Tutorial I: Mechanics of CONFINED Granular Solids

IHPC-IMS Program onAdvances & Mathematical Issues

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

Tutorial I:Mechanics of CONFINED Granular Solids

Alberto M. CuitiñoMechanical and Aerospace Engineering

Rutgers UniversityPiscataway, New Jerseycuitino@jove.rutgers.edu

Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuitiño@rutgers

Collaborators

• Gustavo Gioia • Shanfu Zheng

Singapore 2003 cuitiño@rutgers

Overview

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

Singapore 2003 cuitiño@rutgers

Overview

Die Filling Rearrangement

Large Deformation Localized Deformation

Singapore 2003 cuitiño@rutgers

Pore Structure

Increasing Pressure

PEG 8000

Higher Magnification

Singapore 2003 cuitiño@rutgers

Pore Structure

1mm/sec

100mm/secPEG 8000 Visco-plastic material

Singapore 2003 cuitiño@rutgers

Pore Structure

1mm/sec

100mm/secHPDE Visco-elastic material

Singapore 2003 cuitiño@rutgers

Compact Properties

0

50

100

150

200

0 10 20 30 40 50

Har

dnes

s (N

)

Compaction force (KN)

P

(N

)

HPDE 1 mm/sec

HPDE 100 mm/sec

PEG 100 mm/sec

PEG 1 mm/sec

P

Brazilian Compression

Test

Singapore 2003 cuitiño@rutgers

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)

Singapore 2003 cuitiño@rutgers

No cohesion Cohesion 2 degrees

misalignment

Cohesion Vertical dropping

Die Filling

Singapore 2003 cuitiño@rutgers

Role of Cohesion on Die Filling

Numerical Experimental

Numerical Experimental

Cohesion No Cohesion

Open Configuration Dense Configuration

Singapore 2003 cuitiño@rutgers

Rearrangement Process(a discontinuous process, advancing front)

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

Singapore 2003 cuitiño@rutgers

Physical Description(a theoretical interpretation)

Energy landscape exhibits a Spinoidal Structure (nonconvex)

H H

Convexification implies coexistence of two phases

H

Total

Singapore 2003 cuitiño@rutgers

Energy Landscape

Total Energy

Energy

Relaxation Energy(non-convex part)

Inter-particle Energy (frozen initial configuration)

Wedging & Friction

W = Wt+ Wf+ Ww+ Wb

Singapore 2003 cuitiño@rutgers

Relaxation Mechanism

Particle Rearrangement Mechanism

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

Numerical

Experimental

Singapore 2003 cuitiño@rutgers

Rearrangement

Singapore 2003 cuitiño@rutgers

Non Convex Analysis

Minimization with constrain (Lagrange Multiplier)

Singapore 2003 cuitiño@rutgers

Non Convex Analysis

Effective Energy Density

Singapore 2003 cuitiño@rutgers

Density Evolution

Transformation Front

High Density Phase

Low Density Phase

Singapore 2003 cuitiño@rutgers

Wall Friction

Equilibrium in the current configuration

Generalized Friction Coefficient

Exponential decay from the transformation front

Singapore 2003 cuitiño@rutgers

Particle Deformability

Deformability of High Density Phase

Deformability of Low Density Phase

Singapore 2003 cuitiño@rutgers

Particle Deformability

High Density Phase

Low Density Phase

Singapore 2003 cuitiño@rutgers

Pressure Density Profiles

Singapore 2003 cuitiño@rutgers

Comparison with experiment

Al2O3

Theoretical Experimental

Kong et al., 1999

Singapore 2003 cuitiño@rutgers

Compaction Curves

H = 2L = 0.4

Theoretical Experimental(Deis and Lannutti, 1998)

Singapore 2003 cuitiño@rutgers

Density Histograms

Theoretical Experimental(Deis and Lannutti, 1998)

Incr

easi

ng p

ress

ure

Singapore 2003 cuitiño@rutgers

Effect of RH (low pressure)

p = 0.14 MPa

Experimental(Deis and Lannutti, 1998)

Low pressure range

Theoretical

Unimodal Distribution

Singapore 2003 cuitiño@rutgers

Effect of RH (higher pressure)

Experimental (Deis and Lannutti, 1998) Theoretical

Higher deformability increases the transformed region at constant applied pressure

Bimodal Distribution

Singapore 2003 cuitiño@rutgers

Other Systems with NC Energy

Preferred Term

THIN FILM BUCKLING

Singapore 2003 cuitiño@rutgers

Other Systems with NC Energy

COMPRESSION OF FOAMS

Gibson and Ashby, 1997

Structure Mechanical Response

Singapore 2003 cuitiño@rutgers

Implication: Heterogeneous Deformation

Spinoidal Energy Landscape

Singapore 2003 cuitiño@rutgers

Comparison with Experiment

Theory

Experiment

Material Tested Polyurethane Foam

Theory

Singapore 2003 cuitiño@rutgers

Comparison with Experiment

Singapore 2003 cuitiño@rutgers

Experimental Evidence

Singapore 2003 cuitiño@rutgers

x

y

Surface Measurement

Displacement field measurement using Digital Image Speckle Correlation

(Wang, Gioia and Cuitino, 2001)

Peters & Ranson (1982),

Kahn-Jetter & Chu (1990)

Vendroux & Knauss (1998)

Zhang et al . (1999)

Singapore 2003 cuitiño@rutgers

Digital Image Speckle Correlation

U

V

G(X)

g(x)

where, U, V are the rigid body motion and Ux, Uy, Vx, Vy are the spatial gradients

Actual ImagesUndeformed

Deformed (load step 2)

2

2

,,,,,XG

xgXGVyVxUyUxVUCC

Gray scale values, G and g, characterize point in original and deformed images

Minimization of Correlation function C provides deformation field

Singapore 2003 cuitiño@rutgers

Digital Image Speckle Correlation

Fiber OpticLight Source

CCD Camera

Loading System

Specimen

Computer with Frame Grabber

Setup

Singapore 2003 cuitiño@rutgers

Displacement MapsProgressive Field

Singapore 2003 cuitiño@rutgers

Displacement MapsProgressive Field

Singapore 2003 cuitiño@rutgers

Comparison with Experiment

Singapore 2003 cuitiño@rutgers

Multiscale Modeling

Cascade of length scales

Singapore 2003 cuitiño@rutgers

Simulation