phase field presentation
TRANSCRIPT
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
PHASE FIELD MODEL FORTHE DISLOCATION
(PHASE FIELD SEMINAR)
HUZAIFA SHABBIR
MSS STUDENT
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
NANOSCALE PHASE FIELD MICROELASTICITYTHEORY OF DISLOCATION: MODEL AND 3DSIMULATIONS.
Y.U.WANG, Y.M.JIN, A.M.CUTTINO AND A.G.KHACHATURYAN
2001
RESEARCH ARTICLE
PHASE FIELD MODELING OF DEFECTS ANDDEFORMATION
YUNZHI WANG , JU LI
2009
REVIEW ARTICLE
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
CHALLENGE
Application OfTheory To Real
Material: Need ToConsider
Interaction AmongSeveral
Dislocations. Eachhaving different
Orientation ofBurger vector:
INTERACTION
Long range;determine the
collective behavior
of dislocationsystem
Short range;important in
dislocation reaction
PROBLEMAMPLIFIED
For multi bodyproblem;
Dislocationinteraction
reduced to theirline segmentinteraction.
Not only distancedependent but
also Orientationof the Burger
vector.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
PREVIOUS Work
3D Simulation of Dislocation System
Long range elastic interaction: Describe by the Peach Kohler equation
Short range interaction: Modelled Phenomenologically as Rate Process (Fitting
the rate process parameter to Experimental Obeservation Data)
System Cofiguration: Geometry of all dislocation line and their Burger Vector.
Dynamic of the System: Evolution of Dislocation line
Challenging and Time consuming Part:
Track each segments of all dislocation.
Calculate Forceon each segement from all other segment on each iteration
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Alternate Method:
Phase Field Microelasticity (PFM) Method
Approach to model 3D Evolution of Dislocation System in Elastically
Anisotropic Crystal
No consideration of individual segement of all Dislocations.
PFM Method: deals with Temporal and Spatial Evolution of several
Density Function (Fields)
Number of these Fields = Number of Slip modes determined by the
Crystallography.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Based on the Khachaturyan-Shatalov (KS) of Reciprocal spacetheory
of the strain in Elastically Homogenous System of Misfitting
Coherent Inclusionembedded into Parent Phase.
Application of KS theory: Two Phase Microstructure Evolution driven
by Strain Energy Relaxation.
Formulates
Estrain
[density field] define Spatial arrangement of Inclusion
Evolution of the System: Time Dependent Ginzburg Landau (TDGL)
Kinetic Equation
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Dislocation: A set of Coherent MisfittingPlatelet Inclsuion whose Stress Free
Strain is an Invariant Plane Strain.
oij= bini/d
o
ij is the Strain Tensor, bi is the Burgervector, niis the Unit vector normal to theslip Plane and d is the thickness of theplate.
Figure: Schematic illustration of the platelike coherent Inclusion Imitating theDislocation Loops.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Figure 1: Thin Plate like Coherent Inclusion imitating the Dislocation loops.
It is shown that: If the Habit Plane of the Platelet in an Elastically Anisotropic Body isnormal to nthen (Condition A)
Total Strain Energyproportional to Inclusion Perimeter Lenght.
Coincide with the Strain Energy of Dislocation with b (as Burger Vector)
Contour coincide with the Inclusion Perimeter. Thickness of Inclusion: Play a role of radius of Dislocation Core.
Condition A, Makes it Strain Energy Minimizer.
Reason: Vanishing of Strain Energy Term Proportional to Inclusion Volume.
Therefore, Reduction of Strain Energyresults Spontaneous transformation of all
Inclusion into Plateletswhich correspond to Dislocation Loops.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Formulation of PFMFormulate Dislocation Theory in terms of Phase Transformation:
Consider Displacive Transformation: characterized by Orientation Variants with
stress free strains that are Invariant Plane Strain
oij (,m) = bi (,m)i n ()j/d
= Number all slip planesc
This Kind of Phase Transformation automatically produce Inclusion that transformsthin Plates to Minimize Strain Energy.
Habit Plane coincide with the Slip plane.
These Plate corresponds to Dislocation loops.
Mutual location and Evolution are driven by the Strain Energy Mimizer.
Consider Martensitic Transformation (MT) in FCC System:
Slip mode {111} : 4 Planes (111) and 3 direction [110] in each Plane.
12 possible Orientation Variants: Each describe by Stress free Strain Tensor
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Problem: Dislocation loop modelling in terms of Coherent Inclusions.
Plastic Deformation: Movement of several Dislocations in the same slip plane.
Means: formation of several overlapping inclusion in the same slip plane .
(Impossible in Phase Transformation)
Resolve this Problem; Reformulate the Chemical Free Energy in PFM theory of
Martensitic Transformation
After Modification: Problem can be solved and reduced to the Problem of
Martensitic transformation.
Same results of Strain Energy and Strain Field by PFM and Conventional Dislocation
Theory.
Short Range Interaction automatically take into account such as Annihilation &
Multiplication.10
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Phase Field Description of Arbitrary Dislocation System
Repeatition:
Dislocation Ensemble Number of Density Function
Crystallography and Mode of Plastic Deformation
(Equal to Number of slip plane time the Burger vector in each slip plane)
Density Function is non-zero inside a Dislocation loop and vanish outside it.
Continous description of Burger Vector
Density in Slip plane is given by:
index s from 1,2 ...p; p is the total number of slip planes.
b(,m) = Elementary burger vector in slip plane , m= 1, 2,....q ; q is
total number of Elementary Burger Vectors.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
(,m, r) is the Density function of Dislocation
Total Burger Vector Field: Contribution by all Slip planes (Summation)
Stress free Strain :
Arbitary values of Phase field representing an Initial Non-equilibrium Microstructure.
Evolution towards Equilibrium obtained by Total Energy Minimization.
Formulating Total Energy as a function of Phase field, Derive the Kinetics using TDGL
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Total Energy FunctionalTotal Energy of a crystal with Dislocation in the Phase Field Model consists of Three
Parts:1. The Crystalline Energy (Ecryst)
2. The Elastic Strain Energy (Eelast)
3. Gradient Energy (Egrad)
Total Energy = E = Ecryst + Eelast + Egrad
Crystalline Energy:
Describe: Potential Energy in a Crsytal subjected to general shear produced by
linear combination of Localized simple shear associated with all possible Slip
system characterized by the Phase field
where o (r) is the stress free strain (general inelastic strain)
Ecrystplays a role of Chemical free Energy, this functional is Local.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Elastic Energy
Associated with Elastic Strain or Elastic Dislpacement of the crystal lattice caused
by Dislocation.
Elastic Strain = Total StrainInelastic Strain
Elastic Strain relaxed instantaneously by minimizing the elastic energy under an
inelastic strain through Green Function Solution.
Superscript * define the complex conjugate
f characterize the principle value of the integral (excluding k = 0)
Cijklis the Elastic Modulus Tensor and o
ij(r) is the stress free strain
ik(r) is the green Function
e is the k/k is a unit vector in reciprocal space along k
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Gradient Energy:
Free Energy of Heterogenous system: Not only dependent on Local values ofOrder parameter but on their Spatial Variations. (Gradient Thermodynamics)
Structural non-uniformity exist within the core regions of a Dislocation. Therefore Egradformulates in such a way that it vanish outside the core.
Component of Tensor (1 , 2)ijklare positively defined constant related to the Slipplanes 1and 2.
Important Points:
In general: EgradArea of the Slip plane swept by the Dislocation
In PF Theory; EgradArea of the Dislocation loop.
For Perfect Dislocation; Surface Energy contribution equal to zero
Exist for Partial Dislocation (Not considered Here)
EgradLenght of the Dislocation line (Remaining Part).15
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Phase Field Kinetic Equation Temporal Evolution of density Profile driven by relaxation of the Total Energy
describe the evolution of Dislocation System.
Simplest form of Kinetic Equation is the Stochastic Langevin Equation based on the
TDGL kinetic equation
TDGL equation: Rate of evolution of a Field is a linear function of the
Thermodynamic Driving forces
( , m, r , t) is the Field function Long Range Order Parameter (LRO)
L is the Kinetic Coefficient characterizing the Dislocation Mobility
E is the total Energy Functional
E/( , m, r , t) is the Thermodynamic driving force
is the Langevin Gaussian Noise term reproduce Thermal Fluctuation.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Phase Field Kinetic Equation
Substituting the Value &Differentiating with respect to theDensity Function gives:
This gives a Non-linear IntegroDifferentail Equation
This is the PFM Kinetic Equationgoverning Dilocation Dynamics:
Solution: Completely describe the
Geometry Of each Dislocation ofEvolving Dislocation Ensembleincluding Multiplication andAnnihilation of Dislocation andDislocation Reaction.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Interaction with Defects Crsytalline Material are not Ideal Crsytal.
Have Surface, Linear and Point Defects; which interact with Dislocation and haveProfound Effect on Dislocation Mobility
These Kind of Interaction effect on Mechanical Properties.
PFM Kinetic Equation can be easily modified: Include Interaction of Dislocation with
Defects
Just Need to Introduce Energy terms that Couple Defects with Dislocation.
This kind of Interaction are Short Range therefore taken into Crsytalline Energy.
If Defect generate Strain field. Then Stress Free Strain of Defect is added to Stress
free Strain of Dislocation
By this in Kinetic Equation, Interaction Of Defects Automatically taken into Account.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Application To FCC System
Theoretical Characterization of Dislocation Dynamics reduced to Kinetc Equationfor the Density Function in all Slip Plane.
For FCC: {111} is the Slip Plane and is the Slip direction.
Burger Vector =a/2 where a is lattice parameter.
Total Slip System = 12, Each Having its own Stress Free Strain.
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Application of the Model Model work as efficiently and realistic for Dislocation as it for the Martensitic Phase
Transformation.
LOOP GROWTH
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Figure Shows PFM simulation of Frank
Read Source under Periodic Boundary
Condition, where the Gray Rectangular
Loop (Thin Plate Inclusion) serves as the
Pinned Source Segments
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
LOOP PRECIPITATE INTERACTION
Schematic illustration of SimulatedDislocation interacting with Precipitate.
Defects change the mobility ofDislocation and hence effect MechanicalProperties.
After Modification, PFM automaticallytakes in to account Interaction betweenDislocation and Defects.
CORE CORE INTERPENETRATION
3D Simulation of Annihilation of TwoAttracting Dislocation Segment. TheBlack and Gray distinguish different
intersecting Slip plane
Shows: These segment approach eachother until the Core Interpenetrate andthe Segment Annihilate.
(True with respect to Dislocation Theory)
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
Simulated Stress Strain curve for
Uni-axial loading.
The Dislocation multiplication with
increasing strain can be shown in
the figure.
Simulation give yield stress of 1.8
x10-3G (shear Modulus), Observed
values are 10-3to 10-4G
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
PHASE FIELD MODEL OF
DISLOCATION CLIMB
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PHASE FIELD MODEL FOR DISLOCATION
HUZAIFA (MSS-ICAMS) Seminar - PHASE FIELD - 2014
OUTLOOK Non Conservative Motion of Dilocation (Climb): Plays an Important Role in Processes.
Climb velocity is controlled by long range vacancy diffusion.
Driving Force for the Dislocation Climb:
1. Climb Component of Peach Koehler forces : Arise from Internal/External Stresses
2. Osmotic (Chemical) Force: Due to Deviation of Vacancy Concentration from Equilibrium at a
given Temperature and Stress State.
Most Model for Dislocation Climb;
Take into account Long Range Elastic Interaction of Point defects and Sink/sources of
Vacancies
Donot emphassize the Mesoscopic Short range Interaction and Dynamic motion of
Sources/Sinks with Vancancy Generation/Annihilation ( Important for Mesoscopic Modelling)
Main Advantage of PF: Ability to incorporate Long and Short rangeDislocationDislocation
and Dislocation and Vacancy Interactions, Osmotic force, Diffusionand External Applied
Stress into Single Mathematical Framework Without Need for Simple Analytical Solution.
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