Computer Physics Communications 182 (2011) 1824–1827

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Computer Physics Communications

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Size distribution and waiting times for the avalanches of the Cell Network Modelof Fracture

Gabriel Villalobos a,∗, Ferenc Kun c, Dorian L. Linero b, José D. Muñoz a

a Simulation of Physical Systems Group, CeiBA-Complejidad, Department of Physics, Universidad Nacional de Colombia, Crr 30 # 45-03, Ed. 404, Of. 348, Bogota D.C., Colombiab Analysis, Design and Materials Group, Department of Civil and Environmental Engineering, Universidad Nacional de Colombia, Crr 30 # 45-03, Ed. 404, Of. 348, Bogota D.C., Colombiac Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 August 2010Received in revised form 27 October 2010Accepted 28 October 2010Available online 3 November 2010

Keywords:Statistical models of fractureFinite Element MethodComputational mechanics of solids

The Cell Network Model is a fracture model recently introduced that resembles the microscopicalstructure and drying process of the parenchymatous tissue of the Bamboo Guadua angustifolia. The modelexhibits a power-law distribution of avalanche sizes, with exponent −3.0 when the breaking thresholdsare randomly distributed with uniform probability density. Hereby we show that the same exponent alsoholds when the breaking thresholds obey a broad set of Weibull distributions, and that the humiditydecrements between successive avalanches (the equivalent to waiting times for this model) follow in allcases an exponential distribution. Moreover, the fraction of remaining junctures shows an exponentialdecay in time. In addition, introducing partial breakings and cumulative damages induces a crossoverbehavior between two power-laws in the histogram of avalanche sizes. This results support the idea thatthe Cell Network Model may be in the same universality class as the Random Fuse Model.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The Cell Network Model of Fracture (CNMF [1]), is a two-dimensional statistical model of fracture [2] inspired in the stressfield caused by drying of the bamboo Guadua angustifolia [3,4]. Atthe parenchymatous tissue level, bamboos shrink during drying,causing the detaching of neighboring cells and the appearance offractures. The CNMF models this tissue as a hexagonal array of cellelements (each of them made of six plane frame elements), fixedby rotational springs and joined by brittle springs called the junc-tures (Fig. 1). Shrinking forces – due to drying – acting along theelements distort the structure and cause breaking avalanches ofthe junctures among cells.

Interestingly enough, when a homogeneous distribution ofbreaking thresholds of junctures and fixed Young’s modulii areused, the histogram of avalanche sizes of the CNMF shows powerlaw behavior with an exponent of −2.93(8) [1]. This is by allmeans equal to avalanche size distribution of the random fusemodel, which shows a power law with exponent of −3 [5,6]. Theanalytical solution of both models has been elusive.

Global load sharing fiber bundle systems have provided modelsfor the thoroughly study of critical phenomena. Universality, theeffect of damage and the critical exponents have been found an-alytically (see [7,8] and references therein). Moreover, the random

* Corresponding author.E-mail address: gvillalobosc@bt.unal.edu.co (G. Villalobos).

0010-4655/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.cpc.2010.10.029

Fig. 1. Cell Network Model of Fracture (CNMF). Upper left. The plane frame elementspanning between the nodes i and j, oriented by an angle θ . Each node has twotranslational and one rotational degrees of freedom. Lower left. Two contiguous cells.Right. Structure of the CNMF. The hexagons represent the cells and the junctures arearranged into triangles. Hashing denotes fixed boundary conditions. From [1], not atscale.

fuse model has allowed to investigate the fracture properties of bi-ological materials, as in the case of brittle nacre, [9], describing thetoughness of the material by its microscopical architecture. Beammodels similar to the CNMF have also been used to model fracturein concrete [10].

In the present paper, to characterize the path to global failure,we study both the distribution of humidity decrement among con-

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secutive avalanches (the analogue of waiting times for this model),as well as the fraction of intact fibers as function of the humiditydecrement. This last quantity behaves as an order parameter forthe system (Section 2). Moreover, the universality of the CNMF isexplored numerically by characterizing the histogram of avalanchesizes for two cases: homogeneous distributions of the juncturebreaking thresholds with different widths and Weibull distribu-tions of different shapes, in Section 3. Furthermore, in Section 4a damage function is introduced as follows: Each time a stressthreshold is reached, the stiffness is reduced by a constant factor,until the fiber completely breaks. The main results and commentsare summarized in Section 5.

2. Model

The CNMF is a 2D statistical model of fracture that resemblesthe parenchymatous tissue of the bamboo Guadua angustifolia. Itis composed by two kinds of structures: cell walls and juncturesamong cells. Six cell walls arrange themselves to form hexagons,thanks to an angular springs associated with the rotational degreeof freedom. The cells are arranged like a honeycomb. The junc-tures are placed in sets of three at the common corners of thecells, modeling the silica deposits that glue cells together in Fig. 1.Each kind has a given fixed Young’s modulus for all its elements.Junctures are allowed to break, as a result of the brittle behaviorof the silica, while cell walls are not.

As boundary conditions, all the border nodes are set fixed. Thedeformation of the system comes from shrinking forces acting onevery cell wall and proportional to a global humidity loss param-eter �h. By means of a Finite Element Method (FEM), the result-ing forces and displacements of all the elements are calculated.See Ref. [1] for a detailed description of how the Finite ElementMethod is implemented in this model.

The evolution of the system has three stages. Linear elasticshrinking: local shrinking forces due to humidity losses are ap-plied to the cell walls. The differences between the local strainat the juncture and their individual thresholds are calculated. Thisstep continues until at least one fiber would suffer a strain sur-passing its threshold. Drying induced breaking: By means of a zerofinding algorithm, the exact humidity loss causing the first break-ing is found. The broken element is removed from the structure,changing the stiffness matrix in accordance. Nonlinear avalanche:The breaking of an element calls for a force redistribution overthe whole structure. This redistribution may cause an avalanche ofbreaks. When the avalanche ends, the procedure re-stars from thefirst stage.

2.1. Distributions of humidity decrement between consecutiveavalanches

The distribution of humidity decrements between successiveavalanches provides a description of the path to the global fail-ure of the system. This is the analog to the waiting times betweenavalanches of other models of fracture. Fig. 2 shows the normalizedhistograms of humidity decrements for several values of the max-imum humidity change allowed �h. When the breaking process isdriven until the end, the histograms can be fitted to an exponen-tial, with a fitted humidity constant of −0.074(1), an indication ofa lack of correlation between successive avalanches.

2.2. Fraction of remaining junctures

Let us consider the global load sharing fiber bundle model [8].Given U (σ ) as the fraction of remaining fibers at a given stressand σc the critical stress causing the global breaking of the sys-tem. Thus, U∗(σ ) − U∗(σc) behaves as an order parameter for that

Fig. 2. Humidity decrement between consecutive avalanches histogram. The param-eter that defines a curve is the maximum humidity change (�h, proportional to cellwall strain, shown). The classes (horizontal axis) are the (normalized by the meanwaiting humidity). For a maximum humidity change of 25 (in arbitrary units) theslope is fitted to −0.188(5), for a maximum humidity change of 200 it is −0.074(1).

Fig. 3. Number of intact fibers as function of the humidity change for several ho-mogeneous distribution of the breaking threshold with different widths (semi log).Inset, linear axis.

system. In this section we explore whether the fraction of intactjunctures as function of humidity change behaves as an order pa-rameter. Nonetheless, for the CNMF with homogeneous breakingthresholds at the juncture elements we do not obtain a power lawbut an exponential relaxation as can be seen in Fig. 3.

This exponential behavior is independent of the system size. InFig. 4, the fraction of the population of intact fibers as function ofhumidity is shown for different sizes of the system. The horizontalaxis was rescaled by means of a linear fit on the semi-log data. Allhistograms show an exponential decay.

3. Universality

Fiber bundle models are universal in the sense that the break-ing of the elements follows power law distribution of avalanchesizes irrespective of several system characteristics. For the CNMFwe studied the distribution of avalanche sizes when the thresholdsare generated either from flat distributions of several widths orfrom Weibull distributions with several characteristic parameters.

1826 G. Villalobos et al. / Computer Physics Communications 182 (2011) 1824–1827

Fig. 4. Scaled fraction of intact fibers as function of the humidity change (in unitsof max humidity difference) for different system sizes.

Fig. 5. Histogram of avalanche sizes for several widths of the homogeneous thresh-old distribution centered at 0.35EA.

Flat distributions of the breaking thresholds, all centered at0.35EA but with different widths (spanning on two orders of mag-nitude), show the same power-law distribution of avalanche sizes,with slopes around −3 (Fig. 5). The data shows that narrower dis-tributions show larger fluctuations around this power law thanwider ones.

The probability density function for the Weibull distributionfunction [11] is given by:

f (x;λ,m) ={

mλ(m

λ)m−1e−(x/λ)m

, x � 0,

0, x < 0,(1)

where m and λ are free parameters. It is commonly used to de-scribe the breaking thresholds of fibers by fixing λ = 1 and chang-ing m, which is the main parameter controlling the distributionshape. When we use this distribution for the breaking thresholdsat the junctures, the histogram of avalanche sizes shows also apower law behavior, with an exponent close to −2.9 for smallvalues of m. For larger values the exponential cutoff is more pro-nounced (Fig. 6).

The humidity decrements between consecutive avalanches (thatis, the waiting times) distributes like an exponential, also when the

Fig. 6. Histogram of avalanche sizes for several Weibull distributions of the breakingthresholds, with values of the shape parameters m between 1 and 10. The straightline corresponds to a slope of −2.9.

Fig. 7. Histogram of humidity increments between successive avalanches for severalWeibull distributions of the breaking thresholds, with m between 1 and 10. Thedashed line corresponds to a fit of the series of shape m = 2, with slope −0.45(2).The dash-dot line to that for m = 10, with slope −0.63(2).

thresholds follow a Weibull distribution (Fig. 7). The characteristictime for m = 2 is −0.45(2), very close to the one we gathered forflat distributions of the breaking thresholds.

4. Damage

In order to introduce a degradation for the juncture elementswe reduce the Young’s modulus of the juncture elements by adamage factor 0 < a < 1 each time the juncture fails. When theelement has suffered a maximum number of failures kmax, it is as-sumed to be broken and is removed from the structure.

When a small degradation is introduced (a = 0.1), the distri-bution of avalanche sizes does not a power law with a singleexponent. In fact, avalanches with sizes smaller than 8 seems tofollow a power law of exponent −2.98(7) and for avalanches largerthan 8 a power with exponent −2.1 (Fig. 8). A plausible physi-cal explanation would be that, a damaged juncture (that has notbeen completely removed) enhances the localization of the stress,in contrast with the case when it is completely removed; therefore,allowing for larger avalanches. Of course, much more statistics andlarger system sizes are required to clarify this point. Even smallervalues of a (not shown) cause the maximum humidity loss (and

G. Villalobos et al. / Computer Physics Communications 182 (2011) 1824–1827 1827

Fig. 8. Histogram of avalanche sizes for damage parameter a = 0.1 and several max-imal numbers of failures, kmax. The bolder line (slope −2.98(7)) fits for avalanchesizes between 1 and 7, while the thinner line (slope −2.1(1)) fits for avalanche sizesbetween 7 and 60. The breaking thresholds have a homogeneous threshold distri-bution centered at 0.35EA.

therefore the forces on the elements) to be much larger, creatingnumerical instabilities that end into poor statistics.

5. Conclusions

The numerical evidence of this work indicates that the avalanchesizes for the Cell Network Model of Fracture distribute as a powerlaw with exponent −3.0, for any broad distribution of the brakingthresholds, either if they are flat or Weibull distributed.

The distribution of waiting times shows an exponential decayfor all system sizes evaluated and all tested disorder distributionsof breaking thresholds. Even the characteristic times are similar forall of them. In our opinion, this is related to the fact that the mostcommon failure mode for the system is the softening of the sampleby displacements that would violate the boundary conditions.

Acknowledgements

We thank COLCIENCIAS (“Convocatoria Doctorados Nacionales2008”), Centro de Estudios Interdisciplinarios Básicos y Aplica-dos en Complejidad-CeiBA-Complejidad and Universidad Nacionalde Colombia for financial support. We also thank Professor JorgeA. Montoya and Professor Caori P. Takeuchi for enlightening dis-cussions in the field of Guadua drying.

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