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

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Singapore 2003 cuiti ñ Collaborators Gustavo Gioia Shanfu Zheng

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Singapore 2003 cuiti ñ Context Cascade of Length and Time Scales Particle Dynamics Discrete Element Monte Carlo Mesoscopic Models Continuum Length Scale (particle size) Time Scale (particle size/wave speed) One or less Phase Fields Fast Fourier FEM Atomistics Molecular Dynamics to 10 3 particles to 10 6 particles to 10 9 particles > 10 3 Too many Bonding Energies Activation Barrier Phase Transformations Particle Packing Particle Rearrangement Particle Mixing Granulation Early Consolidation Particle Contact Stress Concentration Internal Localization Fracture Highly Confined Flow Consolidation Particle deformation Sintering Effective Behavior Design Models Sub-particle Resolution Range Supra-particle Resolution Range >

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Singapore 2003 cuiti ñ Supra-Particle Resolution * If a “good”constitutive relation is provided Continuum INPUT From: Parameter fitting with macroscopic experiments OUTPUT Large regions / Long times Global Constrains (shapes, friction) Good for component DESIGN* Particle Dynamics Discrete Element Monte Carlo OUTPUT Limited to small regions and/or Short Times GAP INPUT From sup-particle resolution simulations (Multiscale)sup-particle From DIRECT Experimental Particle-Level Studies (Higashitani, Granick, …)

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Singapore 2003 cuiti ñ Aside: MD/DEM Domain Computation Time/Physical Time PROPORTIONAL pressure*number of particles/particle size Ct/Pt PROPORTIONAL p*n/d P o (mixing range) P = 10 3 P o (consolidation range) Volume Sample = V o Volume Sample = V = V o /1,000 p n Constant Ct/Pt n d Volume Sample = V o e.g. = 1mm 3 Volume Sample = V = V o /1,000,000,000 e.g. V = 1 m 3 Constant Ct/Pt

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Singapore 2003 cuiti ñ Aside: PD/DEM Domain Computation Time/Physical Time PROPORTIONAL pressure*number of particles/particle size Ct/Pt PROPORTIONAL p*n/d PRESSURE Case 1a: Constant Ct/Pt (for example, 1 CPU hour/ 1 second of physical time) Constant particle size If pressure is increased 1,000 the sample size is decreased 1,000 times (for example, from 1mm 3 to 100 m 3 ) Case 1b: Constant sample size (for example, 1mm 3 ) Constant physical time (for example 1 second) Constant particle size If pressure is increased 1,000 the Computational time is increased by 1,000 times (for example, 1 CPU hour to 1.5 CPU month) PARTICLE SIZE Case 2a: Constant Ct/Pt (for example, 1 CPU Hour/ 1 second of physical time) Constant pressure If particle size is reduced by 1,000 the sample size is decreased 10 9 times (for example, from 1mm 3 to 1 m 3 ) Case 2b: Constant sample size (for example, 1mm 3 ) Constant physical time (for example 1 second) Constant pressure If particle size is reduced by 1,000 the Computational time is increased 10 9 times (for example, 1 CPU Hour to 116,000 CPU years) High pressures and small particles conspired against PD/DEM simulations

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Singapore 2003 cuiti ñ Supra-Particle Resolution Particle Dynamics Discrete Element Monte Carlo GRANULAR QUASI CONTINUUM Continuum INPUT From sup-particle resolution simulations (Multiscale)sup-particle From DIRECT Experimental Particle-Level Studies (Higashitani, Granick, …) OUTPUT Large regions / Long times Global Constrains (shapes, friction) Good for component DESIGN* * If a “good”constitutive relation is provided INPUT From: Parameter fitting with macroscopic experiments OUTPUT Limited to small regions and/or Short Times

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Singapore 2003 cuiti ñ 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

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Singapore 2003 cuiti ñ 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 Low forces among particles Short times Transient Early Consolidation Pre- compression Characteristics: Limited relative motion of particles Low particle acceleration Same neighbors Quasi-static Low forces among particles 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

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Singapore 2003 cuiti ñ 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

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Singapore 2003 cuiti ñ Numerical Strategies for assembly of particles Continuum Plasticity models for frictional materials such as Drucker-Prager Discrete Molecular Dynamics Discrete Element Method Continuum with Microstructure

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Singapore 2003 cuiti ñ Continuum with Microstructure Cyclic

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Singapore 2003 cuiti ñ Numerical Strategies for assembly of particles Continuum Plasticity models for frictional materials such as Drucker-Prager Continuum with Microstructure Discrete Molecular Dynamics Discrete Element Method Granular Quasi Continuum PARTICLES POWDERS (discrete) (continuum)

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Singapore 2003 cuiti ñ 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 MeshSet of Particles Combined System Granular Quasi-Continuum

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Singapore 2003 cuiti ñ Nomenclature umum rmrm rnrn r mn RmRm RnRn R mn Particle m Particle n

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Singapore 2003 cuiti ñ Governing Equations PVW Euler Equation Local Equilibrium

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Singapore 2003 cuiti ñ Constrained Kinematics Displacement Field is constrained by the selection of the mesh Displacement of Node a Displacement of particle m Relative Displacement of particle m with respect to particle n

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Singapore 2003 cuiti ñ Discrete Version

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Singapore 2003 cuiti ñ Local/Non-local Formulation u u*( m ) u( n ) u( m ) u*( n ) nn mm d

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Singapore 2003 cuiti ñ Additional Freedom/ Mesh Adaption Indicator Additional relaxation by adding internal nodes Still within the framework of FEM

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Singapore 2003 cuiti ñ Load Paths

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Singapore 2003 cuiti ñ Mueth, Jaeger, Nagel 2000 Comparison with Experiment Experiment Simulation

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Singapore 2003 cuiti ñ Simulations Pre-rearrangement

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Singapore 2003 cuiti ñ Simulations Post-rearrangement

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Singapore 2003 cuiti ñ Macroscopic Behavior

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Singapore 2003 cuiti ñ Macroscopic Behavior RED – Group I BLUE – Group II

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Singapore 2003 cuiti ñ Macroscopic Behavior RED – Group I BLUE – Group II

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Singapore 2003 cuiti ñ Comparison with Experiment Experiment Simulation

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Singapore 2003 cuiti ñ Adjusting Force Fields Elastic Properties Plastic Properties Fracture Properties BULK PROPERTIES SURFACE PROPERTIES Lactose 1.5% PEG 0.5% Glycerin (Courtesy of P&G P. Mort and M. Roddy) Experimental Numerical

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Singapore 2003 cuiti ñ Numerical Studies Unit Cell Approach

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Singapore 2003 cuiti ñ Numerical Studies Yield Stress Effects

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Singapore 2003 cuiti ñ Numerical Studies Young Modulus Effects

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Singapore 2003 cuiti ñ Numerical Studies Hardening Effects

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Singapore 2003 cuiti ñ Materials Solid Materials Granulated Materials HPDE (high density polyethylene) PEG 8000 (Polyethylene glycol) Lactose 1.5% PEG (binder) 0.5% Glycerin (lubricant)

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Singapore 2003 cuiti ñ Experimental Studies PEG 8000 PEG 3350 PEG 1450 Glycerin 0% 0.5% 1.0% 1.5%

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Singapore 2003 cuiti ñ Experimental Studies Glycerin (Yield Stress)

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Singapore 2003 cuiti ñ Experimental Studies PEG ( Elastic Modulus + Hardening )

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Singapore 2003 cuiti ñ Future Emerging trend in the material system design: using computer simulations to probe the space of best candidates for manufacturing. We have developed a basic framework capable of simulating the consolidation process, including: –die filling (with/without cohesion) –particle rearrangement in both 2D and 3D –powder densification in both 2D and 3D, with materials with a wide range of elastic and plastic properties powders with a wide range of size distributions particle different shapes (by agglomeration of spherical particles) wall friction complex shapes dynamic effects FROM TO Specific applications More validation Pre die filling Post ejection behavior, in particular bonding using the GQC This approach will provide estimates of the local and global strength of the material by following the temperature and pressure dependent evolution of the bonding process

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