A lattice Boltzmann wave model applied to fracture phenomena

B.Chopard and S.MarconiCUI, University of Geneva

24, rue General-Dufour1211 Gen`eve 4 Switzerland

fBastien.Chopard,Stephane.Marconi g@cui.unige.ch

Keywords: fracture, lattice Boltzmann, cellular automata,wave modeling.

Abstract: We propose to study generic features of fracturephenomena using a Lattice Boltzmann(LB) wave formalism.This formalism is numerically equivalent to the TransmissionLine Matrix(TLM) and is very similar to a cellular automatadynamics. Fracture of a discrete body is obtained by allow-ing links with local energy higher than a given threshold todisappear. To validate our use of the LB-wave model, sev-eral phenomena have been looked at: stress profile at the tipof a crack, micro-crack formation, propagation of a crack andfragmentation.

1 Introduction

Fracture processes in solids and fragmentation are importantphenomena that are not yet fully understood [1, 2]. There istherefore a clear interest to propose simple theoretical modelsthat can account for the generic properties of these phenom-ena. In particular one would like to understand what governsthe crack propagation speed and how the distribution of frag-ment size scales.

In this article, we address this problem within the so-calledlattice Boltzmann paradigm and propose a simple dynamicalmodel of a solid body at a mesoscopic scale.

Lattice Boltzmann (LB) models [3, 4] are dynamical sys-tems, discrete in space and time, aimed at simulating the be-havior of a real physical system in terms of the local densityof fictitious particles moving and interacting on a regular lat-tice. LB methods have been largely used to simulate systemsof point particles which interact locally, such as complexfluid dynamics [3, 5, 4, 6, 7], reaction-diffusion processesand wave propagation in a heterogeneous media [4].

However, modeling a solid body with this approach (i.e.modeling an object made of many particles that maintainsits shape and coherence over distances much larger than theinter-particle spacing) has remained mostly unexplored. Afirst attempt to model aone-dimensionalsolid body as a cel-lular automata is described in [8]. Then a 2D model hasbeen proposed in [4, 9]. The crucial ingredient of these mod-els is the fact that collective motion is achieved because the“atoms” making up the solid vibrate in a coherent way and

produce an overall displacement. This vibration propagatesas a wave throughout the solid and reflects at the boundary.

In this paper we review the main aspects of this method(i.e. we explain what is the LB model for wave propagationand show why it provides a natural framework to describe asimple mesoscopic model of a solid body). Then, we pro-pose a new generalized model in which the speed of sound isadjustable and which has a much more elegant mathematicalformulation including various shape of boundaries. Finally,we apply our approach to simulate numerically some genericproperties of real solids and, in particular, analyze complexphenomena such as fracture propagation, micro-cracks for-mation and fragmentation.

The paper is organized as follows. Section 2 reviews theLB formalism for the case of wave propagation. Section 3introduces the simple and general versions of our solid bodymodel and shows its relation to the LB-wave dynamics. Sec-tion 4 shows simulation results obtained for various fractureexperiments.

2 The LB-wave model

The dynamics of a LB model is expressed in terms of vari-ablesfi(~r; t) which describe some mesoscopic properties ofthe system. Thisf is the discrete counterpart of the distri-bution function used in the Boltzmann equation of statisticalmechanics. The macroscopic quantities of interest (density,velocity field,..) can be defined from thefis.

Here we shall interpretfi(~r; t) as a flux traveling on thelattice, or a population of pseudo-particles. In our notation,~r

refers to the lattice site,t the iteration time and the subscripti

labels the admissible velocities~vi (e.g. along the main latticedirections). The valuei = 0 corresponds to a population ofrest particles with~v0 = 0.

Lattice BGK models [10] are a particular type of LB mod-els in which the interaction between thefis is written as arelaxation to a local equilibrium distribution. Lattice BGKmodels are the most common implementation of LB systems.They are characterized by the following dynamics

fi(~r+�~vi; t+�)�fi(~r; t) =1

�

�f(0)

i (~r; t)� fi(~r; t)�

(1)

where� is the time step,f (0)i (~r; t) the so-called local equi-

librium distribution and� is a parameter representing a re-laxation time.

The functionsf (0)i s are the key ingredients of the modelfor they actually contain the properties of the physical pro-cess under study: this is the distribution to which the dynam-ics spontaneously relaxes and which is, therefore, intimatelyrelated to the nature of the system.

Wave phenomena, whether mechanical or electro-magnetic derive from two conserved quantities and ~J , to-gether with time reversal invariance and a linear responseof the media. The quantity is a scalar field and~J itsassociated current. The idea behind the LB approach is to“abstract” a physical process to a discrete space and timeuniverse, so that it can be efficiently simulated on a (paral-lel) computer. For waves, this generalization is obtained bykeeping the essential ingredients of real phenomena, namelyconservation of and ~J , linearity and time reversal invari-ance. Thus, in a discrete space-time universe, a generic sys-tem leading to wave propagation is obtained from Eq. (1) byan appropriate choice of the local equilibrium distribution

f(0)

i = a+ b~vi � ~J2v2

if i 6= 0; and f(0)

0 = a0 (2)

wherev is the ratio of the lattice spacing to the time step,a,a0 andb are parameters to be determined and and ~J arerelated to thefis in the standard way:

=Xi

mifi ~J =Xi

mifi~vi

Themi are weights associated with each direction.Note that, here, we make no restriction on the sign of the

fis which may well be negative in order to represent a wave.As opposed to hydrodynamics [3],f (0)i is a linear function

of the conserved quantities, which ensures the superpositionprinciple. The parametersa, b anda0 are chosen such thatthat =

Pimif

(0)

i and ~J =P

imi~vif(0)

i , which ensuresthe conservation of and ~J . For a two-dimensional squarelattice (D2Q5 to use the terminology of [3]) we choosemi = 1 and finda0 + 4a = 1 andb = 1. The freedomon the value ofa0 can be used to adjust locally the wavepropagation speed. Time reversal invariance is enforced bychoosing� = 1=2 as can be easily checked from Eq. (1) with~J ! � ~J and ! in relation (2). Note that the D2Q5lattice is known for giving anisotropic contributions to thehydrodynamic equations. These terms are not present in ourwave model because they appear with a vanishing coefficientwhen� = 1=2 [4].

The multiscale Chapman-Enskog expansion [4] can beused to derive the macroscopic behavior of and ~J whenthe lattice spacing and time step go to zero. We obtain [4, 11]

@2t� 2av2r2 = 0

which is a wave equation with propagation speedc = vp2a

(note thatv is the speed at which information travels).The wave propagation speedc can be adjusted from place

to place by choosing the spatial dependency ofa. Provided

that a0 + 4a = 1 anda0 � 0 (required for numerical sta-bility reasons), the largest possible value isa = 1=4 andcorresponds to a maximum velocityc0 = v=

p2. Therefore

media with different refraction indicesn = c0=c = 1=(2pa)

can be modeled.Perfect reflection on obstacles can be included by modify-

ing the micro-dynamics to befi(~r+ �~vi; t+ �) = �fi0(~r; t)on mirror sites, wherei0 is defined so that~vi0 = �~vi,i.e. the fluxes bounce back to where they came from witha change of sign. Absorption on non-perfect transmittersites can be obtained by modifying the conservation of toP

imif(0)

i = �, where0 � � � 1 is an attenuation factor.This modifiesa ! �a anda0 ! �a0. Finally, by substitut-ing (2) into (1) and using the expression ofa anda0 in termsof c, free propagation with refraction indexn(~r), and partialtransmission and reflection can be expressed as

fi(~r + �~vi; t+ �) =�

2n2� fi+2(~r; t)

f0(~r; t+ �) = 2�n2 � 1

n2� f0(~r; t) (3)

In this equation,� = 0 corresponds to perfect reflection,� =1 to perfect transmission and0 < � < 1 describes a situationwhere the wave is partially absorbed. A particular version ofour LB-wave model has been successfully validated by theproblem of radio wave propagation in a city [12].

In hydrodynamic models,� = 1=2 corresponds to the limitof zero viscosity, which is numerically unstable. In our case,this instability does not show up provided we use an appro-priate lattice. Indeed, in the D2Q5 lattice, our dynamics isalso unitary [13] which ensures that the quantity

E =Xi

f2i (4)

is conserved. This extra property prevents thefis from be-coming arbitrarily large (with positive and negative signs,since is conserved). It also offers us the possibility to de-fine an energy in the model. This will turn out useful in thefollowing sections.

Note finally that the above LB formulation is yet anotherway to express the generalized Transmission Line Matrix(TLM) numerical scheme [14, 15, 16, 17].

3 A LB mesoscopic solid body model

3.1 A model with a simple geometrical interpretation

A cellular automata model of a one-dimensional chain ofparticles has been discussed in [8]. Similarly, in 2D, asolid body can be thought of as an arrangement of particleslinked to their four nearest neighbors with a spring-like in-teraction [4, 9]. An elementary cell of such a solid is dis-played in Fig. 1 where a particle is shown with its four con-nected neighbors. The configuration shown in this figure cor-responds to a situation with a local deformation: the fourneighbors are not at the same distance from the central parti-cle and the positions differ from that of a square lattice.

f1

f2f3

f4time t time t +1

f1

f2f3

f4

Figure 1: Picture of a generic “atom” and its four connectedneighbors in our 2D solid model. The~fis are defined as thelocal deformation along each direction. The cross indicatesthe location of the geometrical center of mass of the fourwhite particles. At the next iteration, the black particle jumpsto a symmetrical position with respect to this point.

The system will react to this deformation in the follow-ing way. First we consider a simple dynamics in which thesolid is made up of two sub-lattices. We term them black andwhite, by analogy to the checkerboard decomposition. Thedynamics consists in moving the black particles as a func-tion of the positions of their white, motionless neighbors, andvice-versa, at every other steps.

As shown in Fig. 1 we denote the location of the black par-ticle by ~ri;j = (xi;j ; yi;j). The surrounding white particlesare at positions~ri�1;j , ~ri+1;j , ~ri;j�1 and~ri;j+1. We definethe separation to the central black particle as

~f1(i; j) = ~ri;j � (~ri�1;j + ~h)

~f2(i; j) = ~ri;j � (~ri;j�1 + ~u)

~f3(i; j) = ~ri;j � (~ri+1;j � ~h)

~f4(i; j) = ~ri;j � (~ri;j+1 � ~u)

(5)

where the~fi are now vector quantities, and~h = (r0; 0) and~u = (0; r0) can be thought of as representing some horizon-tal or vertical equilibrium lengthr0 of the spring connectingadjacent particles.

In this simplest version of our model, the motion of theparticles can be defined by the following geometrical rule(we now assume that the black particles are moving): a bulkparticle such as the black one shown in Fig. 1 moves to asymmetric location with respect to the center of mass of itsfour neighbors, namely(1=4)[~ri�1;j + ~h + ~ri+1;j � ~h +~ri;j�1 + ~u+ ~ri;j+1 � ~u].

Then, in a second step, the white particles move and theblack ones remain fixed. The~fis are recomputed accordingto (5), but this time from the point of view of the white parti-cles and their respective neighbors.

The complete dynamics is obtained by alternating theblack and white motions. It turns out [4, 9] that the above ruleof motion is equivalent to updating the local deformation~fisby the lattice Boltzmann dynamics given in Eq. (3), providedthat we choosen = 1(which implies that~f0 = 0). This isnot surprising since deformation propagates like a wave in asolid.

It is also found that the value of~ =P

imi~fi for particle

with label(k; l) in the 2D lattice corresponds to the momen-tum ~p of that particle. Indeed, simple algebra [4] shows thatthe new location~rkl(t+ 1) of particle(k; l) is given by

~rkl(t+ 1) = ~rkl(t)� (1=Mkl)~

whereMkl is a mass (yet to be determined) associated withthe particle.

Note also that, in this interpretation,~Jx =P

i vix~fi and

~Jy =P

i viy~fi computed for a particle(k; l) respectively

give, the separation between its left-right and up-down neigh-bors. Likewise, the quantityE defined in (4) is interpreted asthe local energy of the particle. It clearly takes the form ofan elastic energy of deformation.

Whereas mass, momentum and energy are naturally con-served in this dynamics which is not the case of angularmomentum. Indeed, in the present model, the coupling be-tween adjacent particles is not given by the Euclidean dis-tance but is decoupled along each coordinate axis (however,a deformation along thex-direction will propagate along they-direction and conversely). Although this technique breaksup the rotational invariance, it allows to work with a squarelattice and is expected not to play a crucial role in the fractureprocesses we shall consider below.

3.2 Boundary Conditions

The updating rule explained above works well for particleshaving exactly four neighbors. However, this is not alwaysthe case when describing a 2D solid body which has someboundaries. For a square-shaped solid body, particles mayhave two or three neighbors. In what follows, we shall con-sider fracture processes which may produce particle withonly one link.

From our geometrical interpretation of the dynamics, weobtain the rule of motion for the particles at the border of thesystem by applying the same rule, but with less neighbors:(1) we consider the locations~r` of the k neighbors of theparticle to be moved; (2) we add to these positions a vectorrepresenting the connecting “spring” at rest; (3) we computethe geometrical center of mass of thesek quantities (4) thenew location of the jumping particle is given by its symmet-rical position with respect to the center of mass.

An algebraic expression for this rule can be found. It takesa simple form provided that a different mass is defined for thebulk or the border particles. Some simple calculations showthat the appropriate massM of a particle should be

M = k1

2

The precise formulation of the dynamics is given below, forthe case of the generalized model.

3.3 Generalized model

As was discussed in the previous section, the above meso-scopic model of a solid is described by a LB dynamics. How-

ever, Eq. (3) allows for more freedom than what we have im-posed geometrically on our model. In particular there is noneed to partition the solid’s atoms in white and black sub-sets. Also, one can add to each atom(k; l) a rest field~f0which allows us to adjust locally the speed of sound of thesystem.

If we relax the checkerboard updating rule and add a fifthfield, the interpretation of the~fis is no longer given consis-tently by the local deformation as expressed in Eq. (5). How-ever, the conserved quantities like the momentum~ and theenergyE are still well defined within this extension and canbe used to interpret the model.

It turns out that Eq. (3) can be rewritten in a quite elegantway to take into account the fact that some particles may haveless than five links

~fi(t+ 1) = �~

M� ~fi+2

~f0(t+ 1) = �

p2M0

M~� ~f0 (6)

where the quantityM0 is the mass associated with the inter-nal link ~f0 and is related to the refraction indexn as

M0 = 2(n2 � 1)

The massM of a particle is then given byM0 and the con-tributions of allk connected links, namely

M = M0 + k1

2

since each spatial link counts for 1/2.If � = 1, it can be shown by direct substitution that the

momentum~ =

p2M0

~f0 +Xi>0

~fi

and the energyE =

Xi�0

f2i

are conserved by Eq. (6), for any value ofk.In the above expression, the summation overi concerns

thek existing links. Note also that in Eq. (6), it is assumedthat, for missing links, the corresponding fieldsfis are al-ways zero.

Since the geometrical interpretation no longer holds withthe dynamics given by (6) (also remember that it does notanymore requires a checkerboard updating rule), the positionof each particle must be computed from its momentum ac-cording to

~r(t+ 1) = ~r(t)�=M

Also, one must specify the initial values of the~fi in a con-sistent way. A possibility is to set them so that the macro-scopic deformation�L =

P~r J, whereJ =

Pi~fi~vi is a

tensor interpreted as the local spatial deformation. Note thatJ is not conserved at the borders.

In this model, total energyE of each particle is thus awell defined quantity as well as its kinetic energyEkin =

1

2

3

4

5

Figure 2: A 100x100 particles solid object bouncing againstwalls. The solid line is the trajectory of the center of mass.The numbers correspond to the labels in the plot of Fig .3

(1=2M)2. From these two quantities an internal energyEint � Etot � Ekin can be introduced. The internal energyEint can be set proportional to a temperatureT using theequipartition theorem. In the initial configuration,T is typi-cally introduced by adding a noise of standard deviation

pT

to the rest position of each atom.

3.4 Fracture rule

We now show how our LB solid model can be used to de-scribe a fracture process. Fracture is a phenomena for whichno definite theory is available [1, 2] and a simple model iscertainly useful to help understanding generic properties.

Since our LB-wave model can easily accommodate borderconditions, only one step is needed to introduce a dynamicalchange in these borders by allowing for the links between theparticles of the lattice describing our solid to be removed. Atthe level of our description, a link may break locally if itsdeformation is too large. Precisely we consider the energystored in a link as the quantity determining the breaking. TheenergyE`(~r; t) of each link` = 1::4 at site~r and timet isdefined asE` = ~f2` . Since, as mentioned above, our dynam-ics is unitary, the total energyEtot =

P~r

�f20 +

P`E`(~r)

�is conserved until a link breaks.

Specifically, the breaking rule we impose is the following:a link ` breaks if the correspondingE` is larger than a giventhreshold�(~r)

E` � �(~r) (7)

which may, in principle, depend on the position (local de-fects). Particles with one or more broken links then behavelike particles at a free boundary.

4 Numerical experiments

In this section, we will show how the LB-wave model can de-scribe the kinematics of an object, the propagation of stressin it and how the extra breaking rule can be applied to sev-eral studies in fracture phenomena. The strength of the modellies in the fact that only initial conditions mimicking experi-mental conditions are needed to apply the model to different

0 500 1000 1500100

200

300

400

500

600

700

800

900

1000

Kinetic EnergyInternal EnergyTotal Energy

Iterations

Ene

rgy

1 2 3 4 5

Figure 3: Conservation of energy for the bouncing objectin Fig. 2. At each bounce, kinetic energy is transformed ininternal energy reflecting the deformation of the object.

applications. Our intention is not as much to go in full simu-lations of these phenomena as illustrate the versatility of theLB-wave model.

4.1 A large moving object

One of the objectives of the LB-wave has been to describe thekinematics of a solid body. The goal was to obtain a macro-scopic behavior from local rules of motion. To illustrate thiscapacity of the model, Fig. 2 shows the trajectory of a square100x100 particles object bouncing against walls (notice thatthe object has conserved its integrity).

Fig 3 illustrates the conservation of energy as defined in(4). The object has an initial temperature, i.e. noise on theinitial positions of particles, which leads to a non-zero ini-tial internal energy. At each bounce, kinetic energy is trans-formed into internal energy which corresponds to the defor-mation of the object. After the bounce, practically all of thekinetic energy is restored, though one can see that the inter-nal energy is slowly increasing with successive bounces.

4.2 Stress at the tip of a crack

However useful, the study of a flawless, homogeneousmedium helps little with the understanding of fracture whoseorigin depends deeply on the length scales where the het-erogeneity of the chemistry and the texture of the mediumcannot be disregarded. However, at an intermediate lengthscale, the texture of a medium can be seen as a continu-ous medium with two kind of randomly distributed hetero-geneities called micro-cavities: “pores” and “micro-cracks”.Pores have roughly the same length in all directions. Micro-cracks on the other hand are much longer than they are wide.

It is useful then to understand how a micro-cavity modifies

x

E

Mode-I load

r

y

Figure 4: Energy surface plot of a 150x150 particles solidwith a slit. A mode-I load is applied in the direction perpen-dicular to the slit.

the mechanical behavior of a medium. To do so we consideran experiment similar to Young’s experiments. We take a2D solid body with a slit, which models a micro-cavity, andapply a mode-I load in the direction perpendicular to the slit.In our simulation, the solid is in fact preconstrained with theload and the borders in the direction of the load are kept fixed.We then measure the energy at each site of the lattice.

Fig. 4 shows how energy is distributed in the solid in suchan experiment. For a 2D solid, analytical calculations arepossible. It is thus known [18] that for a elliptical hole in aninfinite, linearly elastic medium, the stress decays like1=

pr

wherer is the distance to the tip, while far from the hole thedecay is that of a dipole namely� 1=r2.

Fig.5 shows the decay of energy as a function of the dis-tance to the tip of the slit. The plot is taken at the x-coordinate corresponding to x position of the slit. It is thenfitted to the dipole field with a logarithm scale in the verticalaxis to have a better visualization. Fits to1=

pr do not hap-

pen to be as good. This shows that the LB-wave model easilygives a correct behavior at long distance. However the fail-ure to capture the short distance behavior can be accountedfor by the discrete nature of the lattice on which we modelthe stress.

4.3 Crack speed and profile

Mechanical considerations are necessarily of importance tounderstand how much a material is subject to fracture. Het-

0.1346 / 72α 1.961β 29.97

distance from crack tip

Ene

rgy

Energy = α+β/r2

1

10

10 20 30 40 50 60 70

Figure 5: Stress profile at distance r from the tip of a micro-cavity. The stress behaves like a dipole field with decay�1=r2 as shown by the fit.

erogeneities such as micro-cavities can, as we have seen, giverise to high stress in a material which can in turn develop thecavity into a full fracture. However these considerations arestatic in nature and many observations have shown that thedynamics of fracture is in itself as important to understand.Intuitively speaking, any cavity inside a material gives riseto high stress which can lead to the growth of the cavity.Depending on how the growth behaves, we are potentiallyconfronted with a self-entertaining mechanisms.

Figure 6: Illustration of a smooth crack simulated using theLB-wave model on a 150x150 particles solid body with at-tenuation� = 0:90. The original slit in the solid is repre-sented with a solid line.

The study of the dynamics of fracture growth usually takesinto consideration two aspects of a fracture: the cracks shape

Figure 7: Illustration of a branching crack simulated usingthe LB-wave model on a 150x150 particles solid body withattenuation� = 0:98. The original slit in the solid is repre-sented with a solid line.

and its speed with regard to the speed of soundcs in the ma-terial under study. The shape of a crack can be roughly di-vided into two modes: smooth and branching. Phenomeno-logical observations show that cracks travel at speeds whichare lower than the speed of sound. The branching crackstravel at higher speeds than smooth cracks. This can be intu-itively understood in the following way: if the tip of a cracktravels faster than the energy released by it, the energy leftover behind the tip will eventually produce a bifurcation. Thetransition between the two modes has usually been observedat about half the speed of sound.

Despite the simplicity of our model, the simulations doreproduce the two modes of fractures as can be seen on Fig. 6and Fig. 7. The setup for the simulation of the propagationof a fracture is similar to the one described in the previoussection (4.2) with the added breaking rule (7).

There are several parameters one can adjust to change thebehavior of the crack such as the initial load on the body,noise or temperature added to this load, the threshold on en-ergy before a link breaks, a noise on this threshold, an atten-uation� on the LB-wave dynamics and the initial length ofthe slit. However, the main ingredient to distinguish the twomodes in our simulations is the attenuation� .

In addition we are able to adjust the speed of soundcs =c0=n in the lattice. We observe than the rule of the thumb forthe smooth/branching crack transition is conserved by vary-ing cs, see Fig. 8 forn = 1:0 and Fig. 9 forn = 1:5 .

4.4 Micro-cavities

Fracture precursors or micro-cavities have been the subjectof statistical study [19]. It has been suggested by [20] thatthe patterns of micro-cavities might be of fractal nature. Themeasurement suggested is to look at the evolution of the dis-

cs/2

iteration

c/c s

Macro-branching crack

Smooth crack

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

25 50 75 100 125 150 175 200 225

Figure 8: Relative speed of cracks in a solid body with refrac-tion indexn = 1:0, the smooth cracks speed is smaller andmore regular than that of a branching crack. The solid line in-dicates the phenomenological transition between smooth andbranching cracks.

cs/2

iteration

c/c s

0

0.05

0.1

0.15

0.2

0.25

0.3

75 100 125 150 175 200 225 250 275

Figure 9: Relative speed of cracks in a solid nody with refrac-tion indexn = 1:5, the smooth cracks speed is smaller andmore regular than that of a branching crack. The solid line in-dicates the phenomenological transition between smooth andbranching cracks.

(a) (b)

(c) (d)

Figure 10: Micro-cavities pattern in solid bodies of varioussizes with mode-I load: (a) 20x20, (b) 50x50, (c) 100x100and (d) 200x200. For the sake of comparison, the sampleshave been scaled such that they all appear the same size.

tribution of micro-cavities for samples of various size. Ifthe dynamics is scale independent one expects the numberof micro-cavities to be proportional toL2 whereL is the lin-ear size of our lattice.

The simulation consists in applying a mode-I load in thex- and y-direction of our 2D solid. Again, the solid is pre-constrained with the appropriate load and all borders are keptfixed. The pattern of micro-cavities obtained in such a wayfor various size of solid can be seen in Fig.(10) for a solidof size 20x20 up to 200x200. The only measure taken intoaccount is the numberNL of broken links. In fact, this mea-sure yields a distribution which does scale with the size ofthe solid i.e.

NL � L2

A more thorough investigation of the parameter phase spaceor some new ingredient might reveal some fractal behavior.Nonetheless, the next section will show that if the growth ofthese micro-cracks is left to its own, eventually leading tothe destruction of the solid, there is a measurement whichindicates a complex process at work.

4.5 Fragmentation

Fragmentation is a fracture phenomena pushed to its limitsin the sense that it is actually a destruction process. To do so,a very large stress is applied on a body in a very short time.Dissipation will therefore not work to unload the solid andonly the creation of a considerable amount of free surfacesi.e. fractures will do. Due to the destructive nature of frag-mentation, one typically observes the distribution of frag-ment sizes which results from the breakup process. Thesebreaking mechanisms will depend on the material properties,the geometry of the solid body and on the energy input to

347.3 / 987α -0.9556β 1017.

r

M(r) = β r−α

1

10

10 2

10 3

1 10 102

103

Figure 11: Cumulative mass distribution of the fragmenta-tion of a 100x100 particles solid body. The distribution isfitted according to a power-law and gives a value of� 1:0for the exponent.

the system which can be controlled to some extend with theLB-wave model.

In the case of destructive breaking, the distribution of cu-mulative mass of fragmentsM(r), i.e. the number of frag-ments with mass less thanr, are often found to follow a sim-ple power-law

M(r) � r� (8)

with exponent values� to vary between 0.5 and 1.0 . Thispower-law behavior is usually understood to be the signatureof a critical self-organized dynamics. At the least it points tothe fact that the dynamics is taking place at all scales.

Our simulations show that the LB-wave model capturesthis feature of fragmentation, see Fig. 11 for the power-lawdistribution of cumulative mass of fragments and Fig. 12 forthe actual fragmentation of the solid body.

It has been observed in numerical experiments that withvery high stress the distribution starts being exponential [21].This has also been observed in our simulations; we proposethat this effect is due to the finite size of the sample.

5 Conclusion

We have shown that the LB-wave model can model variousgeneric aspects of solid body physics such as kinematics,stress propagation, cracks and fragmentation. We have vali-dated our model by numerical experiments with a solid bodyrepresented by a two-dimensional square lattice. However,the generalization to a three-dimensional model is straight-forward and the use of other lattice topologies should be pos-sible (e.g. hexagonal). The simplicity of the model allowsfor a richness of investigation which can still be augmented;for instance, plasticity could be introduced by allowing dy-namical changes in the rest length of links between particles.

Figure 12: Fragments resulting from the destruction of a100x100 particles solid body. The cumulative mass of thefragments follows a power-law, see Fig. 11

Some local properties of the solid body could also be easilyrelated to the energy.

Finally, we would like to stress that this simplicity allowsfor easy and efficient implementation on any kind of com-puter, personal or parallel, which seems to lack in many cur-rent models [22].

6 Acknowledgements

We acknowledge support from the Swiss funding agency forscience and education (OFES).

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