Efficient Multiscale Plasticity Model for Polycrystalline Materials Based on Micromechanical Homogenization

Ajit Achuthan Clarkson University, Potsdam, NY

Brett A. Bednarcyk and Steven M. Arnold NASA Glenn Research Center, Cleveland, OH

Acknowledgements: Masoud Moghaddam, PhD Candidate, Clarkson U. NASA Glenn Faculty Fellowship Program

Objective: Multiscale Model for Polycrystalline Structures

• Link microstructure to properties to performance

• Key issues:

– Fidelity • FEA of polycrystal

– Computational Efficiency • Iso-stress (Sachs)

• Iso-strain (Taylor)

• Mean-field approaches

• Balanced approach

– Generalized Method of Cells (GMC)

Model Fidelity

Mo

del

Eff

icie

ncy

Analytical

Numerical

Semi-Analytical

Hierarchical (One-Way) Multiscale

Synergistic Multiscale

Concurrent Multiscale

Goal

GMC

HFGMC

FEA

MT

ROM

Science

Engineering

R&T

Seeking a Balance Efficiency vs. Fidelity in Multiscale Modeling

FEA, MD

MD

Micro-Level Field Equations (subcell)

Macro-Level Constitutive Equations (unit cell)

GMC is an Evolving Anisotropic Thermoelastic Inelastic and Damage Constitutive Model

=3

=2

=1

=3 =4=2

d

d

d1

2

3

l

l

1

2 =2

=1

h h h h4321

=1

x2

3x

x1

h

d

l

Repeating Unit Cell (RUC)

The Generalized Method of Cells (GMC) Micromechanics Theory

Subcell ( )

• Fields vary per subcell (as opposed to mean-field methods)

• Need subcell-level crystal plasticity model to provide local inelastic strains

Traditional and Smart Laminate Analysis Module

Effective Properties Job

Standard Multi-Axial Loading Job

Fatigue Damage Cyclic Loading Job

Yield Surface Repeated Loading Job

Problem Type Definition

E*, *, *, *-0.0010

-0.0005

0.0000

0.0005

0.0010

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Time (s)

Lo

ad

-50

-40

-30

-20

-10

0

10

20

30

40

50

-50 -40 -30 -20 -10 0 10 20 30 40 50

Nxx

Ny

y

Integrated multiscale Micromechanics Analysis Code (ImMAC) Software

ABAQUS

Constitutive Model Library

Static Failure Analysis Library

Fiber-Matrix Debonding

Model Library

Fiber Breakage Model Library

Repeating Unit Cell Geometry

Library

Material Property Library

GMC Continuous Composite Analysis

Module

HFGMC Continuous Composite Analysis

Module

GMC Discontinuous Composite Analysis

Module

GMC Smart Piezo-Electro-Magnetic Composite Analysis

Module

Core Micromechanics Analysis Capabilities

HFGMC Discontinuous Composite Analysis

Module

HFGMC Porous Material Analysis Module

Huang1 Open Source Crystal Plasticity Model

Kinematics

Deformation gradient given by multiplicative combination: Velocity gradient: Plastic distortion rate:

Plastic deformation

Elastic deformation

1Huang, Y., “A user-material subroutine incorporating single crystal plasticity in the abaqus finite element program Mech Report 178", Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, June 1999, http://www.columbia.edu/~jk2079/fem/umat_documentation.pdf

nmFFL 1ppp

peFFF

,11 epeee

FLFFFL LFF

Huang Model Constitutive Law

Flow Rule:

Resolved shear stress in -th slip system:

Hardening Rule:

sign, 0

n

ggf

hgg ,

nmIFFC :2

1 eTe

Crystal Plasticity Modeling Framework

Y. Huang , Harvard University, 1991

UMAT

Incremental Strain

estimated, ij( fi,Ji

j) ij

xij

xij xi

j+1

c g

cij+1 c

ij + c

ij+1

ij + c

gij+1 gi

j + gc

J (Jacobian) ij+1 = i

j +

ij+1

Jij+1

xij+1

Abaqus Model or GMC RUC

Load step #i t , f Iteration #j

Single Crystal (Single Element/Subcell) Stress-Strain Results

Abaqus FEA – Single Element

GMC – Single Subcell Compared to FEA

sign, 0

n

ggf

Material: FCC Copper (w/ notional rate-dependence) (Provided by Huang)

Polycrystal Microstructure Generation • Matlab pre-processor developed to generate microstructures based on Voronoi

cell tessellations with random crystal orientations – Random seed locations in 3D space

– Set of points closest to seed assigned to a given grain

– Meshed with 1000 cubic C3D20R elements (treated like RVE) or GMC subcells (treated like RUC)

Abaqus Results for Polycrystals Effects of loading direction and number of grains

X-direction Y-direction

Z-direction

Abaqus Results for Polycrystals Effects of loading direction and number of grains

1 Grain 2 x 2 x 2 Grains 3 x 3 x 3 Grains

4 x 4 x 4 Grains 5 x 5 x 5 Grains

GMC Global Agreement with Abaqus

X-direction Y-direction

Z-direction # of Grains FEA/Abaqus MAC/GMC

Speed up per iteration

1 grain/ 1 element 12 sec

0.021 sec/iter

0.06 sec 0.000235

sec/iteration

90x

3 x 3 x 3 = 27 grains/ 1000 elements

297 sec 4.18 sec/iter

5.062 sec 0.02 sec/iter

209x

5 x 5 x 5 = 125 grains / 2744 elements

1438 sec 16.72 sec/iter

18.63 sec 0.07 sec/iter

239x

Computational Efficiency Comparison

Multiscale Disk Demonstration • Spinning disk – centrifugal load (*DLOAD…CENTRIF)

• Global z-direction plastic strain plotted

Polycrystal Single Crystal

GMC RUC called at 1392 integration pts

Huang model called for 1000 subcells per FEA integration pt.

Synergistic vs. Hierarchical Results

Synergistic Multiscale FEA-Based Hierarchical Multiscale

GMC RUC called at 1392 integration pts

Huang model called for 1000 subcells per FEA integration pt.

Huang model results used to characterize continuum

plasticity model

Conclusions

• Demonstrated viability of GMC micromechanics theory to model polycrystalline materials

• At least 2-orders of magnitude speed-up compared to FEA polycrystal model

• Real pay-off is in multiscale modeling, where GMC polycrystal RUC is called at integration points within a structural FEA

Extension of the model • hcp and multiphase metals (Super alloys) • High temperature • Physics based dislocation model • Length scale effect • Non-cubic crystal structure • Fracture/damage ceramics • Ferroelectric ceramics

• Linkage with information management system

Future Work

Example Two-Phase Material Inclusion VF - 0.512 Constitutive Models:- Inclusions – elastic (Cu) Matrix – single crystal plasticity (Cu)

Inclusion size=0.8 Inclusion size=0.4

GMC based analysis is currently ongoing for 2-phase material. Will use to obtain stress/strain fields in ME3 disc

0 1 2 3 4

x 10-3

0

50

100

150

200

Str

ess (

S3

3)

(MP

a)

strain (E33)

Stress-Strain

Inclusion-size:0.4

Inclusion-size:0.8