Warmth Elevating the Depths Shallower Voids With Warm Dark Matter
Materials with voids
description
Transcript of Materials with voids
Materials with voids
T.A. Abinandanan & R. Mukherjee
Department of Materials EngineeringIndian Institute of Science
Bangalore, India.
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.
Voids
Late Stages of high temperature deformation
Voids
Voids
Nucleation
Growth
Coalescence
Overall Goal
A phase-field model of polycrystals with voids
Applications:
Failure under during temperature deformation
Sintering powder compacts
Features
• Multiple grains: Grain boundaries• Voids: Free surface• Externally applied stress• Enhanced diffusivity at grain boundaries and
surfaces• Most important: vacancy source term.
Atomistic Picture
• Crystal – Void system: Lattice gas model
• Polycrystal with grain boundaries: Potts model
Grain 1, η1,
Grain 2 η2
Void
Atomistic Picture
Approach : Phase Field Model
ρ: Vacancy ConcentrationMaterial & Cavity
η1 ,η2:Order ParameterGrain Orientation
Continuum Analogue
Lattice Gas Model -> Cahn-Hilliard Model with Atoms and VacanciesPotts Model - > Fan-Chen Model
Total Free Energyelch FFF
dVF elij
elijV
el 21=
F : Total Free Energy Fch : Chemical Contribution To Free Energy Fel : Elastic Contribution To Free Energy
dVfFV
ch ])()()(),,([= 222
211
221
Chemical Contribution To Free Energy
f: Bulk Free Energy Densityρ : Vacancy Concentrationη1, η2 : Order Parameters Κρ : Gradient Energy Coefficient for
Gradient in ρΚ η1, Κ η2 : Gradient Energy Coefficient
for Gradient in η1, η2
,])()()(),,([= 222
211
221 dVfF
V
ch
Approach : Phase Field Model
ρ=0, η1=1, η2=0
ρ=1, η1=0, η2=0
ρ=0, η1=0, η2=1
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
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
22222 )()(1)(1=),( iii ZBAf
0.25]2[]24
[=)( 2224
jiii
i
Approach : Phase Field Model
Along AB Along CD
Formulation: Kinetics
Cahn-Hilliard Equation(Vacancy Concentration)
Allen-Cahn Equation(For Grain Orientation)
,.= DMt
)/(= VNF
D
)/(
= VNFLt
J. W. Cahn, Acta Metallurgica, 1961S. M. Allen and J. W. Cahn, Acta Metallurgica, 1979
Vacancies
Conserved during diffusion.
They can also be created and annihilated at GBs.
Existing vacancies – compressive eigenstrain
Created vacancies – dilatational eigenstrain.
Algorithm
At each time-step:Creation / Annihilation: Compute v and create
in proportion to v.Re-scaling: Compute homogeneous strain and
re-scale the system dimensions. Diffusion: Compute diffusion potential, allow
vacancy diffusion.
Variable Mobility
M : Mobility ρ : Vacancy Concentration η1, η2 : Order Parameters P,Q,R,S: Constants
2/122
22
21
21
2222 )]1()1([)1()1( SRQPM
Vacancy Diffusion
Enhanced Mobility at the grain boundary and the surface
Cavity
SurfaceGrain Boundary
Matrix
Dihedral Angle
(Simulation)
I 0.7362 0.7154 61.94 60.00
II 0.5970 0.4125 69.79 69.50
III 0.5387 0.2405 77.10 77.00
s gb
s
gb
2
cos 1
Example: Dihedral Angle
Single Grain With Cavity
Grain Boundary Cavity With Uniaxial Tensile Stress
Void Evolution under stress
cAA c
AA
Note: No vacancy source / sink. Only diffusion.
Analysis of Schmidt and Gross: Elongation direction of second phase under a applied stress in elastically
inhomogeneous system
Very soft inhomogeneity elongates normal to the applied stress
I. Schmidt and D Gross, Proceedings of Royal Society (London) A, 1999
Bicrystal with Cavity
Cavity shape change during grain growth
(No vacancy source / sink; only diffusion)
Void Growth under Tension
Void Shrinkage under Compression
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!
Experimental Results
E.A. Anumol and N. Ravishankar, 2010
Initial Configuration
~400 spherical particlesClosely packed
Fully densified compact
Hollow Polycrystalline Aggregate
Multiple Holes
High Surface Diffusivity
High GB diffusivity
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
Sintering Map
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, …