IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec Mar 2003 & Oct - Nov 2003) Tutorial I: Mechanics of CONFINED Granular Solids Alberto M. Cuitiño Mechanical and Aerospace Engineering Rutgers University Piscataway, New Jersey Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuiti ñ Collaborators Gustavo Gioia Shanfu Zheng

Singapore 2003 cuiti ñ Overview Macroscopic Compaction Curve 1 st Stage 2 nd Stage Compaction Force 3 rd Stage 0 th Stage

Singapore 2003 cuiti ñ Overview Die Filling Rearrangement Large Deformation Localized Deformation

Singapore 2003 cuiti ñ Pore Structure Increasing Pressure PEG 8000 Higher Magnification

Singapore 2003 cuiti ñ Pore Structure 1mm/sec 100mm/sec PEG 8000 Visco-plastic material

Singapore 2003 cuiti ñ Pore Structure 1mm/sec 100mm/sec HPDE Visco-elastic material

Singapore 2003 cuiti ñ Compact Properties P (N) HPDE 1 mm/sec HPDE 100 mm/sec PEG 100 mm/sec PEG 1 mm/sec P Brazilian Compression Test

Singapore 2003 cuiti ñ 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 cuiti ñ Powder Compaction MRS Bulletin, December 1997

Singapore 2003 cuiti ñ No cohesion Cohesion 2 degrees misalignment Cohesion Vertical dropping Die Filling

Singapore 2003 cuiti ñ Role of Cohesion on Die Filling Numerical Experimental Cohesion No Cohesion Open Configuration Dense Configuration

Singapore 2003 cuiti ñ Rearrangement Process (a discontinuous process, advancing front) Video Imaging Glass Beads, Diameter = 1.2 mm Gioia and Cuitino, 1999 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 Maps Al 2 O 3 Granules. Diameter = 30 microns Lannutti, 1997 Punch

Singapore 2003 cuiti ñ Physical Description (a theoretical interpretation) Energy landscape exhibits a Spinoidal Structure (nonconvex) H H Convexification implies coexistence of two phases H Total

Singapore 2003 cuiti ñ Energy Landscape Total Energy Energy Relaxation Energy (non-convex part) Inter-particle Energy (frozen initial configuration) Wedging & Friction W = W t + W f + W w + W b

Singapore 2003 cuiti ñ Relaxation Mechanism Particle Rearrangement Mechanism Snap-Through of Rings (Kuhn et al. 1991) Ring Structures in Cohesive Powders Numerical Experimental

Singapore 2003 cuiti ñ Rearrangement

Singapore 2003 cuiti ñ Non Convex Analysis Minimization with constrain (Lagrange Multiplier)

Singapore 2003 cuiti ñ Non Convex Analysis Effective Energy Density

Singapore 2003 cuiti ñ Density Evolution Transformation Front High Density Phase Low Density Phase

Singapore 2003 cuiti ñ Wall Friction Equilibrium in the current configuration Generalized Friction Coefficient Exponential decay from the transformation front

Singapore 2003 cuiti ñ Particle Deformability Deformability of High Density Phase Deformability of Low Density Phase

Singapore 2003 cuiti ñ Particle Deformability High Density Phase Low Density Phase

Singapore 2003 cuiti ñ Pressure Density Profiles

Singapore 2003 cuiti ñ Comparison with experiment Al 2 O 3 Theoretical Experimental Kong et al., 1999

Singapore 2003 cuiti ñ Compaction Curves  H = 2  L = 0.4 TheoreticalExperimental (Deis and Lannutti, 1998)

Singapore 2003 cuiti ñ Density Histograms Theoretical Experimental (Deis and Lannutti, 1998) Increasing pressure

Singapore 2003 cuiti ñ Effect of RH (low pressure) p = 0.14 MPa Experimental (Deis and Lannutti, 1998) Low pressure range Theoretical Unimodal Distribution

Singapore 2003 cuiti ñ 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 cuiti ñ Other Systems with NC Energy Preferred Term THIN FILM BUCKLING

Singapore 2003 cuiti ñ Other Systems with NC Energy COMPRESSION OF FOAMS Gibson and Ashby, 1997 Structure Mechanical Response

Singapore 2003 cuiti ñ Implication: Heterogeneous Deformation Spinoidal Energy Landscape

Singapore 2003 cuiti ñ Comparison with Experiment Theory Experiment Material Tested Polyurethane Foam Theory

Singapore 2003 cuiti ñ Comparison with Experiment

Singapore 2003 cuiti ñ Experimental Evidence

Singapore 2003 cuiti ñ 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 cuiti ñ Digital Image Speckle Correlation U V G(X)G(X) g(x)g(x) where, U, V are the rigid body motion and Ux, Uy, Vx, Vy are the spatial gradients Actual Images Undeformed Deformed (load step 2) Gray scale values, G and g, characterize point in original and deformed images Minimization of Correlation function C provides deformation field

Singapore 2003 cuiti ñ Digital Image Speckle Correlation Fiber Optic Light Source CCD Camera Loading System Specimen Computer with Frame Grabber Setup

Singapore 2003 cuiti ñ Displacement Maps Progressive Field

Singapore 2003 cuiti ñ Displacement Maps Progressive Field

Singapore 2003 cuiti ñ Comparison with Experiment

Singapore 2003 cuiti ñ Multiscale Modeling Cascade of length scales

Singapore 2003 cuiti ñ Simulation

Singapore 2003 cuiti ñ One vs. Multiple Interfaces One InterfaceMultiple Interfaces Papka and Kyriakidis, 1994

Singapore 2003 cuiti ñ   Metastable branch Stable branch One vs. Multiple Interfaces

Singapore 2003 cuiti ñ Deformation Histograms Increasing pressure Unimodal Bimodal Interfaces

Singapore 2003 cuiti ñ Texture Formation and Evolution Histogram

Singapore 2003 cuiti ñ Texture Formation and Evolution Histogram