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Physica A 370 (2006) 565–572

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n-ary fragmentation model with nearest point flawand maximal net force fracture$

Gonzalo Hernandeza,b,�, Luis Salinasc, Andres Avilad

aUNAB, School of Civil Engineering, Santiago, ChilebUCHILE, Center for Mathematical Modeling, Santiago, Chile

cUSM, Department of Informatics, Valparaiso, ChiledUFRO, Department of Mathematical Engineering, Temuco, Chile

Received 16 October 2005; received in revised form 18 December 2005

Available online 20 March 2006

Abstract

A n-ary fragmentation model is introduced as a generalization of Ref. [G. Hernandez, Two-dimensional model for

binary fragmentation process with random system of forces, random stopping and material resistance, Physica A 323 (1)

(2003) 1–8]. Its main assumptions are: Continuous bi-dimensional material; Uniform and independent random distribution

of the net forces ðf x; f yÞ; Every fragment fracture stops with constant probability p. Furthermore, the material has q

random point flaws that interact with the maximal net forces to produce the fracture. By medium-scale simulations, it was

obtained an approximate power law for the fragment size distribution with an exponent in the range ½1:01; 1:15�. Thesimulations show close resemblance to actual fragmentation of brittle materials.

r 2006 Elsevier B.V. All rights reserved.

Keywords: n-ary Fragmentation; Random stopping; Nearest point flaw; Maximal net force fracture

1. Introduction

Fragmentation processes are common phenomena in Nature (Astronomy, Geology) and Technology(Mining, Industrial Processes). In Refs. [1–3] was given a long enumeration of some natural fragment sizedistributions of high energetic instantaneous breaking of brittle objects. This experimental evidence predicts apower-law behavior for small fragment masses:

FX ðsÞ ¼ PðX ðsÞpsÞ � as�b, (1)

where s is the fragment’s area or volume. The b exponent varies between 1:44 and 3:54, see Refs. [1–3].One-dimensional fragmentation experiments were performed in Ref. [4]. Long thin glass rods were broken

by vertically dropping them. The shape of the fragment size distribution depends on the height from which the

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.02.010

by Grants: FONDECYT 1050808 and 1040366, UNAB DI-07-04 and UTFSM 24.04.21.

ing author. UNAB, School of Civil Engineering, Sazie 2320 second floor, Santiago, Chile.

ess: gjho@vtr.net (G. Hernandez).

ARTICLE IN PRESSG. Hernandez et al. / Physica A 370 (2006) 565–572566

glass rods are dropped. A log-normal shape for smaller heights was obtained, while for increasing heights apower law: NðmÞ � am�b, where m is the fragment’s mass and bffi 1:57.

In Ref. [5] were analytically studied one-dimensional stochastic fragmentation models where the fracturepoints are randomly chosen. Power-law, exponential and log-normal fragment size distributions were obtaineddepending on the fracture point’s distribution. By the introduction of a minimal fragment size mfs (also calledcutoff) and a Poisson distribution for the number of fragments into which pieces are broken, power-lawbehavior was obtained at some stages of the fragmentation process.

A mean-field type approximation to describe the fragmentation process can be formulated by means of therate equations, see Ref. [6]:

qcðx; tÞ

qt¼ �aðxÞcðx; tÞ þ

Z 1x

cðy; tÞaðyÞf ðx=yÞdy, (2)

where cðx; tÞ is the concentration of fragments of mass less than x at time t, aðxÞ is the rate at which fragments ofmass x break into smaller ones and f ðx=yÞ is the conditional probability that a fragment of mass x was producedfrom a fragment of mass yXx. Using scaling and homogeneity assumptions, some exact results were obtained:

cðx; tÞ ¼ 1þt

s

� �2=le�ðsþtÞxl

, (3)

see Ref. [6]. Several one- and two-dimensional fragmentation models have been analytically studied by theapplication of the rate equations. See for instance Refs. [7–10].

The collision phenomena of solid but brittle discs with similar size were numerically studied in Ref. [11]. Todefine the fracture process a molecular network of faults is considered. Using large-scale simulations andmolecular dynamics techniques, the velocity, size and position distributions of the generated fragments wereobtained. By means of a dynamical model of granular solids, a critical point between damage andfragmentation behavior was determined, see Ref. [12]. Large-scale simulations and molecular dynamicstechniques have also been applied to study fracture and fragmentation models in materials that present aprevious distribution of cracks and/or flaws, see Ref. [13].

The statistical properties of a recursive d-dimensional self-similar fragmentation process were studied inRefs. [14,15]. In 2 dimensions, a rectangle breaks up in 4 rectangles with fragmentation probability 0opo1and with probability q ¼ ð1� pÞ becomes ‘‘frozen’’ and never breaks up again. It was analytically determinedthat the volume distribution is a sum of power-laws. An approximate algebraic volume distribution wasobtained for the small volume limit ðv! 0Þ:

FX ðvÞ ¼ PðX ðvÞpvÞ �gð1� pÞ

dv�g, (4)

where g ¼ 2p1=d . Furthermore, it was proved that this fragmentation model is non-self-averaging.A simple model for two-dimensional binary fragmentation was numerically studied in Refs. [16,17]. Its main

assumptions are: random distribution of the values and positions of the net forces that produce the binaryfracture; the fragmentation is generated by a deterministic criterion; every fragment fracture stops at each timestep with probability p; existence of a minimal fragment size mfs. An approximate power-law behavior for thefragment size distribution was determined for a wide range of the parameters.

In Ref. [18], the explosive fragmentation process of two different bi-dimensional systems was studied: adisordered array of elastic beams and a Lennard–Jones liquid. For both models, the cumulative fragment sizedistribution has the form:

NðmÞ / m�ae�ðm=M0Þ (5)

in the long time limit, where M0 is the typical fragment mass. This distribution is generalized for three-dimensional experiments: gypsum disks dynamic brittle fragmentation and elastic materials with crackssubmitted to scalar strain fields, see Ref. [19]:

NðsÞ / s�ðð2D�1Þ=DÞf 1 �2

l

� �D

s

!þ f 2ð�s�10 ðlþ s1=DÞ

DÞ, (6)

ARTICLE IN PRESSG. Hernandez et al. / Physica A 370 (2006) 565–572 567

where NðsÞ is the number of fragments of size s, D is the Euclidean dimension of the system, l is thepenetration depth and f 1 and f 2 can be approximated by exponential functions.

The generalization of Ref. [17] to the case of n-ary fragmentation and the existence of material flaws ispresented and numerically studied in this work. In the new model each fragment is submitted to randomstresses that produce the n-ary fracture. The number of fragments generated is a random variable and thebreaking criterion is deterministic. To model the material flaws, a random distribution of point flaws isconsidered, see Ref. [13]. To stop each fragment iterative fracture a random stopping is evaluated.

In what follows, the model is defined and the numerical results for 4-ary fragmentation are discussed.

2. Definition of the n-ary fragmentation model

At the initial situation, see Fig. 1, there are net forces of traction (or compression), denoted by f x and f y,acting over the material at random positions and directions. At each step, all the fragments will be broken in n

fragments unless they satisfy the stopping condition.The model assumptions and characteristics are:

(a)

Material point flaws: The initial fragment is a continuous square of linear size 1. To model the materialflaws are considered q random point flaws that remains fixed during the fragmentation process, see Ref.[13]. For the numerical study this parameter will be chosen q ¼ 100.

(b)

Fracture forces: For each fragment there are fracture forces (stresses) ðf x; f yÞ that are applied at randompositions of the fragments. They correspond to uniform and independent distributed random numbers in½0; 1�. Moreover, the fragmentation process is self-similar, see for instance Refs. [7,15].

(c)

n-ary fragmentation: At each step all the fragments will be broken in n (even number) fragmentsindependently, like in a cascade process, unless they satisfy the stopping condition:

(c1) n=2 fragments are obtained from the application of the forces, i.e., the fracture or cutting plane in thiscase is the plane perpendicular to the random direction of the larger net force.

(c2) n=2 fragments are obtained due to the existence of flaws in the material. The n=4 cutting planes are theplanes with normal perpendicular to the line defined by the point of application of the larger force andthe position of one of the n=4 nearest point flaws. In this work was chosen n ¼ 4: two fragments aregenerated by the interaction of the nearest point flaw with the position of the application of the largernet force.

An example of the initial step fracture is shown in Fig. 2.The parameter n can be constant or a random variable with Uniform or Poisson distribution.

(d)

Mass conservation assumption: The sum of the new fragments area is the same of the original fragment. (e) Random stopping: There are three situations in which the fragmentation process of a particular fragment

stops:

(e1) If the fragment area is smaller than the minimal fragment size or cutoff: mfs, see Ref. [5].(e2) A fragment iterative fracture stops with probability p, see Refs. [14,16,17].

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fy

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Fig. 1. Initial fragmentation situation with random point flaws.

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fxfy

F3

F2

fx > fy

F1

F4

Nearest Point Flaw Fracture Plane

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Fig. 2. Initial step fracture example.

G. Hernandez et al. / Physica A 370 (2006) 565–572568

(e3) Every fragment has a resistance 0prp1 to the breaking process. A fragment breaks only if themaximum of the net forces acting on it is greater than r. This parameter will be chosen uniform andremains fixed during the fragmentation process.

The random stopping applies for fragments of area less or equal to the critical area ac, introduced in orderto represent the fact that greater fragments have more probability to be broken than the smaller ones.

The model defined in (a)–(e) is a generalization of Ref. [17] to case of n-ary fragmentation and materialpoint flaws. Due to the model complexity a theoretical study is very difficult to develop. For this reason, theresults of the numerical study implemented by medium-scale simulations are presented in the next section. Theobjective of this study was to determine the fragment size distribution.

3. Numerical results: 4-ary fragmentation

The numerical study was performed by medium scale simulations. The methodology for the simulations wasthe following:

(1)

The parameters p, r, ac and mfs are chosen. During all the simulations the value of q was fixed in 100. (2) The results were averaged over 1000 independent random initial conditions, characterized by the initial

stresses and the random distribution point flaws.

(3) It was chosen 4-ary fragmentation since it is the minimal even number to produce non-binary

fragmentation.

(4) The fragmentation process evolves according to the rules (a)–(e) defined in Section 2.

It was determined that the fragment area distribution f ðsÞ follows approximately a power-law distribution,for different values of the parameters of the model:

f ðsÞ ffi as�bðp;r;ac;mfsÞ. (7)

ARTICLE IN PRESSG. Hernandez et al. / Physica A 370 (2006) 565–572 569

The power-law exponent depends on the main parameters of the model: The stopping probability p, theresistance r, the critical area ac and the minimal fragment size mfs. The exponent of the power-law shows anincrease with respect to [17] due to the 4-ary fragmentation and the point flaws random distribution. Thespecific values of b are shown in Table 1.

In Fig. 3 the fragments area distribution f ðsÞ is shown for: p ¼ r ¼ 0:01; 0:02; 0:03 and ac ¼ 0:01,mfs ¼ 0:0001.

The deviation from the power-law behavior for small area fragments is a finite size effect. The deviationfrom the power-law behavior for large area fragments is originated by the parameter ac.

In Figs. 4 and 5 is shown the fragmentation process evolution for different sets of parameters. It can beaffirmed that the visualizations of the model are very complex with patterns of fracture that resemble realfragmentation processes in brittle materials like, for instance, glass. The Fig. 4 shows how the fragmentationprocess changes if the parameters ac;mfs change while p; r remain fixed in the same value. In Fig. 5 shows howthe fragmentation process changes if the parameters p; r have different values while ac;mfs remain fixed.

From Fig. 4 it can be appreciated that:

(1)

Tab

Exp

p

b

If the parameter mfs is decreased, the smaller fragment typical size decreases.

(2) If the parameter p is increased, the larger fragment frequency increases. (3) if the parameter ac is increased, the larger fragment size increases.

From Fig. 5 it can be appreciated that for p; rp0:05, the effect of these parameters combine and they act asa single parameter.

le 1

onent b as a function of p ¼ r

0.01 0.02 0.03 0.04 0.05

1.15 1.08 1.05 1.02 1.00

10-6

10-5

10-4

10-3

10-2

10-1

10010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

fragment's area

frag

men

t are

a di

strib

utio

n

p=0.01p=0.02p=0.03

Fig. 3. Fragment area distribution for ac ¼ 0:01, mfs ¼ 0:0001 and p ¼ r ¼ 0:01; 0:02; 0:03.

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Fig. 4. Fragmentation process evolution for different set of parameters. First row: p ¼ r ¼ 0:1, ac ¼ 0:01, mfs ¼ 0:0005. Second row:

p ¼ r ¼ 0:1, ac ¼ 0:01, mfs ¼ 0:00005. Third row: p ¼ r ¼ 0:15, ac ¼ 0:01, mfs ¼ 0:0005. Fourth row: p ¼ r ¼ 0:15, ac ¼ 0:03, mfs ¼ 0:0005.

G. Hernandez et al. / Physica A 370 (2006) 565–572570

4. Conclusions

In this work, it was presented and numerically studied a model for n-ary fragmentation. The maincharacteristic of this model is a fracture criterion based on the nearest point flaw and maximal net force. Bymedium-scale simulations, it was determined an approximate power-law behavior for the fragments areadistribution for p; rp0:05. We affirm that these results are due to:

(1)

The mass conservation assumption. (2) The existence of fragments that cannot be broken further, i.e., a minimal fragment size.

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Fig. 5. Fragmentation process evolution for different set of parameters. First row: p ¼ 0:025; r ¼ 0:01, ac ¼ 0:01, mfs ¼ 0:0005. Secondrow: p ¼ 0:05; r ¼ 0:01, ac ¼ 0:01, mfs ¼ 0:0005: Third row: p ¼ 0:01; r ¼ 0:025, ac ¼ 0:01, mfs ¼ 0:0005. Fourth row: p ¼ 0:01; r ¼ 0:05,ac ¼ 0:01, mfs ¼ 0:0005. Fifth row: p ¼ 0:01; r ¼ 0:01, ac ¼ 0:01, mfs ¼ 0:0001.

G. Hernandez et al. / Physica A 370 (2006) 565–572 571

(3)

Deterministic criterion for the 4-ary fracture process based on the maximum net force and the nearestpoint flaw.

(4)

The random stopping condition.

The visualizations of this model are very complex with patterns of fracture that resemble real systems.

References

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