumptions of mass conservation and no presence of pressure. Initially, clouds split (condense) into q ¿ 2 equal mass parts. The process continues in a self-similar way such that the dynamic equations can be solved analytically. Further assumptions on the model allow one to prove that the fragment mass distribution follows a power law behavior in the steady state. A mean-Geld type approach to describe the fragmentation process can be formulated through the concentration c(x; t) of fragments of mass less than x at time t by the rate equations, Ref. [8]: ∞ @c(x; t) = − a(x)c(x; t) + c(y; t)a(y)f(x=y) dy ; (1) @t x where a(x) is the rate at which fragments of mass x break into smaller ones (this quantity is supposed not to depend on time) and f(x=y) the conditional probability that a fragment of mass x was produced from a fragment of mass y ¿ x. Using scaling and homogeneity assumptions, some exact results are obtained in Refs. [8,9] if some further assumptions are made about f(x=y). In general, the solutions of the rate equations are very diKcult to obtain. There are some approaches in which some qualitative information is obtained by approximately solving them. Di3erent models for fragmentation have been studied using this kind of approach. Some examples can be found in the following pre-prints, see Refs. [10 –17]. Finally in Refs. [18,19], we proposed simple discrete stochastic processes in twoand three-dimensions which fulGll mass conservation. The fracture process is deGned deterministically over a lattice of linear size 2n , based on microscopic models of random forces. For each direction the net forces acting on the lattice can be computed from these random forces. At each step the most stressed piece is broken into two equal sized (volume or area) fragments. Our objective was to measure the fragment
G. Hernandez / Physica A 300 (2001) 13–24
15
size histogram for our models, and investigate under what circumstances these simple models present power law behavior in the limit of small fragments. The analytical study of the model was very diKcult. For this reason we made a numerical study which involved large scale sequential simulations of our models. Di3erent behaviors were obtained for the fragment size distribution which includes log-normal and power law behavior, depending on the deGnition of the parameters of our models, which are: the scalar net forces acting on the fragments; the selection of the piece to break; the choice of the fracture orientation; the deGnition of how the fragmentation process must continue and Gnally, the deGnition of a criterion to stop the fragmentation. The power law distribution is a non-trivial result, which reproduces empirical results of some highly energetic fracture processes. In Section 2, we deGne our new model as a generalization of the models studied in Refs. [18,19], by the introduction of a new condition for stopping the fragmentation process. In Section 3, we present and discuss the numerical results obtained from large scale parallel and distributed simulations implemented over a small network of high performance computers. 2. A new simple model for fragmentation: random stopping In this section, we present our new model for fragmentation processes, which is a generalization of the model studied in Refs. [18,19]. The model is deGned in two and three dimensions and fulGlls the mass conservation assumption. We will consider twoand three-dimensional discrete lattices of linear size 2n , where 10 6 n 6 14. As the initial situation, see Figs. 1 and 2, there are forces of traction or compression acting over the lattice on every axis denoted by fx , fy and fz , which correspond to uniform and independent distributed random numbers on [0; 1]. These forces are microscopic models of random net forces which act on each direction of the lattice and correspond to an abstraction of the externally imposed load that generates the fragmentation process. At each step all the pieces will be broken into two pieces unless they satisfy the stopping condition. These new pieces will be of the same shape and size.
Fig. 1. Initial situation for the two-dimensional lattice.
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Fig. 2. Initial situation for the three-dimensional lattice.
Now, we will deGne the basic characteristics of our new model. We will concentrate in three-dimensions, since two-dimensions is a particular case. In each time step the following considerations must be taken into account: (a) De/nition and interpretation of the forces acting on every fragment on each axis. These random and independent uniformly distributed forces will represent the elastic deformations or strains, following the Hooke’s law. (b) A deterministic criteria to choose the piece to be broken. At each time step each one of the pieces or fragments will be broken, unless it veriGes a stopping criterion, that will be deGned below. (c) A criterion to choose the fracture orientation for the pieces to be broken. The fracture or cutting plane is the plane perpendicular to the direction of the larger net force. For instance, if the cutting plane is YZ then the maximum net force is fx . Two new pieces are generated, with equal shape and size (volume or area) corresponding to half of the volume or area of the original piece. This ensures mass conservation. (d) A criterion to de/ne how the breaking process continues. The idea of our model is to continue the fragmentation process in a self-similar way. In order to do that we have to deGne the new scalar forces acting on each axis of the pieces that were generated. The new forces of the pieces generated will be uniform and independent random numbers. (e) A stopping criterion for the fragmentation process. At this point we have to consider the discrete nature of the model. This implies that we cannot continue the breaking process for ever. It seems natural to impose that it is not possible to break a piece perpendicular to the direction into which it has a length of unity. Furthermore, to generalize our model, we will introduce a new stopping criterion: The Random Stopping. There are two situations in which the fragmentation process of a particular piece stops. The Grst one: if the piece to be broken is fractured in a perpendicular plane with respect to a “unity side” length. The second one: there is a probability p (called the stopping probability) that the fragmentation process stops, for a piece of size less or equal to the parameter called area crit or vol crit depending on the dimension. As
G. Hernandez / Physica A 300 (2001) 13–24
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typical values for the area crit and vol crit, we have chosen the size of a piece with each side equal to a half of the linear side of the initial situation, i.e., area crit = 2n=2 × 2n=2 = 2n ; vol crit = 2n=2 × 2n=2 × 2n=2 = 23n=2 :
(2)
The parameters area crit, vol crit and p are Gxed for the fragmentation process. We will introduce this new stopping rule in order to represent the fact that greater pieces have more probability to be broken than the smaller ones. Our model is based on microscopic and local deGnitions of random forces which generates the fragmentation process. Hence, the fragment size distribution and the shape of the fragments are determined by these forces in a quantitative way. The dynamics of the fragmentation process and how it ends is deGned by the stopping criterion: a “natural” stopping situation based on the discrete deGnition of the material and how the forces act on it, and a random stopping situation, in which large fragments have larger probability to be broken than the smaller ones. The characteristics stated before deGne a stochastic process discrete in time and space that satisGes the Markov property, since, at each time step the knowledge of the sizes (or shapes) of the pieces and its stresses, determines which piece must be broken and the shape of the new fragments. The exact analytical study of the model is very diKcult. For this reason, in the next section we will present a numerical study which involves large scale parallel and distributed simulations. 3. Numerical results In this section, we will present the results of the numerical study implemented by large scale parallel and distributed simulations. Mainly, we are interested in determining the fragment size (area or volume) distribution, that we will denote f.s.d., and the dynamical properties of this model. This study was performed in a very small and simple network of heterogeneous and high performance computers connected by fast ethernet protocols, as it is shown in Fig. 3. The applications were programmed in C language and PVM, see Ref. [20], a very well-known software libraries for parallel and distributed computing in a message passing environment. They interacted as master– slave processes. The methodology for the simulations was the following: we averaged our results over many independent random initial conditions, characterized by the initial force conGguration, i.e., the random force acting on each axis. Since the simulations are rather time consuming we considered between 1000 and 10,000 initial conditions for the 2-d models, while for the 3-d models we could only average our results over 500–2000 initial conditions. The system linear size was N = 2n where N , for our simulations, was equal to N = 1024, 4096 and 16; 384 (n = 10, 12 and 14) in the 2-d case N = 1024 and 4096 (n = 10 and 12) for the three-dimensional case. These lattice sizes and number of initial conGgurations are signiGcantly larger than used in the study for the more
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Fig. 3. Heterogeneous networks of computers used for computations.
particular models deGned in Refs. [18,19]. We explain this fact due to three main reasons: (1) The increase in computing power of today’s computer’s CPU and main memory in comparison to 5 years ago. (2) A much better design and implementation of the data structures and programs. (3) The use of parallel and distributed processing in a small but fast network of high performance computers. For each model we computed the fragment size (area or volume) distribution obtained from the fragmentation process, f.s.d. The maximum number of iterations is N 2 = 22n for 2-d and N 3 = 23n for 3-d, which gives an exponential computing time, that was observed only in the models with stopping probability less or equal to p = 0:35; since the number of fragments which continues in the fragmentation process grows exponentially. For larger probability values, the computing time is very small, since the number of fragments generated is very small. First, we will discuss in detail the 2-d case. In this dimension the larger size considered was N = 16; 384 (n = 14). The main results can be observed in Figs. 4 –9 and Table 1: we obtain a power law behavior for the fragment size distribution, F(s) ∼ s− , for a wide range of the parameter of the model p, the stopping probability, speciGcally for: p ∈ (0:00; 0:25 ± 0:05) and area crit ∈ (n; 2n ). As we will show, the variation of the exponents with respect to the stopping probability is almost linear, and their values were obtained taking into account the Gnite size e3ects appearing for small area limits, which are not signiGcantly larger. The values of the exponents were obtained from classical techniques of non-linear regression. All the correlation coeKcients were greater than 0:98. The dependence of the 2d exponents with respect to the stopping probability values can be expressed approximately by a linear function = 1:46 − 4:05p
(3)
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Fig. 4. F.s.d. for probabilities p 6 0:25 and N = 16; 364.
Fig. 5. Non-linear exponents dependence for N = 16; 364.
which was obtained from linear regression techniques. In this case, the correlation coeKcient was 0:92. For probabilities greater than p = 0:25 ± 0:05, the f.s.d. changes smoothly from an approximate power law with positive exponent to a power law with negative exponent. Because area crit is a parameter of the model, a sensitivity analysis was performed for larger and smaller area crit values than what was considered at Grst, i.e., area crit = 2n . We implement this study for p = 0:25 since it was a value considered as an upper limit for the very accurate power law behavior and N = 4096 (n = 12). Then, the typical area in this case was area crit = 212 = 4096 and there is a very wide range for values of this parameter: [2; 22n ]. Therefore, for the sensitivity study we chose very large values and very small ones. The results are shown in Figs. 6 and 7. For large values of the parameter, Fig. 6, the results are exactly the same as those area crit = 212 = 4096, i.e., all f.s.d. corresponds to a power law with: = 1:705. The results for small values of area crit present an increasing and smooth deviation from the results obtained for the typical value, as area crit increase. This fact is shown in Fig. 7.
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Fig. 6. F.s.d. large values of area crit (N = 4096, p = 0:25).
Fig. 7. F.s.d. small values of area crit (N = 4096, p = 0:25).
G. Hernandez / Physica A 300 (2001) 13–24
Fig. 8. Fragmentation processes examples for N = 8: p = 0:05 (left), p = 0:10 (right).
Fig. 9. Fragmentation processes examples for N = 8: p = 0:15 (left), p = 0:25 (right).
Table 1 2d—Exponents for the power law behaviors for p 6 0:25 Stopping probability p
2d—Exponent
0:00 0:05 0:10 0:15 0:20 0:25
1:705 1:072 0:914 0:787 0:669 0:554
21
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Fig. 10. F.s.d. for probabilities less or equal to p = 0:25 and N = 1024. Table 2 3d—Exponents for the power law behaviors for p 6 0:25 Probability p
3d—Exponent
0:00 0:05 0:10 0:15 0:20 0:25
1:457 1:408 1:306 1:259 1:154 1:077
We can conclude that the variation of the area crit parameter mainly a3ects the constant value in the power law relation F(s) = s−
(4)
and presents only very small inQuence on the value of the exponent . In Figs. 8 and 9, we can appreciate how the fragmentation process looks for different probability values. The black lines correspond to the boundaries of the fragments. They show that the visualization of our model is very complex, for a wide range of probability values p, with patterns of fracture that resemble the real process. The numerical results for the three-dimensional case are very similar to the twodimensional ones, as shown in Fig. 10 and Table 2, but they have more signiGcant Gnite size e3ects deviations in the limit of small volume than in the two-dimensional case. If we analyze the large volume results, they show power law behaviors for the f.s.d. with a range for exponents = 1:457 for p = 0:00 to = 1:077 for p = 0:25: In this range, the exponents decrease linearly. The values of the exponents were obtained from classical techniques of non-linear regression. All the correlation coeKcients were greater than 0:87. The dependence of the 3d exponents with respect to the probability values can be expressed approximately
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by a linear function: = 1:47 − 1:55p
(5)
which was obtained from classical linear regression techniques. The correlation coeKcient was 0:99: As in the two-dimensional case, for probabilities greater than p = 0:25 ± 0:05, the f.s.d. changes smoothly from an approximate power law with positive exponent to a power law with negative exponent.
4. Conclusions In this work we have presented the results obtained from the numerical study of a simple model for two- and three-dimensional discrete fragmentation. The main features are: uniform and independent random local distribution of the forces acting on each piece, deterministic criterion for the fracture process at each step of the fragmentation based on these forces and a random stopping criterion based on the size of the fragment together with stopping conditions determined by the discrete nature of the material. The numerical study was implemented by large scale simulations over a small network of heterogeneous high performance computers. Di3erent power law behaviors were obtained for the fragment size distribution which includes power law behavior with positive exponents, for a wide range of the main parameter of the model: the stopping probability p. This parameter was included in the model in order to simulate large fracture probabilities for larger pieces. The dependence between this parameter and the power law exponent can be approximated by linear functions. For probability values below 0:25 ± 0:05, the number of fragments increases very fast at each step, producing the power law behavior. On the other hand, the area crit and vol crit parameters only generate small variations on this non-linear dependence. The power law distribution is a non-trivial result which reproduces empirical results of some highly energetic fracture processes.
Acknowledgements This work was supported by Conicyt-Chile, through Proyecto Fondecyt 199 –1026. References [1] B.R. Lawn, T.R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, Cambridge, 1975. [2] D.L. Turcotte, Fractals and fragmentation, J. Geophys. Res. 91 (1986) 1921–1926. [3] D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, 1992.
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[4] T. Ishii, M. Matsushita, Fragmentation of long thin glass rods, J. Phys. Soc. Japan 61 (10) (1992) 3474–3477. [5] M. Matsushita, K. Sumida, How do thin glass rods break? Stochastic models for one-dimensional fracture, Chuo University 31 (1988) 69 –79. [6] S. Steacy, C. Sammis, An automaton for fractal patterns of fragmentation, Nature 353 (1991) 250–252. [7] W. Newmann, I. Wasserman, Hierarchical fragmentation model for the evolution of self-gravitating clouds, The Astrophys. J. 354 (1990) 411–417. [8] S. Redner, Statistical models for the fracture of disordered media, in: H. Herrmann, S. Roux (Eds.), Random Materials and Processes Series, Elsevier, Amsterdam, 1990. [9] P.L. Krapivsky, E. Ben-Naim, Scaling and multiscaling in models of fragmentation, Phys. Rev. E 50 (5) (1994) 3502–3507. [10] R.F. Machado, J. Kamphorst Leal de Silva, Fragmentation–Inactivation: A Scaling Approach, Departamento de F$Tsica, Instituto de Ciencias Exactas, Universidade Federal de Minais Gerais, CP 702, 30161-970, Belo Hoizonte, Brazil. [11] G.J. Rodgers, M.K. Hassan, Stable Distributions in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [12] G.J. Rodgers, M.K. Hassan, Multifractality and Multiscaling in Two Dimensional Fragmentation, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [13] P. Singh, On the Kinetics of Multidimensional Fragmentation, Department of Physics Shahjalal Science and Technology University, Sylhet, Bangladesh. [14] M.K. Hassan, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [15] M.K. Hassan, Multifractality and the Shattering Transition in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [16] P.L. Krapivsky, I. Grosse, E. Bean-Naim, Scale Invariance and Lack of Self-Averaging in Fragmentation, Center for Polymer Studies and Department of Physics, Boston University, Boston MA 02215, USA. [17] G.J. Rodgers, M.K. Hassan, Distributions in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, United Kingdom. [18] Some simple models for fragmentation, G. Hern$andez, H.J. Herrmann, in: D. Les Houches, X. Beysens, Campi, E. Pfe3erkkovn (Eds.), Proceedings Workshop in Fragmentation Phenomena, April 1994, pp. 259. [19] G. Hern$andez, H.J. Herrmann, Discrete models for 2- and 3-dimensional fragmentation, Physica A 215 (1995) 420–430. [20] http:==www.epm.ornl.gov=pvm=
www.elsevier.com/locate/physa
Discrete model for fragmentation with random stopping Gonzalo Hern$andez School of Engineering, Andres Bello University, Sazie 2320, Santiago, Chile Received 27 October 2000; received in revised form 15 June 2001
Abstract In this work, we present the numerical results obtained from large scale parallel and distributed simulations of a model for two- and three-dimensional discrete fragmentation. Its main features are: (1) uniform and independent random distribution of the forces that generate the fracture; (2) deterministic criteria for the fracture process at each step of the fragmentation, based on these forces and a random stopping criteria. By large scale parallel and distributed simulations, implemented over a heterogeneous network of high performance computers, di3erent behaviors were obtained for the fragment size distribution, which includes power law behavior with positive exponents for a wide range of the main parameter of the model: the stopping probability. Also, by a sensitive analysis we prove that the value of the main parameter of the model does not a3ect these results. The power law distribution is a non-trivial result which reproduces empirical c 2001 Published by Elsevier Science B.V. results of some highly energetic fracture processes. Keywords: Discrete fragmentation; Maximum force fracture; Random stopping
1. Introduction Fracture and fragmentation processes are common phenomena in nature and technology. In Refs. [1–3] Turcotte, Lawn and Wilshaw gave a long enumeration of natural fragment size distributions (fragments from weathering, asteroids, coal heaps, rock fragments from nuclear and chemical explosions, projectile collisions, etc.) for which power laws were measured with exponents ranging from 1.9 to 2.6 concentrating around 2.4. There are careful experiments in one dimension, see Ref. [4], in which long thin glass rods are broken by vertically dropping them. Depending on the height from which the glass rods are dropped, the fragment size distribution varies from a log-normal shape, for smaller heights, to a power law for increasing heights. E-mail address: [email protected] (G. Hern$andez). c 2001 Published by Elsevier Science B.V. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 3 4 3 - 0
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G. Hernandez / Physica A 300 (2001) 13–24
Continuous stochastic models for one dimensional fragmentation can be found in Ref. [5], where the fracture points are chosen randomly. Power law, exponential and log-normal fragment size distributions can be obtained depending on the a priori fracture points distribution. By the introduction of a minimal fragment size (a piece that cannot be broken further) and a Poisson distribution for the number of fragments in which pieces are broken, a power law behavior is obtained at some stages of the fragmentation process. Stochastic processes which are discrete in space and time have also been studied as models for fragmentation phenomena using cellular automata. In Ref. [6] two and three dimensional cellular automata are proposed to model the power law distribution resulting from shear experiments of a layer of uniformly sized fragments. The fracture probability of a fragment is determined by the relative size of its neighbors. A larger probability is obtained for fragments with a larger number of equal size neighbors. The process continues until no blocks of equal size are neighbors. In all cases a power law fragment size distribution is obtained, with an exponent that depends on the parameters of the model. In Ref. [7], a model for the fragmentation of gas clouds via gravitational condensation is developed, under the assumptions of mass conservation and no presence of pressure. Initially, clouds split (condense) into q ¿ 2 equal mass parts. The process continues in a self-similar way such that the dynamic equations can be solved analytically. Further assumptions on the model allow one to prove that the fragment mass distribution follows a power law behavior in the steady state. A mean-Geld type approach to describe the fragmentation process can be formulated through the concentration c(x; t) of fragments of mass less than x at time t by the rate equations, Ref. [8]: ∞ @c(x; t) = − a(x)c(x; t) + c(y; t)a(y)f(x=y) dy ; (1) @t x where a(x) is the rate at which fragments of mass x break into smaller ones (this quantity is supposed not to depend on time) and f(x=y) the conditional probability that a fragment of mass x was produced from a fragment of mass y ¿ x. Using scaling and homogeneity assumptions, some exact results are obtained in Refs. [8,9] if some further assumptions are made about f(x=y). In general, the solutions of the rate equations are very diKcult to obtain. There are some approaches in which some qualitative information is obtained by approximately solving them. Di3erent models for fragmentation have been studied using this kind of approach. Some examples can be found in the following pre-prints, see Refs. [10 –17]. Finally in Refs. [18,19], we proposed simple discrete stochastic processes in twoand three-dimensions which fulGll mass conservation. The fracture process is deGned deterministically over a lattice of linear size 2n , based on microscopic models of random forces. For each direction the net forces acting on the lattice can be computed from these random forces. At each step the most stressed piece is broken into two equal sized (volume or area) fragments. Our objective was to measure the fragment
G. Hernandez / Physica A 300 (2001) 13–24
15
size histogram for our models, and investigate under what circumstances these simple models present power law behavior in the limit of small fragments. The analytical study of the model was very diKcult. For this reason we made a numerical study which involved large scale sequential simulations of our models. Di3erent behaviors were obtained for the fragment size distribution which includes log-normal and power law behavior, depending on the deGnition of the parameters of our models, which are: the scalar net forces acting on the fragments; the selection of the piece to break; the choice of the fracture orientation; the deGnition of how the fragmentation process must continue and Gnally, the deGnition of a criterion to stop the fragmentation. The power law distribution is a non-trivial result, which reproduces empirical results of some highly energetic fracture processes. In Section 2, we deGne our new model as a generalization of the models studied in Refs. [18,19], by the introduction of a new condition for stopping the fragmentation process. In Section 3, we present and discuss the numerical results obtained from large scale parallel and distributed simulations implemented over a small network of high performance computers. 2. A new simple model for fragmentation: random stopping In this section, we present our new model for fragmentation processes, which is a generalization of the model studied in Refs. [18,19]. The model is deGned in two and three dimensions and fulGlls the mass conservation assumption. We will consider twoand three-dimensional discrete lattices of linear size 2n , where 10 6 n 6 14. As the initial situation, see Figs. 1 and 2, there are forces of traction or compression acting over the lattice on every axis denoted by fx , fy and fz , which correspond to uniform and independent distributed random numbers on [0; 1]. These forces are microscopic models of random net forces which act on each direction of the lattice and correspond to an abstraction of the externally imposed load that generates the fragmentation process. At each step all the pieces will be broken into two pieces unless they satisfy the stopping condition. These new pieces will be of the same shape and size.
Fig. 1. Initial situation for the two-dimensional lattice.
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Fig. 2. Initial situation for the three-dimensional lattice.
Now, we will deGne the basic characteristics of our new model. We will concentrate in three-dimensions, since two-dimensions is a particular case. In each time step the following considerations must be taken into account: (a) De/nition and interpretation of the forces acting on every fragment on each axis. These random and independent uniformly distributed forces will represent the elastic deformations or strains, following the Hooke’s law. (b) A deterministic criteria to choose the piece to be broken. At each time step each one of the pieces or fragments will be broken, unless it veriGes a stopping criterion, that will be deGned below. (c) A criterion to choose the fracture orientation for the pieces to be broken. The fracture or cutting plane is the plane perpendicular to the direction of the larger net force. For instance, if the cutting plane is YZ then the maximum net force is fx . Two new pieces are generated, with equal shape and size (volume or area) corresponding to half of the volume or area of the original piece. This ensures mass conservation. (d) A criterion to de/ne how the breaking process continues. The idea of our model is to continue the fragmentation process in a self-similar way. In order to do that we have to deGne the new scalar forces acting on each axis of the pieces that were generated. The new forces of the pieces generated will be uniform and independent random numbers. (e) A stopping criterion for the fragmentation process. At this point we have to consider the discrete nature of the model. This implies that we cannot continue the breaking process for ever. It seems natural to impose that it is not possible to break a piece perpendicular to the direction into which it has a length of unity. Furthermore, to generalize our model, we will introduce a new stopping criterion: The Random Stopping. There are two situations in which the fragmentation process of a particular piece stops. The Grst one: if the piece to be broken is fractured in a perpendicular plane with respect to a “unity side” length. The second one: there is a probability p (called the stopping probability) that the fragmentation process stops, for a piece of size less or equal to the parameter called area crit or vol crit depending on the dimension. As
G. Hernandez / Physica A 300 (2001) 13–24
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typical values for the area crit and vol crit, we have chosen the size of a piece with each side equal to a half of the linear side of the initial situation, i.e., area crit = 2n=2 × 2n=2 = 2n ; vol crit = 2n=2 × 2n=2 × 2n=2 = 23n=2 :
(2)
The parameters area crit, vol crit and p are Gxed for the fragmentation process. We will introduce this new stopping rule in order to represent the fact that greater pieces have more probability to be broken than the smaller ones. Our model is based on microscopic and local deGnitions of random forces which generates the fragmentation process. Hence, the fragment size distribution and the shape of the fragments are determined by these forces in a quantitative way. The dynamics of the fragmentation process and how it ends is deGned by the stopping criterion: a “natural” stopping situation based on the discrete deGnition of the material and how the forces act on it, and a random stopping situation, in which large fragments have larger probability to be broken than the smaller ones. The characteristics stated before deGne a stochastic process discrete in time and space that satisGes the Markov property, since, at each time step the knowledge of the sizes (or shapes) of the pieces and its stresses, determines which piece must be broken and the shape of the new fragments. The exact analytical study of the model is very diKcult. For this reason, in the next section we will present a numerical study which involves large scale parallel and distributed simulations. 3. Numerical results In this section, we will present the results of the numerical study implemented by large scale parallel and distributed simulations. Mainly, we are interested in determining the fragment size (area or volume) distribution, that we will denote f.s.d., and the dynamical properties of this model. This study was performed in a very small and simple network of heterogeneous and high performance computers connected by fast ethernet protocols, as it is shown in Fig. 3. The applications were programmed in C language and PVM, see Ref. [20], a very well-known software libraries for parallel and distributed computing in a message passing environment. They interacted as master– slave processes. The methodology for the simulations was the following: we averaged our results over many independent random initial conditions, characterized by the initial force conGguration, i.e., the random force acting on each axis. Since the simulations are rather time consuming we considered between 1000 and 10,000 initial conditions for the 2-d models, while for the 3-d models we could only average our results over 500–2000 initial conditions. The system linear size was N = 2n where N , for our simulations, was equal to N = 1024, 4096 and 16; 384 (n = 10, 12 and 14) in the 2-d case N = 1024 and 4096 (n = 10 and 12) for the three-dimensional case. These lattice sizes and number of initial conGgurations are signiGcantly larger than used in the study for the more
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Fig. 3. Heterogeneous networks of computers used for computations.
particular models deGned in Refs. [18,19]. We explain this fact due to three main reasons: (1) The increase in computing power of today’s computer’s CPU and main memory in comparison to 5 years ago. (2) A much better design and implementation of the data structures and programs. (3) The use of parallel and distributed processing in a small but fast network of high performance computers. For each model we computed the fragment size (area or volume) distribution obtained from the fragmentation process, f.s.d. The maximum number of iterations is N 2 = 22n for 2-d and N 3 = 23n for 3-d, which gives an exponential computing time, that was observed only in the models with stopping probability less or equal to p = 0:35; since the number of fragments which continues in the fragmentation process grows exponentially. For larger probability values, the computing time is very small, since the number of fragments generated is very small. First, we will discuss in detail the 2-d case. In this dimension the larger size considered was N = 16; 384 (n = 14). The main results can be observed in Figs. 4 –9 and Table 1: we obtain a power law behavior for the fragment size distribution, F(s) ∼ s− , for a wide range of the parameter of the model p, the stopping probability, speciGcally for: p ∈ (0:00; 0:25 ± 0:05) and area crit ∈ (n; 2n ). As we will show, the variation of the exponents with respect to the stopping probability is almost linear, and their values were obtained taking into account the Gnite size e3ects appearing for small area limits, which are not signiGcantly larger. The values of the exponents were obtained from classical techniques of non-linear regression. All the correlation coeKcients were greater than 0:98. The dependence of the 2d exponents with respect to the stopping probability values can be expressed approximately by a linear function = 1:46 − 4:05p
(3)
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Fig. 4. F.s.d. for probabilities p 6 0:25 and N = 16; 364.
Fig. 5. Non-linear exponents dependence for N = 16; 364.
which was obtained from linear regression techniques. In this case, the correlation coeKcient was 0:92. For probabilities greater than p = 0:25 ± 0:05, the f.s.d. changes smoothly from an approximate power law with positive exponent to a power law with negative exponent. Because area crit is a parameter of the model, a sensitivity analysis was performed for larger and smaller area crit values than what was considered at Grst, i.e., area crit = 2n . We implement this study for p = 0:25 since it was a value considered as an upper limit for the very accurate power law behavior and N = 4096 (n = 12). Then, the typical area in this case was area crit = 212 = 4096 and there is a very wide range for values of this parameter: [2; 22n ]. Therefore, for the sensitivity study we chose very large values and very small ones. The results are shown in Figs. 6 and 7. For large values of the parameter, Fig. 6, the results are exactly the same as those area crit = 212 = 4096, i.e., all f.s.d. corresponds to a power law with: = 1:705. The results for small values of area crit present an increasing and smooth deviation from the results obtained for the typical value, as area crit increase. This fact is shown in Fig. 7.
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Fig. 6. F.s.d. large values of area crit (N = 4096, p = 0:25).
Fig. 7. F.s.d. small values of area crit (N = 4096, p = 0:25).
G. Hernandez / Physica A 300 (2001) 13–24
Fig. 8. Fragmentation processes examples for N = 8: p = 0:05 (left), p = 0:10 (right).
Fig. 9. Fragmentation processes examples for N = 8: p = 0:15 (left), p = 0:25 (right).
Table 1 2d—Exponents for the power law behaviors for p 6 0:25 Stopping probability p
2d—Exponent
0:00 0:05 0:10 0:15 0:20 0:25
1:705 1:072 0:914 0:787 0:669 0:554
21
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Fig. 10. F.s.d. for probabilities less or equal to p = 0:25 and N = 1024. Table 2 3d—Exponents for the power law behaviors for p 6 0:25 Probability p
3d—Exponent
0:00 0:05 0:10 0:15 0:20 0:25
1:457 1:408 1:306 1:259 1:154 1:077
We can conclude that the variation of the area crit parameter mainly a3ects the constant value in the power law relation F(s) = s−
(4)
and presents only very small inQuence on the value of the exponent . In Figs. 8 and 9, we can appreciate how the fragmentation process looks for different probability values. The black lines correspond to the boundaries of the fragments. They show that the visualization of our model is very complex, for a wide range of probability values p, with patterns of fracture that resemble the real process. The numerical results for the three-dimensional case are very similar to the twodimensional ones, as shown in Fig. 10 and Table 2, but they have more signiGcant Gnite size e3ects deviations in the limit of small volume than in the two-dimensional case. If we analyze the large volume results, they show power law behaviors for the f.s.d. with a range for exponents = 1:457 for p = 0:00 to = 1:077 for p = 0:25: In this range, the exponents decrease linearly. The values of the exponents were obtained from classical techniques of non-linear regression. All the correlation coeKcients were greater than 0:87. The dependence of the 3d exponents with respect to the probability values can be expressed approximately
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by a linear function: = 1:47 − 1:55p
(5)
which was obtained from classical linear regression techniques. The correlation coeKcient was 0:99: As in the two-dimensional case, for probabilities greater than p = 0:25 ± 0:05, the f.s.d. changes smoothly from an approximate power law with positive exponent to a power law with negative exponent.
4. Conclusions In this work we have presented the results obtained from the numerical study of a simple model for two- and three-dimensional discrete fragmentation. The main features are: uniform and independent random local distribution of the forces acting on each piece, deterministic criterion for the fracture process at each step of the fragmentation based on these forces and a random stopping criterion based on the size of the fragment together with stopping conditions determined by the discrete nature of the material. The numerical study was implemented by large scale simulations over a small network of heterogeneous high performance computers. Di3erent power law behaviors were obtained for the fragment size distribution which includes power law behavior with positive exponents, for a wide range of the main parameter of the model: the stopping probability p. This parameter was included in the model in order to simulate large fracture probabilities for larger pieces. The dependence between this parameter and the power law exponent can be approximated by linear functions. For probability values below 0:25 ± 0:05, the number of fragments increases very fast at each step, producing the power law behavior. On the other hand, the area crit and vol crit parameters only generate small variations on this non-linear dependence. The power law distribution is a non-trivial result which reproduces empirical results of some highly energetic fracture processes.
Acknowledgements This work was supported by Conicyt-Chile, through Proyecto Fondecyt 199 –1026. References [1] B.R. Lawn, T.R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, Cambridge, 1975. [2] D.L. Turcotte, Fractals and fragmentation, J. Geophys. Res. 91 (1986) 1921–1926. [3] D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, 1992.
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[4] T. Ishii, M. Matsushita, Fragmentation of long thin glass rods, J. Phys. Soc. Japan 61 (10) (1992) 3474–3477. [5] M. Matsushita, K. Sumida, How do thin glass rods break? Stochastic models for one-dimensional fracture, Chuo University 31 (1988) 69 –79. [6] S. Steacy, C. Sammis, An automaton for fractal patterns of fragmentation, Nature 353 (1991) 250–252. [7] W. Newmann, I. Wasserman, Hierarchical fragmentation model for the evolution of self-gravitating clouds, The Astrophys. J. 354 (1990) 411–417. [8] S. Redner, Statistical models for the fracture of disordered media, in: H. Herrmann, S. Roux (Eds.), Random Materials and Processes Series, Elsevier, Amsterdam, 1990. [9] P.L. Krapivsky, E. Ben-Naim, Scaling and multiscaling in models of fragmentation, Phys. Rev. E 50 (5) (1994) 3502–3507. [10] R.F. Machado, J. Kamphorst Leal de Silva, Fragmentation–Inactivation: A Scaling Approach, Departamento de F$Tsica, Instituto de Ciencias Exactas, Universidade Federal de Minais Gerais, CP 702, 30161-970, Belo Hoizonte, Brazil. [11] G.J. Rodgers, M.K. Hassan, Stable Distributions in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [12] G.J. Rodgers, M.K. Hassan, Multifractality and Multiscaling in Two Dimensional Fragmentation, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [13] P. Singh, On the Kinetics of Multidimensional Fragmentation, Department of Physics Shahjalal Science and Technology University, Sylhet, Bangladesh. [14] M.K. Hassan, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [15] M.K. Hassan, Multifractality and the Shattering Transition in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, UK. [16] P.L. Krapivsky, I. Grosse, E. Bean-Naim, Scale Invariance and Lack of Self-Averaging in Fragmentation, Center for Polymer Studies and Department of Physics, Boston University, Boston MA 02215, USA. [17] G.J. Rodgers, M.K. Hassan, Distributions in Fragmentation Processes, Department of Physics Brunel University, Uxbridge, Middlesex, US8 3PH, United Kingdom. [18] Some simple models for fragmentation, G. Hern$andez, H.J. Herrmann, in: D. Les Houches, X. Beysens, Campi, E. Pfe3erkkovn (Eds.), Proceedings Workshop in Fragmentation Phenomena, April 1994, pp. 259. [19] G. Hern$andez, H.J. Herrmann, Discrete models for 2- and 3-dimensional fragmentation, Physica A 215 (1995) 420–430. [20] http:==www.epm.ornl.gov=pvm=