1. IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec 2002 - Mar 2003 & Oct - Nov 2003) Tutorial I: Mechanics of CONFINED Granular Solids Alberto M. Cuitiño Mechanical and Aerospace Engineering Rutgers University Piscataway, New Jersey cuitino@jove.rutgers.edu Institute of High Performance Computing Institute for Mathematical Sciences, NUS

2. Singapore 2003 cuiti ñ o@rutgers Collaborators Gustavo Gioia Shanfu Zheng

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

4. Singapore 2003 cuiti ñ o@rutgers Overview Die Filling Rearrangement Large Deformation Localized Deformation

5. Singapore 2003 cuiti ñ o@rutgers Pore Structure Increasing Pressure PEG 8000 Higher Magnification

6. Singapore 2003 cuiti ñ o@rutgers Pore Structure 1mm/sec 100mm/sec PEG 8000 Visco-plastic material

7. Singapore 2003 cuiti ñ o@rutgers Pore Structure 1mm/sec 100mm/sec HPDE Visco-elastic material

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

9. Singapore 2003 cuiti ñ o@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)

10. Singapore 2003 cuiti ñ o@rutgers Powder Compaction MRS Bulletin, December 1997

11. Singapore 2003 cuiti ñ o@rutgers No cohesion Cohesion 2 degrees misalignment Cohesion Vertical dropping Die Filling

12. Singapore 2003 cuiti ñ o@rutgers Role of Cohesion on Die Filling Numerical Experimental Cohesion No Cohesion Open Configuration Dense Configuration

13. Singapore 2003 cuiti ñ o@rutgers 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

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

15. Singapore 2003 cuiti ñ o@rutgers 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

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

17. Singapore 2003 cuiti ñ o@rutgers Rearrangement

18. Singapore 2003 cuiti ñ o@rutgers Non Convex Analysis Minimization with constrain (Lagrange Multiplier)

19. Singapore 2003 cuiti ñ o@rutgers Non Convex Analysis Effective Energy Density

20. Singapore 2003 cuiti ñ o@rutgers Density Evolution Transformation Front High Density Phase Low Density Phase

21. Singapore 2003 cuiti ñ o@rutgers Wall Friction Equilibrium in the current configuration Generalized Friction Coefficient Exponential decay from the transformation front

22. Singapore 2003 cuiti ñ o@rutgers Particle Deformability Deformability of High Density Phase Deformability of Low Density Phase

23. Singapore 2003 cuiti ñ o@rutgers Particle Deformability High Density Phase Low Density Phase

24. Singapore 2003 cuiti ñ o@rutgers Pressure Density Profiles

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

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

27. Singapore 2003 cuiti ñ o@rutgers Density Histograms Theoretical Experimental (Deis and Lannutti, 1998) Increasing pressure

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

29. Singapore 2003 cuiti ñ o@rutgers Effect of RH (higher pressure) Experimental (Deis and Lannutti, 1998) Theoretical Higher deformability increases the transformed region at constant applied pressure Bimodal Distribution

30. Singapore 2003 cuiti ñ o@rutgers Other Systems with NC Energy Preferred Term THIN FILM BUCKLING

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

32. Singapore 2003 cuiti ñ o@rutgers Implication: Heterogeneous Deformation Spinoidal Energy Landscape

33. Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment Theory Experiment Material Tested Polyurethane Foam Theory

34. Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment

35. Singapore 2003 cuiti ñ o@rutgers Experimental Evidence

36. Singapore 2003 cuiti ñ o@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)

37. Singapore 2003 cuiti ñ o@rutgers 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

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

39. Singapore 2003 cuiti ñ o@rutgers Displacement Maps Progressive Field

40. Singapore 2003 cuiti ñ o@rutgers Displacement Maps Progressive Field

41. Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment

42. Singapore 2003 cuiti ñ o@rutgers Multiscale Modeling Cascade of length scales

43. Singapore 2003 cuiti ñ o@rutgers Simulation

44. Singapore 2003 cuiti ñ o@rutgers One vs. Multiple Interfaces One InterfaceMultiple Interfaces Papka and Kyriakidis, 1994

45. Singapore 2003 cuiti ñ o@rutgers   Metastable branch Stable branch One vs. Multiple Interfaces

46. Singapore 2003 cuiti ñ o@rutgers Deformation Histograms Increasing pressure Unimodal Bimodal Interfaces

47. Singapore 2003 cuiti ñ o@rutgers Texture Formation and Evolution Histogram

48. Singapore 2003 cuiti ñ o@rutgers Texture Formation and Evolution Histogram