probabilistic 3d fracture
DESCRIPTION
Fracture reliability of 3D graded composite using novel statistical methodTRANSCRIPT
Sharif RahmanThe University of Iowa
Iowa City, IA 52245
Stochastic Multiscale Fracture Analysis of 3D Functionally Graded Media
2009 ASME PVP Conference, Prague, Czech Republic, July 2009
Work supported by NSF (CMS-0409463)
Arindam ChakrabortyStructural Integrity Associates
San Jose, CA 95138
OUTLINE
Introduction Moment-Modified Polynomial
Dimensional Decomposition (PDD)
Example Conclusions & Future Work
Two Challenging Problems Modeling random microstructure Predicting tail probabilities of fracture
response
W/Cu FGM(Zhou et al., JNM,
2007)
INTRODUCTION
FGM Fracture
crack
on uu
on tt
D
Mosaic or Level-Cut Poisson random
field(Grigoriu, JAP, 2003; Rahman, IJNME, 2008)
Objective: Develop a probabilistic, concurrent, multiscale model for calculating crack-driving forces in 3D FGM under mixed-mode deformations
INTRODUCTION
Various Multiscale Analyses
Sequential
D
D
D
D
Invasive
D
D
D
D
Concurrent(Chakraborty &
Rahman, EFM, 2008)
PDD METHOD
A Crack in a Two-Phase FGM
D
D
D
D
weak form
elasticity tensor
Output Crack-driving forces (SIFs) Fracture reliability
Crack-propagation path
random particle vol. fraction random
microstructurerandom constituent
properties
Mosaic or level-cut
RF
Input
PDD METHOD
Polynomial Dimensional Decomposition
NONLINEARSYSTEM
S-variate PDD of y (Rahman, IJNME, 2008)
PDD METHOD
Expansion Coefficients by MCS/CV
Two sets of coefficients needed for two distinct crack-tip conditions
PDD METHOD
Moment-modified PDD (each crack-tip cond.)
D
D
microscale elements
(microstructure)
macroscale elements (nomicrostructu
re)
2σ = =1 kN/cm
16 cm
= 8 cma
8 cm
8 cm
PDD METHOD
2D Verification
Edge-cracked SiC-Al FGMRandom microstructure
and constituent propertiesDet. crack location and
size/BCs
0 10 20 30 40 50 60 70
KIc, MPa m1/2
10 -3
10 -2
10 -1
10 0
Prob
abil
ity
of f
ract
ure
init
iati
on (PF(K
Ic))
Concurrent Multiscale(Monte Carlo) Microscale
(Monte Carlo)
Concurrent Multiscale
Concurrent Multiscale
(Univariate)
(Bivariate)
Univariate PDD requires five times fewer FEA than crude MCS (2000 vs. 10,000 FEA)
EXAMPLE
Edge-Cracked SiC-Al FGM
A
B
16 cm
16
cm
8 cm
4 cm
C
o
i
Crack
1x
2x
3x
D
DD
D
Particle vol. fraction 1D, inhomogeneous, Beta RFParticle location Mosaic RF, spatially-varying Poisson
intensityConstituent properties of SiC & Al indep. LN variables(Means of E: 419.2, 69.7 GPa; Means of : 0.19, 0.34)
= i =1 kN/cm2; o = 0.6 kN/cm2
Part. rad. = 0.48 cm
Nearly700 RVs( = 0.4)
EXAMPLE
Global Responses (Two Samples)
Sample 1 Sample 2
EXAMPLE Mode-I SIFs (Univariate)
0 5 10 15 20
KI, MPa m1/2
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
10 20 30 40 50
KI, MPa m1/2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
10 20 30 40 50 60 70
KI, MPa m1/2
0.00
0.02
0.04
0.06
0.08
0.10
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
AB
16 cm
16
cm
8 cm
4 cm
C
o
i
Crack
1x
2x
3x
EXAMPLE Modes-II and –III SIFs (Univariate)
-16 -14 -12 -10 -8 -6 -4 -2 0
KII, MPa m1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
0 5 10 15 20
KII, MPa m1/2
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
-6 -4 -2 0 2 4 6 8
KII, MPa m1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Prob
abil
ity
dens
ity
func
tion
Concurrent (=0.4)
Concurrent (=0.2)
0 2 4 6 8 10
KIII, MPa m1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40P
roba
bili
ty d
ensi
ty f
unct
ion
Concurrent (=0.4)
Concurrent (=0.2)
0 2 4 6 8 10 12 14
KIII, MPa m1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
0 2 4 6 8 10
KIII, MPa m1/2
0.00
0.10
0.20
0.30
0.40
0.50
0.60
Pro
babi
lity
den
sity
fun
ctio
n
Concurrent (=0.4)
Concurrent (=0.2)
0 10 20 30 40 50 60 70
KIc, MPa m1/2
10 -3
10 -2
10 -1
10 0
PF(K
Ic)
Tip C
Tip A
Tip B
=0.2
=0.4
EXAMPLE
Conditional Probability of Fracture Initiation
A
B
16 cm
16
cm
8 cm
4 cm
C
o
i
Crack
1x
2x
3x
CONCLUSIONS/FUTURE WORK
A moment-modified polynomial dimensional decomposition method was developedFourier-polynomial expansionsMCS/control variatemoment-modified random output
Efficiently generates SIF distributions
Probability of fracture initiation varies significantly along the crack front
Future work: Crack growth, cohesive zone models, particle-matrix debonding, dynamic & thermal fracture, etc.