Phase Field Presentation

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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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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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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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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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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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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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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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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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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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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 &

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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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(,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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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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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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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 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 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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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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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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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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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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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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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 OF

DISLOCATION CLIMB

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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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