sharif rahman the university of iowa iowa city, ia 52245
DESCRIPTION
NSF Workshop on Probability & Materials: From Nano-to-Macro Scale. STOCHASTIC FRACTURE OF FUNCTIONALLY GRADED MATERIALS. Sharif Rahman The University of Iowa Iowa City, IA 52245. January 2005. OUTLINE. Introduction Fracture of FGM Shape Sensitivity Analysis Reliability Analysis - PowerPoint PPT PresentationTRANSCRIPT
Sharif RahmanThe University of Iowa
Iowa City, IA 52245
January 2005
STOCHASTIC FRACTURE OF FUNCTIONALLY GRADED MATERIALS
NSF Workshop on Probability & Materials: From Nano-to-Macro Scale
OUTLINE
Introduction Fracture of FGM Shape Sensitivity
Analysis Reliability Analysis Ongoing Work Conclusions
INTRODUCTION
Fracture Toughness
Thermal Conductivity
Temperature Resistance
Compressive strength
Metal Rich
CrNi Alloy
Ceramic RichPSZ
The FGM Advantage
FGMs avoid stress concentrations at sharp material interfaces and can be utilized as multifunctional materials
Ilschner (1996)
INTRODUCTION
FGM Microstructure and Homogenization
Emetal
Eceramicceramic
metal
Elastic Modulus,Poisson’s Ratio,
etc.
Micro-Scale
Local Elastic Field
Averaged Elastic Field
Homogenization
Macro-Scale
EffectiveElasticity
Volume fraction,
Porosity, etc.
Ceramic matrix with metallic inclusions
Metallic matrix with ceramic inclusions
Transition zone
Gradation Direction
(0 m1)
Ceramic matrix with metallic inclusions
Metallic matrix with ceramic inclusions
Transition zone
Gradation Direction
(0 m1)
INTRODUCTION
Objective
Develop methods for stochastic fracture-mechanics analysis of functionally graded materials
Material
Resistance
CrackDrivingForce
>
Tensile Properties
Fracture
Toughness
Temperature
Radiation
Fatigue
Properties
Applied Stress
Crack Size and
ShapeGeometry of
Cracked Body
Loading Rate
Loading Cycles
Material Resistance Crack-Driving ForceSP P
Work supported by NSF (Grant Nos: CMS-0409463; DMI-0355487; CMS-9900196)
FRACTURE OF FGM
Crack-Tip Fields in Isotropic FGM
2a
crack
S1
S2
S3
r
E(x), (x)
x1x2
1 2,E E x x E x
1 2,x x x
tip11 11 11
1
2I II
I IIK f K fr
tip22 22 22
1
2I II
I IIK f K fr
tip12 12 12
1
2I II
I IIK f K fr
tip1 1 1
1
2I II
I IItip
rz K g K g
tip2 2 2
1
2I II
I IItip
rz K g K g
FRACTURE OF FGM
J-integral for FGM
1 11 1
i iij j ij j
j jA A
z zqJ W dA W qdA
x x x x
J-integral for Two Superimposed States 1 & 2
1 2 1 2
1 2 1 21 1
1 1
i i i iS S S
ij ij j ij ij jj j
A A
z z z zqJ W dA W qdA
x x x x
Superscript 1 Actual Mixed-Mode StateSuperscript 2 Auxiliary State with SIF = 1
( ) (1) (2) (1,2)SJ J J M
FRACTURE OF FGM
2 11 2(1,2) (1,2)
11 1
2 2 1 11 1 2 2
1 1 1 1
1 +
2
i iij ij j
jA
ij ij ij ijij ij ij ij
A
z z qM W dA
x x x
qdAx x x x
2 11 2(1,2) (1,2)
11 1
2 21 1 2
1 1 1
+
i iij ij j
jA
ij ij ijklij ij kl
A
z z qM W dA
x x x
DqdA
x x x
New Interaction Integral Methods
Both isotropic (Rahman & Rao; EFM; 2003) and orthotropic (Rao & Rahman, CM; 2004) FGMs can be analyzed
K KE
MI II or 2
1 2( , )
Method I: Homogeneous Auxiliary Field
Method II: Non-
Homogeneous Auxiliary Field
FRACTURE OF FGM
1 0.522 1, 2 xx L Ee
-0.50-0.30-0.100.100.300.50-1.00
-0.60
-0.20
0.20
0.60
1.00
2b2b
a
W
x2
x1
= 450
L2
L2
-0.50-0.30-0.100.100.300.50-1.00
-0.60
-0.20
0.20
0.60
1.00
(N = 370)
0.4 2a W
L=2, W=1
=0.3
1 1
1exp
2E x E x
ProposedMethod-I
1,2M
ProposedMethod-II
1,2M
Kim andPaulino
*( )kJ
IK
E a IIK
E a IK
E a IIK
E a IK
E a IIK
E a
0 1.448 0.610 1.448 0.610 1.451 0.6040.1 1.392 0.585 1.391 0.585 1.396 0.5790.25 1.313 0.549 1.312 0.549 1.316 0.5440.5 1.193 0.495 1.190 0.495 1.196 0.4910.75 1.086 0.447 1.082 0.446 1.089 0.4431 0.990 0.405 0.986 0.404 0.993 0.402
Plane Stress Condition
Example 1 (Slanted Crack in a Plate)
Gradation Direction
SHAPE SENSITIVITY ANALYSIS
Velocity Field & Material Derivative
,, xTxTx,xV
d
d
d
d
Zzzzz, allfor ,ΩΩ a
Zzzzzzz VV ,,, aa
Need a numerical method (FEM) to solve these two
equations for
x
x
V(x)
z z and
Governing and Sensitivity Equations
, ;, ;, TTxTx
0
( ) ( )lim
z x V x z xz
SHAPE SENSITIVITY ANALYSIS
Performance Measure
Shape Sensitivity
( , )g d
z zÑ
, ,, , , , , , , div
i i i j i jz i z i j j z i j z i jk kg z g z V g z g z V g d
V
,, , , , ,/ ; / ; / ; /i i ji j i j i j i j z i z i jz z x z z x g g z g g z
SHAPE SENSITIVITY ANALYSIS
Sensitivity of Interaction Integral Method
Method I : Homogeneous Auxiliary Field
1 2
1 2
( ) ( )+i i
i i
pV pVP p
x x
26 26
(1,2)
1 1
div( )i i iA Ai i
M p p dA PdA
V
Method II : Non-Homogeneous Auxiliary Field
1 2
1 2
( ) ( )+i i
i i
sV sVS s
x x
14 14
(1,2)
1 1
div( )i i iA Ai i
M s s dA S dA
V
Rahman & Rao; CM; 2004 and Rao & Rahman, CMAME; 2004
SHAPE SENSITIVITY ANALYSIS
W
L
2b
x1
2a
2b
L
W
x2
122 1,
xx L Ee 2L=2W=20, 2a=2, =0.3 Plane Stress Conditions
1 1expE x E x
Sensitivity of SIF Values /
( )IK a
a
( )IK a
a
( )IIK a
a
( )IIK a
a
0 1.5549 0.5018 0.0030 0.0033 0.1 1.3970 0.4599 0.3485 0.1739 0.2 0.9900 0.3471 0.5483 0.2992 0.3 0.5126 0.1870 0.5184 0.3304 0.4 0.1513 0.0407 0.2981 0.2288
Proposed Method-I
1,2M
0.5 0.0007 0.0008 0.0007 0.0007 0 1.5549 0.5018 0.0030 0.0033
0.1 1.3970 0.4599 0.3485 0.1739 0.2 0.9900 0.3471 0.5483 0.2992 0.3 0.5126 0.1870 0.5184 0.3304 0.4 0.1513 0.0407 0.2981 0.2288
Finite Difference
0.5 0.0007 0.0008 0.0007 0.0007
Sensitivity of SIF Values /
( )IK a
a
( )IK a
a
( )IIK a
a
( )IIK a
a
0 1.5549 0.5018 0.0030 0.0033 0.1 1.3970 0.4599 0.3485 0.1739 0.2 0.9900 0.3471 0.5483 0.2992 0.3 0.5126 0.1870 0.5184 0.3304 0.4 0.1513 0.0407 0.2981 0.2288
Proposed Method-I
1,2M
0.5 0.0007 0.0008 0.0007 0.0007 0 1.5549 0.5018 0.0030 0.0033
0.1 1.3970 0.4599 0.3485 0.1739 0.2 0.9900 0.3471 0.5483 0.2992 0.3 0.5126 0.1870 0.5184 0.3304 0.4 0.1513 0.0407 0.2981 0.2288
Finite Difference
0.5 0.0007 0.0008 0.0007 0.0007
Example 2 (Plate with an Internal Crack)
Gradation Direction
FRACTURE RELIABILITY
FGM System
;
;
EE E
x V
x V
{ : ( ) 0}; NF y v v v Failure Criterion
Stochastic Fracture Mechanics
2a
crack
S1
S2
S3
r
E(x), (x)
x1x2
Random Input
1
2
3
2
E
S
a
S
S
VV
V
Load
Material & gradation propertiesGeometry
Failure Probability
0FP P y V
Fracture initiation and propagation
FRACTURE RELIABILITY
Multivariate Function Decomposition
Univariate Approximation
1 1 11
ˆ ˆ( ) ( , , ) (0, ,0, ,0, ,0) ( 1) (0, ,0)N
N ii
y y v v y v N y
v
Bivariate Approximation
1 2
1 2
21
( 1)( 2)ˆ ( ) (0, ,0, ,0, ,0, ,0, ,0) ( 2) (0, ,0, ,0, ,0) ( )
2
N
i i ii i i
N Ny y v v N y v y
v 0
General S-Variate Approximation
0
1ˆ ( ) ( 1) ( )
Si
S S ii
N S iy y
i
v v
At most 1 variable in a term
At most 2 variables in a term
At most S variables in a
term
Reliability Analysis
FRACTURE RELIABILITY
Performance Function Approximations
( ) 0
0 ( )F yP P y f d
VvV v v
1 2
1 2
1 21 2
1 2 1 2 1 21 1 , 0 1 2
1 1( ) ( )
! ! !
j jjNj jj
i i ij jjj i j j i ii i i
y yy y v v v
j v j j v v
v 0 0 0
11 1
1ˆ ( ) ( )
!
jNj
ijj i i
yy y v
j v
v 0 0
Univariate
1 2ˆ( ) ( )y y v v R
Bivariate
2 3ˆ( ) ( )y y v v R
1 2
1 2
1 21 2
1 2 1 2 1 2
21 1
, 0 1 2
1ˆ ( ) ( )
!
1
! !
jNj
ijj i i
j jj j
i ij jj j i i i i
yy y v
j v
yv v
j j v v
v 0 0
0
Terms with dimensions 2 & higher
Terms with dimensions 3 & higher
Lagrange Interpolation
FRACTURE RELIABILITY
1 2
1 2 1 2 1 1 2 2 1 2 1 2
2 1
( )
1
1 1
( ) ( )
( , ) ( ) ( ) ,
nj
i i j i i ij
n nj j
i i i i j i j i i i i ij j
y v v y v
y v v v v y v v
Monte Carlo Simulation
Lagrange shape
functions
( ),1 1
1
1ˆ 0
SNi
FiS
P yN
vI
( ),2 2
1
1ˆ 0
SNi
FiS
P yN
vI
UnivariateApproximatio
n
Bivariate Approximati
on
FRACTURE RELIABILITY Example 3 (Probability of Fracture Initiation)
(a)
W
åì
a
L/2
L/2
Crack2b2
2b1
Integral Domain
çì
1
2
Random Variable
Mean
Standard Deviation
Probability Distribution
a 3.5 0.404 Uniform
W 7.5 0.289 Uniform
1 0.1 Gaussian
1 0.1 Gaussian
0 0.3 Gaussian
E1 1 0.1 Lognormal
E2 3 0.3 Lognormal
5 0.5 Lognormal
1 2 1 21 2 1 2( , ) tanh ( cos sin )
2 2
E E E EE
2 ( ) 3 ( )( ) ( )cos ( )sin ( ) cos
2 2 2Ic I IIy K K K V V
V V V V
Performance Function (Maximum Hoop Stress Criterion)
Gra
dation
Direc
tion
FRACTURE RELIABILITY
Example 3 (Results)
40 52 64 76 88 100
Fracture Toughness ( KIc)
10 -4
10 -3
10 -2
10 -1
10 0
Pro
babi
lty o
f F
ract
ure
Initi
atio
n
Monte Carlo Simulation(105 samples)
Univariate Approximation
(25 FEA)
BivariateApproximation
(277 FEA)
FORM(50-100 FEA)
ONGOING WORK
Stochastic Micromechanics
Nonhomogeneous Random Field
Volume FractionPorosity
Micromechanics
Rule of MixturesMori-Tanaka Theory
Self-Consistent TheoryEshelby’s Inclusion Theory
Particle InteractionGradients of Volume Fraction
Stochastic Material Properties
Elastic ModulusPoisson’s RatioYield Strength
etc.
Nonhomogeneous Random Field
Spatially-varying FGMMicrostructure
ONGOING WORK Level-Cut Random Field for FGM Microstructure
Phase 1 { : ( ) 1}
Phase 2 { : ( ) 0}
n
n
Y
Y
x x
x x
Translation Random Field
1,( ) ( ( )); ( )
0,
z aY I Z I z
z a
x x
Second-Moment Properties
1
1 2 1 2 11 1 2
( ) ( ) ( )
( ) ( ) ( ) ( ) ( , )
Y P Z a p
Y Y P Z a Z a p
x x x
x x x x x x
E
E
Filtered Non-Homogeneous Poisson Field
Find probability law of Z(x) to match target statistics p1 and p11
Grigoriu (2003)Homogeneous microstructure
( )
1
( ) ( ) , D
ni i i
i
Z Z h D D
x x x N
Φ Γ
( ) = Non-homogeneous Poisson field with intensity measure ( )D DD d x xN
volume fraction
two-point correlation function
ONGOING WORK
Multi-Scale Model of FGM Fracture
Force fieldof particle
(CrNi)
Force fieldof inter-
layer
Force fieldof matrix
(PSZ)
MDsimulation (particle)
MDsimulation (interlayer)
MDsimulation
(matrix)
Stochasticnano-
structure
Nano-mechanics of cluster
Homo-genization
Stochasticmicro-
structure
Micro-mechanics
of FGM
Homo-genization
Fracture of FGM
Sensitivityanalysis
Reliabilityof FGM
Interatomicpotential
Constituentproperties
Clusterproperties
FGMproperties
Molecular-scale Nano-scale Micro-scale Macro-scale
QuantumMechanics
MolecularMechanics
ContinuumMechanics
Force fieldof particle
(CrNi)
Force fieldof inter-
layer
Force fieldof matrix
(PSZ)
MDsimulation (particle)
MDsimulation (interlayer)
MDsimulation
(matrix)
Stochasticnano-
structure
Nano-mechanics of cluster
Homo-genization
Stochasticmicro-
structure
Micro-mechanics
of FGM
Homo-genization
Fracture of FGM
Sensitivityanalysis
Reliabilityof FGM
Interatomicpotential
Constituentproperties
Clusterproperties
FGMproperties
Molecular-scale Nano-scale Micro-scale Macro-scale
QuantumMechanics
MolecularMechanics
ContinuumMechanics
CONCLUSIONS
New interaction integral methods for linear-elastic fracture under mixed-mode loading conditions
Continuum shape sensitivity analysis for first-order gradient of crack-driving force with respect to crack geometry
Novel decomposition methods for accurate and computationally efficient reliability analysis
Ongoing work involves stochastic, multi-scale fracture of FGMs
REFERENCES
• Rao, B. N. and Rahman, S., “A Mode-Decoupling Continuum Shape Sensitivity Method for Fracture Analysis of Functionally Graded Materials,” submitted to International Journal for Numerical Methods in Engineering, 2004.
• Rahman, S., “Stochastic Fracture of Functionally Graded Materials,” submitted to Engineering Fracture Mechanics, 2004.
• Xu, H. and Rahman, S., “Dimension-Reduction Methods for Structural Reliability Analysis,” submitted to Probabilistic Engineering Mechanics, 2004.
• Rahman, S. and Rao, B. N., “A Continuum Shape Sensitivity Method for Fracture Analysis of Isotropic Functionally Graded Materials,” submitted to Computational Mechanics, 2004.
• Rao, B. N. and Rahman, S., “A Continuum Shape Sensitivity Method for Fracture Analysis of Orthotropic Functionally Graded Materials,” accepted in Mechanics and Materials, (In Press).
• Rahman, S. and Rao, B. N., “Continuum Shape Sensitivity Analysis of a Mode-I Fracture in Functionally Graded Materials,” accepted in Computational Mechanics, 2004 (In Press).
• Rao, B. N. and Rahman, S., “Continuum Shape Sensitivity Analysis of a Mixed-Mode Fracture in Functionally Graded Materials,” accepted in Computer Methods in Applied Mechanics and Engineering, 2004 (In Press).
• Rao, B. N. and Rahman, S., “An Interaction Integral Method for Analysis of Cracks in Orthotropic Functionally Graded Materials,” Computational Mechanics, Vol. 32, No. 1-2, 2003, pp. 40-51.
• Rao, B. N. and Rahman, S., “Meshfree Analysis of Cracks in Isotropic Functionally Graded Materials,” Engineering Fracture Mechanics, Vol. 70, No. 1, 2003, pp. 1-27.