Computation of Probability Distribution of Strengthof Quasibrittle Structures Failing at Macrocrack Initiation

Jia-Liang Le, M.ASCE1; Jan Eliáš2; and Zdeněk P. Bažant, Hon.M.ASCE3

Abstract: Engineering structures must be designed for an extremely low failure probability, Pf < 10�6. To determine the correspondingstructural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures thatfail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs.It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength dis-tribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of theprobability about 10�4 to 10�3. The model implies that the strength distribution of quasibrittle structures depends on the structure size,varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull dis-tribution applicable for large-size structures. Compared with the classical Weibull strength distribution, which is limited to perfectly brittlestructures, the grafted Weibull-Gaussian distribution of the RVE strength makes the computation of the strength distribution of quasibrittlestructures inevitably more complicated. This paper presents two methods to facilitate this computation: (1) for structures with a simple stressfield, an approximate closed-form expression for the strength distribution based on the Taylor series expansion of the grafted Weibull-Gaussian distribution; and (2) for structures with a complex stress field, a random RVE placing method based on the centroidal Voronoitessellation. Numerical examples including three-point and four-point bend beams, and a two-dimensional analysis of the ill-fated Malpassetdam, show that Method 1 agrees well with Method 2 as well as with the previously proposed nonlocal boundary method. DOI: 10.1061/(ASCE)EM.1943-7889.0000396. © 2012 American Society of Civil Engineers.

CE Database subject headings: Probability distribution; Statistics; Cracking; Size effect; Computation.

Author keywords: Finite weakest link model; Strength statistics; Representative volume element; Structural safety; Fracture; Concretestructures; Composites.

Introduction

The design of various engineering structures, such as buildings,infrastructure, aircraft, ships, etc., must be targeted at an extremelylow failure probability Pf < 10�6 [Nordic Committee for BuildingStructures (NKB) 1978; Melchers 1987; Duckett 2005]. Since it isimpossible to determine the design strength for such a low failureprobability by experiment, a physically based model of probabilitydistribution of structural strength is of paramount importance. Thetype of strength distribution is well understood for structures withtwo extreme failure behaviors:1. Perfectly ductile behavior, for which, according to the central

limit theorem, the structural strength must have a Gaussian (ornormal) cumulative distribution function (cdf) because it is a

weighted sum of the random strengths of material elementsalong the failure surface; and

2. Perfectly brittle behavior, for which, according to the extremevalue statistics and the weakest-link model for an infinite chainof representative volume elements (RVEs), the structuralstrength must have a Weibull cdf because the structural failureis triggered by the failure of a single RVE, the size of which isnegligible compared with the structure size.Recent research efforts have been directed to structures made

of quasibrittle (i.e., brittle heterogeneous) materials, which includeconcrete, fiber composites, coarse-grained or toughened ceramics,rocks, sea ice, wood, rigid foams, bones, etc., and most brittle ma-terials at micrometer and submicrometer scales. An important featureof quasibrittle structures is that the size ofmaterial inhomogenieties isnot negligible compared with the structure size, which causes thebehavior of quasibrittle structures to be size dependent. Small-sizestructures exhibit a quasi-plastic failure behavior whereas large-sizestructures exhibit a brittle failure behavior. Such a size-dependentfailure behavior has by now beenwell documented by numerous sizeeffect tests (Bažant and Chen 1997; Bažant 2004, 2005).

The present study is limited to a broad class of structures that fail(under controlled load) as soon as a macrocrack initiates from oneRVE. The failure corresponds to the maximum load and representsthe loss of stability under controlled load, after which the crackpropagates dynamically. Such behavior characterizes the structuresof positive geometry, defined as structures of a geometry for whichthe derivative of the stress intensity factor with respect to the cracklength at constant load is positive. Statistically, such structure maybe modeled as a chain of RVEs, i.e., by the weakest-link model.Therefore, the RVE must be defined as the smallest volume whose

1Assistant Professor, Dept. of Civil Engineering, Univ. of Minnesota,Minneapolis, MN 55455; formerly Graduate Research Assistant, North-western Univ., Evanston, IL 60208.

2Masaryk-Fulbright Fellow, Northwestern Univ., Evanston, IL 60208;Assistant Professor on leave from Institute of Structural Mechanics, BrnoUniv. of Technology, Czech Republic 60100.

3McCormick Institute Professor and W. P. Murphy Professor of CivilEngineering and Materials Science, Northwestern Univ., Evanston,IL 60208 (corresponding author). E-mail: z-bazant@northwestern.edu

Note. This manuscript was submitted on July 19, 2011; approved onDecember 19, 2011; published online on December 23, 2011. Discussionperiod open until December 1, 2012; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Engineering Me-chanics, Vol. 138, No. 7, July 1, 2012. ©ASCE, ISSN 0733-9399/2012/7-888–899/$25.00.

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failure causes the failure of the whole structure (Bažant and Pang2007). Recently, the random particle model indicated that the RVEsize is approximately equal to the autocorrelation length of the spa-tial variation of material strength (Grassl and Bažant 2009). There-fore, RVE strengths can be assumed to be mutually independent.

Physically, the macrocrack initiation represents the formation ofan opened stress-free crack, which occurs at the moment at whichthe fracture process zone (FPZ) reaches its full size and starts totravel forward, trailed by the stress-free crack. The transition tothe opened stress-free crack occurs by localization of microcracksin the FPZ (Bažant and Planas 1998; Eliáš and Bažant 2011; Linand Labuz 2011). Mechanically, the FPZ can usually be adequatelydescribed by the cohesive crack model, although the multidimen-sional nonlocal damage models can provide a more realisticdescription. In quasibrittle materials it is next to impossible todetermine the location of the tip of the stress-free crack optically,but for practical purposes it is not necessary. The crucial pointfor adoption of the present study’s weakest-link model is that thefailure of one RVE triggers the failure of the entire structure.

Bažant and Pang (2006, 2007) recently developed amodel for thecdf of the strength of one RVE. The model was later refined on thebasis of fracture mechanics of nano-cracks propagating by small,activation energy controlled, random jumps through a nano-scaleelement such as an atomic lattice or a disordered nano-structure(Bažant et al. 2009; Le et al. 2011). Based on the transition ratetheory for the fracture of nano-element and a hierarchical series-parallel coupling model for the statistical multiscale transition, ithas been shown that the strength cdf of one RVE can be approxi-mately described as Gaussian distribution whose far left tail is modi-fied by a power law (which is the tail of Weibull distribution).

Because, for quasibrittle structures, the chain underlying theweakest-link model is not infinite but finite, the strength distribu-tion must depend on the structure size and geometry, varying fromthe Gaussian distribution with a remote Weibull tail for small-sizestructures, to the Weibull distribution for large-size structures. Thisfinite weakest-link model agrees well with the observed strengthhistograms of various quasibrittle materials including concrete,fiber composites, industrial ceramics, and dental ceramics (Bažantand Pang 2007; Pang et al. 2008; Bažant et al. 2009; Le andBažant 2009).

Note that what the joint probability theorem gives is only thestructural strength distribution. To predict the failure probabilityand reliability of actual structures under random loading, the struc-tural strength distribution must be in engineering designs combined(according to the joint-probability theorem) with the load distribu-tion. Because this subject is well understood (e.g., Haldar andMahadevan 2000), it is not included in this paper.

The weakest-link model with a grafted Gauss-Weibull distribu-tion complicates the computation of the cdf of structural strength.For large enough structures, what matters is only the power-law tailof the cdf of strength of one RVE, which causes the entire cdf ofstructural strength to follow the Weibull distribution (e.g., Bažantand Pang 2006, 2007; Le et al. 2011). In this case, it is convenientto adopt the concept of equivalent number of RVEs, Neq, whichrepresents the number of RVEs that gives the same failure proba-bility for a specimen subjected to uniform uniaxial tension (Bažantand Pang 2007). It can be shown that Neq explicitly depends on thestress field, and it is possible to obtain a closed-form solution forthe cdf of strength of large-size structures (Neq > 5;000).

However, for small- and intermediate-size structures, the cdf ofstructural strength is governed by both the Weibull and Gaussianparts of the grafted distribution of RVE strength, and then Neq de-pends on both the stress field and the stress magnitude. In this case,

a closed-form expression for the cdf of strength seems to be impos-sible (Le et al. 2011).

This paper presents two methods to calculate the strength cdf ofgeneral quasibrittle structures that fail at the macrocrack initiationfrom one RVE: (1) an approximate analytical formulation of the cdfof strength, which is developed on the basis of the Taylor seriesexpansion of the grafted Weibull-Gaussian distribution; and(2) a general numerical scheme based on random placing of theRVEs, which resembles the Monte-Carlo type integration.

Review of Weakest-Link Model and Its NonlocalExtension

This study focuses on the quasibrittle structures, which fail at theinitiation of a macrocrack from one RVE. Statistically, the failure ofthis class of structures is equivalent to a chain of RVEs and followsthe weakest-link model, in which each link corresponds to one RVEand has a statistically independent strength. Based on atomisticfracture mechanics and a statistical multiscale transition model,it has been shown that the strength distribution of one RVEP1ðσÞ can be modeled as a Gaussian distribution onto which apower-law tail (Weibull distribution) is grafted from the far-leftend at a point of probability Pgr ≈ 10�3 � 10�4 (Bažant and Pang2007; Bažant et al. 2009; Le et al. 2011)

P1ðσÞ ¼ rf ð1� e�hσ∕s1imÞ≈ rf hσ∕s1im ¼ PWðσÞ ðσ ≤ σgrÞð1a Þ

P1ðσÞ ¼ Pgr þrf

δGffiffiffiffiffiffi2π

pZ

σ

σgr

e�ðσ0�μGÞ2∕2δ2Gdσ ¼ PGðσÞ

ðσ > σgrÞð1b Þ

where hxi ¼ maxðx; 0Þ; m = Weibull modulus; s1 ¼ s0r1∕mf ; s0 =

scale parameter of the Weibull tail; μG and δG = the mean and stan-dard deviation, respectively, of the Gaussian core if consideredextended to �∞; rf is a scaling parameter required to normalizethe grafted cdf such that P1ð∞Þ ¼ 1; Pgr ¼ grafting probability ¼rf f1� exp½�ðσgr∕s1Þm�g; and σgr = grafting stress. Finally,continuity of the probability density function at the grafting pointrequires that ðdP1∕dσÞjσþgr ¼ ðdP1∕dσÞjσ�gr .

Based on the weakest-link model, the structure survives if andonly if all the RVEs survive. Therefore, the strength distribution ofthe structure can be calculated by the following joint probabilitytheorem:

Pf ðσNÞ ¼ 1�YNi¼1

f1� P1½σNsðxiÞ�g ð2Þ

or

ln½1� Pf ðσNÞ� ¼XNi¼1

lnf1� P1½σNsðxiÞ�g ð3Þ

where N = number of RVEs in the structure = V∕V0; V and V0 = thevolumes of the entire structure and one RVE, respectively; Pf =failure probability of structure (1� Pf = survival probability);and σN = nominal strength of structure, which is a load parameterof the dimension of stress. In general, σN ¼ cnPm∕bD or cnPm∕D2

for two—or three-dimensional scaling; Pm = maximum load of thestructure; cn = parameter chosen such that σN represents the maxi-mum principal stress in the structure; b = structure thickness in thethird dimension; D = characteristic structure dimension or size; andsðxiÞ = dimensionless stress field such that the actual stress σðxiÞ inthe i-th RVE centered at coordinates xi is equal to σNsðxiÞ. σðxiÞmay be interpreted as the maximum principal stress σI .

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More generally, stress σ may be replaced by a suitable stresstensor function, cðσÞ, capturing the triaxiality of failure criterion(Bažant and Planas 1998; Tsai and Wu 1971). This function canbe simplified into the condition of independent survival of theRVE under each of the three principal stresses, σI , σII , and σIII(Bažant and Xi 1991), provided that a crack normal to any principalstress can cause failure. In the following, the second and third prin-cipal stresses are assumed to be small compared with the first one,σðxiÞ ¼ c½σðxiÞ�≈ σIðxiÞ:

lnð1� Pf Þ ¼XNi¼1

lnð1� P1fc½σðxiÞ�gÞ

≈XNi¼1

XIIIj¼I

lnf1� P1 ½σj ðxiÞ�g

≈XNi¼1

lnf1� P1 ½σI ðxiÞ�g ð4Þ

It is clear that Eq. (4) requires the subdivision of a structure intoa number of RVEs. In a general case, such a subdivision is subjec-tive, typically nonunique, and often excessively refined. Therefore,it is preferable to replace the sum over a finite number of RVEs byan integral over the structure volume as follows:

lnð1� Pf Þ ¼ZVlnf1� P1½σNsðxÞ�g

dVðxÞV0

ð5Þ

What governs the strength cdf of large-size structures is the far-left tail of the strength cdf of one RVE, i.e., P1ðσÞ ¼ hσ∕s0im.Eq. (5) yields the following:

Pf ¼ 1� exp½�NeqðσN∕s0Þm� ð6Þand

Neq ¼ZVhsðxÞim dVðxÞ

V0ð7Þ

By contrast, a closed-form expression of Pf is not possible forsmall- and intermediate-size structures (i.e., Le et al. 2011). There-fore, one must still rely on the primary model, the weakest-linkmodel [Eq. (5)].

For quasibrittle structures, the weakest-link model can furtherbe generalized to include the nonlocality of the material due toits inhomogeneity (Bažant and Xi 1991; Bažant and Novák 2000).In such a generalization, the failure probability at a particularmaterial point depends not only on the local stress but also on thestress in its neighborhood of a size approximately equal to theRVE size l0. In deterministic calculations, the nonlocal concept isnecessary for regularizing the boundary value problem with strain-softening distributed damage and ensuring the convergence of frac-ture energy dissipation, which prevents spurious localization withfailure at zero energy dissipation, and avoids spurious mesh sensi-tivity. In statistical calculations, nonlocal averaging is a convenientway to introduce a spatial correlation (Breysse and Fokwa 1992;Bažant and Novák 2000) and makes it possible to avoid directuse of the autocorrelation function.

The grafted Gauss-Weibull distribution of the RVE strength, de-rived from atomistic fracture mechanics and multiscale transitionbased on a hierarchy of series and parallel couplings, accountsfor the statistical effect of material inhomogeneities (Bažant andPang 2006, 2007; Bažant et al. 2009). However, standing alone,this distribution does not include the material characteristic lengthscale required for the finite weakest-link model. This characteristiclength is introduced into the mathematical formulation [Eq. (4)] by

associating the grafted distribution with the RVE size, and treatingthe structure as a system of discrete RVEs. In the continuumversion with nonlocal averaging, the material characteristic lengthis imposed by making the effective size of the nonlocal averagingzone coincide with the RVE size. This effective size at the sametime represents the autocorrelation length of the random strengthfield.

Following Bažant and Novák (2000), the nonlocal stress is cal-culated from the weighted average stress of the nonlocal zone asfollows:

�σðxÞ ¼ZVαðkx� x0kÞσðx0ÞdVðx0Þ ð8Þ

where α = weighting function. There are various possiblechoices for an appropriate weighting function α (Bažant andPijaudier-Cabot 1988; Bažant and Jirásek 2002), and the resultsdo not depend too much on the choice (Bažant and Novák2000). For the sake of simplicity, a stepwise constant weightfunction α is considered in the following:

∝ ðx0 � xÞ ¼�1∕V0 ðkx0 � xk ≤ l0∕2Þ0 ðkx0 � xk > l0∕2Þ

ð9Þ

where l0 = average size of RVE; and V0 ¼ πl20∕4 for two-dimensional (2D) problems. Note that the nonlocal principalstresses should be calculated from the nonlocal stress tensor, whichcan be obtained by nonlocal averaging of each stress component (adirect averaging of the local principal stresses would not be correctbecause these stresses occur in different directions at every materialpoint). Incorporating the nonlocal concept into the weakest-linkmodel [Eq. (5)] yields the following:

Pf ¼ 1� exp

�ZVlnf1� P1½�σðxÞ�g

dVðxÞV0

�ð10Þ

One important unsolved problem in the conventional nonlocalmodel is the ambiguity in the treatment of the weighting functionwhen the nonlocal zone protrudes outside the structural boundary.Various approaches have been proposed, including these two:1. Scale the part of weighting function that is inside the body so

that the volume under the weighting function would revert to 1(Bažant and Jirásek 2002); or

2. Use the original weighting function but add to it at some pointa Dirac delta function of the same weight as the protruding part(Borino et al. 2003).However, there is a lack of physical justifications for either

approach.In a recent conference article (Bažant et al. 2010), a nonlocal

boundary layer model was proposed to overcome the ambiguityin adjusting the nonlocal weighting function when the averagingdomain protrudes outside the boundary of the solid. In this model,the structure is divided into two parts: a boundary layer Vb and aninterior domain VI . The boundary layer, which completely sur-rounds the interior domain, has a thickness equal to the RVE sizel0. In the interior domain, one can use the conventional nonlocalmodel to evaluate the stresses (Bažant and Novák 2000; Bažantand Jirásek 2002). Since the boundary layer thickness is equalto l0, the nonlocal domain for the points in the interior domain willnever protrude outside the structure boundary.

For the boundary layer, the continuum stress used to determinethe failure probability Pf is calculated from the deformation aver-aged over the thickness of the boundary layer l0. Since, in mostcases, the strain profile across the boundary layer is nearly linear,the averaged strain can approximately be taken as the continuum

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strain at the middle surface ΩM of the boundary layer. Therefore,the failure probability, as determined by the weakest-link model,Eq. (10), can be rewritten as

Pf ¼ 1� exp

�ZVI

lnf1� P1½�σðxÞ�gdVIðxÞV0

þ l0

ZΩM

lnf1� P1½σðxÞ�gdΩMðxÞ

V0

�ð11Þ

Direct numerical integration of elastic stress field according toEq. (11) will be referred to subsequently as the nonlocal boundarylayer method.

Although the stress in the boundary layer is treated as local, it isactually a nonlocal stress since it corresponds to the average strainin the boundary layer, which is a feature encompassing spatialcorrelation. The difference from the nonlocal stress at the pointsof the interior domain is that it is defined only for boundary layerthickness as a whole and not for an arbitrary generic point acrossthe thickness. Further note that the local treatment of the stress inthe boundary layer has previously been advocated for the determin-istic nonlocal models, to introduce diminishing nonlocality as theboundary is approached (Krayani et al. 2009). It has been shownthat the proposed nonlocal boundary layer model agrees well withthe original weakest-link model for structures with regular geom-etries, e.g., rectangular beams under pure bending (Bažantet al. 2010).

For very small structures for which the RVE size or the boun-dary layer exceeds about one-third of the cross-section dimension,the nonlocal concepts, including their present form with the boun-dary layer, lose physical meaning. In such cases, it is better to usethe random discrete lattice-particle models. The geometrical ran-domness of discrete models is insufficient to simulate the sizeeffect. To that end, one must include an autocorrelated random fieldof local strength, with the autocorrelation length equal to the RVEsize (Carmeliet and de Borst 1995; Vořechovský and Sadílek 2008;Grassl and Bažant 2009).

For large-size structures, the boundary layer and the nonlocalzone have a negligible effect compared with the structure sizeeffect. The nonlocal weakest-link model then converges to the localmodel [Eq. (6)].

Approximate Closed-Form Expression for StrengthDistribution

In the present theory, the cdf of strength of one RVE is separatedinto two parts: a Weibull tail [Eq. (1a)], and a Gaussian core[Eq. (1b)]. Within the framework of nonlocal boundary layermodel, one can likewise subdivide both the boundary layer ΩMand the interior part VI into two parts:

1. The Weibullian region, where the principal stress is lessthan the grafting stress, causing the failure probability to begoverned by the Weibull tail, Eq. (1a); and

2. The Gaussian region, where the principal stress is larger thanthe grafting stress, causing the failure probability to be gov-erned by the Gaussian core, Eq. (1b). Fig. 1 shows such a divi-sion for beams under three-point bending and pure bending.Eq. (11) may then be rewritten as follows:

lnð1� Pf Þ ¼ZVW

lnf1� PW ½�σðxÞ�gdVWðxÞ

V0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}IVW

þZVG

lnf1� PG½�σðxÞ�gdVGðxÞV0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

IVG

þ l0

ZΩW

lnf1� PW ½σðxÞ�gdΩW ðxÞ

V0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}IΩW

þ l0

ZΩG

lnf1� PG½σðxÞ�gdΩGðxÞV0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

IΩG

ð12Þ

where VW = Weibullian region of the interior part of structure =fxjx ∈ VI∧�σðxÞ ≤ σgrg; VG = Gaussian region of the interior partof structure = fxjx ∈ VI∧�σðxÞ > σgrg; ΩW = Weibullian region ofthe boundary layer = fxjx ∈ ΩM∧σðxÞ ≤ σgrg; and ΩG = Gaussianpart of the boundary layer = fxjx ∈ ΩM∧σðxÞ > σgrg. Clearly, onehas VW∩VG ¼ ∅, VW∪VG ¼ VI , ΩW∩ΩG ¼ ∅, and ΩW∪ΩG ¼ ΩM .

The integrals for the Weibullian region can easily be computedby using the concept of Neq [Eq. (6)] as

IVW¼ �

�σN

s0

�mNV

eq ð13Þ

where

NVeq ¼

ZVW

h�sðxÞim dVWðxÞV0

ð14Þ

IΩW¼ �

�σN

s0

�mNΩ

eq ð15Þ

where

NΩeq ¼

ZΩW

hsðxÞim dΩWðxÞl0V0

ð16Þ

yx

V

VG

W

= gr

y

DxV

VG

W

= gr

D

=-2 /y DNmax

=N

xam

=-2 /y DNmax

=/

21

||

Dx

Nxa

m

D (b)(a)

G Wl0l0 G

Fig. 1. Stress field in beam and its division into Weibullian and Gaussian regime: (a) pure bending; (b) three-point bending

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where sðxÞ and �sðxÞ = the dimensionless local and nonlocal stressfields for the Weibullian parts of the boundary layer and the interiorpart of structure, respectively. Note that the size of Weibullian re-gions VW and ΩW explicitly depends on σN .

In contrast to integrals IVWand VΩW

for the Weibullian region,closed-form expressions for integrals IVG

and IΩGover the Gaussian

region could not be obtained. The present method approximates theintegrands of IVG

and IΩGby means of the Taylor series expansion

of the grafted Weibull-Gaussian distribution of the RVE strength.The weakest-link model implies that the material elements with

sufficiently small principal stress make negligible contributions tothe failure of the entire structure. Thus, for the purpose of calcu-lating the failure probability, one could neglect the part of structurethat is subjected to relatively small stress. For example, consider theWeibull distribution with Weibull modulus m ¼ 24. If the failureprobability of the element with principal stress σ is denoted as p,then the failure probabilities of the elements with 0:8σ, 0:6σ, and0:4σ are about 4:7 × 10�3p, 4:7 × 10�6p, and 2:8 × 10�10p,respectively, the last two being negligible.

Therefore, when evaluating integrals IVGand IΩG

, one can limitconsideration to the elements in which the principal stress is withinthe range σ ∈ ½μσN ; σmax�, where σmax = maximum principal stressused in the calculation and μ ¼ maxðσgr∕σN ; 0:6σmax∕σNÞ. Forsuch a limited stress range, one can approximate the integrands ofIVG

and IΩGby the Taylor series expansion with respect to the maxi-

mum principal stress σmax ¼ tσN .The number of terms that must be retained in the Taylor series

expansion depends on σN , t, μ and the parameters of the graftedWeibull-Gauss cdf. Using too many terms would defeat the goalof attaining a simple analytical solution. Thus, one can usuallytruncate the Taylor series expansion after its third derivative term.However, as will be shown subsequently, the truncation after thethird derivative is insufficient for an accurate approximation ofthe function ln½1� P1ðσÞ� for the range of σ ∈ ½μσN ; tσN �.

To further improve the approximation, one can consider a linearcombination of the Taylor series expansions of ln½1� P1ðσÞ� atσ ¼ μσN and σ ¼ tσN as follows:

ln½1� P1ðσÞ� ¼ ωðσÞX3k¼0

f ðkÞðμσNÞk!

ðσ � μσNÞk

þ ½1� ωðσÞ�X3k¼0

f ðkÞðtσNÞk!

ðσ � tσNÞk ð17Þ

where f ðkÞðσÞ ¼ dk ln½1� PGðσÞ�∕dσk; and the detailed expres-sions for f ðkÞðσÞ are given in the appendix. To make the approxi-mated function match the asymptotic properties of functionln½1� P1ðσÞ� at σ ¼ μσN and σ ¼ tσN , it is clear that the functionωðσÞ must decay from 1 at σ ¼ μσN to 0 at σ ¼ tσN . The presentmethod introduces the following quadratic function:

ωðσÞ ¼ 1��σ � μσN

tσN � μσN

�2

ð18Þ

The performance of this approximation is evaluated for differ-ent values of tσN corresponding to the failure probabilitiesP1ðtσNÞ ¼ 0:01, 0.10, 0.50, and 0.99. Fig. 2 presents the compari-son between the exact solution of ln½1� P1ðσÞ�, the three-termTaylor series expansion of ln½1� P1ðσÞ� at σ ¼ tσN , the three-termTaylor series expansion of ln½1� P1ðσÞ� at σ ¼ μσN , and thepresent approximation [Eqs. (17) and (18)] for the stress rangeσ ∈ ½μσN ; tσN �. It can be seen that the three-term Taylor seriesexpansion at either σ ¼ tσN or μσN is unable to provide a closeapproximation [Eqs. (17) and (18)]. In contrast, the proposedapproximation function agrees well with the exact value ofln½1� P1ðσÞ�. Consequently, integrals IVG

and IΩGcan be rewritten

as

IVGðt;μÞ ¼

X3k¼0

σkN

k!f ðkÞðμσNÞΔVG;1ðk; t;μÞ

þX3k¼0

σkN

k!f ðkÞðtσNÞΔVG;2ðk; t;μÞ ð19Þ

-0.10

-0.08

-0.06

-0.04

-0.02

0

2.01.81.61.4

3.53.02.52.01.5

0

-1

-2

-3

-4

0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

2.21.81.4 2.6

1.81.71.61.5

0

-0.002

-0.004

-0.006

-0.008

-0.010

·s

P gr

-4

[MPa][MPa]

))(

-1(nlP 1

))(

-1(nlP 1

t Nt N

t N t N2.2

0.9t

N

0.7t

N

0.8t

N

0.9 t

N

9.0t

N

8 .0t

N

7.0t

N

6.0t

N

6.0t

N

7.0t

N

0.9t

N

grgr

grgr

1P t1 N

P t1 N

P t1 N

P t1 N

N N

NN

tN

N

8.0t

N

Fig. 2. Approximation of ln½1� P1ðσÞ� by Taylor series expansion

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IΩGðt;μÞ ¼

X3k¼0

σkN

k!f ðkÞðμσNÞΔΩG;1ðk; t;μÞ

þX3k¼0

σkN

k!f ðkÞðtσNÞΔΩG;2ðk; t;μÞ ð20Þ

where:

ΔVG ;1ðk; t;μÞ ¼1V0

ZVGðt;μÞ

�1�

��sðxÞ � μt � μ

�2�½�sðxÞ � μ�kdVG

ð21Þ

ΔVG;2ðk; t;μÞ ¼1

V0ðt � μÞ2ZVGðt;μÞ

½�sðxÞ � μ�2½�sðxÞ � t�kdVG

ð22Þ

ΔΩG ;1ðk; t;μÞ ¼l0V0

ZΩGðt;μÞ

�1�

�sðxÞ � μt � μ

�2�½sðxÞ � μ�kdΩG

ð23Þ

ΔΩG;2ðk; t;μÞ ¼l0

V0ðt � μÞ2ZΩGðt;μÞ

½sðxÞ � μ�2½sðxÞ � t�kdΩG

ð24Þ

For structures with some simple stress fields, such as thelinear and bilinear stress profiles, ΔVG;i

ðk; t;μÞ and ΔΩG;iðk; t;μÞ

(i ¼ 1, 2) can be integrated analytically. This is likely possiblefor many structures, for which the stress field near the point withthe largest maximum principal stress can be approximated by a lin-ear or bilinear function.

Computation of Strength Distribution byPseudorandom Placing of RVEs

For complicated structures, the stress field in the region near themaximum principal stress point could be highly nonlinear. In sucha case, the use of the aforementioned Taylor series expansionmethod would not lead to a closed-form expression. One must thenresort to some numerical method to calculate the cdf of structuralstrength. So, besides the use of Eq. (11), the present study proposesa general numerical scheme based on a pseudorandom placingof the RVEs, referred to as the direct RVE placing method. Thismethod directly corresponds to the original weakest-link model[Eq. (3)].

There are many possible ways of placing the RVEs in the struc-ture, and each of them would yield a different strength cdf. There-fore, for statistically isotropic materials, it is preferable to makerepeated pseudorandom choices of the RVE locations, and thenaverage the results of all the realizations. The main advantage ofthe pseudorandom placing of the RVEs is that it eliminates thedirectional bias, which inevitably arises with a regular placing. Forsmall-size structures where the RVE size is comparable to the struc-ture size, one would need to generate many realizations of the ran-dom placing of RVEs. For large-size structures where the RVE sizeis negligible compared with the structure size, the RVE convergesto a point in the structure and the random placing of RVEs wouldnot have a significant effect on the resulting cdf of strength. There-fore, fewer realizations are needed for large-size structures.

The present method employs the centroidal Voronoi tessellationto generate a set of random locations of N points, whereN ¼ V∕V0. Each of these points represents the center of oneRVE. This tessellation produces a set of points corresponding to

the centroids of the Voronoi cells, which satisfy two essentialrequirements:1. The distance between any two adjacent points is approximately

equal to the RVE size l0; and2. The minimum distance of the point from the structure bound-

ary is close to l0∕2 (Fig. 3).Once the centers of the assumed RVEs are fixed, the average

stress for each RVE can be obtained based on the stress field cal-culated by the standard finite element method. The RVE center ρgenerally does not coincide with the integration point of the finiteelements, x. So the nonlocal stress for each RVE is calculated byaveraging the stresses at the integration points that are enclosedwithin the RVE (see Fig. 4), i.e.,

�σðρiÞ ¼P

Nj¼1 σðxjÞαðxj � ρiÞκjP

Nj¼1 αðxj � ρiÞκj

ð25Þ

where αðxj � ρiÞ = weighting function; and κj = the volume asso-ciated with integration point xj. To compute the average stressin one RVE, consider only the integration points that lie withinthat RVE. Therefore, the function α must have a cut-off that ex-cludes all the integration points outside the RVE, i.e., α ¼ 0, ifjjxj � ρijj > l0∕2. For the sake of simplicity, a piecewise constantfunction [Eq. (9)] can be used. By combining the weakest-linkmodel with the centroidal Voronoi tessellation, the cdf of structuralstrength can be calculated as follows:

lnð1� Pf Þ ¼XNi¼1

lnf1� P1½�σðρiÞ�g ð26Þ

Note that this approach closely resembles the Monte-Carlointegration of Eq. (5) with a fixed number N of integration points(e.g., Dimov 2008), although the present method of placing theintegration points is more refined.

Fig. 3. Example of centroidal Voronoi tessellation (left) and obtainedimaginary locations of RVEs (right)

0l

j

jx

i

Fig. 4. Averaging of stress obtained by FEM in imaginary i-th RVEvolume

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In the centroidal Voronoi tessellation, there is always a layer ofRVEs placed along the structure boundary (Fig. 3), which agreeswith the concept of the nonlocal boundary layer model. One differ-ence, though, is that the nonlocal boundary layer model uses thestress at the middle surface of the layer while the current approachuses the average stress across the RVE. Therefore, in the nonlocalboundary layer model, switching between the cubical RVE shape,V0 ¼ l30 (squares V0 ¼ l20) and the spherical one, V0 ¼ πl30∕6(circles V0 ¼ πl20∕4), has no effect, while in the direct RVE placingmethod, the RVE shape affects the number N of the RVEs and con-sequently their mutual distances as well as the minimal distance tothe boundary.

The present study uses spherical RVEs. Since an assembly ofnon-overlapping spheres would occupy a volume larger than thestructure volume V , the spheres must overlap each other to fit thisassembly into the structure volume V . Thus the mutual average dis-tance between the RVE centers must be slightly less than l0. It alsoaffects the distance between the centers of circumferential RVEsand the structure boundary, which would again be slightly less thanl0∕2. Therefore, in general, the present method predicts higher fail-ure probabilities than the nonlocal boundary layer method does.

Analysis of Failure Statistics of QuasibrittleStructures

Beams under Three-Point Bending and Pure Bending

Three-point and four-point bending tests are commonly adopted tomeasure the strength histograms of quasibrittle structures (Weibull1939; Munz and Fett 1999; Tinschert et al. 2000; Lohbauer et al.2002; dos Santos et al. 2003). For beams with a large span-to-depthratio, the engineering beam theory may be used to calculate themaximum principal stress field. Accordingly, for three-point bend-ing tests, the stress field can be represented by a two-way linearstress gradient along both the beam depth and span.

In four-point bending tests in which the two loading points arefar apart, the failure is governed by the middle portion between theloading points, which is essentially experiencing a constant bend-ing moment, i.e., pure bending. The stress field of the middle por-tion of the beam can be represented by a one-way stress gradientalong the beam depth. Nevertheless, if the two loading points arenot far apart, the contribution of the end portion of beam betweenthe loading point and the support, which has a two-way linear stressdistribution, can simply be added.

For the case of pure bending, the stress varies linearly in they-direction (normal to the beam axis). Consider a prismatic beamof depth D (y-direction) and length λD (x-direction). 2D analysispresumes that the failure occurs simultaneously along the wholeprism thickness (in the z-direction). The elastic stress in the bodycan be described as

σðxÞ ¼ σxðx; yÞ ¼ � 2yDσN ð27Þ

Since the stress varies linearly, the nonlocal principal stress fieldcoincides with the local principal stress field. The maximum stressin the boundary layer is σΩ

max ¼ ð1� l0∕DÞσN and the maximumstress in the interior part is σV

max ¼ ð1� 2l0∕DÞσN . Within theframework of aforementioned Taylor series expansion method,the boundary between the Weibull and Gaussian regions is givenby the straight line y ¼ �σgrD∕ð2σNÞ [Fig. 1(a)]. For theTaylor series expansion method, the following quantities areneeded:

NVeq ¼

λD2

2ðmþ 1ÞV0min

�σgr

σN; 1� 2l0

D

�mþ1

ð28Þ

NΩeq ¼

(λDl0V0

1� l0

D

m

1� l0

D ≤ σgrσN

0 otherwise

ð29Þ

ΔVG;1ðk; t;μÞ ¼

8><>:

λD2

2V0

�1

k þ 1

�1� 2l0

D� μ

�kþ1

� 1ðt � μÞ2ðk þ 3Þ

�1� 2l0

D� μ

�kþ3

� 1� 2l0

D > μ

0 otherwise

ð30Þ

ΔVG;2ðk; t;μÞ ¼

8><>:

λD2

2V0ðt � μÞ2Xkþ2

i¼0

αi

iþ 1

�ð�μÞiþ1 �

�2l0D

� 1

�iþ1

� 1� 2l0

D > μ

0 otherwise

ð31Þ

ΔΩG;1ðk; t;μÞ ¼8<:

λDl0V0

�1� l0

D� μ

�k� λDl0ðt � μÞ2V0

�1� l0

D� μ

�kþ2

1� l0D >

σgrσN

0 otherwise

ð32Þ

ΔΩG;2ðk; t;μÞ ¼8<:

λDl0ðt � μÞ2V0

�1� l0

D� μ

�2�1� l0

D� t

�k

1� l0D >

σgr

σN

0 otherwise

ð33Þ

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where αi = coefficients of binomial expansion of ðx� μÞ2ðx� tÞk. Substituting Eqs. (28)–(33) into Eqs. (12), (19), and (20), one can calculatethe corresponding strength cdf analytically.

For beams under three-point bending, the stress field σðx; yÞ can be assumed to be linear in two directions (Fig. 1(b)). The stress fieldcan be written as

σ ¼ 2yD

�2jxjDλ

� 1

�σN ð34Þ

Similar to the case of pure bending, the quantities that are required for the Taylor series expansion method can be obtained as follows:

NVeq ¼

λD2 min

�σgr

σN; 1� 2l0

D

�mþ1

2ðmþ 1Þ2V0

2641� ðmþ 1Þ ln

0@Dmin

�σgr

σN; 1� 2l0

D

�D� 2l0

1A375 ð35Þ

NΩeq ¼

λD2 min

�σgr

σN; 1� l0

D

�mþ1

l0

ðD� l0Þðmþ 1ÞV0ð36Þ

ΔVG ;1ðk; t;μÞ ¼8<:

D2λ2V0

�f ðkÞ � 1

ðt � μÞ2 f ðk þ 2Þ� �

1� 2l0D

> μ�

0 otherwise

ð37Þ

ΔVG;2ðk; t;μÞ ¼

8><>:

1V0ðt � μÞ2

Xkþ2

i¼0

αigðiÞ1� 2l0

D > μ

0 otherwise

ð38Þ

ΔΩG;1ðk; t; τÞ ¼8<:

λD2l0ðD� l0ÞV0

�1

k þ 1

�1� l0

D� μ

�kþ1

� 1ðk þ 3Þðt � μÞ2

�1� l0

D� μ

�kþ3

� �1� l0

D>

σgr

σN

�0 otherwise

ð39Þ

ΔΩG;2ðk; t;μÞ ¼

8><>:

ð�1Þkþ2λD2l02ðt � μÞ2ðD� l0ÞV0

Xkþ2

i¼0

βi

iþ 1

�ð1� μÞD� l02D

�iþ1

1� l0D >

σgrσN

0 otherwise

ð40Þ

where βi = coefficients of the binomial expansion of ½xþ μþ ðl0 � DÞ∕D2λ�2½xþ t þ ðl0 � DÞ∕D2λ�k , and

f ðkÞ ¼ ð�1Þkk þ 1

�μkþ1 ln

�Dμ

D� 2l0

�þXki¼0

ti�ð2l0∕D� 1þ μÞk�iþ1

k � iþ 1

�ð41Þ

gðiÞ ¼ λμiþ1D2

4ðiþ 1Þ ln½1� 2Dð1� μÞ þ 4l0�

� ð�1Þiþ1

ðiþ 1Þ2�2D

�k�1

ðl0 � D∕2Þiþ1

× f½4l0 þ 1� 2Dð1� μÞ�iþ1 � 1g ð42Þ

For the interior region, the present method uses the local stressto approximate the nonlocal stress, which is exact for the most partof the structure except for the midspan. Such an approximation willnot cause any error in the calculation of the cdf of both the small-size and large-size beams because: (1) for small-size beams, thefailure probability is entirely governed by the boundary layer;

and (2) for large-size beams, the nonlocal stress field convergesto the local stress field. For intermediate-size beams, which liebetween these two extreme cases, the error in calculating the cdfof strength is expected to be insignificant.

Figs. 5 and 6 present the comparison of the calculated strengthcdfs of beams under three-point and four-point bending by usingthe Taylor series expansion method, the direct RVE placingmethod, and the nonlocal boundary layer method. For the four-point bend beam, the two loading points are placed at a quarterlength from the supports. Consider a linear one-way stress distri-bution for the middle portion of the beam, and a linear two-waystress distribution for the two end portions of the beam. In thecalculation, fix the beam dimension, D ¼ 0:1 m and λ ¼ 4, andconsider four different RVE sizes l0 ¼ 3:75, 7.5, 15, and

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30 mm. The statistical parameters used in the calculations areas follows: s1 ¼ 1 MPa, m ¼ 24, μG ¼ 2:0956 MPa, and δG ¼0:5361 MPa. For the direct RVE placing method, 10, 50, 100,and 100 simulations are performed for beams with RVE sizel0 ¼ 3:75, 7.5, 15, and 30 mm, respectively, and the calculated cdfsare averaged for each size.

As seen from Figs. 5 and 6, the strength cdfs calculated by theTaylor series expansion method and the nonlocal boundary layermethod agree with each other for all the sizes. The direct RVE plac-ing method predicts a higher failure probability because the averagedistance to the boundary is smaller than l0∕2, which implies that thestresses in the boundary RVEs are higher than the stresses in thecenter of the boundary layer.

The aforementioned calculations demonstrate that the Taylorseries expansion method can accurately calculate the failure prob-ability. Note that the present formulation is generally applicableto the strength cdf of structures whose critical stress region,

i.e., σ ∈ ½μσmax; tσmax�, has approximately a linear one-way ortwo-way stress profile.

Example: Analysis of Failure Statistics of MalpassetDam

The Malpasset dam in the French Maritime Alps was built in 1954and failed at its first complete filling in 1959 (Bartle 1985; Levyand Salvadori 1992; Pattison 1998). It is believed that the failurewas caused by vertical flexural cracks engendered by lateral dis-placement of abutment because of a slip of thin clay-filled seamin schist. Following the previous study (Bažant et al. 2007), onecould model the midheight cross-section of dam as a horizontalcircular two-hinge arch of constant depth H ¼ 6:78 m. The radiusand central angle of the arch are R ¼ 92:68 m and 2β ¼ 133°. Thedam is loaded by an enforced displacement caused by slip of itsright support (Fig. 7).

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.80

0.2

0.4

0.6

0.8

1

0

1·10-6

gr

N

N

l l l=

30m

m

l

gr

0

0 00

l l=

15m

m

l

l

mm

57.3= 00 0

0

Fig. 5. Comparison of cdfs of four geometrically similar beams under three-point bending

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.80

0.2

0.4

0.6

0.8

1

0

1·10-6

gr

gr

l l=

15m

m

ll

mm

57.3= 00 0

0

l l ll 00 00

N

N

Fig. 6. Comparison of cdfs of four geometrically similar beams under four-point bending

displacement

R

H

D

rD

(a) (b)

Fig. 7. Simplified 2D model of Malpasset dam

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Stochastic finite element simulation in the previous study(Bažant et al. 2007) demonstrated a strong energetic-statistical sizeeffect on the strength of the dam. The present study uses thenumerical nonlocal boundary layer method, the Taylor series ex-pansion method, and the direct RVE placing method to investigatethe failure statistics of the dam and the corresponding size effect onits mean strength.

To examine the size effect on the failure statistics, considergeometrically similar arches where D = depth of the arch,which is considered as the characteristic size to be scaled, andr ¼ radius of arch ¼ 13:67D. For such a 2D arch model, it is con-venient to express the uniaxial stress in the polar coordinates ξ, θ.Consider that the right support of the arch is subjected to a hori-zontal slip u. From Castigliano’s theorem and the classical theory ofbending, the bending stress in the arch can be calculated as

σ ¼ σN2ðcos β � cos θÞξ

1� cos βð43Þ

where

σN ¼ 6uð1� cos βÞCbD

rD

ð44Þ

Here C = compliance = ½R β�βðcos θ� cos βÞ2dθ�r3∕E; E = elastic

modulus; ξ ¼ y∕D; and y = distance from the neutral axis; andσN is the maximal tensile stress at the boundary (ξ ¼ �1∕2) ofthe central cross-section (θ ¼ 0).

The nonlocal boundary layer method is first used to calculate thestrength cdf of the arch structure, where the elastic stress field istaken from an elastic finite element analysis. The RVE size l0 istaken to be 0.28 m (Bažant et al. 2007). Considering different archsizes, one can obtain the size effect curve of the mean strength. Thepresent study uses the following statistical parameters for the cdf ofRVE strength: s1 ¼ 2:115 MPa, m ¼ 24, μG ¼ 2:9053 MPa, andδG ¼ 0:4369 MPa.

The mean size effect curve calculated by the nonlocal boundarylayer method matches well the mean size effect calculated by usingthe microplane model (Bažant et al. 2007) for intermediate- andlarge-size arches (Fig. 8). These two models begin to deviate whenthere are only 3–4 RVEs across the arch thickness. This is because,for small-size structures, the boundary layer occupies too large aportion of the thickness. The averaging of elastic strain withinthe boundary layer cannot realistically represent the actual stressand strain redistribution in deterministic calculation.

The cdf of the arch strength can also be calculated by the Taylorseries expansion method. Based on the analytical stress field[Eq. (43)] with the small-angle approximation cos θ ¼ 1þ θ2∕2,

one can obtain closed-form expressions for NVeq, NΩ

eq, ΔVGand

ΔΩG. The detailed expressions (which are omitted here) yield an

analytical equation for the cdf of the arch strength.Finally, the direct RVE placing method is used to calculate the

cdfs of strength of geometrically similar arches for four differentsizes: D ¼ H, H∕2, H∕4, and H∕8. The arch volume is filledby the RVEs based on the centroidal Voronoi tessellation. Forarches with D ¼ H, H∕2, H∕4, and H∕8, 3, 30, 50, and 50 real-izations are performed, respectively, and the average cdf of nominalstrength is calculated. Fig. 9 presents typical RVE placings forarches of each size. The stress field from an elastic finite elementanalysis is used to compute the average stress for each RVE[Eq. (25)]. The weakest-link model then yields the cdf of structurestrength.

Fig. 10 presents the cdfs of structural strength calculated by thenonlocal boundary layer model, by the Taylor series expansionmethod, and by the direct RVE placing method. Here, the nonlocalboundary layer model agrees well with the Taylor series expansionmethod because the stress obtained by elastic finite element analy-sis is very close to the engineering stress field from Eq. (43). Again,the direct RVE placing method predicts a higher failure probabilitybecause of the fact that the distance of the circumferential RVEsfrom the boundary is smaller than l0∕2. For the scaled large-sizearches, for which the RVE size becomes negligible compared withthe arch size, all the three methods converge to essentially thesame cdf.

Fig. 8. Comparison of mean size effect curves including probability density functions of three different dam sizes

l0 l0

D H= /8

D H= /4

D H= /2 D H=

Fig. 9. Example of pseudorandom location of RVEs in dam obtainedby centroidal Voronoi tessellation for four sizes D

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Summary and Conclusions

This study proposes an approximate analytical formulation for thestrength distribution of quasibrittle structures failing at the initiationof a macrocrack from one RVE. This formulation yields a closed-form expression of strength distribution for structures in whichthe stress distribution in the high stress region can be approximatedby some simple functions. The formulation agrees well withtwo numerical approaches, namely, the random RVE placingmethod and the nonlocal boundary layer method. The closed-form

expression for the strength distribution provides an efficient way tocalibrate the statistical parameters by optimum fitting of the ob-served strength histograms, and it also creates a basis for the ana-lytical formulation of size- and geometry-dependent safety factors.

Appendix. Derivatives of P1(σ) and ln [1−P1(σ)]

Derivatives of the grafted Weibull-Gaussian distribution P1ðσÞ areas follows:

p1ðσÞ ¼∂P1ðσÞ∂σ ¼

8<:

pW ¼ rf ms hσ∕s1im�1e�hσ∕s1im ; σ ≤ σgr

pG ¼ rf1ffiffiffiffi2π

pδGe�ðx�μGÞ2

2δ2G ; σ > σgr

p01ðσÞ ¼∂p1ðσÞ∂σ ¼

8>><>>:

rfms1

2Dσs1

Em�2

eh�σs1imh1� 1

m � hσ∕s1imi; σ ≤ σgr

�rfx�μG

δ3Gffiffiffiffi2π

p e�ðx�μGÞ2

2δ2G ; σ > σgr

p001ðσÞ ¼∂2p1ðσÞ∂σ2 ¼

8>><>>:

rfms1

3Dσs1

Em�3

e�h σs1imh1� 3

m þ 2m2 þ

3m � 3

Dσs1

Em þ hσ∕s1i2m

i; σ ≤ σgr

�rf 1δ3G

ffiffiffiffi2π

p e�ðx�μGÞ2

2δ2G

h1� 2ðx�μGÞ2

2δ2G

i; σ > σgr

Derivatives of ln½1� P1ðσÞ� are as follows:

∂ ln½1� P1ðσÞ�∂σ ¼ � p1ðσÞ

1� P1ðσÞ

∂2 ln½1� P1ðσÞ�∂σ2 ¼ �

�p1ðσÞ

1� P1ðσÞ�2� p01ðσÞ1� P1ðσÞ

∂3 ln½1� P1ðσÞ�∂σ3 ¼ �2

�p1ðσÞ

1� P1ðσÞ�3� 3p1ðσÞp01ðσÞ½1� P1ðσÞ�2

� p″1ðσÞ1� P1ðσÞ

Acknowledgments

Financial support by the U.S. National Science Foundationunder Grant CMS-0556323 to Northwestern University is grate-fully acknowledged. The work was started during J.E.’s visitingappointment at Northwestern University, half of which was sup-ported by a Fulbright-Masaryk grant from the Fulbright Foundationand half by the aforementioned grant. Additional financial support

was provided by Ministry of Education, Youth and Sports of theCzech Republic under project number ME10030.

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1

0

0.2

0.4

0.6

0.8

1

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.6

0

1·10-6

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