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.