Extreme value theory and the St. Petersburg paradox in the failure statistics of wires
CComment to Phys. Rev. Lett. 124, 245501 (2020)
Alessandro Taloni ∗ and Stefano Zapperi
2, 3, † CNR-ISC, Institute for Complex Systems, Via dei Taurini 19, 00185 Roma, Italy. Center for Complexity and Biosystems and Department of Physics,University of Milano, Via Celoria 16, 20133 Milano, Italy CNR-ICMATE, Institute of Condensed Matter Chemistry andTechnologies for Energies, Via Roberto Cozzi 53, 20125 Milano, Italy (Dated: February 8, 2021)
In a recent letter [1], Fontana and Palffy-Muhoray pro-posed a connection between materials failure statisticsand the St. Petersburg paradox by linking the averageload carrying capacity of a wire to its length. The result,however, was derived assuming that ”the force requiredto fracture the fiber is a linear function of the defect size”[1], which is in glaring contrast with fracture mechanics.Here we address the problem combining extreme valuetheory (EVT) [2] with the Griffith’s stability crack crite-rion [3]. According to the Griffith’s assumption, the fail-ure stress should be inversely proportional to the squareroot of the largest defect size. We also show that in theasymptotic limit, the wire strength follows the Gumbel’sdistribution, in full agreement with the data reportedin [1], as we demonstrate using the maximum likelihoodmethod. We thus conclude that the load carrying capac-ity of the wires studied in [1] follows EVT, in agreementwith previous observations for different materials [2].We consider a wire of length L which we divide in N = L/L independent elements of size L . FollowingRef. [1], we want to relate the statistics of the micro-cracks present in the wire with its failure strength. Defin-ing P ( n ) as the probability density function (pdf) ofmicro-cracks of length w ≡ nL , F ( z ) = (cid:82) z dnP ( n ) isthe probability that no micro-crack larger than z will befound in the wire. We then define n max as the largestmicro-crack in the wire, with the only constraint that w max = n max L (cid:28) L [4]. If ρ N ( n max ) is the pdf for thelargest micro-crack, then F N ( z ) = (cid:82) z dn max ρ N ( n max ) =[ F ( z )] N . The Fisher-Gnedenko-Tippet theorem ensuresthat F N ( z ) → G ( z ) for large N , where G belongs toone of three families only: Weibull, Fr´echet or Gumbel[5, 6]. The convergence to either one of these univer-sal distributions depends on the asymptotic propertiesof P ( n ) [7, 8]. If the distribution of micro-cracks hasan exponential tail [4, 8, 9], G ( z ) converges asymptot- ically to the Gumbel distribution [10]. To derive thefracture strength, the authors of Ref. [1] assume thatit is linearly dependent on the size of a defect, obtainedthrough the St. Petersburg model, but this assumptionis not justified by fracture mechanics. A relation be-tween crack length and fracture strength in an elasticmedium is provided by the Griffith’s stability criterion,for which a crack of length w subject to a normal stress σ is stable as long as σ < K C /Y w − / [3], where K C is the critical stress intensity factor and Y is a geometricfactor. In our context, the wire should break when thelargest micro-crack becomes unstable, hence the proba-bility that a wire of length L does not fail under a stress σ is given by Σ L ( σ ) ∼ exp (cid:104) − L/L e − ( σ /σ ) (cid:105) . This is theDuxbury-Leath-Beale distribution [4], which was shownto converge to the Gumbel distribution as L (cid:29) L , i.e.Σ L ( σ ) → exp (cid:2) − L/L e ( σ − µ ) /β (cid:3) [8]. The average break-ing stress is then given by (cid:104) σ (cid:105) = β [ γ − ln ( L/L )] + µ which recovers Eq.(1) of [1]: γ is the Euler-Mascheroniconstant and β and µ are Gumbel’s parameters.The tensile experiments performed in [1] on polyesterand polyamide wires, corroborate the extreme valuestatistics over 6 order of magnitude. We fitted the datausing the Gumbel form of the generalized EVT, whichis purposely designed to account for strain and thermaleffects (Fig.1 main panels) [2, 11]. Since the experimentswere performed with different strain rate for each samplesize, the statistical analysis can only be performed usingthe maximum likelihood method for parameters estima-tion [2]. Our fit would indicate a very small strain ratedependence in the parameters in agreement with the ex-periments on polyester (inset Fig.1a). The experimentson polyamide indicate a strain-rate dependence that isnot captured by the fit. A possible reason for this dis-crepancy is that the precise value of the strain-rate is notknown and can only be estimated indirectly by 1 /t rup , asalso acknowledged by the authors [12]. ∗ [email protected] † [email protected] [1] J. Fontana and P. Palffy-Muhoray, Phys. Rev. Lett. , 245501 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.245501 .[2] A. Taloni, M. Vodret, G. Costantini, and S. Zapperi, Na-ture Reviews Materials , 211 (2018).[3] A. A. Griffith, Philosophical transactions of the royal so- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ciety of london. Series A, containing papers of a mathe-matical or physical character , 163 (1921).[4] P. Duxbury, P. Leath, and P. D. Beale, Physical ReviewB , 367 (1987).[5] R. A. Fisher and L. H. C. Tippett, in Mathematical Pro-ceedings of the Cambridge Philosophical Society (Cam-bridge University Press, 1928), vol. 24, pp. 180–190.[6] B. Gnedenko, Annals of mathematics pp. 423–453 (1943).[7] M. R. Leadbetter, G. Lindgren, and H. Rootz´en,
Ex-tremes and related properties of random sequences andprocesses (Springer Science & Business Media, 2012).[8] C. Manzato, A. Shekhawat, P. K. Nukala, M. J. Alava,J. P. Sethna, and S. Zapperi, Physical review letters ,065504 (2012).[9] H. Kunz and B. Souillard, Physical Review Letters ,133 (1978).[10] E. J. Gumbel, Statistics of extremes (Courier Corpora-tion, 2004).[11] A. L. Sellerio, A. Taloni, and S. Zapperi, Physical ReviewApplied , 024011 (2015).[12] J. Fontana, personal communication. FIG. 1. Main panels: fracture stresses of polyester (a) andpolyamide (b) fibers from the experiments in [1] (Fig.2). Themaximum likelihood estimates were used to calculate (cid:104) σ (cid:105) (reddashed line). Insets: breaking stresses as a function of therupture times (Fig.3 in [1]). Dashed blue lines are (cid:104) σ (cid:105) , eval-uated using the same parameters values fitted in the mainpanels, and using 1 /t ruprup