We study the sample size dependence of the strength of disordered materials
with a flaw, by numerical simulations of lattice models for fracture. We find a
crossover between a regime controlled by the fluctuations due to disorder and
another controlled by stress-concentrations, ruled by continuum fracture
mechanics. The results are formulated in terms of a scaling law involving a
statistical fracture process zone. Its existence and scaling properties are
only revealed by sampling over many configurations of the disorder. The scaling
law is in good agreement with experimental results obtained from notched paper
samples.Comment: 4 pages 5 figure

E. J. Gumbel

H. Kettunen

L. da Vinci

Mikko J. Alava

Phani K. V. V. Nukala

Stefano Zapperi

W. Weibull

Z. P. Bazant

English

arXiv

We study the sample-size dependence of the strength of disordered materials with a flaw, by numerical simulations of lattice models for fracture. We find a crossover between a regime controlled by the disorder and another controlled by stress concentrations, ruled by continuum fracture mechanics. The results are formulated in terms of a scaling law involving a statistical fracture process zone. Its existence and scaling properties are revealed only by sampling over many configurations of the disorder. The scaling law is in good agreement with experimental results obtained from notched paper sample

AIR Universita degli studi di Milano

Role of Disorder in the Size Scaling of Material StrengthMikko J. AlavaLaboratory of Physics, Helsinki University of Technology, FIN-02015 HUT, Espoo, FinlandPhani K. V. V. NukalaComputer Science and Mathematic Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6359, USAStefano ZapperiINFM-CNR, S3, Dipartimento di Fisica, Universita` di Modena e Reggio Emilia, via Campi 213/A, I-41100, Modena, Italyand ISI Foundation, Viale S. Severo 65, 10133 Torino, Italy(Received 22 August 2007; published 5 February 2008)We study the sample-size dependence of the strength of disordered materials with a flaw, by numericalsimulations of lattice models for fracture. We find a crossover between a regime controlled by the disorderand another controlled by stress concentrations, ruled by continuum fracture mechanics. The results areformulated in terms of a scaling law involving a statistical fracture process zone. Its existence and scalingproperties are revealed only by sampling over many configurations of the disorder. The scaling law is ingood agreement with experimental results obtained from notched paper samples.DOI: 10.1103/PhysRevLett.100.055502 PACS numbers: 62.20.M, 05.40.a, 81.40.NpThe fracture strength of materials depends on variouscharacteristic length scales of the specimen and representsa fundamental open problem of science and engineering.Probably the oldest scientific study of this issue was per-formed by Leonardo da Vinci, who measured the carryingcapacity of metal wires of varying length [1]. The simpleobservation was that the longer the wire, the less weight itcould sustain. This size effect can be understood consider-ing that some form of disorder, such as dislocations, grains,or microcracks, is always present in materials. Hence,different parts of the sample should have a differentstrength depending of the local microstructure. In addition,the strength is not a self-averaging quantity since it isdominated by the weakest spot, where global failure islikely to initiate. Longer wires will, in general, containmore weak parts and are thus bound to fail earlier onaverage. While the physical mechanism behind thisextreme-value based statistical size effect is clear, obtain-ing mathematical laws in realistic situations is still a for-midable task [2].In quasibrittle materials, such as concrete and manyother composites, size effects are particularly complicatedbecause of the significant damage accumulation precedingsample failure. The limitations of extreme-value statisticsstem from long-ranged elastic interactions: cracks interactand it is not always obvious how to isolate the weakestspot. To overcome this problem, it is customary to considera specimen containing a preexisting flaw, a notch. If thenotch is sufficiently large, the stress enhancement aroundthe crack tip will be sufficient to localize the failure. In thislimit, disorder can be treated as a small perturbation [3] bydefining a fracture process zone (FPZ) around the crack tip,where all the damage accumulation is confined. Severalformulations to account for the size effects have beenproposed in the literature [4–9]. These approaches aremainly based on the ad hoc extensions of linear elasticfracture mechanics (LEFM). This is a well establishedframework to understand cracks in homogeneous media,but it encounters fundamental problems when disorder isstrong and homogenization methods are not applicable.This is particularly true for small notches, when failure isinfluenced by statistical effects and may, for instance,initiate far from the preexisting notch due to nucleatedmicrocracks.In LEFM the stability of a flaw against failure is given bythe Griffith energy criterion, analogous to the classicaltheory of nucleation in first order phase transitions.Considering the elastic energy released by the crack andthe interfacial energy gained in creating it, one can showthat a crack of length a0 becomes unstable for stresseslarger than c  Kc= a0p , where the critical stress inten-sity factor Kc EGcpis a function of the fracture tough-ness Gc and the elastic modulus E [10]. The Griffithargument has been generalized by Bazant to quasibrittlematerials [4], postulating that when the FPZ is present, thecrack length a0 should increase by an additional lengthscale , yielding c  Kc= a0p: (1)Equation (1) incorporates two important effects: First, inthe large notch limit =a0  0 one should recover anexpression that follows the LEFM scaling, in which thestrength is inversely proportional to 1= a0p . Second, for avanishing external flaw size a0 ! 0, the average strengthshould still remain finite. In this limit, however, disordercannot be considered as a small perturbation, and theassumptions underlying Eq. (1) are not valid.PRL 100, 055502 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending8 FEBRUARY 20080031-9007=08=100(5)=055502(4) 055502-1 © 2008 The American Physical SocietyHere we investigate the role of the disorder in the failureof notched quasibrittle specimens, providing a microscopicjustification and establishing the limits of validity ofEq. (1). We study the size scaling of strength by theextensive numerical simulations, a difficult task due tothe different length scales involved and to the need ofsignificant statistical averaging. We vary the disorder,which we model as a locally varying random failure thresh-old and show that it plays a crucial role in determining thesize effect, influencing the fracture toughness Kc.Furthermore, the length scale  naturally emerges fromthe simulations and can be shown to be directly related tothe FPZ size. Finally, for notch sizes smaller than a criticallength ac, we observe a crossover to the inherent, sample-size-dependent strength of the unnotched sample. Wepresent a scaling formula that incorporates all these effectsand confirm its validity by comparing the simulations toexperiments on notched paper samples.To simulate a disordered elastic solid, we consider thesimplest case where disorder and LEFM-like stress en-hancements can be incorporated. We perform extensivesimulations of the random fuse model (RFM) [11,12].The RFM represents a quasibrittle failure process by anelectrical analog composed of a network of fuses. Weconsider a triangular lattice of linear size L with a centralnotch of length a0. The fuses have unit conductance (whichwould correspond to E  1 in the elastic system) andrandom breaking thresholds ic. These represent a locallyvarying fracture toughness or strength. The ic lie between 0and 1, with a cumulative distribution Pic  i1=Dc , whereD represents a quantitative measure of disorder. The largerD is, the stronger the disorder [13]. In the simulation, theburning of a fuse occurs irreversibly, whenever the electri-cal current in the fuse exceeds its threshold ic. An externalcurrent is increased applying a voltage difference betweenthe top and the bottom lattice bus bars and applyingperiodic boundary conditions along the other direction.Damage accumulates until a connected fracture path dis-connects the network and one can define the strength c asthe total peak current divided by the length of the bus bar.We also perform numerical simulations for the randomspring model (RSM) [14], similar to the mesoscale modelsused routinely for concrete [15]. The RSM is similar to theRFM, but the fuses are replaced by elastic springs thatbreak when their elongation reaches a random threshold.While the RSM represents more faithfully the elastic con-tinuum, the statistical properties of the fracture process areanalogous to those of the RFM [14].The inset of Fig. 1 reports the strength, averaged overdifferent configurations, with varying a0, D, and L. Themost instructive way of plotting is to consider the invertedsquare strength, 1=2c. Assuming Eq. (1), it is clear that1=2c should become a linear function of a0 for largeenough notches. Plotting the data in such a manner inFig. 1 reveals four interesting features: (i) for a0  1,the scaling of Eq. (1) is recovered asymptotically; (ii) thelinear part of the data when extrapolated toward a0  0reveals a disorder-dependent intercept D, that should berelated to the size of the FPZ; (iii) the slope of the linearpart of the data [1=K2cD] is disorder dependent, whichimplies a disorder-dependent fracture toughness GcD;and finally, (iv) a careful observation (see Fig. 2) revealsthat for small a0 less than a critical crack size ac, thestrength scaling crosses over from a stress concentrationdominated LEFM scaling [Eq. (1)] to a disorder-dominatedscaling. In particular, the data presented in the inset ofFig. 2 indicate that the strength of the unnotched system(for a0  0) is smaller than the LEFM limit Kc=p [where1=K2c is the slope and  is the intercept of the lines in Fig. 1based on Eq. (1)]. Hence, in order to extrapolate the typicalsample-size-dependent strength from notched experiments0 20 40 60 80 100a001020304050607080901001/σc2D=0.1D=0.3D=0.5D=0.6D=0.75Spring network1 10 100a01σcFIG. 1 (color online). Inset: The strength c as a function ofthe notch size a0 for several disorders D. Results for differentsystem sizes L are plotted together. The main figure shows thescaling plot of the data according to Eq. (1).0 10 20a034567891011121/σc2L=64L=128L=192L=256L=32010 100 1000L0.30.350.40.450.50.550.6σc(a0=0)extrapolation for a0=0FIG. 2 (color online). A close-up of the strength data for D 0:6 and various a0 and L. The inset compares fracture strength inunnotched samples with that predicted by Eq. (1) for a0  0.PRL 100, 055502 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending8 FEBRUARY 2008055502-2it is necessary to know ac. As can be seen, these featuresare also exhibited by RSM results.The crossover at a critical crack size ac marks theimportant role of structural disorder or internal damageon the size effect and relates to the size effect in unnotchedsamples. For the RFM, the size effect without a notch hasbeen shown by simulations and theoretical arguments tohave a logarithmic dependence on the linear system size, L[12,16]. For many engineering materials, one resorts to theWeibull theory [17] as an empirical starting point.This crossover from LEFM dominated strength scalingto disorder-dominated strength scaling is illustrated inFig. 3, where we compare the simulation data to experi-mental results on paper samples with varying center notchsizes. The paper data are from Ref. [18]. The two data setspresented are for strips of L  15 cm cut from laboratory-made handsheets, of fine paper-type, with center defects ofnominal sizes of a0  0:5, 1, 1.5, 2, and 2.5 mm. Tensiletests were performed on 100 samples, in order to averagethe results. In Fig. 3, we observe clearly the presence of ac,below which the notches have a negligible role on c, andfor crack sizes a0 larger than ac fracture is ruled by LEFM.We have also fitted the simulation data to these experi-ments, by considering similar a0=L ratios and varying theD to obtain reasonable agreement with the experimentaldata.The observations from Figs. 1 and 2 can be summarizedinto a single scaling theory by noticing that there must be ascale ac above which the LEFM holds and c followsEq. (1). For a0  ac the strength scaling deviates signifi-cantly from Eq. (1) and saturates to a value that dependsonly on disorder and the sample size, L;D. This is thestrength of the unnotched system of size L and disorder D.The crossover from LEFM scaling [Eq. (1)] occurs at anotch size ac which can be obtained from Eq. (1) as1=L;D2 ’ ac  =K2c . It is possible to describe thiscrossover by the following scaling form, valid for all a0such that K2c2c  a0fac=a0; (2)where the scaling function fy fulfills the limits fy ’1 if y  1;y if y  1: (3)The length scale ac ’ 	KcD=L;D2  D corre-sponds to a crossover scale below which material strengthis governed by disorder strength and system size. When thenotch size exceeds this crossover length scale (a0 > ac),fracture is governed by LEFM. At fixed L, stronger dis-order will increase ac since the strength of unnotchedsamples decays faster than the decrease in Kc. This isunderstandable since the disorder masks more efficientlythe stress concentration due to the notch. Likewise, at fixedD, ac increases with increasing L.Simulations of the RFM allow us to access the damageevolution prior to failure and can thus be used to visualizethe development of the FPZ. As shown in Fig. 4, for asingle realization of the disorder, at maximum stress we seeonly diffuse damage, without apparent localization so thatthe FPZ cannot be observed. When we average the damageover different configurations, however, a clear FPZ0.05 0.1 0.15a0/L00.10.20.3(Kc / σc)2D = 0.6 L=128paper sample 1paper sample 2FIG. 3 (color online). Comparison of numerical results withthe experimental data for two kinds of paper. The figure presentsthe size effect for small notch sizes using a RFM simulation of asystem of size L  128 and a disorder of D  0:6.FIG. 4 (color online). The damage (fraction of failed elementsor fuses) at maximum stress c for L  128 and D  0:6. Theblack markers illustrate the broken fuses in one single, randomlychosen sample whereas the color background represents thedamage averaged over N  2000 (number of samples). Thecolor code indicates the intensity of damage. A damage cloudthat increases in density is clearly visible close to the initialnotch. The figure demonstrates the screening of damage due tofree crack faces. It is clear that in the disordered samples, FPZ isa statistical zone, visible only when damage profiles are aver-aged over many samples.PRL 100, 055502 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending8 FEBRUARY 2008055502-3emerges in front of the crack tip (see Fig. 4). Hence theFPZ should be considered a statistical concept, visible onlywhen averaging over disorder, while the effect of the FPZis nevertheless seen in the size effect (). To measure theFPZ size, we consider a projection of the average damagealong the crack direction and obtain a profile that is decay-ing exponentially toward a homogeneous backgroundvalue: dx  A B expx=FPZ [see Fig. 5(a)]. Wehave analyzed the data for different values of D andchecked that the profiles do not depend on a0 as long asthis is not too far from ac. The LEFM stress intensity factorwould indicate a 1=rp-like divergence of the stress at thecrack tip. It is evident that the observed exponential shapeof the damage profile d is in contrast to a 1=rp-like decayand should be naturally interpreted as a screening of thecrack tip caused by the disorder. In Fig. 5(b), we plot thededuced fracture process size FPZD against the intrinsicscale  that one obtains from the fits of the strength data tothe Eq. (1) and which also is an important part of thescaling theory presented in Eq. (2). It can be seen thatthese are linearly proportional indicating that  is indeed adirect measure of the FPZ size. Notice that an exponentialdamage zone has indeed been measured in paper samplesand the corresponding length scale was compared with theone obtained from Eq. (1) [19].In conclusion, we have resorted to simulations of statis-tical fracture models to analyze the problem of the sizeeffect in the failure of materials. For large notches, thesimulations recover the expected scaling of LEFM [4] andallow us to relate the effective FPZ size  to the actualaverage damage profiles. As the notch size is decreased weobserve a crossover at a novel scale ac to a disorder-dominated size-dependent regime that is not described byLEFM and is furthermore seen in experiments. All theregimes are summarized in a generalized scaling expres-sion [Eq. (2)] for the strength of disordered media.M. J. A. acknowledges the support of the Center ofExcellence program of the Academy of Finland, discus-sions with Dr. R. Wathe´n, and the access to the data ofRef. [18]. M. J. A. and S. Z. gratefully thank the financialsupport of the European Commissions NEST Pathfinderprogramme TRIGS under contract No. NEST-2005-PATH-COM-043386. P. K. V. V. N. acknowledges support fromMathematical, Information and Computational SciencesDivision, Office of Advanced Scientific ComputingResearch, U.S. Department of Energy under ContractNo. DE-AC05-00OR22725 with UT-Battelle, LLC.P. K. V. V. N. also acknowledges the use of IBM BG/Lresources made available to him at Argonne NationalLaboratory through INCITE. We are grateful toM. Gro¨hn and CSC, Finnish IT Center for Science, forassistance.[1] L. da Vinci, I libri di Meccanica (Hoepli, Milan, Italy,1940).[2] E. J. Gumbel, Statistics of Extremes (Columbia UniversityPress, New York, 2004).[3] Z. P. Bazant and J. Planas, Fracture and Size Effect inConcrete and Other Quasibrittle Materials (CRC Press,Boca Raton, FL, 1997).[4] Z. P. Bazant, Proc. Natl. Acad. Sci. U.S.A. 101, 13 400(2004).[5] X. Hu and F. Wittmann, Mater. Struct. 25, 319 (1992).[6] B. Karihaloo, Int. J. Fract. 95, 379 (1999).[7] S. Morel, J. Schmittbuhl, E. Bouchaud, and G. Valentin,Phys. Rev. Lett. 85, 1678 (2000).[8] S. Morel, E. Bouchaud, and G. Valentin, Phys. Rev. B 65,104101 (2002).[9] A. Carpinteri and N. Pugno, Nat. Mater. 4, 421 (2005).[10] A. A. Griffith, Phil. Trans. R. Soc. A 221, 163 (1921).[11] L. de Arcangelis, S. Redner, and H. J. Herrmann, J. Phys.(Paris), Lett. 46, 585 (1985).[12] M. J. Alava, P. Nukala, and S. Zapperi, Adv. Phys. 55, 349(2006).[13] While this particular distribution has no deep physicaljustification, it is a convenient one to tune the disorderstrength through a single parameter (D). The results,however, should be independent of the particular distribu-tion. Bond dilution disorder yields similar results.[14] P. K. V. V. Nukala, S. Zapperi, and S. Simunovic, Phys.Rev. E 71, 066106 (2005).[15] G. Lilliu and J. G. M. van Mier, Eng. Fract. Mech. 70, 927(2003).[16] P. M. Duxbury, P. D. Beale, and P. L. Leath, Phys. Rev.Lett. 57, 1052 (1986).[17] W. Weibull, A Statistical Theory of the Strength ofMaterials (Generalstabens Litografiska Anstalts Fo¨rlag,Stockholm, 1939).[18] R. Wathe´n, Lic. thesis, HUT, Espoo, Finland, 2003.[19] H. Kettunen and K. Niskanen, J. Pulp Pap. Sci. 26, 35(2000).0 50 100x0.0010.010.11d(x)D=0.3D=0.5D=0.6D=0.750 5 10 15 20ξFPZ051015202530ξ(a) (b)FIG. 5 (color online). (a) Damage profiles along the crack axisfor various disorders D. Damage profiles follow an exponentialdecay on a uniform damage background, i.e., dx A B expx=FPZ, where A and B are constants and x isthe distance from the crack tip along the crack axis. (b)  vs FPZfor various disorders D.PRL 100, 055502 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending8 FEBRUARY 2008055502-4

Role of disorder in the size scaling of materials strength

We study the sample-size dependence of the strength of disordered materials with a flaw, by numerical simulations of lattice models for fracture. We find a crossover between a regime controlled by the disorder and another controlled by stress concentrations, ruled by continuum fracture mechanics. The results are formulated in terms of a scaling law involving a statistical fracture process zone. Its existence and scaling properties are revealed only by sampling over many configurations of the disorder. The scaling law is in good agreement with experimental results obtained from notched paper samples.Peer reviewe

Alava, Mikko J.

Nukala, Phani K. V. V.

Zapperi, Stefano

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