# Energetic ( Quasibrittle ) Mean Size Effect Laws and Statistical Generalization

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NANO-MECHANICS BASED ASSESSMENT OF FAILURE RISK, SIZE EFFECT AND LIFETIME OF QUASIBRITTLE STRUCTURES AT DIFFERENT SCALES ZDEN K P. BAANT COLLABORATOR: JIALIANG LE, SZE-DAI PANG SPONSORS: DoT, NSF, BOEING, CHRYSLER, DoE - PowerPoint PPT PresentationTRANSCRIPT

NANO-MECHANICS BASED ASSESSMENT OF FAILURE RISK, SIZE EFFECT AND LIFETIME OF QUASIBRITTLE STRUCTURES AT DIFFERENT SCALES

ZDENK P. BAANT

COLLABORATOR: JIALIANG LE, SZE-DAI PANG SPONSORS: DoT, NSF, BOEING, CHRYSLER, DoE

CapeTown, 3rd Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007Bangalore, IIS, 11/7/07; Milan 11/12/07

Energetic ( Quasibrittle ) Mean Size EffectLaws and Statistical Generalization1c, 2c, 3c based on cohesive crack model, 1s statistical generalizationlog (Size D)Type 2Type 3log ( Nom. Strength N )10.10.11101002c3cLEFM21Type 111010010.1mnWeibullr1Statistical1s1c

Failure at Crack Initiation:

Type 1 Energetic-Statistical Size Effect on Strength and Lifetime

Importance of Tail Distribution of Pf Prob. of Failure= function of Pf andPf = 1 exp[-(/m)m]Tail Offset Ratio TW / TG Load

Definition:

physical justification of

Flaw Size Distribution:Why? Merely relates macro-level to micro-level hypotheses: 1. Noninteracting flaws, one in one volume element 2. Griffith (not cohesive!) theory holds on micro-level. 3. Cauchy distribution of flaw sizes:

Both fatigued polycrystalline metal and concrete are brittle, follow Weibull pdf, yet the flaws cannot be identified concrete.Weibull statistics? NO

1) pdf of One RVE

RVE = smallest material volume whose failure causes the whole structure to fail

Atomistic Basis of cdf of Quasibrittle RVEMaxwell-Boltzmann distribution frequency of exceeding activation energy Q3)cdf of break surface:2)Critical fraction of broken bondsreached within stress duration Pf10Fs(s)sTail =-assumedlinear Interatomic pot.

Power Law Tail of cdf of Strengtha) Series CouplingIf each link has tail m, the chain has the same tail mIf each fiber has tail p, the bundle has tail np additive exponentsb) Parallel CouplingThe reach of power-law tail is decreased drastically by parallel coupling, increased by series coupling.Parallel coupling produces cdf with Gaussian core.Power-law tail with zero threshold is indestructible!sslong chains(c)ssss

Power Tail Length for Bundles & Chains1)Brittle bundle with n = 24 fibers (Daniels' model, 1945) having Weibull cdf with p = 1 Gaussian core down to 0.3 Power tail up to 10-45irrelevant! (D (l.y.)3)2)Plastic bundle with n = 24 fibers (Central Limit Theorem) having Weibull cdf with p = 1 ... Gaussian core down to 0.01 Power tail up to 10-45irrelevant!4)Plastic bundle with n = 2 fibers having Weibull cdf with p = 12 ... Gaussian core down to 0.3 Power tail up to 3x10-3 Plastic fibers extend Weibull tail to 3x10-3 . OK!3)Brittle bundle with n = 2 fibers having Weibull cdf with p = 12 ...Gaussian core down to 0.3 Power tail up to 5x10-5 - longer but not enough Hence, a hierarchy of parallel-series couplings is required! Chains tend to extend the power tail!

2) pdf of Structure as a Chain of RVEs, with Size and Shape Effects1 exp[-(/m)m](Infinite chain - Weibull)(finite chain)Neq = equivalent N, modified by stress field (geometry effect)

Nano-Mechanics Based Chain-of-RVEs Model of Prob. Distribution of Structural Strength, Including Tail Chain model (structure of positive geometry)cdf for 1 RVEGaussianWeibull(power tail)1Pf0RVE strength10-3cdf for 104 RVEs brittle1Pf0Large structure strengthcdf for 500 RVE quasibrittleGaussianWeibull1Pf0 Structure strengthw = 0.150 w = 0.061 99.9%GaussianWeibullw = 0.0519 1 RVE causes the structure to fail (Type 1 size effect)grafting pt.1 RVEStructureNote: If power-law tail reaches only up to Pf = 10-12, a chain of 1047 RVEs would be needed to produce Weibull cdf.ss

5.3Quasi-Brittleness or Threshold Strength?Despite using threshold to optimize fit, Weibull theory can only fit tailOptimum fit by Weibull cdf with finite threshold13.614.61Optimum fit by chainofRVEs, zero threshold Age 2 days1m=161m=20Weibull (1939) tests of Portland cement mortar1m=24ln ln(-u)7 days28 daysWeibull scale ln[-ln(1-Pf)]ndata (2 days) = 680ndata (7 days) = 1082ndata (14 days) = 1106Pf 0.65RVE size 0.6-1.0 cm Specimen vol. 100-3000 cm3Weibull cdf with finite threshold:KINK - classical Weibull theory cant explain Pgr 0.0001-0.01

cdf of Structure Strength in Weibull Scaleln( / S0)Neq= 1101105102103Kink used to determine size of RVE and PgrIncreasing sizeWeibullGaussian1 RVE12NStructure1 exp[-(/m)m](Infinite chain - Weibull)(finite chain)Neq = equivalent N, modified by stress field (geometry effect)Now: Fit by chainofRVEs, zero threshold 1161201m=24ln Pf 0.65KINK - classical Weibull theory cant explain

Consequences of Chain-of-RVEs Model for Structural Strength1) Threshold of power-law tail must be 0, i.e.cdf of strength have kinks at the grafting points, moving up with size (# of RVEs)2)Failure Prob.Neq=1101105102103Strength3)log(strength)log (size)Mean size effect - deviation from power law sets PgrPgr= 0.0010.0030.0050.010log (size)C.o.V0 = 0.30 = 0.20 = 0.1C.o.V. of strength decreases with structural size (# of RVEs)4)5)Calculate safety factor for Pf = 10-6 as a function of equiv. # of RVEslog (size)log(strength)m110-610-6Weibull Asymptote can increase or decrease

Consequences of Chain-of-RVEs Model for Structural Strength1) Threshold of power-law tail must be 0, i.e.cdf of strength have kinks at the grafting points, moving up with size (# of RVEs)2)Failure Prob.Neq=1101105102103Strength3)log(strength)log (size)Mean size effect - deviation from power law sets PgrPgr= 0.0010.0030.0050.010log (size)C.o.V0 = 0.30 = 0.20 = 0.1C.o.V. of strength decreases with structural size (# of RVEs)4)5)Calculate safety factor for Pf = 10-6 as a function of equiv. # of RVEslog (size)log(strength)m110-610-6Weibull Asymptote can increase or decrease

Reinterpretation of Jacksons (NASA) Tests of Type 1 Size Effect on Flexural Strength of Laminates - Energetic-Statistical TheoryType I Size EffectEnergetic-Statistical Size Effect Law:Nominal Strength:= constants,= char. size of structure,= Weibull modulus,= no. of dimension for scaling

Best Fits of Jacksons (NASA) Individual Data Sets of Laminates Stat. Theory Alone

Weibull theory m = 3 and CoV = 3 % ? Weibull theory m = 30 and CoV = 23 % ?

Laminate stacking sequence:1.angle-ply2.cross-ply3.quasi-isotropic4.unidirectionalm = 3m = 30Energetic

Optimum Fit of Existing Test DataNumerical Simulations by Nonlocal Weibull Theorylog(D/Db) (Size)After Bazant, Xi, Novak (1991, 2000)

Classical (local) theory: weakest-link model if one RVE is a continuum point: Weibull Size effect:Nonlocal generalization (finite RVE):

= nonlocal strain over one RVE.localaveragedNonlocal Weibull Theory= way to combine statistical & energetic size effects(Baant and Xi, 1991) failure probability of structure--- to capture stress redistribution approximately (1991):= spatial density of failureprobability of continuum point

RVE defined by homogenization? averaging ~ central limit theorem captures only low-order statistical moments - misses the crucial cdf tailcaptured by homogenizationmatters for softening damage & failure of large structureNew RVE definition: Smallest material volume whose failure causes failure of the whole structure (of positive geometry). homogenization theory is useless for tail

Can RVE be largely or mostly Weibullian? NO!10RVE strengthPf Assume RVE to be largely Weibullian But then the RVE must behave as a chain But then damage must localize into one sub-RVE So the sub-RVE must be the true RVEAssumed WeibullRVE? NO!=sThis must be true RVE!Proof:

ln{ln[ 1/(1 Pf )]}Weibull distribution (finite threshold)ln sChain of Gaussian RVEs GumbelWeibull distribution(finite threshold)ln(s - su)ln(mean strength)ln(Neq)3.5xPf = 10-6Present Theory (zero threshold)ln{ln[ 1/(1 Pf )]}ln sPresent TheoryWeibull distribution(finite threshold)

Neq=500104105Neq=500104105Dental CeramicsAlumina-glass Composite Lohbauer et al. 2002Comparison of Present Theory (No Threshold) to Weibull Model with Finite Threshold1mPresent Theory

ln{ ln[ 1/(1 Pf ) ] }Optimum Fit by Weibull Theory with Finite Threshold ln(s - su)Pf =10-1100mm18.6mm3.81328662.711.7143.119.610.410.41.71ndata = 21ndata = 2710-610-510-410-310-2sdesign =196su = 19010-110-610-510-410-310-2su = 586sdesign =586ndata = 107ndata = 27su = 577sdesign =577su = 588sdesign =588S.F = 1.84m = 361m = 733S.F = 1.25m = 691S.F = 1.18m = 662S.F = 1.15su = 13.4S.F = 1.22m = 16.4ndata = 102sdesign =13.5(incorrect)-4-2024sdesign =230.1ndata = 27su = 230m = 398S.F = 1.73

1.8114.3913-pt Bend Test on Porcelain (Weibull 1939)4-pt Bend Test on Dental Alumina-Glass Composite (Lohbauer et al., 2002) 4-pt Bend Test on Sintered aSiC(Salem et al., 1996)4-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives(Santos et al., 2003)4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 Additives(Santos et al., 2003)4-pt Bend Test on Sintered Si3N4(Gross, 2003)

ln{ ln[ 1/(1 Pf )]}Optimum Fit by ChainofRVEs, Zero Threshold ln(s) (stress)ndata = 102Pf 0.80100mm18.6mm3-pt Bend Test, Porcelain241ndata = 1071614-pt Bend Test, Sintered aSiCPf 0.2032866ndata = 27Pf 0.303214-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 AdditivesPf 0.254014-pt Bend Test on Sintered Si3N44.03.119.610.410.4301Pf 0.25ndata = 21ndata = 27(correct)4-pt Bend Test on DentalAlumina-Glass Composite ndata = 27Pf 0.402012.512.581

ln{ln[1/(1 Pf )]}Optimum Fits by Chain-of-RVEs (zero thr

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