On the analysis of finite deformations and continuum damage in materials with random properties Materials Process Design and Control Laboratory Swagato.
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On the analysis of finite deformations On the analysis of finite deformations and continuum damage in materials with and continuum damage in materials with
random propertiesrandom properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Swagato Acharjee and Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: swagato@cornell.edu zabaras@cornell.eduURL: http://mpdc.mae.cornell.edu/
peoplepeople
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
RESEARCH SPONSORS
U.S. AIR FORCE PARTNERS
Materials Process Design Branch, AFRL
Computational Mathematics Program, AFOSR
CORNELL THEORY CENTER
ARMY RESEARCH OFFICE
Mechanical Behavior of Materials Program
NATIONAL SCIENCE FOUNDATION (NSF)
Design and Integration Engineering Program
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
OUTLINE
•Motivation
•Overview of GPCE
•GPCE Solution methodology
•GPCE based Applications
•Merits and pitfalls of GPCE
•Overview of Support space/Stochastic Galerkin method
•Solution scheme using Support space method
•Extension to Continuum Damage
•Applications
•Conclusions/Future work
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CCOORRNNEELLLL U N I V E R S I T Y
Two w
ay flow
of sta
tistica
l infor
mation
1 1e2 1e4 1e6 1e9
Eng
inee
ring
Length Scales ( )
Phy
sics
Che
mis
tryM
ater
ials
0 A
Informati
on flo
w
Statistical filter
Electronic
Nanoscale
Microscale
Mesoscale
Continuum
MOTIVATION: THE BIG PICTURE
Material information – inherently statistical in nature.
•Atomic scale – Kinetic theory, Maxwell’s distribution etc.
•Microstructural features – correlation functions, descriptors etc.
Information flow across scales
Need to develop efficient tools for incorporating statistical information for a complete characterization of material behavior.
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
MOTIVATION:UNCERTAINTY IN FINITE DEFORMATION PROBLEMS
Metal formingForging velocity
Lubrication – friction at die-workpiece interface
Intermediate material state variation over a multistage sequence –residual-stresses, temperature, change in microstructure, expansion/contraction of the workpiece
Die shape – is it constant over repeated forgings ?
Damage evolution through processing stages
Preform shapes (tolerances)
Composites – fiber orientation, fiber spacing, constitutive model
Biomechanics – material properties, constitutive model, fibers in tissues
Material heterogeneity
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CCOORRNNEELLLL U N I V E R S I T Y
OVERVIEW OF FINITE DEFORMATION DETERMINISTIC PROBLEM
Linearized principle of virtual work equation
0 0
. .B B
dP dFdV P dFdV
BB0 BBFF eFF p
FFInitial configurationInitial configuration Deformed configurationDeformed configuration
B
Governing equationGoverning equation
. 0P
(1) Multiplicative decomposition framework(1) Multiplicative decomposition framework
(2) State variable based rate-dependent (2) State variable based rate-dependent constitutive modelsconstitutive models
(3) Total Lagrangian formulation(3) Total Lagrangian formulation
(4) Semi-implicit stress update scheme (4) Semi-implicit stress update scheme (Ortiz,1990)(Ortiz,1990) F
xX
X x
x
Strain measure – Green strainStrain measure – Green strain
Conjugate stress measure – Conjugate stress measure – PKII stressPKII stress
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CCOORRNNEELLLL U N I V E R S I T Y
GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW
n
iii txWtxW
0)(),(~),,(
Stochastic Stochastic processprocess
Chaos Chaos polynomialspolynomials
(random (random variables)variables)Reduced order representation of a stochastic processes.
Subspace spanned by orthogonal basis functions from the Askey series.
Chaos polynomialChaos polynomial Support spaceSupport space Random variableRandom variable
LegendreLegendre [[]] UniformUniform
JacobiJacobi [[]] BetaBeta
HermiteHermite [-[-∞,∞]∞,∞] Normal, LogNormalNormal, LogNormal
LaguerreLaguerre [0, [0, ∞]∞] GammaGamma
Number of chaos polynomials used to represent output uncertainty depends on Number of chaos polynomials used to represent output uncertainty depends on
- Type of uncertainty in input- Type of uncertainty in input - Distribution of input uncertainty- Distribution of input uncertainty- Number of terms in KLE of input - Degree of uncertainty propagation desired- Number of terms in KLE of input - Degree of uncertainty propagation desired
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CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY ANALYSIS USING SSFEM
Key features
Total Lagrangian formulation – (assumed deterministic initial configuration)
Spectral decomposition of the current configuration leading to a stochastic deformation gradient
Bn+1(θ)
xn+1(θ)=x(X,tn+1, θ,)
B0
Xxn+1(θ)
F(θ)
1 1( ) ( )n ni i B B
10 1
( , )( ) ( , )
nn
tt
x X,F x X,
X1
1( ) ( , ) ( ) nx QF P X
1 1( )i i i iQ Q F F = P P
11
( )( , )
n
nP xx
1 1( ) ( )n ni i x x
11( )
i
i i
nnP
xx
( )
Q XX
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CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY ANALYSIS USING SSFEM
Linearized PVW
On integration (space) and further simplification
( ) ( ) ( )i j i jP
f d Galerkin projection Inner product
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
State variable based power law model.
State variable – Measure of deformation resistance- mesoscale property
Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel.
Eigen decomposition of the kernel using KLE.
0
n
fs
21 1( ,0, ,0) exp
r
bp pR
2
01
( ) (1 ( ))i n ii
s s v
p p
V20.3398190.2390330.1382470.0374605
-0.0633257-0.164112-0.264898-0.365684-0.466471-0.567257
V10.4093960.3958130.382230.3686460.3550630.3414790.3278960.3143130.3007290.287146
Eigenvectors Initial and mean deformed config.
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CCOORRNNEELLLL U N I V E R S I T Y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Displacement (mm)
SD
Loa
d (N
)
Homogeneous materialHeterogeneous material
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12
14
Displacement (mm)
Load
(N)
Mean
Load vs Displacement SD Load vs Displacement
Dominant effect of material heterogeneity on response statistics
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS
MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM
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EFFECT OF UNCERTAIN FIBER ORIENTATION
Aircraft nozzle flap – composite material, subjected to pressure on the free end
Equiv Stress: 0.006 0.026 0.047 0.068 0.088SD Equiv Stress: 0.003 0.005 0.008 0.011 0.013
Deterministic
Stochastic-mean
Orthotropic hyperelastic material model with uncertain angle of orthotropy modeled using KL expansion with exponential
covariance and uniform random variables
Two independent random variables with order 4 PCE (Legendre Chaos)
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EFFECT OF UNCERTAIN FIBER ORIENTATION – MC COMPARISON
Nozzle tip displacement
MC results from 1000 samples generated using Latin Hypercube Sampling
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CCOORRNNEELLLL U N I V E R S I T Y
Bn+1(θ)B0
X(θ) xn+1(θ)F(θ)
xn+1(θ)=x(XR,tn+1, θ,)
XR
F*(θ)
MODELING INITIAL CONFIGURATION UNCERTAINTY
BR
FR(θ)
Introduce a deterministic reference configuration BR which maps onto a stochastic initial configuration by a stochastic reference deformation gradient FR(θ). The deformation problem is then solved in this reference configuration.
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Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7 . Power law constitutive model.
Plastic strain 0.7
Eq. strain1.266561.194461.122351.050240.9781360.9060290.8339220.7618160.6897090.617602
Eq. strain0.7495560.7081580.6667590.6253610.5839620.5425640.5011650.4597670.4183680.37697
Initial configuration assumed to vary uniformly between two extremes with strain maxima in different regions in the stochastic simulation.
STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY
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Stochastic simulation
Plastic strain 0.7
SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747
Eq. strain0.7566080.7497910.7429750.7361580.7293420.7225250.7157090.7088920.7020760.69526
SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747
SD uy0.1178790.1060910.09430280.0825150.07072710.05893930.04715140.03536360.02357570.0117879
SD ux0.05046060.04541450.04036850.03532240.03027640.02523030.02018420.01513820.01009210.00504606
Results plotted in mean deformed configuration
STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY
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Point at top
Plastic strain 0.7
Outer boundary plot
00.5
11.5
2
2.53
3.54
0.264 0.266 0.268 0.27 0.272 0.274 0.276
x (mm)
y (m
m)
Mean-MC
Mean - SSFEM - o4
Mean -Deterministic
STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY
Point at centerline
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MERITS AND PITFALLS OF GPCE
• Reduced order representation of uncertainty
• Faster than mc by at least an order of magnitude
• Exponential convergence rates for many problems
• Provides complete response statistics
But….
• Behavior near critical points.
• Requires continuous polynomial type smooth response.
• Performance for arbitrary PDF’s.
• How do we represent inequalities spectrally ?
• How do we compute eigenvalues spectrally ?
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SUPPORT SPACE METHOD - INTRODUCTION
Finite element representation of the support space.
Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support.
Provides complete response statistics.Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).
Easily extend to updated Lagrangian formulations.
Constitutive problem fully deterministic – deterministic evaluation at quadrature points – trivially extend to damage problems.
True PDF
Interpolant
FE Grid
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SUPPORT SPACE METHOD – SOLUTION SCHEME
Linearized PVW
Galerkin projection0 0
. .j mi i j k k m
B B
dP dV P dV
u uX X
. .B B
dP dFdV P dFdV
( ) ( )
( ) ( )( ). ( ).( ) ( )
n nn nB B
dP dV P dV
u ux x
( ) ( )
( ) ( )( ). ( ) ( ). ( )( ) ( )
n nn nP B P B
dP dVd P dVd
u ux x
Galerkin projection
GPCE
Support space
1 ( )
1 ( )
( )( ). ( )( )
( ) ( ). ( )( )
s
e n
s
e n
nel
e nP B
nel
e nP B
dP dVd
P dVd
ux
ux
2i j i j l l lu K B
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CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY IN INITIAL CONFIGURATION- NONPOROUS MATERIAL
SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747
SD uy0.1219070.1097160.09752570.0853350.07314430.06095360.04876290.03657220.02438140.0121907
SD Eq. strain0.3044580.2757960.2471350.2184740.1898120.1611510.132490.1038280.07516710.0465058
Eq. strain0.7958130.7851450.7744770.7638090.753140.7424720.7318040.7211360.7104680.699799
SD ux0.05562520.05006270.04450020.03893760.03337510.02781260.02225010.01668760.0111250.00556252
Using 5x1 uniform support space grid
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0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.310
5
10
15
20
25
30 PDF of xMCSupport Space 10x1
FURTHER VALIDATION
Comparison of statistical parameters
Parameter Monte Carlo (1000 LHS samples)
Support space 5x1 uniform grid
Support space 10x1 uniform grid
Mean 0.2693 0.26934 0.269352
SD 0.0185 0.018399 0.0184991
Skewness 3.106e-06 2.8086e-06 3.0909e-06
Kurtosis 2.474e-07 2.3045e-07 2.4645e-07
Parameter Monte Carlo (1000 LHS samples)
Support space 5x1 uniform grid
Support space 10x1 uniform grid
Mean 0.2502 0.250583 0.250223
SD 0.0613 0.0591207 0.0611877
Skewness -1.8664e-04 -0.00012534 -0.0001836
Kurtosis 3.8112e-005
2.61928e-05 3.7323e-05
x-coordinate (top of sample)
x-coordinate (center of sample)PDF of x-coordinate (top of sample)
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EXTENSION TO CONTINUUM DAMAGE
Stochastic finite deformation damage evolution based on Gurson-Tvergaard-Needleman model.
Updated Lagrangian formulation (Anand, Zabaras et. al.).
Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel.
Constitutive model
ˆ1det ˆ1
P iff
F
2
01
ˆ ˆ( ) (1 ( ))i n ii
f f v
p p
01exp 1fC s
( ) ps t A B
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PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747
Mean
Initial Final
Using 6x6 uniform support space grid
SD-Void fraction0.01860.01720.01580.01430.01290.01150.0101
Void fraction0.04190.03880.03570.03250.02940.02630.0231
SD-Void fraction0.00980.00960.00940.00920.00910.00890.0087
Uniform 0.02
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PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Load displacement curves
Displacement (mm)
Load
(N)
0.1 0.2 0.3 0.4
1
2
3
4
5
6Mean
Mean +/- SD
Displacement (mm)
SD
Load
(N)
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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FURTHER VALIDATION
Comparison of statistical parameters
Parameter Monte Carlo (1000 LHS samples)
Support space 6x6 uniform grid
Support space 7x7 uniform grid
Mean 6.1175 6.1176 6.1175
SD 0.799125 0.798706 0.799071
Skewness 0.0831688 0.0811457 0.0831609
Kurtosis 0.936212 0.924277 0.936017
Final load values
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CCOORRNNEELLLL U N I V E R S I T Y
Demonstration of two non-statistical methods for modeling uncertainty in finite deformation problems.
• Both provide complete response statistics and convergence in distribution.
• The support-space approach incurs a larger computation cost in comparison to the GPCE approach for a given stochastic simulation of systems with smooth inputs.
• GPCE fails for systems with sharp discontinuities. (inequalities).
• Easier to integrate the support space method into existing codes. Only change global assembly routine. Ideal for complex simulations with strong nonlinearities. (Finite deformations – eigen strains, inequalities, complex constitutive models).
• GPCE needs explicit spectral expansion and repeated Galerkin projections.
IN CONCLUSION
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• The support-space approach can handle completely empirical probability density functions due to local support with no change in the convergence properties (convergence is based on number of elements used to discretize the support-space and the order of interpolation).
• GPCE on the other hand loses its convergence properties if the Askey chaos chosen does not correspond to the input distribution.
• Curse of dimensionality – both methods are susceptible. More research needed on intelligent approximations.
IN CONCLUSION
Relevant Publication
S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, in press
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FUTURE WORK
Linkage?
Information Theory
• Field of mathematics founded by Shannon in 1948
•Try to transfer as much information as possible about parameters of interest (displacements, stresses, strains etc)
• Extend to metal forming simulations. (Strong nonlinearities – contact)
• Examine effect of process parameters/ material randomness on design objectives.
• Incorporate microscale statistical information.
Information theoretic correlation kernels
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