NANO-MECHANICS BASED ASSESSMENT OF NANO-MECHANICS BASED ASSESSMENT OF FAILURE RISK, SIZE EFFECT AND LIFETIME FAILURE RISK, SIZE EFFECT AND LIFETIME
OF QUASIBRITTLE STRUCTURES AT OF QUASIBRITTLE STRUCTURES AT DIFFERENT SCALESDIFFERENT SCALES
ZDENZDENĚĚK P. BAŽANTK P. BAŽANT
COLLABORATOR: JIALIANG LE, SZE-DAI PANG SPONSORS: DoT, NSF, BOEING, CHRYSLER, DoE
CapeTown, 3CapeTown, 3rdrd Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007 Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007
Bangalore, IIS, 11/7/07; Milan 11/12/07Bangalore, IIS, 11/7/07; Milan 11/12/07
NANO-MECHANICS BASED ASSESSMENT OF NANO-MECHANICS BASED ASSESSMENT OF FAILURE RISK, SIZE EFFECT AND LIFETIME FAILURE RISK, SIZE EFFECT AND LIFETIME
OF QUASIBRITTLE STRUCTURES AT OF QUASIBRITTLE STRUCTURES AT DIFFERENT SCALESDIFFERENT SCALES
ZDENZDENĚĚK P. BAŽANTK P. BAŽANT
COLLABORATOR: JIALIANG LE, SZE-DAI PANG SPONSORS: DoT, NSF, BOEING, CHRYSLER, DoE
CapeTown, 3CapeTown, 3rdrd Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007 Int. Conf. on Struct. Eng., Mech, & Computation (SEMC), 9/10/2007
Bangalore, IIS, 11/7/07; Milan 11/12/07Bangalore, IIS, 11/7/07; Milan 11/12/07
,1 1
0
rN rk
Energetic ( Quasibrittle ) Mean Size EffectLaws and Statistical Generalization
1c, 2c, 3c – based on cohesive crack model, 1s – statistical generalization
log (Size D)
Type 2
Type 3
log
( Nom
. Str
engt
h N
)
1
0.1
0.1 1 10 100
2c 3c
LEFM
2
1
21
01
1
0
D
D
DD
DN
21
00
1
D
DN
Type 1
1 10 100
1
0.1m
nWeibull
r
1
rmnN rk1
0
Statistical
1s
1cDD
D
b
b
Failure at Crack Failure at Crack Initiation:Initiation:
Type 1 Energetic-Type 1 Energetic-Statistical Size Effect Statistical Size Effect
on Strength and on Strength and LifetimeLifetime
Importance of Tail Distribution of Pf Pr
ob. o
f Fai
lure
0
1
Tolerable failure prob.Pf = 10-6
Gaussian cdf
Weibull cdf
Same mean, same
σ/Mean1
TG
TW
and
Pf = 1 – exp[-(σ/m]
Load
I. 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.
Hypotheses:
A structure is quasibrittle if Weibull cdf applies for sizes > 104 equivalent RVEs
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
pua )/(e
1) pdf of 1) pdf of One RVEOne RVE
RVE = smallest material RVE = smallest material volume whose failure causes volume whose failure causes
the whole structure to failthe whole structure to fail
Atomistic Basis of cdf of Quasibrittle RVEMaxwell-Boltzmann distribution
frequency of exceeding activation energy Q/e Q kTf
3) cdf of break surface:
2) Critical fraction of broken bonds
breaks restorations 1) Net rate of breaks ( ) / ( ) /= e eQ kT Q kT
/= 2e sinhQ kT
kT
reached within stress duration
min sinh , 1is bF C
kT
2 ebbQ kTC
b
Q
E = 0
x
1
Pf1
0
Fs
bC kT
Tail =
-assumedlinear
Inte
rato
mic
pot
.
Power Law Tail of cdf of Strength
a) Series Coupling• 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
1 2 n
12
N
softening(reality)
plastic
brittle
b) Parallel Coupling
• 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
cha
ins
(c)
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 10-45…irrelevant! (D (l.y.)3)
2) Plastic bundle with n = 24 fibers (Central Limit Theorem) havingWeibull cdf with p = 1 Gaussian core down to 0.01
Power tail up to 10-45…irrelevant!
4) Plastic bundle with n = 2 fibers havingWeibull 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 havingWeibull cdf with p = 12 Gaussian core down to 0.3
Power tail up to 5x10-5 longer but not enough
Hence, a hierarchy of parallelseries couplings is required!
Chains tend to extend the power tail!
2) pdf of 2) pdf of Structure Structure
as a Chain of RVEs, as a Chain of RVEs,
with with Size and Size and Shape EffectsShape Effects 1 1 eqN
f RVEP P 1 – exp[-(σ/m]eqN
(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 RVE
GaussianWeibull(power tail)
1Pf
0RVE strength
10-3
cdf for 104 RVEs brittle
1Pf
0Large structure strength
cdf for 500 RVE quasibrittle
Gaussian
Weibull
1Pf
0 Structure strength
= 0.150 = 0.061
99.9%
GaussianWeibull
= 0.0519
• 1 RVE causes the structure to fail (Type 1 size effect)
grafting pt.
1 RVE
Stru
ctur
e
Note: If power-law tail reaches only up to Pf = 10-12, a chain of 1047 RVEs would be needed to produce Weibull cdf.
12
N
5.3
-6
-2
2
2.4 2.8 3.2 3.6 4.0
ln( )
ln[
ln(1 P
f)]
Quasi-Brittleness or Threshold Strength?
-6
-2
2
0.5
1.5
2.5
3.5
ln u
Despite using threshold to optimize fit, Weibull theory can only fit tail
Optimum fit by Weibull cdf with finite threshold
1
3.6
1
4.6
1
Optimum fit by chain–of–RVEs, zero threshold
Age2 d
ays
1
m=16
1
m=20
Weibull (1939) tests of Portland cement mortar
1
m=24
ln ln(u)
7 day
s
28 d
aysW
eibu
ll sca
leln
[ln
(1P
f)]
ndata (2 days) = 680ndata (7 days) = 1082ndata (14 days) = 1106 Pf 0.65
RVE size 0.61.0 cmSpecimen vol. 1003000 cm3 01 e
mN u S
fP Weibull cdf with finite threshold:
KINK - classical Weibull theory can’t explain
Pgr 0.00010.01
cdf of Structure Strength in Weibull Scale
ln( / S0)
-8
-6
-4
-2
0
2
-1.0 -0.6 -0.2 0.2
ln( N /S 0 )
Neq= 1101
105 102103
Kink used to determine size of RVE and Pgr
Increasing size
Weib
ull
Gaussian
1 RVE
12
N
Stru
ctur
e 1 – exp[-(σ/m](Infinite chain - Weibull)
1 1 eqN
f RVEP P
(finite chain)
Neq = equivalent N, modified by stress field (geometry effect)
Now: Fit by chain–of–RVEs, zero threshold
-6
-2
2
2.4 2.8 3.2 3.6 4.0
ln( )
ln[
ln(1 P
f)]
116
1
20
1
m=24
ln
Pf 0.65
KINK - classical Weibull theory can’t explain
Previously: Fit by Weibull cdf with finite threshold
ln(u)
ln[
ln(1P
f)]
5.3
13.6
14.6
1
Age2 d
ays
Weibull (1939) tests of Portland cement mortar
7 day
s28
day
s
-6
-2
2
0.5
1.5
2.5
3.5
eqN
Consequences of Chain-of-RVEs Model for Structural Strength
1) Threshold of power-law tail must be 0, i.e. nufP )(~
! 0u
cdf of strength have kinks at the grafting points, moving up with size (# of RVEs)
2)
Failu
re P
rob.
-8
-6
-4
-2
0
2
-1.0 -0.6 -0.2 0.2
ln( N /S 0 )
Neq=1
101
105
102103
Strength
3)
log(
stre
ngth
)
log (size)
Mean size effect - deviation from power law sets Pgr
-0.2
0.0
0.2
0.4
0 1 2 3 4log N eq
log(
/S0
)
Pgr= 0.0010.003
0.0050.010
0.0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6log(RVE)
CoV
log (size)
C.o.
V
0 = 0.3
0 = 0.2
0 = 0.1
C.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 RVEs
log (size)
log(
stre
ngth
)
m1
10-6
10-6
Weibull Asymptote
can increase or decrease
Consequences of Chain-of-RVEs Model for Structural Strength
1) Threshold of power-law tail must be 0, i.e. nufP )(~
! 0u
cdf of strength have kinks at the grafting points, moving up with size (# of RVEs)
2)
Failu
re P
rob.
-8
-6
-4
-2
0
2
-1.0 -0.6 -0.2 0.2
ln( N /S 0 )
Neq=1
101
105
102103
Strength
3)
log(
stre
ngth
)
log (size)
Mean size effect - deviation from power law sets Pgr
-0.2
0.0
0.2
0.4
0 1 2 3 4log N eq
log(
/S0
)
Pgr= 0.0010.003
0.0050.010
0.0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6log(RVE)
CoV
log (size)
C.o.
V
0 = 0.3
0 = 0.2
0 = 0.1
C.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 RVEs
log (size)
log(
stre
ngth
)
m1
10-6
10-6
Weibull Asymptote
can increase or decrease
Reinterpretation of Jackson’s (NASA) Tests of Type 1 Size Effect on Flexural Strength of Laminates - Energetic-Statistical Theory
f r / f
r0
D/Db
0.10.1
10
1
1 10 100 1000
r = 0.8,m = 35, = 0.135
Type I Size Effect
, 10 rmrn
rNdf
Energetic-Statistical Size Effect Law:
b
b
rsDD
rD
Nominal Strength:
bdr Dsmnrf , , , , ,0 = constants,
D = char. size of structure,= Weibull modulus,m
dn = no. of dimension for scaling
Best Fits of Jackson’s (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. unidirectional
m = 3 m = 30Energetic
Optimum Fit ofExisting Test Data
Numerical Simulations by Nonlocal WeibullTheory
log(D/Db) (Size)
Db D
1 10 100 10000.1D/Db
asymptote-smallasymptote-large
Nielson 1954
Statistical formula, m=24
Tests :
1
2
0.5
3
4
nm
3 point Wright 19524 point Wright 19521inch Walker&Bloem 19572inch Walker&Bloem 1957
Lindner 1956
Sabnis&Mirza 1979
Rocco 1995Rokugo 1995
Reagel&Willis 1931
t r/tr,
1 10 100 10000.1D/Db
asymptote-small
asymptote-large
Nonlocal Weibull (III)
Statistical formula, m=24
Numerical :
1
2
0.5
3
4
nm
log(
f r/fr,)
After Bazant, Xi, Novak (1991, 2000)
Classical (local) theory: – weakest-link model if one RVE is a continuum point:
fP
Nonlocal generalization (finite RVE):
= nonlocal strain over one RVE.
loca
l
aver
aged
Nonlocal Weibull Theory
= way to combine statistical & energetic size effects
(Bažant and Xi, 1991)
failure probability of structure --- to capture stress redistribution approximately (1991):
= spatial density of failureprobability of continuum point
RVE – defined by homogenization?RVE – defined by homogenization?–– averaging ~ central limit theorem …captures only low-order statistical moments - misses the crucial cdf tail
captured by homogenization
1
0RVE strength
Pf
matters for softening damage & failure of large structure
New 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 Can RVE be largely or mostly Weibullian? NO!Weibullian? NO!
1
0RVE strength
Pf
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
Assumed WeibullRVE? NO!
=
This must be true RVE!
Proof:
ln{l
n[ 1
/(1–
Pf )
]}
Weibull distribution (finite threshold)
- 4 0 4 8
- 1 2
- 8
- 4
0
4
ln
Chain of Gaussian RVEs – Gumbel
Weibull distribution(finite threshold)
ln(u)
ln(m
ean
stre
ngth
)
ln(eq)
4 5 6 7
-12
-8
-4
0
4
3.5x Pf = 10-6
4 6
2
2.2
2.4
2.6
Present Theory (zero threshold)
3 4 5 6 7
-12
-8
-4
0
4
ln{l
n[ 1
/(1–
Pf )
]}
ln
Present Theory
Weibull distribution(finite threshold) Neq=500
104105
Neq=500104105
Dental CeramicsAlumina-glass Composite
Lohbauer et al. 2002
Comparison of Present Theory (No Threshold) to Weibull Model with Finite Threshold
1m
Present Theory
2.1 2.3 2.5 2.7 2.9
-4 0 4-2 0 2 4
1 2 3 4 5 6
ln{
ln[ 1
/(1–
Pf )
] }
Optimum Fit by Weibull Theory with Finite Threshold
ln(u)
Pf =10-1
100mm18.6mm
3.8
1
3
2
8 66
2.7
1
4 3 20 1010 4 3 20 1010
1.71
4 3.1
19.6 10.410.4
1.7
1ndata = 21 ndata = 27
10-6
10-5
10-4
10-3
10-2
design
=196
u = 190
10-1
10-6
10-5
10-4
10-3
10-2
u = 586
design
=586
ndata = 107
ndata = 27
u = 577
design
=577
u = 588
design
=588
S.F = 1.84 = 361
= 733S.F = 1.25
= 691S.F = 1.18
= 662S.F = 1.15
u = 13.4
S.F = 1.22 = 16.4
ndata = 102
design
=13.5
(incorrect)
-4 -2 0 2 4
design
=230.1- 4 0 4 8
ndata = 27
u = 230
4 3 2012.5 12.5
= 398S.F = 1.73
1.81
1
4.39
1
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 –SiC(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
– P
f )]}
Optimum Fit by Chain–of–RVEs, Zero Threshold
ln() (stress)
-3
-2
-1
0
1
2.7 2.8 2.9
ndata = 102
Pf 0.80
100mm18.6mm
3-pt Bend Test, Porcelain
24
1-6
-4
-2
0
2
5.5 5.7 5.9 6.1
ndata = 107
16
1
4-pt Bend Test, Sintered –SiC
Pf 0.20
3
2
8 66
-4
-2
0
2
6.4 6.5 6.6 6.7
ndata = 27Pf 0.30
321
4 3 20 1010
4-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives
-4.0
-2.0
0.0
2.0
6.35 6.45 6.55 6.65
4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 Additives
Pf 0.25
4 3 20 1010
40
1
-4
-2
0
2
6.4 6.5 6.6 6.7 6.8
4-pt Bend Test on Sintered Si3N4
4.0
3.1
19.6 10.410.4
30
1
Pf 0.25ndata = 21 ndata = 27
(correct)4-pt Bend Test on Dental
Alumina-Glass Composite
5.5 6 6.5
-4
-2
0
2
ndata = 27
Pf 0.40
20 12.512.54 38
1
4.5 5.5 6.5
6.0 6.4 6.8
2.2 2.6 3.0
ln{l
n[1/
(1–
Pf )
]}Optimum Fits by Chain-of-RVEs (zero threshold),
Weibull cdf with Finite Threshold, and Gaussian cdf
ln (strength in expanded scale)
3-pt Bend Test on Porcelain
16
1
4-pt Bend Test on Sintered –SiC
32
1
4-pt Bend Test on Sintered Si3N4 with Y2O3/Al2O3 Additives
4-pt Bend Test on Sintered Si3N4 with CTR2O3/Al2O3 Additives
44
1
4-pt Bend Test on Sintered Si3N4
30
1
Pf =10-1
10-6
10-5
10-4
10-3
10-2
des,G
=115
Pf =10-1
10-6
10-5
10-4
10-3
10-2
des,G
=453
24
1
S.FG = 1.30S.FR = 1.70S.FW = 1.22
Chain-of-RVEs
Gaussian cdfWeibull cdf, finite threshold
Asymptotic cdfs
des,R
=170des,W
=196
S.FG = 3.14S.FR = 2.12S.FW = 1.84
S.FG = 1.53S.FR = 1.57S.FW = 1.18
des,R
=440des,W
=587
S.FG = 1.55S.FR = 1.42S.FW = 1.15
des,G
=426des,R
=465des,W
=577des,G
=323des,R
=419des,W
=588
S.FG = 2.27S.FR = 1.75S.FW = 1.25
des,R
=9.70des,G
=12.62des,W
=13.51
With Threshold(wrong)
NoThreshold(correct)
Gaussian
6.0 6.4 6.8 5.6 6.2 6.8
4 5 6 7
S.FG = N.A.S.FR = 5.53S.FW = 1.73
81
des,W
=68des,R
=230.1
4-pt Bend Test on DentalAlumina-Glass Composite
LifetimeLifetime
Lifetime cdf via Morse Interatomic PotentialLifetime cdf via Morse Interatomic Potential• Mean time between interatomic scission under a constant stress:
]/)(exp[)( 0 kTQ Unstressed and stressed energy well
Atomic vibration Period
Nonl. stress dependence of activation energy barrier
• Morse interatomic potential for 1 bond:2)(
0 ]e1[)( 0rraQrE Dissociation energy barrier
• Energy barrier as function of stress (Phoenix): )~/ln(
~00 QQ
• Failure probability of atomic lattice:
s
t
tF
0 )]([
dexp1
log(
Pf)
log( )(1/s 10 to 50)1/
~0 kTsQ
Unstressed bond
r
E
Q
Q0
Morse Potential
kT
sQs 0
~
00tail ~F
Stressed bond
Distribution of Lifetime for 1 RVE
12
N
a) Series Coupling• If each link has
tail m, the chain has the same tail m
b) Parallel Coupling1 2 n
• If each fiber has tail p, the bundle has tail np additive exponents
Extent of power-law tail is shortened by parallel coupling:
n = 2: Ptail 10-310-2
n = 3: Ptail 10-510-4
c) Hierarchical Model of Lifetime Distribution
long
cha
ins • Parallel coupling produces cdf with Gaussian
core.
• Series coupling increases the power-tail reach. • Power-law tail with zero threshold is
indestructible!
Implications for Lifetime cdf at Macro-ScaleImplications for Lifetime cdf at Macro-ScaleImplications for Lifetime cdf at Macro-ScaleImplications for Lifetime cdf at Macro-Scale1) Weibull moduli for strength cdf ms and lifetime cdf m
1 n kT
Q
m
m
nsm
kTsQnm ss 00
~/
~
2) Dependence of cdf of lifetime on structure size
1m
Weibull asymptote
3) Size effect on mean structural lifetime
log D (Size)
ms/m 10~50
Much stronger size effect!
Pf1
0 Structural lifetime
Neq=1Neq=500Neq=104
GaussianWeibull
proportional to activation energy barrier
Increasing size
)log(
Failure probability distribution as a function of applied stress and load duration
0 2 4 60
0.5
1
Pf
log(/ref)
/s0 =1.2
1.1
0.950.8
Weibull
Grafted Weibull - Gaussian
/s0
Pf
0 1 2 30
0.5
1
10102
103
Weibull
Grafted
/ref=1
log( / ref) / s
0
Pf
Effect of loading duration on mean structural
strength µ
log(Neq ) log( / ref)
log(
mea
n st
reng
th, µ
)
log(
/0 )
log(Neq)
log(
/µ0 )
log(/ref)0 2 4 6-0.5
-0.4
-0.3
-0.2
-0.1
0
0 1 2 3 4-0.5
-0.4
-0.3
-0.2
-0.1
0
241
501
50 1
Neq =1
10
10 3
50
s = 1/50
/ref=110
10 2
10 3
10 6
241m =24
Simple Simple Probabilistic Probabilistic Analysis via Analysis via AsymptoticsAsymptotics
Type 1 Size Effect on Mean and
pdf via Asymptotic Matching
Small (D 0)
log D ( Size ) lo
g N
( N
om. S
tren
gth
)
Gaussian pdfIntermediate Asymptote
mnd
Weibull pdf
Large D D
Larger D
Small Size Asymptote
Large Size Asymptote
Higher T Longer
Each RVE = onehierarchical
model
1
0
rn rm
s bN r
s o
l rDf
l D l D
Malpasset Dam, failed 1959 —size effect must have contributed
Ruins of Malpasset DamRuins of Malpasset DamPhotos by Hubert Chanson
and Alain PasquetFailed 1959, at Failed 1959, at FrejusFrejusFrench Maritime French Maritime AlpsAlps
Cause of Failure of Malpasset DamSudden LocalizedSudden Localized
FractureFracture
Tolerable movement of abutment would today be Tolerable movement of abutment would today be 77% smaller77% smaller..
Deterministic Computations by ATENA with Microplane Model for
Scaled Dams of Various Sizes
- progressive - progressive distributeddistributed crackingcracking
- sudden - sudden localized localized fracture fracture
REALREAL
MODELMODEL
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:
sl
0, ,b rr D f
1
0
rn rm
s bN r
s o
l rDf
l D l D
( ( ll00 neglected )neglected )
olololol
Verification by Energetic-Probabilistic Finite Element Simulations
assumed
Energetic
-probabilistic
formula
matches
perfectly!
b sD l
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.
Google “Bazant”, then download
455.pdf ,464.pdf, 465.pdf, 470.pdf .
CONCLUSION
