Seminar: Multiscale Modeling of Heterogeneous Granular Systems

IHPC-IMS Program onAdvances & Mathematical Issues

in Large Scale Simulation(Dec 2002 - Mar 2003 & Oct - Nov 2003)

Seminar:Multiscale Modeling of Heterogeneous

Granular Systems

Alberto M. CuitiñoMechanical and Aerospace Engineering

Rutgers UniversityPiscataway, New [email protected]

Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuitiño@rutgers

Collaborators

• Gustavo Gioia • Shanfu Zheng


Rutgers1. Harvard University

2. William and Mary

3. Yale University

4. Princeton University

5. Columbia University

6. University of Pennsylvania

7. Brown University

8. Rutgers University (1766)

9. Dartmouth

RutgersUniversit

y

1766 Rutgers Founded as

Queen’s C

ollege

1864 Named New

Jerse

y’s Land Gran

t Colle

ge

1989 Rutgers is

electe

d into

Associa

tion of A

merican

Universitie

s

RU

1970 University

of Medici

ne and Dentist

ry Founded

2003

Richard L. M

cCormick

19th president of R

utgers

50,000 Students

10,000 Faculty

and Staf

f

175 Academ

ic Depart

ments

1869 First A

merican

College F

ootball Gam

e,

(6) Rutgers

vs. Prin

ceton (4

)

1914 School of E

ngineering Nam

ed Separate S

chool

1956 The C

olleges

and Schools of R

utgers

Become th

e Stat

e Univers

ity of N

ew Je

rsey


Rutgers

RutgersUniversity

Philadelphia

NYC

New Brunswick

Camden

Newark


College AveLivignstonDouglassBusch

Rutgers,New Brunswick

Cook


Rutgers,Busch

StadiumGolfScience and Engineering

Hairston Leads Rutgers Past Navy 48-27SEPTEMBER 27, 2003


Rutgers,Mechanical and Aerospace

Rutgers,Engineering

Entrance

Doyle D. Knight

Michael R. Muller

Timothy Wei

Abdelfattah M.G. Zebib

Norman J. Zabusky

Jerry Shan

Tobias Rossman

S. Bachi

Fluid Mechanics

Zhixiong (James) Guo

Yogesh Jaluria

Constantine E. Polymeropoulos

Kyung T. Rhee

Stephen D. Tse

Thermal Sciences

Haim Baruh

Hae Chang Gea

Noshir A. Langrana

Constantinos Mavroidis

Madara M. Ogot

Dajun Zhang

Design and Dynamics

Haym Benaroya

William Bottega

Alberto Cuitiño

Mitch Denda

Ellis Dill

Andrew Norris

Kook Pae

Assimina Pelegri

George Weng

Solids, Materials and Structures


Current Research

• Granular Systems (G. Gioia and S. Zheng)• Crystal Plasticity• Multiscale Modeling • Foam Mechanics• Folding of Thin Films• Microelectronics• Digital Image Correlation• Computational Material Design

(Ferroelectric Polymers)

Support from NSF, DOE, DARPA, FAA, NJCST, IFPRI, CAFT is gratefully acknowledged


Damage due to Electromigration in Interconnect Lines

Sequence of pictures showing void and hillock formation in an 8µm wide

Al interconnect due to electromigration

(current density 2x107 A/cm², temperature 230°C)

Thomas Göbel (t.goebel@ifw-dresden .de), 18.04.02


E 0,

V

T

Schimschak and Krug, 2000

Schimschak and Krug, 2000


Grain Boundary Effects

Grain 1 Grain 2

VOID TRAPPINGby GRAIN BOUNDARY

Initial DefectVOID MOTION

@ GRAIN BOUNDARY

VOID RELEASEFrom GRAIN BOUNDARY

e-Atkinson and Cuitino ‘03


Goal

Understand and quantitatively predict the MACROSCOPIC

behavior of powder systems under compressive loading based on

MICROSCOPIC properties such as particle/granule behavior and spatial arrangement

Load

Need for MULTISCALE Study PARTICLES POWDERS (discrete) (continuum)


Background

10-4 10-3 10-2 10-1

Normalized Compaction Force

0.4

0.5

0.6

0.7

0.8

0.9

1

Re

lativ

eD

en

sity

MacroscopicCompaction Curve

1st Stage 2nd Stage

Compaction Force

3rd Stage

0th Stage


Stages

Mixing Die Filling Rearrangement

Large Deformation Localized Deformation


Identifying Processes and Regimes

Mixing

Transport

Granulation

Characteristics:

• Large relative motion of particles

• Differential acceleration between particles

• Large number of distinct neighbors

• Low forces among particles

• Long times, relatively slow process

• Quasi steady state

Discharge

Die Filling

Vibration

Characteristics:

• Large relative motion of particles

• Differential acceleration between particles

• Large number of distinct neighbors


• Short times

• Transient

Early Consolidation

Pre-compression

Characteristics:

• Limited relative motion of particles

• Low particle acceleration

• Same neighbors

• Quasi-static


• Small particle deformation (elastic)

Consolidation

Characteristics:

• No relative motion of particles

•Low acceleration

• Same neighbors

•Quasi-static

• Sizable forces among particles

•Small particle deformation (elastic + plastic)

Compact Formation

Characteristics:

• No relative motion of particles

• Low acceleration

• Same neighbors

•Quasi-static

• Large forces among particles

• Large particle deformation


Identifying Numerical Tools (which can use direct input from finer scale)

Mixing

Transport

Granulation

PD/DEM/MC

Discharge

Die Filling

Vibration

PD/DEM/MC

Ballistic Deposition

Early Consolidation

Pre-compression

PD/DEM/MC

Consolidation

GCC

Compact Formation

GQC

OUR SCOPE

Numerical tools appropriate for process



Mixing

Transport

Granulation

PD/DEM/MC

Discharge

Die Filling

Vibration

PD/DEM/MC


Early Consolidation

Pre-compression

PD/DEM/MC

Consolidation

GCC

Compact Formation

GQC

OUR SCOPE



Die Filling

Numerical Experimental

Numerical Experimental

Cohesion No Cohesion

Open Configuration Dense Configuration



Mixing

Transport

Granulation

PD/DEM/MC

Discharge

Die Filling

Vibration

PD/DEM/MC


Early Consolidation

Pre-compression

PD/DEM/MC

Consolidation

GCC

Compact Formation

GQC

OUR SCOPE



Rearrangement

Video ImagingGlass Beads, Diameter = 1.2 mmGioia and Cuitino, 1999

Increasing Pressure Increasing Pressure

Process by which open structures collapse into dense configurations• Cohesive Powders are susceptible to rearrangement while• Non-Cohesive Powders are not

X-Ray Tomography-Density MapsAl2O3 Granules. Diameter = 30 micronsLannutti, 1997

Punch


A physical description

Energy landscape exhibits a Spinoidal Structure (nonconvex)

H H

Convexification implies coexistence of two phases

H

Total


A relaxation mechanism

Particle Rearrangement Mechanism

Snap-Through of Rings (Kuhn et al. 1991) Ring Structures in Cohesive Powders

Numerical

Experimental


Relaxation process

Numerical

Experimental


Experiments and Theory

Al2O3

Theoretical Experimental

Kong et al., 1999


2D: simulation and experiment


Rearrangement Front

Experiment Simulation


“Grains”


2D Simulations (Size Distribution)


3D Simulations


Mueth, Jaeger, Nagel 2000

Comparison with Experiment



Further Predictions



Particle Rearrangement 3D

• Homogeneous particle size;

• r = 0.5 mm;

• Particles = 120,991

mmr 5.0


Quantitative Predictions

Nc

0 10 20 30 40 504

4.5

5

5.5

6

6.5

u=2.5u=5.0u=7.5

u=9.1(d)

u=0.25

Nc = 6.27

h/

0 10 20 30 40 50

0

0.5

1

u=2.5u=5.0u=7.5

u=9.1

(c)

u=0.25

v/

0 10 20 30 40 50

0

0.5

1

u=2.5u=5.0u=7.5u=9.1

(b)

u=0.25

0 10 20 30 40 500.45

0.5

0.55

0.6

0.65 = 0.627

=0.51

u=2.5u=5.0u=7.5u=9.1

(a)

•Rearrangement front;

•Density increase;

•Relative movement stops;

•Contact number increase;


Heterogeneous System(Same Material)

Without rearrangement After rearrangement

•Log-normal distribution; d = 2.16 ~ 9.10 mm; particles=13,134


Multiphase Systems



Mixing

Transport

Granulation

PD/DEM/MC

Discharge

Die Filling

Vibration

PD/DEM/MC


Early Consolidation

Pre-compression

PD/DEM/MC

Consolidation

GCC

Compact Formation

GQC

OUR SCOPE



Constrain kinematics of the particles by overimposing a displacement field described by a set of the displacements in a set of points (nodes) and a corresponding set of interpolation functions (a FEM mesh)

A quasi-continuum approach

FEM Mesh Set of Particles Combined System

Granular Quasi-Continuum


Governing Equations

PVW

Euler Equation

P

8m2P

P

8n2Vm21î wmn +

P

8m2Pfm áî um = 0

P

8n2Vm21

drmndî wmn

+ fm = 0

Local Equilibrium


Force Fields

Indentation ()

Fo

rce

0 0.1 0.2 0.3

-1200

-1000

-800

-600

-400

-200

0

d

r1 r2

= r1 + r2 - d

Hertziancontact

Similarity solution

volumechanged

• Elastic contact follows Hertz contact law;

• Plastic deformation follows similarity solution; Contacts on each particle are independent.

• Volume change after inter-particle voids are filled in.


Role of FF parameters - y

Force (kN)

Den

sity

(g/c

m3)

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

HDPE 100

E = 1 GPa

n = 3

= 0.3

0 = 1.1 g/cm3

y = 1 (in MPa)

y = 0.1

y = 0.01• Effect of yielding stress is significant;

• Lower y yield higher deformation under the same pressure and thus higher density;

• Solidification force differ significantly but the solidification density relative unchange.


Role of FF parameters: hardening

Force (kN)

Den

sity

(g/c

m3)

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

HDPE 100

E = 1 GPa

y = 1 MPa

= 0.3

0 = 1.1 g/cm3

n = 3

n = 10

n = • Effect of hardening

parameter n is significant;

• Soft material (n) is easy to be solidified.


Role of FF parameters:Poisson’s ratio

Force (kN)

Den

sity

(g/c

m3)

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

HDPE 100

E = 1 GPa

y = 1 MPa

n = 3

0 = 1.1 g/cm3

= 0.3 = 0.2


Case of Study: Multiphase System


Spatial Distribution

Phase I

Variant 1

Phase I

Variant 2

Phase I

Variant 3



Phase II

Variant 1-4



Phase III

Needs an uniform distribution


Multiphase SystemRearrangement

Full mixed + Cohesion force


Multiphase SystemPost-Rearrangement

Input for GQC


Calibration of FF

stress (MPa)

De

nsi

ty(g

/cc)

0 1 2 3 4 5 6 70.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Sample:mass = 0.75 gsize = 12126.6 (red)

12125.8 (blue)density = 0.789 g/cc (red)

0.898 g/cc (blue)Particles:

size = 0.216 ~ 0.91 mmnumber = 13,134

Detergent Granule 1


Multiphase SystemComparison with Experiment

stress (MPa)

density

(g/cc)

0 0.5 1 1.5 2 2.5 30.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Sample :mass = 0.075 (g)size = 553.3 (mm3)density = 0.91 (g/cc)particle size = 0.1~0.5 (mm)particles = 7,986

• Density diversity at initial state is mainly due to the irregular shape of real particles;

• At early stage of experiment the deformation is the mainly from the particle rearrangement.

MACROSCOPIC Behavior


Multiphase System:density evolution

0

2

4

6

Z

0

2

4

6

8

10

X

0

2

4

6

8

10

Y

X Y

Z

dens: 0.68 0.78 0.87 0.97 1.06 1.16 1.25 1.35

Movie Here

Full Field Predictions


Multiphase System:pressure evolution

X

0

2

4

6

8

10

Y

0

2

4

6

8

10

Z

0

2

4

6

X Y

Z

Movie Here


Granular Quasi-Continuum

• Allows for explicit account of the particle level response on the effective behavior of the powder

• Provides estimates of global fields such as stress, strain density

• Is numerically efficient, can also be improved by using stochastic integration

• Provides variable spatial resolution

• Is not well posed to handle large non-affine motion of particles

• Particle deformation is only considered in an approximate manner (as in PD/DEM)

GOOD BAD


Towards Computationally Aided Material Design

100 nm

CASCADE OF SCALES

MICRO NANO ATOMISTIC

20 m 0.1 nm

FORCE PARAMETERS

NANO COMPOSITE PARAMETERS

MICRO SCALE

ACCEPATLE MAXIMUM PORE SIZE

PORE SIZE

ACCEPTABLERANGE OF FORCE PARAMTERS

ACCEPTABLE RANGE OF

NANOCOMPOSITE PARAMETERS

TO NANO SCALE


Summary and conclusions

• Powder compaction is a complex process where many dissimilar entities (particles) consolidate by various concurrent mechanisms.

• In the low pressure regime, rearrangement and localized particle deformation dominates the mechanical response.

• In this initial regime, compaction proceeds in a discontinuous fashion by an advancing front.

• The physics of the rearrangement can be traced to a spinoidal structure in the energy density of the system.

• This process can be theoretically described using the framework of non-convex analysis.

• The effect of particle deformability and die wall roughness are incorporated into the analysis in a clear and physical manner.

• Numerical simulations verify the theoretical model• Experimental studies validate the model and simulations• 3D simulations show a similar behavior that 2D ones, indicating the same physics

operates in 2D and 3D cases.

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Seminar: Multiscale Modeling of Heterogeneous Granular Systems

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