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

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The Cell Network Model is a fracture model recently introduced that resembles the microscopical structure and drying process of the parenchymatous tissue of the Bamboo Guadua angustifolia. The model exhibits a power-law distribution of avalanche
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  Size distribution and waiting times for the avalanches of the Cell NetworkModel of Fracture Gabriel Villalobos a ∗ , Ferenc Kun c , Dorian L. Linero b , Jos´e D. Mu˜noz 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., Colombia., b 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., Colombia., c Department of Theoretical Physics. University of Debrecen, H-4010 Debrecen, P.O.Box 5, Hungary. August 4, 2010 Abstract The Cell Network Model is a fracture model recently in-troduced that resembles the microscopical structure anddrying process of the parenchymatous tissue of the Bam-boo  Guadua angustifolia  . The model exhibits a power-lawdistribution of avalanche sizes, with exponent  − 3 . 0 whenthe breaking thresholds are randomly distributed with uni-form probability density. Hereby we show that the sameexponent also holds when the breaking thresholds obey abroad set of Weibull distributions, and that the humiditydecrements between successive avalanches (the equivalentto waiting times for this model) follow in all cases an ex-ponential distribution. Moreover, the fraction of remaining junctures shows an exponential decay in time. In addition,introducing partial breakings and cumulative damages in-duces a crossover behavior between two power-laws in theavalanche size histograms. This results support the ideathat the Cell Network Model may be in the same univer-sality class as the Random Fuse Model.Statistical models of fracture Finite Element MethodComputational mechanics of solids.PACS 02.50.-r,05.90.+m,46.50.+a,62.20.F-, 62.20.M- 1 Introduction The Cell Network Model of Fracture (CNMF [1]), is a twodimensional statistical model of fracture [2] inspired in thestress field caused by drying of the bamboo  Guadua angus-tifolia   [3, 4]. At the parenchymatous tissue level, bamboosshrink during drying, causing the detaching of neighboringcells and the appearance of fractures. The CNMF mod-els this tissue as an hexagonal array of cell elements (eachof them made of six beams), fixed by angular springs and joined by brittle springs called the junctures. (Figure: 1). ∗ Corresponding author, gabrielvc@gmail.com Shrinking forces -due to drying- acting along the elementsdistort the structure and cause breaking avalanches of the junctures among cells. Cell N Cell M CellWalls Junctures Nodes θ v  j u  j u   i v i ϕϕ i j PFE Figure 1: Cell Network Model of Fracture (CNMF).  Upper left  , The plane frame element spanning between the nodes i  and  j , oriented by an angle  θ . Each node has two trans-lational and one rotational degrees of freedom.  Lower left  .Two contiguous Cells.  Right   Structure of the CNMF. Thehexagons represent the cells and the junctures are arrangedinto triangles. Hashing denote fixed boundary conditions.From [1] Not at scale.Interestingly enough, when an homogeneous distributionof breaking thresholds of junctures and fixed Young mod-ulii are used, the histogram of avalanche sizes of the CNMFshows power law behavior with an exponent of   − 2 . 93(8)([1]). This is by all means equal to avalanche size distribu-tion of the random fuse model, which shows a power lawwith exponent of  − 3 [5, 6]. The analytical solution of bothmodels has been elusive.One dimensional fiber bundle systems have providedmodels for the thoroughly study of critical phenomena.Universality, the effect of damage and the critical expo-1   a  r   X   i  v  :   1   0   0   8 .   0   6   0   9  v   1   [  c  o  n   d  -  m  a   t .  s  o   f   t   ]   3   A  u  g   2   0   1   0  nents have been found analytically (see [7, 8] and referencestherein.). The logical extension of those models to 2D, therandom fuse model, has allowed to investigate the fractureproperties of biological materials, as in the case of brittlenacre, ([9]), describing the toughness of the material byits microscopical architecture. Beam models similar to theCNMF has also been used to model fracture in concrete[10].In the present paper, to characterize the path to globalfailure, we study both the distribution of humidity decre-ment among between consecutive avalanches (the analogueof waiting times for this model), as well as the fraction of intact fibers as function of the humidity decrement. Thislast quantity behaves as an order parameter for the system(see  (Sec: 2 )). Moreover, the universality of the CNMFis explored numerically by characterizing the histogram of avalanche sizes for two cases: homogeneous distributions of the juncture breaking thresholds with different widths andWeibull distributions of different shapes,  (Sec: 3 ). Fur-thermore, in section  (Sec: 4 ) a damage function is intro-duced as follows: Each time a stress threshold is reached,the stiffness is reduced by a constant factor, until the fibercompletely breaks. The main results and comments aresummarized in Sec  (Sec: 5 ). 2 Model The CNMF is a 2D statistical model of fracture that re-sembles the parenchymatous tissue of the bamboo  Guadua augustifolia  . It is composed by two kinds of structures:cell walls and junctures among cells. Six cell walls arrangethemselves to form hexagons, thanks to a angular springsassociated with the rotational degree of freedom. The cellsare arranged like a honeycomb. The junctures are arrangedin sets of three at the common corners of the cells, mod-eling the silica deposits that glue cells together. (Figure:1) Each kind has a given fixed Young modulus for all itselements. Junctures are allowed to break, as a result of thebrittle behavior of the silica, while cell walls are not.As boundary conditions, all the border nodes are setfixed. The deformation of the system comes from shrink-ing forces acting on every cell wall and proportional to aglobal humidity loss parameter ∆ h . By means of a FiniteElement Method, the resulting forces and deformations of all the elements are calculated.The evolution of the system has three stages.  Linear Elastic Shrinking  : local shrinking forces due to humiditylosses are applied to the cell walls. The differences betweenthe local strain at the juncture and their individual thresh-olds are calculated. This step continues until at least onefiber would suffer an strain surpassing its threshold.  Dry-ing induced Breaking  : By means of a zero finding algo-rithm, the exact humidity loss causing the first breakingis found. The broken element is removed from the struc-ture, changing the stiffness matrix in accordance.  Nonlin-ear avalanche  : The breaking of an element calls for a forceredistribution over the whole structure. This redistributionmay cause an avalanche of breaks. When the avalancheends, the procedure re-stars from the first stage. 2.1 Distributions of humidity decrementbetween consecutive avalanches The distribution of humidity decrements between succes-sive avalanches provides a description of the path to theglobal failure of the system. This is the analog to the wait-ing times between avalanches of other models of fracture.Fig. (Figure: 2) shows the normalized histograms of hu-midity decrements for several values of the maximum hu-midity change allowed ∆ h . When the breaking process isdriven until the end, the histograms can be fitted to an ex-ponential, with a fitted humidity constant of 13 . 5(2), whichis and indication of lack of correlation between successiveavalanches. 110100100010000024681012Parameter: Maximum h. loss= 5= 10= 15= 20= 25= 100= 200= 500 Figure 2: Humidity decrement between consecutiveavalanches histogram. The parameter that defines a curveis the maximum humidity change (∆ h , proportional to CellWall strain, shown). The classes (horizontal axis) are the(normalized by the mean waiting humidity). 2.2 Fraction of remaining junctures Let us consider the one-dimensional global load sharingfiber bundle model [8]. Given  U  ( σ ) as the fraction of remaining fibers at a given stress and  σ c  the criticalstress causing the global breaking of the system. Thus, U  ∗ ( σ ) − U  ∗ ( σ c ) behaves as an order parameter. For theCNMF with homogeneous breaking thresholds at the junc-ture elements we obtain an exponential relaxation in thenumber of remaining junctures ((Figure: 3)).2  0.00010.0010.010.1120406080100120140160180200 Homogeneous threshold distribution half width 'h' 00.10.20.30.40.50.60.70.80.9120406080100120140160180200Homogeneous threshold distribution half width 'h' w = 0.01 w = 0.05 w = 0.1 w = 0.2 w = 0.5  w = 0.01  w = 0.05  w = 0.1  w = 0.2  w = 0.5 Figure 3: Number of intact fibers as function of the hu-midity change for several homogeneous distribution of thebreaking threshold with different widths (semi log).  Inset  ,linear axis.This exponential behavior is independent of the systemsize. In (Figure: 4), the fraction of the population of intactfibers as function of humidity is shown for different sizes of the system. The horizontal axis was rescaled by means of a linear fit on the semi-log data. All histograms show anexponential decay. 1 e -050.00010.0010.010.1100.511.52       N      (      h      ) h (rescaled)Size of the system162 El. 66 J262 El. 112 J386 El. 170 J534 El. 240 J706 El. 322 J Figure 4: Scaled fraction of intact fibers as function of thehumidity change (in units of max humidity difference) fordifferent system sizes. 3 Universality Fiber bundle models are universal in the sense that thebreaking of the elements follows power law distributionof avalanche sizes irrespective of several system charac-teristics. For the CNMF we studied the distribution of avalanche sizes when the thresholds are generated eitherfrom flat distributions of several widths or from Weibulldistributions with several characteristic parameters.Flat distributions of the breaking thresholds, all centeredat 0 . 35 EA  but with different widths (spanning on two or-ders of magnitude), show the same power-law distributionof avalanche sizes, with slopes around − 3 (Fig. (Figure: 5)).The data shows that narrower distributions show largerfluctuations around this power law than wider ones. 1 e -0 61 e -0 50.00010.0010.010.11110100       N      (      S      i     z     e      ) SizeWidth of distributionHalf width. = 0.01Half width. = 0.02Half width. = 0.05Half width. = 0.1Half width. = 0.2Half width. = 0.5Half width. = 1slope = -3.0(1) Figure 5: Histogram of avalanche sizes for several widthsof the homogeneous threshold distribution centered at0 . 35 EA The probability density function for the Weibull distri-bution function [11] is given by: f  ( x ; λ,k ) =   kλ  kλ  k − 1 e − ( x/λ ) k x ≥ 00  x <  0  ,  (1)where  k  and  λ  are free parameters. It is commonly usedto describe the breaking thresholds of fibers by fixing  λ =1and changing  k , which is the main parameter controllingthe distribution shape. When we use this distribution forthe breaking thresholds at the junctures, the histogram of avalanche sizes shows also a power law behavior, with anexponent close to -2.9 for small values of k. For largervalues the exponential cutoff is more pronounced. (Figure:6).The humidity decrements between consecutiveavalanches (that is, the waiting times) distributeslike an exponential, also when the thresholds follow aWeibull distribution ((Figure: 7)).The characteristic timefor  k =5 is 16.1(8), very close to the one we gathered forflat distributions of the braking thresholds.3  1e-050.00010.0010.010.111 10        N       (       S       ) SK=1K=1.5K=2K=2.5K=3K=5K=7K=10f1(x) Figure 6: Histogram of avalanche sizes for several Weibulldistributions of the breaking thresholds, with values of theshape parameters  k  between 1 and 10. The straight linecorresponds to an slope of   − 2 . 9. 4 Damage In order to introduce a degradation for the juncture ele-ments we reduce the Young modulus of the juncture ele-ments by a damage factor 0  < a <  1 each time the juncturefails. When the element has suffered a maximum numberof failures  k max , it is assumed to be broken and is removedfrom the structure.When a small degradation is introduced ( a =0 . 1), thepower law distribution of avalanche sizes seems to exhibit acrossover from an exponent  − 3 to an exponent  − 2 at sizesaround 8 (Fig. (Figure: 8)). This may indicate that theremaining elasticity helps to sustain the structure. How-ever, more statistics and larger system sizes are requiredto clarify this point. Even smaller values of   a  (not shown)cause the maximum humidity loss (and therefore the forceson the elements) to be much larger, creating numerical in-stabilities that end into poor statistics. 5 Conclusions The numerical evidence of this work indicates that theavalanche sizes for the Cell Network Model of Fracture dis-tribute as a power law with exponent  − 3 . 0, for any broaddistribution of the braking thresholds, either if they are flator Weibull distributed.The distribution of waiting times show an exponentialdecay for all system sizes evaluated and all tested disorderdistributions of breaking thresholds. Even the characteris-tic times are similar for all of them. In our opinion, this isrelated to the fact that the most common failure mode forthe system is the softening of the sample by displacementsthat would violate the boundary conditions. 11 010010002 4 6 8 10 12 14 16 18 20    N   (   h   /  o  v  e  r   l   i  n  e   h h/overline hK=1K=1.5K=2K=2.5K=3K=5K=7K=10f2(x)f8(x) Figure 7: Histogram of humidity increments between suc-cessive avalanches for several Weibull distributions of thebreaking thresholds, with  k  between 1 and 10. The blueline corresponds to a fit of the series of shape  k  = 10, withslope  − 2 . 1. The green line to that for  k  = 3, with slope − 2 . 4 Acknowledgments:  We thank  COLCIENCIAS   (“Con-vocatoria Doctorados Nacionales 2008”),  Centro de Estudios Interdisciplinarios B´ asicos y Aplicados en Complejidad- CeiBA - Complejidad   and  Universidad Na-cional de Colombia   for financial support. We alsothank Professor Jorge A. Montoya and Professor Caori P.Takeuchi for enlightening discussions in the field of Guaduadrying. References [1] G. Villalobos, D. L. Linero, J. D. Mu˜noz, A statisti-cal model of fracture for a 2d hexagonal mesh: thecell network model of fracture for the bamboo guaduaangustifolia, Computer Physics Communications doi:http://dx.doi.org/10.1016/j.cpc.2010.06.015 .URL  arXiv:1002.3417 [2] M. J. Alava, P. K. V. V. Nukala, S. Zapperi, Statisticalmodels of fracture, Advances in Physics 55 (2006) 349.URL  arXiv.org:cond-mat/0609650 [3] J. A. Montoya Arango, Trocknungsverfahren furr diebambusart guadua angustifolia unter tropischen be-dingungen, Ph.D. thesis, Universitat Hamburg (2006).[4] C. Takeuchi, J. F. Rivera, Structural behaviour of braced guadua frames., in: 11th International Confer-ence on Non-conventional Materials and TechnologiesNOCMAT, 2009.[5] S. Pradhan, A. Hansen, P. C. Hemmer, Crossover be-havior in failure avalanches, Phys. Rev. E 74 (1) (2006)016122.  doi:10.1103/PhysRevE.74.016122 .4  1 e -0 61 e -0 50.00010.0010.010.11110100       N      (      S      i     z     e      ) Size Max 1Damage 0.1 Max 10Damage 0.1 Max 20Fit Mx20, 1 - 6Fit Mx20, 7 - 60 Figure 8: Histogram of avalanche sizes for damage param-eter  a =0 . 1 and several maximal numbers of failures,  k max .The bolder line (slope  − 2 . 98(7) fits for avalanche sizes be-tween 1 and 7, while the thinner line (slope  − 2 . 1(1)) fitsfor avalanche sizes between 7 and 60.[6] A. Hansen, P. Hemmer, Phys. Lett. A. 184 (394).[7] F. Kun, F. Raischel, R. C. Hidalgo, H. J. Herrmann,Modelling Critical and Catastrophic Phenomena inGeoscience, 2006, Ch. Extensions of fiber bundle mod-els.  doi:10.1007/b11766995 .URL  http://www.ica1.uni-stuttgart.de/~hans/fibres.html [8] S. Pradhan, A. Hansen, B. K. Chakrabarti, Failureprocesses in elastic fiber bundles, Rev. Mod. Phys.82 (1) (2010) 499–555.  doi:10.1103/RevModPhys.82.499 .[9] P. K. V. V. Nukala, S. ˇSimunovi´c, Statistical physicsmodels for nacre fracture simulation, Phys. Rev. E72 (4) (2005) 041919.  doi:10.1103/PhysRevE.72.041919 .[10] S. A. Galindo Torres, J. D. Mu˜noz Casta˜no, Simula- tion of the hydraulic fracture process in two dimen-sions using a discrete element method, Phys. Rev. E75 (6) (2007) 066109.  doi:10.1103/PhysRevE.75.066109 .[11] W. Weibull, A statistical distribution function of wideapplicability, ASME Journal of Applied Mechanics(1951) 293 – 297.URL  http://www.barringer1.com/wa_files/Weibull-ASME-Paper-1951.pdf 5
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