Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


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]: http://mpdc.mae.cornell.edu/

An information-theoretic approach for property prediction of

random microstructures

mailto:[email protected]




NEED FOR UNCERTAINTY ANALYSIS

Variation in properties, constitutive relations

Imprecise knowledge of governing physics, surroundings

Simulation based uncertainties (irreducible)

Uncertainty is everywhere

Porous media

Silicon wafer

Aircraft engines

Material process

From DOEFrom Intel websiteFrom NIST From GE-AE website




UNCERTAINTY AND MULTISCALING

MacroMesoMicro

Uncertainties introduced across various length scales have a non-trivial interaction

Current sophistications – resolve macro uncertainties

Use micro averaged models for resolving physical scales

Imprecise boundary conditions Initial perturbations

Physical properties, structure follow a statistical description




Initial preform shape

Material properties/models

Forging velocity

Texture, grain sizes

Die/workpiece friction

Die shapeSmall change in preform shape

could lead to underfill

Material ModelForging rate

Die/Billet shape

Friction

Cooling rate

Stroke length

Billet temperature

Stereology/Grain texture

Dynamic recrystallization

Phase transformation

Phase separation

Internal fracture

Other heterogeneities

Yield surface changes

Isotropic/Kinematic hardening

Softening laws

Rate sensitivity

Internal state variables

Dependance Nature and degree

of correlation

Process

UNCERTAINTY IN METAL FORMING PROCESSES




RANDOM VARIABLES = FUNCTIONS ? Math: Probability space (, F, P)

Sample space Sigma-algebra Probability measure

F

W : Random variableW

: ( )W

Random variable

A stochastic process is a random field with variations across space and time

: ( , , )X x t




SPECTRAL STOCHASTIC REPRESENTATION

: ( , , )X x t

A stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE

PROBABILITY SPACE

HYPERGEOMETRIC ASKEY POLYNOMIALS

PIECEWISE POLYNOMIALS (FE TYPE)

SPECTRAL DECOMPOSITION

COLLOCATION, MC (DELTA FUNCTIONS)

GENERALIZED POLYNOMIAL CHAOS EXPANSION

SUPPORT-SPACE REPRESENTATION

KARHUNEN-LOÈVE EXPANSION

SMOLYAK QUADRATURE, CUBATURE, LH




KARHUNEN-LOEVE EXPANSION

1

( , , ) ( , ) ( , ) ( )i ii

X x t X x t X x t

Stochastic process

Mean function

ON random variablesDeterministic functions

Deterministic functions ~ eigen-values , eigenvectors of the covariance function

Orthonormal random variables ~ type of stochastic process

In practice, we truncate (KL) to first N terms

1( , , ) fn( , , , , )NX x t x t




GENERALIZED POLYNOMIAL CHAOS Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

0

( , , ) ( , ) (ξ( ))i ii

Z x t Z x t

Stochastic output

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NX x t x t

Askey polynomials ~ type of input stochastic process

Usually, Hermite, Legendre, Jacobi etc.




SUPPORT-SPACE REPRESENTATION

Any function of the inputs, thus can be represented as a function defined over the support-space

1( , , ) : ( ) 0NA f ξ ξ

JOINT PDF OF A TWO RANDOM

VARIABLE INPUT

FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS

2

2

1

ˆ

ˆ( ( ) ( )) ( )d

L

A

q

X X

X X f

Ch

ξ ξ ξ ξ

OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS

– SMOLYAK QUADRATURE

– IMPORTANCE MONTE CARLO




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 2( ,0, ,0) exp

r

b

p pR

2

01

( ) (1 ( ))i i 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.

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY




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




Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF.

For a stochastic process

Definition of moments

NISG - Random space discretized using finite elements to

Output PDF computed using local least squares interpolation from function evaluations at integration points.

( , , ) ( , , ) , ,g x t g x t x X t T

( ( , , )) ( )ppM g x t f d

h

1 1

( ( , , )) ( ) ( ( , , )) ( )h

nel nh h p h h p h

p i e ie i iee i

M g x t f d w g x t f

1 1

( ( , ))nel nint

h h p hp i ei ei

e i

M w g x t f

ie

Deterministic evaluations at fixed points

NISG - FORMULATION




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.

Decoupled function evaluations at element integration points.

True PDF

Interpolant

FE Grid

Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).

NISG - DETAILS




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

Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution

2

01

ˆ ˆ( ) (1 ( ))i n ii

f f v

p p

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR




Load displacement curves

Displacement (mm)

Lo

ad

(N)

0.1 0.2 0.3 0.4

1

2

3

4

5

6

Mean

Mean +/- SD

Displacement (mm)

SD

Lo

ad

(N)

0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR




Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm)

Random initial radius – 10% variation about mean (1 mm)– uniformly distributed

Random die workpiece friction U[0.1,0.5]

Power law constitutive model

Using 10x10 support space grid Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016

Random ? Shape

Random ? friction

PROCESS UNCERTAINTY




Force SD Force

PROCESS STATISTICS




0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

3.50E-04

0 2 4 6 8 10 12 14

Grid resolution (Number of elements per dimension)

Re

lativ

e e

rro

r

Parameter Monte Carlo (20000 LHS samples)

Support space 10x10

Mean 2.2859e3 2.2863e6

SD 297.912 299.59

m3 -8.156e6 --9.545e6

m4 1.850e10 1.979e10

Final force statistics

Convergence study

PROCESS STATISTICS

Relative Error




As the number of random variables increases, problem size rises exponentially.

1

10000

1E+08

1E+12

1E+16

1E+20

0 5 10 15 20 25

No. of variables

Fu

nct

ion

eva

luat

ion

s

(assume 10 evaluations per random dimension)

CURSE OF DIMENSIONALITY




ADAPTIVE DISCRETIZATION BASED ON OUTPUT STOCHASTIC FIELD

• Refine/Coarsen input support space grid based on output defined control parameter (Gradients, standard deviations etc.)

• Applicable using standard h,p adaptive schemes.

Support-space of input Importance spaced grid

PROPOSED SOLUTIONS




DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003)

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Full grid Scheme Sparse grid Scheme Dimension adaptive Scheme

Very popular in computational finance applications.

Has been used in as high as 256 dimensions.

PROPOSED SOLUTIONS




Idea Behind Information Theoretic ApproachIdea Behind Information Theoretic Approach

Statistical Mechanics

InformationTheory

Rigorously quantifying and modeling

uncertainty, linking scales using criterion

derived from information theory, and

use information theoretic tools to predict parameters in the face

of incomplete Information etc

Linkage?

Information Theory

Basic Questions:1. Microstructures are realizations of a random field. Is there a principle by which the underlying pdf itself can be obtained.2. If so, how can the known information about microstructure be incorporated in the solution.3. How do we obtain actual statistics of properties of the microstructure characterized at macro scale.




MAXENT as a tool for microstructure reconstructionMAXENT as a tool for microstructure reconstruction

Input: Given average and lower moments of grain sizes and ODFs

Obtain: microstructures that satisfy the given properties

Constraints are viewed as expectations of features over a random field. Problem is viewed as finding that distribution whose ensemble properties match those that are given.

Since, problem is ill-posed, we choose the distribution that has the maximum entropy.

Microstructures are considered as realizations of a random field which comprises of randomness in grain sizes and orientation distribution functions.




The MAXENT principleThe MAXENT principle

The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown.

E.T. Jaynes 1957

MAXENT is a guiding principle to construct PDFs based on limited information

There is no proof behind the MAXENT principle. The intuition for choosing distribution with

maximum entropy is derived from several diverse natural phenomenon and it works in practice.

The missing information in the input data is fit into a probabilistic model such that

randomness induced by the missing data is maximized. This step minimizes assumptions about

unknown information about the system.




MAXENT : a statistical viewpointMAXENT : a statistical viewpoint

MAXENT solution to any problem with set of features is ( )ig I

Parameters of the distributioniInput features of the microstructure

Fit an exponential family with N parameters (N is the number of features given), MAXENT reduces to a parameter estimation problem.

Mean provided

( )ig I

1-parameter exponential family(similar to Poisson distribution)

Gaussian distribution

Mean, variance givenNo information provided(unconstrained optimiz.)The uniform distribution

Commonly seen distributions

-2 0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1




Microstructural feature: Grain sizesMicrostructural feature: Grain sizes

Grain size obtained by using a series of equidistant, parallel

lines on a given microstructure at different angles. In 3D, the size

of a grain is chosen as the number of voxels (proportional to volume) inside a particular grain.




Cubic crystal

Microstructural feature : ODF Microstructural feature : ODF

RODRIGUES’ REPRESENTATIONRODRIGUES’ REPRESENTATIONFCC FUNDAMENTAL REGIONFCC FUNDAMENTAL REGION

Crystal/lattice

reference frame

e2^

Sample reference

frame

e1^ e’1

^

e’2^

crystalcrystal

e’3^

e3^

ORIENTATION SPACEEuler angles – symmetries

Neo Eulerian representation

n

Rodrigues’ Rodrigues’ parametrizationparametrization

CRYSTAL SYMMETRIES?Same axis of rotation => planes

Each symmetry reduces the space by a pair of planes

Particular crystal

orientation




Subject to

Lagrange Multiplier optimization

Lagrange Multiplier optimization

feature constraints

features of image I

MAXENT as an optimization problemMAXENT as an optimization problem

Partition Function

Find




Equivalent log-linear modelEquivalent log-linear model

Find that minimizes

Equivalent log-likelihood problem

Kuhn-Tucker theorem: The that minimizes the dual function L also maximizes the system entropy and satisfies the constraints posed by

the problem

Direct modelsDirect models Log-linear modelsLog-linear models

ConcaveConcave ConcaveConcave

Constrained (simplex)Constrained (simplex) UnconstrainedUnconstrained

““Count and normalize” Count and normalize”

(closed form solution)(closed form solution)Gradient based methodsGradient based methods

A A comparisoncomparison




Gradient EvaluationGradient Evaluation

• Objective function and its gradients: Objective function and its gradients:

• Infeasible to compute at all points in one conjugate gradient iterationInfeasible to compute at all points in one conjugate gradient iteration

• Use sampling techniques to sample from the distribution evaluated Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler)at the previous point. (Gibbs Sampler)




Improper pdf (function of lagrange multipliers)

Start from a random microstructure.

Go through each grain of the microstructure and

sample an ODF according to the conditional probability distribution (conditioned on

the other grains)

continue till the samples converge to the distribution

Processor 1 Processor r

…Each processor

goes through only a subset of the

grains.

Parallel Gibbs sampler algorithmParallel Gibbs sampler algorithm




Optimization SchemesOptimization Schemes

Convergence analysis with stabilization Convergence analysis w/o stabilization

Noise in function evaluation increases as step size for the next minima increases. This ensures that the impact on the next evaluation is reduced.




Voronoi structureVoronoi structure

Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space.

Voronoi cell tessellation :

{p1,p2,…,pk} : generator points.

1, 2{ ,..., } kn nS p p p

{ : , ( , ) ( , )}ki i jC x j i d x p d x p

Division of into subdivisions so that for each point, pi

there is an associated convex cell,

kCell division of k-dimensional space :

Voronoi tessellation of 3d space. Each cell is a microstructural grain.




Mathematical representationMathematical representation

OFF file representation (used by Qhull package) Initial lines consists of keywords (OFF), number of vertices and volumes. Next n lines consists of the coordinates of each vertex. The remaining lines consists of vertices that are contained in each volume.

Brep (used by qmg, mesh generator)Dimension of the problem. A table of control points (vertices). Its faces listed in increasing order of dimension (i.e., vertices first, etc) each associated with it the following: 1.The face name, which is a string. 2.The boundary of the face, which is a list of faces of one lower dimension. 3.The geometric entities making up the face. its type (vertex, curve, triangle, or quadrilateral), • its degree (for a curve or triangle) or degree-pair (for a quad), and • its list of control points

Volumes need to be hulled to obtain consistent

representation with commercial packages

Convex hulling to obtain a triangulation of surfaces/grain boundaries




Preprocessing: stage 1Preprocessing: stage 1

Growth of big grains to accommodate small grains entrenched in-between

Compute volumes of all grains Adjust vertices of neighboring grains so that the new voronoi tessellation fills the volume of initial grain Recompute surfaces and planes of the new geometry




Steps Obtain input voronoi representation in OFF format. Obtain the convex hull of the volumes/grains so that each surface is a triangle (triangulation of surfaces). Use ANSYSTM to convert this representation to the universal IGES (Initial Graphics Exchange specification) format.

• Surface database : To ensure non-duplication of surfaces, a database consisting of previously encountered hyper-planes is searched. When a new surface is created, if it is already in the database and if all the vertices of the surface were not present in a previous grain, no new surface is made.

Domain smoothing: The regions of the microstructure inside the region [0 1]3 is chosen. Edges are smoothed so that the boundaries represent edges of a k-dimensional cube of unit side.

Preprocessing: stage 2Preprocessing: stage 2




MeshingMeshing

X Y

ZFrame 001 17 Nov 2005

Pixel based meshing scheme. Boundary is distorted since element shapes and sizes are fixed.

Tetrahedral element meshed. Grain boundaries conform with the mesh shapes.




Mesh refinementMesh refinement

Tetrahedral mesh Hexahedral mesh

Input to homogenization tool to obtain plastic property and eventually property statistics




(First order) homogenization scheme(First order) homogenization scheme

(a) (b)

How does macro loading affect the microstructure

1. Microstructure is a representation of a material point at a smaller scale

2. Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)




Homogenization of deformation gradientHomogenization of deformation gradient

How does macro loading affect the microstructure

Microstructure without cracks

(a) (b)

Use BC: = 0 on the boundary

Note w = 0 on the volume is the Taylor assumption, which is the upper bound




ImplementationImplementation

Largedef formulation for macro scale

Update macro displacements

Boundary value problem for microstructure

Solve for deformation field

Consistent tangent formulation (macro)

Integration of constitutive equations

Continuum slip theory

Consistent tangent formulation (meso)

Macro-deformation gradient

Homogenized (macro) stress, Consistent tangent

Mesoscale stress, consistent tangent

meso deformation gradient

(a) (b)

Macro

Meso

Micro

Homogenized propertiesHomogenized properties

X

Y

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28

(a)

(c)

(b)

XY

Z

Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)

X Y

Z

Equivalent Stress (MPa): 20 30 40 50 60 70 80

XY

Z


XY

Z0

10

20

30

40

50

60

0.000 0.010 0.020 0.030 0.040 0.050 0.060

Equivalent plastic strain

Equ

ival

ent s

tres

s (M

Pa)

Simple shear

Plane strain compression

(a)

(c)

(b)

(d)







Problem definition: Given an experimental image of an aluminium alloy

(AA3302), properties of individual components and given the expected

orientation properties of grains, it is desired to obtain the entire variability

of the class of microstructures that satisfy these given constraints.

Polarized light micrograph of aluminium alloy AA3302 (source Wittridge NJ et al. Mat.Sci.Eng. A, 1999)

2D random microstructures: evaluation of property statistics2D random microstructures: evaluation of property statistics




Grain sizes: Heyn’s intercept method. An equidistant network of parallel lines drawn on a microstructure and intersections with grain boundaries are computed.

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Grain Size( m)

prob

abili

ty

<Gsz>=10.97

<Gsz2>=124.90

Input constraints in the form of first two moments. The corresponding MAXENT distribution is shown on the right.

MAXENT distribution of grain sizesMAXENT distribution of grain sizes




Assigning orientation to grainsAssigning orientation to grains

Given: Expected value of the orientation distribution function.

To obtain: Samples of orientation distribution function that satisfies the given ensemble

properties

-2 -1 0 1 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Orientation angle (in radians)

Orie

nta

tion

dis

trib

utio

n fu

nctio

n

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Orientation angle (in radians)

Ori

en

tatio

n d

istr

ibu

tion

fun

ctio

n

Input ODF (corresponds to a pure shear deformation, Zabaras et al. 2004)

Ensemble properties of ODF from reconstructed distribution




0 0.05 0.1 0.15 0.230

40

50

60

70

80

Equivalent Stress

Eq

uiv

ale

nt

Str

ain

(M

Pa

)

Bounding plastic curves over a setof microstructural samples

Evaluation of plastic property boundsEvaluation of plastic property bounds

Orientations assigned to individual grains from the ODF samples obtained using MAXENT.

Bounds on plastic properties obtained from the samples of the microstructure




MotivationMotivation

Uncertainties induced due to non-uniformities in grain growth

patterns




Input uncertaintiesInput uncertainties

Problem inputs: Microstructures obtained using monte-carlo grain growth model Problem inputs: Microstructures obtained using monte-carlo grain growth model at different stages of the growth.at different stages of the growth.

Sources of uncertainty: Anything that Sources of uncertainty: Anything that changes the driving force for grain changes the driving force for grain growth (curvature driven, reduction in growth (curvature driven, reduction in surface energy) (e.g) ambient surface energy) (e.g) ambient conditions not exactly same in conditions not exactly same in microstructures near surface and in the microstructures near surface and in the bulk.bulk.

Problem parameters:Problem parameters:1.1. 10 input microstructures used that 10 input microstructures used that

constraint the input informationconstraint the input information2.2. Time lag of ~50 MC steps between Time lag of ~50 MC steps between

each sample.each sample.3.3. Simulated on a 9261 point gridSimulated on a 9261 point grid




Maximum-entropic distribution of grain sizesMaximum-entropic distribution of grain sizes

0 100 200 300 400 500 6000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Grain size ( m3)

Pro

babi

lity

<Gsz>=383.4967<std(Gsz)>=41.4490




0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Grain size

Pro

ba

bili

ty

Sampling technique employedSampling technique employed

Weakly consistent scheme




Input ODF

Some representative ODF samples

from the MaxEnt

distribution

ODF reconstruction using MAXENTODF reconstruction using MAXENT

Problem inputs/algorithm parameters:

1. 145 degrees of freedom2. MaxEnt algorithm using

Brent’s line search method3. Eighty Gibbs iteration through

each grain of the microstructure




Input ODF

Ensemble properties of reconstructed samples of

microstructures

Ensemble propertiesEnsemble properties




Final uncertainty representationFinal uncertainty representation

pdf

Grain size

OD

F (a

func

tion

of 1

45 ra

ndom

par

amet

ers)

Microstructures sampled as points from the joint pdf space




Microstructure models & meshesMicrostructure models & meshes

Tetrahedral meshes




Obtaining statistics of non-linear propertiesObtaining statistics of non-linear properties

Different microstructural models of a Different microstructural models of a polycrystal Aluminiumpolycrystal Aluminium

microstructuremicrostructure is obtained by sampling the resultant distribution. Each is obtained by sampling the resultant distribution. Each

of these specimens is subject to a of these specimens is subject to a pure axial tensionpure axial tension along the x along the x

direction. Plots of the resultant stress-contour and the resulting direction. Plots of the resultant stress-contour and the resulting

homogenized stress-strain curves are plotted for different realizationshomogenized stress-strain curves are plotted for different realizations




Homogenized stress fields on the microstructureHomogenized stress fields on the microstructure

Equivalent Stress (MPa)84.881980.753676.625472.497168.368964.240660.112455.984151.855947.727643.5994

Equivalent Stress (MPa)1251151059585756555453525

Pixel based meshing Hexahedral meshing




Equivalent Stress (MPa)1251151059585756555453525

Equivalent Stress (MPa)84.469180.753677.038273.322869.607465.891962.176558.461154.745751.030247.314843.5994

Homogenized stress fields on the microstructureHomogenized stress fields on the microstructure




Comparison of pixel based versus hexahedral meshing schemesComparison of pixel based versus hexahedral meshing schemes

Equivalent strain

Eq

uiv

alen

tstr

ess

(MP

a)

0 0.001 0.002 0.003

10

20

30

40

50

Hexahedral mesh

Pixel based mesh

The pixel based meshing scheme distorts grain

boundaries and not only increases their area but also twists their shape which leads to a higher

degree of stress localization as viewed in

previous plot.




Plots of homogenized stress-strain curvesPlots of homogenized stress-strain curves

Equivalent strain

Equiv

ale

ntst

ress

(MP

a)

0 0.0005 0.001 0.0015 0.002

10

15

20

25

30

35

40

45

A plot showing three different samples of

the stress-strain plots obtained for different statistical models of the microstructure

generated using the MaxEnt scheme.




Stress contours across grain boundaries and triple junctionsStress contours across grain boundaries and triple junctions

Orientation 0.4142 -0.2071 -0.0858

Orientation-0.2929 -0.4142 0.2929

Orientation0.4142 0.0858 -0.2071

Orientation0.2071 -0.4142 0.0858

Orientation0.4142 0.0858 -0.2071

Extreme sharp variation in texture

across the triple junction. Hence, leads

to a large degree of stress localization




Applications (many …)Applications (many …)

X Y

Z


X Y

Z


X Y

Z


Statistics of plastic

properties




DiscussionDiscussion

• A statistical distributions of mictrostructure was obtained A statistical distributions of mictrostructure was obtained incorporating variability in grain sizes and grain orientations. incorporating variability in grain sizes and grain orientations.

• Stress field distributions show a significant difference between the Stress field distributions show a significant difference between the pixel based mesh and the hexahedral mesh. One possible reason pixel based mesh and the hexahedral mesh. One possible reason may be attributed to the fact that grain boundaries are distorted as a may be attributed to the fact that grain boundaries are distorted as a result of which the localized stresses near the grain boundaries are result of which the localized stresses near the grain boundaries are felt in some regions in the bulk of the grain. Also, for the hexahedral felt in some regions in the bulk of the grain. Also, for the hexahedral grid grid 21960 elements21960 elements were used while for the pixel based grid, were used while for the pixel based grid, 13824 elements13824 elements were used. We are currently performing were used. We are currently performing convergence studiesconvergence studies with respect to the mesh sizes but the number with respect to the mesh sizes but the number of elements used were roughly equivalent. Also, sharp changes in of elements used were roughly equivalent. Also, sharp changes in the field were noticed in the vicinity of the grain boundaries due to the field were noticed in the vicinity of the grain boundaries due to steep variations in texture. steep variations in texture.

• Statistical samples of microstructure model were used to obtained Statistical samples of microstructure model were used to obtained different samples of homogenized stress-strain curves.different samples of homogenized stress-strain curves.




MODELING GRAIN BOUNDARY PHYSICS

Equivalent stress contours –Include failure mechanisms

–Grain boundary properties

–Local stress concentrations develop to cause the emission of a few partial dislocations from grain boundaries, and

these high stresses drive the partial dislocations across the grain interiors

–MD studies indicate that this is the major mechanism

of the limited inelastic deformation in the grain interiors of nanocrystalline

materials.

Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and...

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