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Materials with voids

Materials with voids. T.A. Abinandanan & R. Mukherjee Department of Materials Engineering Indian Institute of Science Bangalore, India. Outline. V oids, cavities, cracks Void growth and shrinkage Key feature: Vacancies are both conserved and non-conserved. Void evolution under stress

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Materials with voids

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  1. Materials with voids T.A. Abinandanan & R. Mukherjee Department of Materials Engineering Indian Institute of Science Bangalore, India.

  2. Outline • Voids, cavities, cracks • Void growth and shrinkage • Key feature: Vacancies are both conserved and non-conserved. • Void evolution under stress • Void growth under stress • Sintering of nanoparticle clusters.

  3. Voids Late Stages of high temperature deformation

  4. Voids

  5. Voids Nucleation Growth Coalescence

  6. Overall Goal A phase-field model of polycrystals with voids Applications: Failure under during temperature deformation Sintering powder compacts

  7. Features • Multiple grains: Grain boundaries • Voids: Free surface • Externally applied stress • Enhanced diffusivity at grain boundaries and surfaces • Most important: vacancy source term.

  8. Atomistic Picture • Crystal – Void system: Lattice gas model • Polycrystal with grain boundaries: Potts model

  9. Atomistic Picture Grain 1, η1, Grain 2 η2 Void

  10. Approach : Phase Field Model ρ: Vacancy Concentration Material & Cavity η1 ,η2: Order Parameter Grain Orientation Continuum Analogue Lattice Gas Model -> Cahn-Hilliard Model with Atoms and Vacancies Potts Model - > Fan-Chen Model

  11. Total Free Energy • F : Total Free Energy • Fch : Chemical Contribution To Free Energy • Fel : Elastic Contribution To Free Energy

  12. Chemical Contribution To Free Energy f: Bulk Free Energy Density ρ : Vacancy Concentration η1, η2: Order Parameters Κρ: Gradient Energy Coefficient for Gradient in ρ Κη1, Κη2: Gradient Energy Coefficient for Gradient in η1,η2

  13. Approach : Phase Field Model ρ=0, η1=1, η2=0 ρ=1, η1=0, η2=0 ρ=0, η1=0, η2=1

  14. Free energy plots near equilibrium phases Minima are located at (η1,η2)=(1,0) And (0,1), for ρ=0.0 Matrix Minima are located at (η1,η2)=(0,0), for ρ=1.0 Void

  15. Bulk Free Energy Density • Grain I : ρ=0, η1=1, η2=0 • Cavity:ρ=1, η1=0, η2=0 • Grain I I:ρ=0, η1=0, η2=1

  16. Approach : Phase Field Model Along AB Along CD

  17. Formulation: Kinetics Cahn-Hilliard Equation (Vacancy Concentration) Allen-Cahn Equation (For Grain Orientation) J. W. Cahn, ActaMetallurgica, 1961 S. M. Allen and J. W. Cahn, ActaMetallurgica, 1979

  18. Vacancies Conserved during diffusion. They can also be created and annihilated at GBs. Existing vacancies – compressive eigenstrain Created vacancies – dilatational eigenstrain.

  19. Algorithm At each time-step: Creation / Annihilation: Compute mv and create in proportion to mv. Re-scaling: Compute homogeneous strain and re-scale the system dimensions. Diffusion: Compute diffusion potential, allow vacancy diffusion.

  20. Variable Mobility Vacancy Diffusion Cavity Matrix Surface Grain Boundary • M : Mobility • ρ : Vacancy Concentration • η1, η2 : Order Parameters • P,Q,R,S: Constants • Enhanced Mobility at the grain boundary and the surface

  21. Dihedral Angle Example: Dihedral Angle

  22. Single Grain With Cavity Void Evolution under stress Grain Boundary Cavity With Uniaxial Tensile Stress

  23. Note: No vacancy source / sink. Only diffusion.

  24. Analysis of Schmidt and Gross: Elongation direction of second phase under a applied stress in elastically inhomogeneous system Bicrystalwith Cavity Very soft inhomogeneity elongates normal to the applied stress I. Schmidt and D Gross, Proceedings of Royal Society (London) A, 1999

  25. Cavity shape change during grain growth (No vacancy source / sink; only diffusion)

  26. Void Growth under Tension

  27. Void Shrinkage under Compression

  28. A final example Sintering of Nanoparticle Clusters The small size of the cluster allows us to study sintering without worrying about vacancy source/sink terms. The small size of the cluster also allows 3D simulations!

  29. Experimental Results E.A. Anumol and N. Ravishankar, 2010

  30. Initial Configuration ~400 spherical particles Closely packed

  31. Fully densified compact

  32. Hollow Polycrystalline Aggregate

  33. Multiple Holes

  34. High Surface Diffusivity

  35. High GB diffusivity

  36. Nanoparticle Sintering Full densification is always the end result. Hollow structures of various forms (one compact hole, one interconnected hole, multiple holes) are intermediate configurations. Hollow: High surface diffusivity

  37. Sintering Map

  38. Conclusions A comprehensive model for a polycrystalline material with voids is being developed. It incorporates enhanced diffusivity at surfaces and grain boundaries. Vacancies are conserved and non-conserved. It is being used for studying a wide variety of phenomena –high temperature deformation, void growth, sintering, hot pressing, …

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