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Energetic ( Quasibrittle ) Mean Size Effect Laws and Statistical Generalization

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

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Energetic ( Quasibrittle ) Mean Size Effect Laws and Statistical Generalization

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  1. NANO-MECHANICS BASED ASSESSMENT OF FAILURE RISK, SIZE EFFECT AND LIFETIME OF QUASIBRITTLE STRUCTURES AT DIFFERENT SCALES ZDENĚK P. BAŽANT COLLABORATOR: JIALIANG LE, SZE-DAI PANG SPONSORS: DoT, NSF, BOEING, CHRYSLER, DoE CapeTown, 3rd Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007 Bangalore, IIS, 11/7/07; Milan 11/12/07

  2. Energetic ( Quasibrittle ) Mean Size EffectLaws and Statistical Generalization 1c, 2c, 3c – based on cohesive crack model, 1s – statistical generalization Type 1 1 1 3c 2c Type 3 r LEFM 1 Type 2 2 log ( Nom. Strength N) 1c 1 0.1 0.1 1s m n Weibull Statistical log (Size D) 0.1 1 10 100 1 10 100

  3. Failure at Crack Initiation:Type 1 Energetic-Statistical Size Effect on Strength and Lifetime

  4. 1 Same mean, same w Gaussian cdf Tolerable failure prob.Pf= 10-6 Weibull cdf σ/m Mean1 0 TG TW Importance of Tail Distribution ofPf Pf = 1 – exp[-(σ/m)m] Prob. of Failure Load = function of Pf and Tail Offset RatioTW / TG

  5. Hypotheses: • Interatomic bond energies have • Maxwell-Boltzmann distribution, and • activation energy depends on stress. • II. Tests of lab specimens < 5 RVEs • do not disagree with Gaussian pdf. A structure is quasibrittle if Weibull cdf applies for sizes > 104 equivalent RVEs Definition:

  6. Flaw Size Distribution: —physical justification of Weibull statistics? …NO 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.

  7. 1) pdf of One RVERVE = smallest material volume whose failure causes the whole structure to fail

  8. E 1) Net rate of breaks 1   = 0  breaks restorations Q x Maxwell-Boltzmann distribution frequency of exceeding activation energyQ Atomistic Basis of cdf of Quasibrittle RVE -assumed linear Interatomic pot. 2) Critical fraction of broken bonds reached within stress duration 3) cdf of break surface: Pf 1 Tail = Fs(s) 0 s

  9. s n 1 2 plastic  1 2 softening(reality) brittle e N s b)Parallel Coupling s Power Law Tail of cdf of Strength a)Series Coupling • If each link has tail m, the chain has the same tail m s • If each fiber has tail p, the bundle has tail np additive exponents s s (c) • The 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! long chains s

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

  11. 2) pdf of Structure as a Chain of RVEs, with Size and Shape Effects  1 – exp[-(σ/m)m] (Infinite chain - Weibull) (finite chain) Neq= equivalent N, modified by stress field (geometry effect)

  12. Nano-Mechanics Based Chain-of-RVEs Model of Prob. Distribution of Structural Strength, Including Tail 1 2 N s Structure • Chain model (structure of positive geometry) • 1 RVE causes the structure to fail (Type 1 size effect) 1 RVE s cdf for 104 RVEs brittle cdf for 500 RVE quasibrittle Pf cdf for 1 RVE Pf Pf 1 1 1 99.9% Gaussian Gaussian Gaussian Weibull (power tail) grafting pt. Weibull Weibull 10-3 w = 0.150 w = 0.061 w = 0.0519 0 0 0 RVE strength Structure strength Large structure strength Note: If power-law tail reaches only up to Pf= 10-12, a chain of 1047 RVEs would be needed to produce Weibull cdf.

  13. Quasi-Brittleness or Threshold Strength? Optimum fit by Weibull cdf with finite threshold Optimum fit by chain–of– RVEs, zero threshold Weibull (1939) tests of Portland cement mortar ndata (2 days) = 680 ndata(7 days) = 1082 ndata(14 days) = 1106 Pf  0.65 KINK - classical Weibull theory can’t explain 1 Weibull scaleln[-ln(1-Pf)] 3.6 1 1 1 Despite using threshold to optimize fit, Weibull theory can only fit tail Age2 days m=24 m=16 4.6 m=20 28 days 5.3 7 days 1 1 ln  ln(-u) RVE size  0.6-1.0 cmSpecimen vol.  100-3000 cm3 Weibull cdf with finite threshold:  Pgr 0.0001-0.01

  14. Previously: Fit by Weibull cdf with finite threshold Weibull (1939)tests of Portland cement mortar 2 1 ln[-ln(1-Pf)] 3.6 -2 1 Age2 days 4.6 28 days 5.3 7 days 1 -6 ln(-u) 0.5 1.5 2.5 3.5 cdf of Structure Strength in Weibull Scale 1 – exp[-(σ/m)m] 1 Structure 2 (Infinite chain - Weibull) (finite chain) 1 RVE Neq= equivalent N, modified by stress field (geometry effect) N Now: Fit by chain–of– RVEs, zero threshold Gaussian 105 103 102 Pf  0.65 101 Neq= 1 KINK - classical Weibull theory can’t explain 1 1 Weibull m=24 20 16 Kink used to determine size of RVE and Pgr Increasing size 1 ln  ln( / S0)

  15. Consequences of Chain-of-RVEs Model for Structural Strength 1) Threshold of power-law tail must be 0, i.e. 2) 3) 105 103 Mean size effect - deviation from power law sets Pgr 102 Pgr= 0.001 cdf of strength have kinks at the grafting points, moving up with size (# of RVEs) 101 0.003 0.005 log(strength) 0.010 Failure Prob. Neq=1 log (size) Strength 5) Calculate safety factor for Pf= 10-6 as a function of equiv. # of RVEs 4) 0= 0.3 C.o.V. of strength decreases with structural size (# of RVEs)  can increase or decrease 0=0.2 C.o.V 0= 0.1 log(strength) 10-6 Weibull Asymptote 1 m 10-6 log (size) log (size)

  16. Consequences of Chain-of-RVEs Model for Structural Strength 1) Threshold of power-law tail must be 0, i.e. 2) 3) 105 103 Mean size effect - deviation from power law sets Pgr 102 Pgr= 0.001 cdf of strength have kinks at the grafting points, moving up with size (# of RVEs) 101 0.003 0.005 log(strength) 0.010 Failure Prob. Neq=1 log (size) Strength 5) Calculate safety factor for Pf= 10-6 as a function of equiv. # of RVEs 4) 0= 0.3 C.o.V. of strength decreases with structural size (# of RVEs)  can increase or decrease 0=0.2 C.o.V 0= 0.1 log(strength) 10-6 Weibull Asymptote 1 m 10-6 log (size) log (size)

  17. 10 r = 0.8,m = 35,w = 0.135 fr / fr0 1 0.1 0.1 1 10 100 1000 D/Db Reinterpretation of Jackson’s (NASA) Tests of Type 1 Size Effect on Flexural Strength of Laminates - Energetic-Statistical Theory Energetic-Statistical Size Effect Law: Nominal Strength: Type I Size Effect = constants, = char. size of structure, = Weibull modulus, = no. of dimension for scaling

  18. Best Fits of Jackson’s (NASA) Individual Data Sets of LaminatesStat. Theory Alone m = 3 m = 30 Energetic • Weibull theory m = 3 and CoV =3 % ? • Weibull theory m = 30 and CoV =23 % ? Laminate stacking sequence: 1. angle-ply 2. cross-ply 3. quasi-isotropic 4. unidirectional

  19. Optimum Fit ofExisting Test Data 4 4 Tests : Numerical : Nielson 1954 3 3 point Wright 1952 3 4 point Wright 1952 1inch Walker&Bloem 1957 Nonlocal Weibull (III) 2inch Walker&Bloem 1957 Reagel&Willis 1931 2 2 Statistical formula, m=24 Sabnis&Mirza 1979 tr/tr, Rokugo 1995 asymptote-large Rocco 1995 log(fr/fr,) Lindner 1956 asymptote-small Statistical formula, m=24 n n asymptote-small m m asymptote-large 1 1 0.5 0.5 0.1 1 10 0.1 1 100 10 1000 100 1000 D/Db D/Db Db D Numerical Simulations by Nonlocal WeibullTheory log(D/Db) (Size) After Bazant, Xi, Novak (1991, 2000)

  20. Nonlocal Weibull Theory (Bažant and Xi, 1991) Classical (local) theory: – weakest-link model if one RVE is a continuum point: = spatial density of failure probability of continuum point averaged local Nonlocal generalization (finite RVE): =way to combine statistical & energetic size effects = nonlocal strain over one RVE.  failure probability of structure --- to capture stress redistribution approximately (1991):  Weibull Size effect:

  21. 1 Pf RVE strength 0 RVE – defined by homogenization? –averaging ~ central limit theorem …captures only low-order statistical moments - misses the crucial cdf tail  matters for softening damage & failure of large structure captured by homogenization  homogenization theory is useless for tail New RVE definition: Smallest material volume whose failure causes failure of the whole structure (of positive geometry).

  22. Can RVE be largely or mostly Weibullian? NO! s Proof: RVE? NO! This must be true RVE! 1 Assumed Weibull = Pf RVE strength 0 • 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 RVE

  23. Comparison of Present Theory (No Threshold) to Weibull Model with Finite Threshold Chain of Gaussian RVEs – Gumbel Dental Ceramics Alumina-glass Composite Present Theory ln{ln[ 1/(1– Pf )]} Weibull distribution (finite threshold) Lohbauer et al. 2002 3.5x Pf= 10-6 ln{ln[ 1/(1– Pf )]} ln(s - su) ln s 105 104 Weibull distribution (finite threshold) Neq=500 Present Theory ln(mean strength) 105 Weibull distribution (finite threshold) 1 104 Present Theory (zero threshold) Neq=500 m ln s ln(Neq)

  24. 3 4 12.5 20 12.5 3 4 10 20 10 3 4 10 20 10 Optimum Fit by Weibull Theory with Finite Threshold (incorrect) 3-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 a–SiC (Salem et al., 1996) su = 13.4 su = 190 su = 230 m = 16.4 m = 361 m = 398 Pf =10-1 S.F = 1.22 S.F = 1.84 S.F = 1.73 10-2 4.39 ndata = 102 ndata = 27 ndata = 107 1.81 10-3 3.8 1 1 1 10-4 2 10-5 18.6mm 100mm 3 6 8 6 10-6 ln{ ln[ 1/(1– Pf ) ]} sdesign=230.1 sdesign=13.5 sdesign=196 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) su = 586 su = 577 su = 588 m = 691 m = 662 m = 733 10-1 S.F = 1.18 S.F = 1.15 S.F = 1.25 10-2 1.7 1.7 2.7 ndata = 27 ndata = 21 ndata = 27 1 10-3 1 1 10-4 10-5 3.1 4 10-6 10.4 19.6 10.4 sdesign=586 sdesign=577 sdesign=588 ln(s - su) -4 -2 0 2 4

  25. 3 4 3 4 10 20 10 3 4 10 20 10 Optimum Fit by Chain–of–RVEs, Zero Threshold (correct) 3-pt Bend Test, Porcelain 4-pt Bend Test, Sintered a–SiC 4-pt Bend Test on Dental Alumina-Glass Composite Pf  0.80 ndata = 102 Pf  0.40 Pf  0.20 ndata = 107 ndata = 27 16 8 2 18.6mm 24 1 12.5 20 12.5 100mm ln{ ln[ 1/(1– Pf )]} 1 3 6 8 6 1 4-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives 4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 Additives 4-pt Bend Test on Sintered Si3N4 Pf  0.30 Pf  0.25 Pf  0.25 ndata = 27 ndata = 21 ndata = 27 40 32 30 1 1 1 3.1 4.0 10.4 19.6 10.4 ln(s) (stress)

  26. Optimum Fits by Chain-of-RVEs (zero threshold), Weibull cdf with Finite Threshold, and Gaussian cdf Chain-of-RVEs Asymptotic cdfs Gaussian cdf Weibull cdf, finite threshold 4-pt Bend Test on Sintered a–SiC 3-pt Bend Test on Porcelain 4-pt Bend Test on Dental Alumina-Glass Composite S.FG = 1.30 S.FR = 1.70 S.FW = 1.22 S.FG = 3.14 S.FR = 2.12 S.FW = 1.84 Pf =10-1 10-2 10-3 S.FG = N.A. S.FR = 5.53 S.FW = 1.73 16 24 10-4 8 1 1 10-5 1 ln{ln[1/(1– Pf )]} 10-6 sdes,G=115 sdes,R=170 sdes,W=196 sdes,R=9.70 sdes,G=12.62 sdes,W=13.51 sdes,R=230.1 sdes,W=68 4-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives 4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 Additives 4-pt Bend Test on Sintered Si3N4 S.FG = 1.53 S.FR = 1.57 S.FW = 1.18 S.FG = 1.55 S.FR = 1.42 S.FW = 1.15 S.FG = 2.27 S.FR = 1.75 S.FW = 1.25 With Threshold (wrong) Pf =10-1 10-2 10-3 32 No Threshold (correct) 44 30 Gaussian 10-4 1 1 1 10-5 10-6 sdes,R=440 sdes,G=453 sdes,W=587 sdes,G=426 sdes,R=465 sdes,W=577 sdes,G=323 sdes,R=419 sdes,W=588 ln s (strength in expanded scale)

  27. Lifetime

  28. Lifetime cdf via Morse Interatomic Potential (1/s 10 to 50) • Mean time between interatomic scission under a constant stress: Unstressed and stressed energy well E Morse Potential Atomic vibration Period Unstressed bond Nonl. stress dependence of activation energy barrier Q0 • Morse interatomic potential for 1 bond: Q Stressed bond r Dissociation energy barrier • Energy barrier as function of stress • (Phoenix): log(Pf) • Failure probability of atomic lattice: log( )

  29. 1 n 1 2 2  N s long chains s Distribution of Lifetime  for 1 RVE a)Series Coupling b)Parallel Coupling Extent of power-law tail is shortened by parallel coupling: n = 2: Ptail 10-310-2 n = 3: Ptail10-510-4 • If each link has tail m, the chain has the same tail m • If each fiber has tail p, the bundle has tail np additive exponents c)Hierarchical Model of Lifetime Distribution • Parallel coupling produces cdf with Gaussian core. • Series coupling increases the power-tail reach. • Power-law tail with zero threshold is indestructible!

  30. Implications for Lifetime cdf at Macro-Scale n 1 s 1) Weibull moduli for strength cdf ms and lifetime cdf m proportional to activation energy barrier 3) Size effect on mean structural lifetime 2) Dependence of cdf of lifetime  on structure size Pf ms/m10~50  Much stronger size effect! 1 Neq=500 Neq=104 Neq=1 Gaussian Weibull Weibull asymptote 1 m Increasing size 0 Structural lifetime  log D (Size)

  31. Failure probability distribution as a function of applied stress  and load duration  Weibull 103 102 Pf Grafted Pf 10  /ref=1 /s0 Weibull /s0 =1.2 Grafted Weibull - Gaussian Pf  / s0 log( /ref) 1.1 0.95 0.8 log(/ref)

  32. Effect of loading duration  on mean structural strength µ 24 106 1 103 log( /0 ) 102 10  /ref=1 1 m =24 24 log(meanstrength, µ) log(Neq) 103 50 1 50 log( /µ0 ) 10 log( /ref) log(Neq) Neq=1 1 s = 1/50 50 log(/ref)

  33. Simple Probabilistic Analysis via Asymptotics

  34. Type 1 Size Effect on Mean and pdf via Asymptotic Matching Each RVE = one hierarchical model Small (D 0) Small Size Asymptote Larger D log N ( Nom. Strength ) Intermediate Asymptote Gaussian pdf Weibull pdf Longer Large Size Asymptote  Higher T m nd Large D D log D( Size )

  35. Malpasset Dam, failed 1959 —size effect must have contributed

  36. Ruins of Malpasset Dam Failed 1959, at Frejus French Maritime Alps c Photos by Hubert Chanson and Alain Pasquet

  37. Cause of Failure of Malpasset Dam Sudden Localized Fracture Tolerable movement of abutment would today be 77% smaller.

  38. Deterministic Computations by ATENA with Microplane Model for Scaled Dams of Various Sizes MODEL - progressive distributed cracking - sudden localized fracture REAL

  39. Predicting Energetic-Probabilistic Scaling without Nonlocal Analysis • Fit of deterministic computations for at least 3 sizes gives • One evaluation of Weibull probability integral gives … asymptotic matching formula fixed: ( l0neglected )

  40. Verification by Energetic-Probabilistic Finite Element Simulations assumed Energetic -probabilistic formula matches perfectly!

  41. For quasibrittle materials, not only the mean nominal strength, but also the strength distribution, the safety factors and lifetime distribution, depend on structure size and shape. CONCLUSION Google “Bazant”, then download 455.pdf ,464.pdf, 465.pdf, 470.pdf .

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