on the analysis of finite deformations and continuum damage in materials with random properties...

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: [email protected] [email protected]: http://mpdc.mae.cornell.edu/

peoplepeople



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



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



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


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



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



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



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



UNCERTAINTY ANALYSIS USING SSFEM

Linearized PVW

On integration (space) and further simplification

( ) ( ) ( )i j i jP

f d Galerkin projection Inner product



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.



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



UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS

MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM



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)



EFFECT OF UNCERTAIN FIBER ORIENTATION – MC COMPARISON

Nozzle tip displacement

MC results from 1000 samples generated using Latin Hypercube Sampling



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.



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



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




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


Point at centerline



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 ?



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



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



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



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


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




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)



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



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



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



FURTHER VALIDATION

Comparison of statistical parameters




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



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



• 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



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

on the analysis of finite deformations and continuum damage in materials with random properties...

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