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 [email protected]

Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuitiño@rutgers

Collaborators

• Gustavo Gioia • Shanfu Zheng


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


Overview

Die Filling Rearrangement

Large Deformation Localized Deformation


Pore Structure

Increasing Pressure

PEG 8000

Higher Magnification


Pore Structure

1mm/sec

100mm/secPEG 8000 Visco-plastic material


Pore Structure

1mm/sec

100mm/secHPDE Visco-elastic material


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


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)


No cohesion Cohesion 2 degrees

misalignment

Cohesion Vertical dropping

Die Filling


Role of Cohesion on Die Filling

Numerical Experimental

Numerical Experimental

Cohesion No Cohesion

Open Configuration Dense Configuration


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


Physical Description(a theoretical interpretation)

Energy landscape exhibits a Spinoidal Structure (nonconvex)

H H

Convexification implies coexistence of two phases

H

Total


Energy Landscape

Total Energy

Energy

Relaxation Energy(non-convex part)

Inter-particle Energy (frozen initial configuration)

Wedging & Friction

W = Wt+ Wf+ Ww+ Wb


Relaxation Mechanism

Particle Rearrangement Mechanism

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

Numerical

Experimental


Rearrangement


Non Convex Analysis

Minimization with constrain (Lagrange Multiplier)


Non Convex Analysis

Effective Energy Density


Density Evolution

Transformation Front

High Density Phase

Low Density Phase


Wall Friction

Equilibrium in the current configuration

Generalized Friction Coefficient

Exponential decay from the transformation front


Particle Deformability

Deformability of High Density Phase

Deformability of Low Density Phase


Particle Deformability

High Density Phase

Low Density Phase


Pressure Density Profiles


Comparison with experiment

Al2O3

Theoretical Experimental

Kong et al., 1999


Compaction Curves

H = 2L = 0.4

Theoretical Experimental(Deis and Lannutti, 1998)


Density Histograms

Theoretical Experimental(Deis and Lannutti, 1998)

Incr

easi

ng p

ress

ure


Effect of RH (low pressure)

p = 0.14 MPa

Experimental(Deis and Lannutti, 1998)

Low pressure range

Theoretical

Unimodal Distribution


Effect of RH (higher pressure)

Experimental (Deis and Lannutti, 1998) Theoretical

Higher deformability increases the transformed region at constant applied pressure

Bimodal Distribution


Other Systems with NC Energy

Preferred Term

THIN FILM BUCKLING


Other Systems with NC Energy

COMPRESSION OF FOAMS

Gibson and Ashby, 1997

Structure Mechanical Response


Implication: Heterogeneous Deformation

Spinoidal Energy Landscape


Comparison with Experiment

Theory

Experiment

Material Tested Polyurethane Foam

Theory




Experimental Evidence


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)


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


Digital Image Speckle Correlation

Fiber OpticLight Source

CCD Camera

Loading System

Specimen

Computer with Frame Grabber

Setup


Displacement MapsProgressive Field


Displacement MapsProgressive Field




Multiscale Modeling

Cascade of length scales


Simulation

Documents

Tutorial I: Mechanics of CONFINED Granular Solids