1.3cm numerical modelling of phase transformation...
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Numerical Modelling of PhaseTransformation-Mechanics Coupling Using a
Phase Field Method
Kais AMMAR
Thesis supervisor : Georges CAILLETAUD: Samuel FOREST
Co-supervision : Benoît APPOLAIRE
Mines ParisTech, CNRS UMR 7633Centre des Matériaux
20th January 2010
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2 / 97 Why ?
Phase transformation-mechanics coupling:
Different morphological evolutions
Base Ni Co-Pt [Le Bouar] Cu-Al-Ni
Different kinetics
Different behaviours (TRIP)
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3 / 97 Why ?
Phase transformation-mechanics coupling:
Different morphological evolutions
Ni-Al[Diologent,2004] Cu-Cd [Sullonen]
Different kinetics
Different behaviours (TRIP)
2/56
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4 / 97 Existing Approach
Two approaches to describe phase transformation-mechanicscoupling:
1. Macroscopic approach [Leblond et al., 1986, Fischer et al.,2000]
2. Microscopic approach
Sharp interface description (front tracking techniques)
Diffuse-interface models (phase field) [Khachaturyan,1983]
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5 / 97 Numerical Methods for Phase Field
B. Appolaire I. Steinbach A. Gaubert
Many numerical methods have been proposed to solve the coupled mechanics-phase field problems:
Finite volume scheme [Appolaire and Gautier, 2003, Furtado et al., 2006].
Mixed finite difference-finite element scheme [Steinbach and Apel, 2006].
Fourier method [Khachaturyan, 1983, Gaubert et al., 2008].
Finite element method[Ubachs et al., 2005, Schrade et al., 2007, Ammar et al., 2009].
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6 / 97 Aims of the Thesis
The aims of this work is to develop a general framework thatcombines standard phase field approaches with a different complexmechanical behaviour for each phase:
To develop a finite element formulation of a fully coupled phasefield/diffusion/mechanical problem.
To implement the non-linear elastoplastic phase field model inthe finite element code Zebulon.
To solve some elementary initial boundary value problems incoupled diffusion-elasto-plasticity and validate againstcorresponding sharp interface analytical solutions.
5/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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10 / 97 Principle of Phase Field Method
A. Karma G. Abrivard A. Khatchaturyan
The phase field approach is suitable for modeling free boundary problems(Grain boundaries, interfaces).
It avoids to explicitly track the usually geometrically complicated interfacesduring microstructure evolution.
It can be applied to a wide range of microstructural evolution problemsrelated to various materials processes (solidification, precipitation,coarsening and grain growth and polycrystalline materials...)
Phase-field method has been extended and coupled with generalprocesses of materials science that include dissipation, such that diffusion,mechanics, dislocation dynamics and fracture.
Formulation of a Phase Field/Diffusion/Mechanical Model 9/56
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11 / 97 Principle of Phase Field MethodIntroduction of a phase field variable:
Physical motivated (order parameter...).
Artificial order to locate the various phases.
This variable is uniform inside a phase or domainaway from the interface (for example 0 and 1) andvaries continuously across the diffuse interfacebetween different phases.
Introduction of a phase field variable.Construction of total free energy.Determination of the evolution equation of phase fieldIdentification of phase field parameters.
Formulation of a Phase Field/Diffusion/Mechanical Model 10/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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13 / 97 Mechanical Phase-Field CouplingTotal Free Energy Functional
F (φ,∇φ, c) =∫V
f(φ,∇φ, c) dv =∫V
[fch(φ, c) +
α
2|∇φ|2
]dv
Chemical Energy Density fch(c, φ) (Binary alloys) [Kim et al., 1998]
fch(c, φ) =
h(φ)
fα +
(1− h(φ))
fβ +
Wg(φ)
fα,β(c) =12kα,β(c− aα,β)2 + bα,β
where h(φ) = φ2(3− 2φ)
g(φ) = φ2(1− φ)2
Formulation of a Phase Field/Diffusion/Mechanical Model 12/56
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14 / 97 Mechanical Phase-Field CouplingTotal Free Energy Functional
F (φ,∇φ, c) =∫V
f(φ,∇φ, c) dv =∫V
[fch(φ, c) +
α
2|∇φ|2
]dv
Chemical Energy Density fch(c, φ) (Binary alloys) [Kim et al., 1998]
fch(c, φ) =
h(φ)
fα +
(1− h(φ))
fβ +
Wg(φ)
fα,β(c) =12kα,β(c− aα,β)2 + bα,β
where h(φ) = φ2(3− 2φ)
g(φ) = φ2(1− φ)2
Formulation of a Phase Field/Diffusion/Mechanical Model 12/56
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15 / 97 Mechanical Phase-Field CouplingTotal Free Energy Functional
F (φ,∇φ, c) =∫V
f(φ,∇φ, c) dv =∫V
[fch(φ, c) +
α
2|∇φ|2
]dv
Chemical Energy Density fch(c, φ) (Binary alloys) [Kim et al., 1998]
fch(c, φ) = h(φ) fα + (1− h(φ)) fβ +
Wg(φ)
fα,β(c) =12kα,β(c− aα,β)2 + bα,β where h(φ) = φ2(3− 2φ)
g(φ) = φ2(1− φ)2
Formulation of a Phase Field/Diffusion/Mechanical Model 12/56
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16 / 97 Mechanical Phase-Field CouplingTotal Free Energy Functional
F (φ,∇φ, c) =∫V
f(φ,∇φ, c) dv =∫V
[fch(φ, c) +
α
2|∇φ|2
]dv
Chemical Energy Density fch(c, φ) (Binary alloys) [Kim et al., 1998]
fch(c, φ) = h(φ) fα + (1− h(φ)) fβ + Wg(φ)
fα,β(c) =12kα,β(c− aα,β)2 + bα,β where h(φ) = φ2(3− 2φ)
g(φ) = φ2(1− φ)2
Formulation of a Phase Field/Diffusion/Mechanical Model 12/56
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17 / 97 Mechanical Phase-Field CouplingSpecific free energy
fmech(ε∼e, V ) = fe(ε∼
e) + fp(V )
where V set of internal variables and ε∼ = ε∼e + ε∼
? + ε∼p.
Mechanical dissipation potential
Ω(σ,A) = g(σ∼, A) + Ωp(A)
A: set of thermodynamical force associated with V
State laws Complementary laws
σ∼ =∂fmech
∂ε∼e
ε∼∼p =
∂Ω
∂σ∼
A =∂fmech
∂VV = −∂Ω
∂A
Formulation of a Phase Field/Diffusion/Mechanical Model 13/56
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18 / 97 Mechanical Phase-Field CouplingTotal free energy functional:
F (φ,∇φ, c, ε∼e, Vk) =
∫V
f(φ,∇φ, c, ε∼e, Vk) dv
=∫V
[fch(φ, c) + fmech(φ, c, ε∼
e, Vk) +α
2|∇φ|2
]dv
Mechanical free energy contribution
fmech(φ, c, ε∼e, Vk) = fe(φ, c, ε∼
e) + fp(φ, c, Vk)
Mechanical dissipation potential
Ω(φ, c,σ∼ , Ak)
Formulation of a Phase Field/Diffusion/Mechanical Model 14/56
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19 / 97 Mechanical Phase-Field CouplingBalance Equations:
Local static mechanical equilibrium:
∇.
(∂f
∂ε∼e
)= 0
Balance of mass:
c−∇.
[L(φ)
(∇∂f
∂c
)]= 0
Evolution equation of order parameter:
−βφ− ∂f
∂φ+ ∇.
∂f
∂∇φ= 0
Constitutive equations (Clausius-Duhem dissipation inequality)
σ∼ =∂f
∂ε∼e
J = −L(φ)
(∇∂f
∂c
)π = −βφ− ∂f
∂φ
ξ =∂f
∂∇φ= α∇φ
Formulation of a Phase Field/Diffusion/Mechanical Model 15/56
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20 / 97 Mechanical Phase-Field CouplingBalance Equations:
Local static mechanical equilibrium:
∇.
(∂f
∂ε∼e
)= 0
Balance of mass:
c−∇.
[L(φ)
(∇∂f
∂c
)]= 0
Evolution equation of order parameter:
−βφ− ∂f
∂φ+ ∇.
∂f
∂∇φ= 0
Constitutive equations (Clausius-Duhem dissipation inequality)
σ∼ =∂f
∂ε∼e
J = −L(φ)
(∇∂f
∂c
)π = −βφ− ∂f
∂φ
ξ =∂f
∂∇φ= α∇φ
Formulation of a Phase Field/Diffusion/Mechanical Model 15/56
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21 / 97 Mechanical Phase-Field CouplingBalance Equations:
Local static mechanical equilibrium:
∇. σ∼ = 0
Balance of mass:
c−∇.
[L(φ)
(∇∂f
∂c
)]= 0
Evolution equation of order parameter:
−βφ− ∂f
∂φ+ ∇.
∂f
∂∇φ= 0
Constitutive equations (Clausius-Duhem dissipation inequality)
σ∼ =∂f
∂ε∼e
J = −L(φ)
(∇∂f
∂c
)π = −βφ− ∂f
∂φ
ξ =∂f
∂∇φ= α∇φ
Formulation of a Phase Field/Diffusion/Mechanical Model 15/56
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22 / 97 Mechanical Phase-Field CouplingBalance Equations:
Local static mechanical equilibrium:
∇. σ∼ = 0
Balance of mass:c−∇. J = 0
Evolution equation of order parameter:
−βφ− ∂f
∂φ+ ∇.
∂f
∂∇φ= 0
Constitutive equations (Clausius-Duhem dissipation inequality)
σ∼ =∂f
∂ε∼e
J = −L(φ)
(∇∂f
∂c
)
π = −βφ− ∂f
∂φ
ξ =∂f
∂∇φ= α∇φ
Formulation of a Phase Field/Diffusion/Mechanical Model 15/56
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23 / 97 Mechanical Phase-Field CouplingBalance Equations:
Local static mechanical equilibrium:
∇. σ∼ = 0
Balance of mass:c−∇. J = 0
Balance of generalized stresses (Gurtin’s balance of microforces):π + ∇. ξ = 0
Constitutive equations (Clausius-Duhem dissipation inequality)
σ∼ =∂f
∂ε∼e
J = −L(φ)
(∇∂f
∂c
)π = −βφ− ∂f
∂φ
ξ =∂f
∂∇φ= α∇φ
Formulation of a Phase Field/Diffusion/Mechanical Model 15/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
![Page 25: 1.3cm Numerical Modelling of Phase Transformation ...matperso.mines-paristech.fr/Donnees/data02/295-Presentation_these.pdf · 5/97 Numerical Methods for Phase Field B. Appolaire I](https://reader030.vdocuments.net/reader030/viewer/2022040715/5e1cc168543a9b36d426342b/html5/thumbnails/25.jpg)
25 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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26 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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27 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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28 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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29 / 97 Finite Element Implementation
-u?, c?, φ?
-
-
6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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30 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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31 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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32 / 97 Finite Element Implementation
-u?, c?, φ?
-
- 6
??
?
Balance Equations∇.σ∼ = 0
c−∇.J = 0
π + ∇. ξ = 0
Variational Formulations=(u ?)=(c?)=(φ?)
Degrees of Freedom(ui, c, φ)
Finite Element Methodto Discretize Space
Euler Implicite Methodto Discretize Time
Residual VectorsRuRcRφ
[K] ==
[Ke
t11] [
Ket12] [
Ket13]
[Ke
t21] [
Ket22] [
Ket23]
[Ke
t13] [
Ket32] [
Ket33]
Newton− Raphson′s Method
Compute∆u,∆c,∆φ
Formulation of a Phase Field/Diffusion/Mechanical Model 17/56
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33 / 97
Formulation of a Phase Field/Diffusion/Mechanical Model 18/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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35 / 97 Phase-Field and "Homogenization"Total Free energy
f(φ,∇φ, c, ε∼e, Vk) = fch(φ, c) + fe(φ, c, ε∼
e) + fp(φ, c, Vk) +α
2|∇φ|2
Two appraoch of of introducing linear and nonlinear mechanical constitutiveequations into the standard phase field approach:
1. Standard model [Khachaturyan, 1983, Gaubert et al., 2008].
The material behaviour is described by a unified set ofconstitutive equations.
Each parameter is usually interpolated between the limitvalues known for each phase.
2. Homogenization method [Steinbach and Apel, 2006]
One distinct set of constitutive equations is attributed toeach individual phase k at any material point.
Phase-Field and "Homogenization" 20/56
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36 / 97 Phase-Field and "Homogenization"Standard Khachaturyan Model
ε∼? = φ ε∼
?α + (1− φ) ε∼
?β
C≈
(φ, c) = φC≈ α
(c) + (1− φ)C≈ β
(c)
Elastic energy f(ε∼e, φ):
fe(φ, c, ε∼e) =
12
(ε∼− ε∼?) : C
≈: (ε∼− ε∼
?) =12ε∼e : C
≈: ε∼
e
Phase-Field and "Homogenization" 21/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
![Page 38: 1.3cm Numerical Modelling of Phase Transformation ...matperso.mines-paristech.fr/Donnees/data02/295-Presentation_these.pdf · 5/97 Numerical Methods for Phase Field B. Appolaire I](https://reader030.vdocuments.net/reader030/viewer/2022040715/5e1cc168543a9b36d426342b/html5/thumbnails/38.jpg)
38 / 97 Multiphase Approach
va
ria
ble
ΨΨα
Ψβ
α phase β phaseinterface
να
νβ
ν
distance
ν = να
ν = νβ
ε∼ = χ ε∼α + (1− χ) ε∼β and σ∼ = χσ∼α + (1− χ)σ∼β
fe(φ, c, ε∼e, Vk) = χ feα + (1− χ) feβ
For instance:
χ(x , t) = φ(x , t) or χ(x , t) =c− cβcα − cβ
Phase-Field and "Homogenization" 23/56
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39 / 97 Voigt/Taylor Model
Voigt/Taylor assumptions
ε∼α = ε∼β = ε∼σ∼ = φσ∼α + (1− φ)σ∼β
Effective elasticity tensor
C≈
= φC≈ α
+ (1− φ)C≈ β
Plastic strain and Eigenstrain:
ε∼? = C
≈−1 : (φC
≈ α: ε∼
?α + (1− φ)C
≈ β: ε∼
?β)
ε∼p = C
≈−1 : (φC
≈ α: ε∼
pα + (1− φ)C
≈ β: ε∼
pβ)
Phase-Field and "Homogenization" 24/56
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40 / 97 Reuss/Sachs Model
Reuss/Sachs assumptions
ε∼ = φ ε∼α + (1− φ) ε∼βσ∼α = σ∼β = σ∼
Effective compliance matrix
S≈
= φS≈α
+ (1− φ)S≈ β
Plastic strain and Eigenstrain:
ε∼?(φ) = φ ε∼
?α + (1− φ)ε∼
?β
ε∼p = φ ε∼
pα + (1− φ)ε∼
pβ
Phase-Field and "Homogenization" 25/56
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41 / 97 Phase-Field Model and "Homogenization"
Elastic energy fe(ε∼, φ):
fe = φ fα + (1− φ)fβ
=12
(ε∼− ε∼?) : C
≈: (ε∼− ε∼
?)
wherefα =
12
(ε∼− ε∼?α) : C
≈ α: (ε∼− ε∼
?α)
fβ =12
(ε∼− ε∼?β) : C
≈ β: (ε∼− ε∼
?β)
Voigt/Taylor model Reuss/Sachs model
C≈
= φC≈ α
+ (1− φ)C≈ β
C≈
= (φS≈α
+ (1− φ)S≈β
)−1
ε∼? = C
≈−1 : (φC
≈ α: ε∼
?α + (1− φ)C
≈ β: ε∼
?β) ε∼
? = φ ε∼?α + (1− φ)ε∼
?β
Phase-Field and "Homogenization" 26/56
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42 / 97 Phase-Field Model and "Homogenization"
Elastic energy fe(ε∼, φ):
fe = φ fα + (1− φ)fβ
=12
(ε∼− ε∼?) : C
≈: (ε∼− ε∼
?)
wherefα =
12
(ε∼− ε∼?α) : C
≈ α: (ε∼− ε∼
?α)
fβ =12
(ε∼− ε∼?β) : C
≈ β: (ε∼− ε∼
?β)
Voigt/Taylor model Reuss/Sachs model
C≈
= φC≈ α
+ (1− φ)C≈ β
C≈
= (φS≈α
+ (1− φ)S≈β
)−1
ε∼? = C
≈−1 : (φC
≈ α: ε∼
?α + (1− φ)C
≈ β: ε∼
?β) ε∼
? = φ ε∼?α + (1− φ)ε∼
?β
Phase-Field and "Homogenization" 26/56
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43 / 97 Phase-Field Model and "Homogenization"
Evolution equations for order parameter and concentration:
∇.ξ + π = −βφ+ ∇.(α∇φ)− ∂fch
∂φ− ∂fu
∂φ= 0
c = −∇.(−L(φ)∇µ) = −∇.
[−L(φ)
(∇∂fch
∂c+ ∇∂fu
∂c
)]Identification of parameters:
γ =√αW/(3
√2)
δ = 2.94√
2α/W
Phase-Field and "Homogenization" 27/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
![Page 45: 1.3cm Numerical Modelling of Phase Transformation ...matperso.mines-paristech.fr/Donnees/data02/295-Presentation_these.pdf · 5/97 Numerical Methods for Phase Field B. Appolaire I](https://reader030.vdocuments.net/reader030/viewer/2022040715/5e1cc168543a9b36d426342b/html5/thumbnails/45.jpg)
45 / 97 Elastoplastic Phase-Field CouplingPlastic free energy density fp(φ, Vα, Vβ):
f(φ,∇φ, c, ε∼e, Vα, Vβ) = fch(φ, c)+fe(φ, c, ε∼
e)+fp(φ, Vα, Vβ)+α
2|∇φ|2
withfp = φ fαp (Vα) + (1− φ)fβp (Vβ)
where Vα,β are the set of internal variables of both phases
Dissipation pseudo-potential Ω :
Ω = φΩα(Aα) + (1− φ)Ωβ(Aβ)
Aα,β are the set of thermodynamic forces associated with Vα,β.
Classical Von Mises yield function gα,β defined by :
gα,β = σ∼eqα,β −Rα,β
Phase-Field and "Homogenization" 29/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
![Page 47: 1.3cm Numerical Modelling of Phase Transformation ...matperso.mines-paristech.fr/Donnees/data02/295-Presentation_these.pdf · 5/97 Numerical Methods for Phase Field B. Appolaire I](https://reader030.vdocuments.net/reader030/viewer/2022040715/5e1cc168543a9b36d426342b/html5/thumbnails/47.jpg)
47 / 97 Coherent Equilibrium
Bitter-Crum assumptions:
1. The interfaces between α and β are coherent.
2. The eigenstrains are spherical tensors independent ofconcentration.
3. Homogeneous isotropic linear elasticity is considered.
Elastic Energy [Bitter and crum theorem]:
fe = z(1− z)E (ε∼?)2
1− ν
where z is the volume fraction of α.
Energy ratio
A =1
k (Aα −Aβ)2E (ε?)2
(1− ν)
Phase-Field and "Homogenization" 31/56
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48 / 97 Coherent Equilibrium
Phase-Field and "Homogenization" 32/56
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49 / 97 Coherent Equilibrium
0.3 0.4 0.5 0.6 0.7c0
0.0
0.5
1.0
1.5
2.0A
Williams point
β α
α + βincoherent β field
boundary
incoherent α field
boundary
Phase-Field and "Homogenization" 33/56
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50 / 97 Coherent Equilibrium
0.3 0.4 0.5 0.6 0.7c0
0.0
0.5
1.0
1.5
2.0A
Williams point
β α
α + βincoherent β field
boundary
incoherent α field
boundary
Phase-Field and "Homogenization" 33/56
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51 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α
A=0
A=0
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 34/56
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52 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α
A=0
A=0
A=0.1125
A=0.1125
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 34/56
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53 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α
A=0
A=0
A=0.1125
A=0.1125
A=0.613
A=0.613
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 34/56
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54 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α
A=0
A=0
A>1
A>1
A=0.1125
A=0.1125
A=0.613
A=0.613
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 34/56
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55 / 97 Coherent Equilibrium
Eigenstrain is linear function of composition (Vegard’s law)
ε∼? = ε?(cα − cβ)1∼
Elastic Energy:
fe = z(1− z)E (ε?)2
1− ν(cα − cβ)2
Energy ratio
A =E (ε?)2
k(1− ν)
Phase-Field and "Homogenization" 35/56
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56 / 97 Coherent Equilibrium
0.3 0.4 0.5 0.6 0.7c0
0.0
0.5
1.0
1.5
2.0
2.5
3.0A β α
α + βincoherent β field
boundaryincoherent α field
boundary
Phase-Field and "Homogenization" 36/56
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57 / 97 Coherent Equilibrium
0.3 0.4 0.5 0.6 0.7c0
0.0
0.5
1.0
1.5
2.0
2.5
3.0A β α
α + βincoherent β field
boundaryincoherent α field
boundary
Phase-Field and "Homogenization" 36/56
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58 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α A=0
A=0
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 37/56
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59 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α A=0
A=0
A=0.15
A=0.15
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 37/56
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60 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α A=0
A=0
A=0.15
A=0.15
A=0.525
A=0.525
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 37/56
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61 / 97 Coherent Equilibrium
0.2 0.3 0.4 0.5 0.6 0.7 0.8c0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
c βc α A=0
A=0
A=0.15
A=0.15
A=0.525
A=0.525
A>1.5
A>1.5
βsin
glephase
αsin
glephase
Phase-Field and "Homogenization" 37/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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64 / 97 Initial/Boundary Conditions
Schematic of the misfitting planar oxide layer (α) growing at the surface of a purezirconium slab (β).
The interface/boundary conditions :
c(∀x, y = 0,∀z, t) = csα = 0.68c(∀x, y = h,∀z, t) = c∞ = 0.22
Choosing the Zr phase as the stress free reference state, the eigenstrain, inthe phase α, is a spherical tensor independent of concentration:
ε∼?α = δZrO21∼ and ε∼
?β = 0∼
where 1∼ the unit second order tensor.Plasticity and Phase Transformation Kinetics 40/56
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65 / 97 Growth Kinetics
The growth kinetics of the oxide layer
∆e = K
√t
with K = 7.5.10−10 m. s−1/2 and Kexp= 7.10−10 m. s−1/2
Plasticity and Phase Transformation Kinetics 41/56
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66 / 97
Plasticity and Phase Transformation Kinetics 42/56
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67 / 97
Evolution vs. time of the concentration field in oxygen during the growth of anoxide with an interface initially destabilized by a sine (left) and of a sinusoidal
oxide layer (right).
Plasticity and Phase Transformation Kinetics 43/56
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68 / 97 Effect of the Misfit Generated Stress
0 200 400 600 800 1000
time (h)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Δe(μm)
chemical
δZrO2= ±2%
δZrO2= ±4%
δZrO2= ±6%
δZrO2= ±7%
Growth kinetics of the oxide layerfor different dilatation misfits in the
oxide layer
Interfacial equilibrium concentration[Johnson and Alexander, 1986]
cintα = aα + ∆c
cintβ = aβ + ∆c
with
∆c =Ecoh −∆fel + κγ
k(aβ − aα)
where
Ecoh = (ε∼α − ε∼β) : σ∼β
∆fe = feα − feβ
Plasticity and Phase Transformation Kinetics 44/56
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69 / 97 Effect of the Misfit Generated Stress
0 200 400 600 800 1000
time (h)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Δe(μm)
chemical
δZrO2= ±2%
δZrO2= ±4%
δZrO2= ±6%
δZrO2= ±7%
Growth kinetics of the oxide layerfor different dilatation misfits in the
oxide layer
y
c sα
c∞β
(c intα )ch
(c intα )el
(c intβ )ch
(c intβ )el
Composition
c
Schematic illustration of the compositionprofiles in both matrix and oxide layer
Plasticity and Phase Transformation Kinetics 44/56
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70 / 97 Effect of the Misfit Generated Stress
0 200 400 600 800 1000
time (h)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Δe(μm)
chemical
δZrO2= ±2%
δZrO2= ±4%
δZrO2= ±6%
δZrO2= ±7%
Growth kinetics of the oxide layerfor different dilatation misfits in the
oxide layer
−10 −5 0 5 10
δZrO2(%)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
K/K0
Kinetic constants K normalized bythe pure chemical case K0 versus
the dilation misfit.
Plasticity and Phase Transformation Kinetics 44/56
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71 / 97 Effect of the Oxide Elasticity Moduli
0 100 200 300 400 500 600 700
time (h)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Δe(μm)
chemical
EZr02=70GPa
EZr02=120GPa
EZr02=160GPa
EZr02=220GPa
0 50 100 150 200 250
EZr02(GPa)
0.70
0.75
0.80
0.85
0.90
0.95
1.00
K/K0
Growth kinetics of the oxide layer for different oxide Young’s moduli EZrO2 . Inset
shows the dependency of the corresponding kinetic constants KPlasticity and Phase Transformation Kinetics 45/56
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72 / 97 Effect of Plastic Accommodation Processes
0 100 200 300 400 500time (h)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Δe(μm)
chemical
elastic
plastic
c sα
c∞β
(c intα )ch
(c intα )el
(c intα )pl
(c intβ )ch
(c intβ )el
(c intβ )pl
Composition
c
The growth kinetics of ZrO2 oxide layer in the infinite unstressed Zr matrix assuming a
chemical, elastic and ideal plastic behaviour for oxide layer.Plasticity and Phase Transformation Kinetics 46/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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74 / 97Growth of an Elastic Particle in an Elastioplastic Matrix
The α precipitate of radius rint is embedded in the β matrix with an infinite radius.
Boundary conditions :
ξ .n = 0, J .n on all boundariesσ∼(r = R, θ) = 0 0 ≤ θ ≤ θ0 : free surface conditionuθ(r, θ = 0) = 0 0 ≤ r ≤ Ruθ(r, θ = θ0) = 0 0 ≤ r ≤ R : symmetric boundary condition
Plasticity and Phase Transformation Kinetics 48/56
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75 / 97Growth of an elastic particle in an elastioplastic matrix
0.0 0.5 1.0 1.5 2.0 2.5
r/rint
−1.5
−1.0
−0.5
0.0
0.5
σij/σ0
σrr,σθθ
σθθ
σrr
analytic
elastic
Normal σrr, tangential σθθ stress distributions in a radial direction
Plasticity and Phase Transformation Kinetics 49/56
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76 / 97Growth of an elastic particle in an elastioplastic matrix
0.0 0.5 1.0 1.5 2.0 2.5
r/rint
−1.5
−1.0
−0.5
0.0
0.5
σij/σ0
σrr,σθθ
σrr,σθθ
σθθ
σrr
analytic
elastic
elasto-plastic
Normal σrr, tangential σθθ stress distributions in a radial direction
Plasticity and Phase Transformation Kinetics 49/56
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77 / 97Growth of an elastic particle in an elastioplastic matrix
0.0 0.5 1.0 1.5 2.0 2.5
r/rint
−1.5
−1.0
−0.5
0.0
0.5
σij/σ0
σrr,σθθ
σrr,σθθ
σθθ
σrr
analytic
elastic
elasto-plastic
σ0 1.25σ
01.5σ
01.75σ
02σ
0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
r p/r 0
analytic
numeric
Normal σrr, tangential σθθ stress distributions in a radial direction
Plasticity and Phase Transformation Kinetics 49/56
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78 / 97Growth of an Elastic Particle in an Elastioplastic Matrix
0 10 20 30 40 50
r/r0
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
composition
Analytic
Numeric
Time evolution of the concentration profiles
Interfacial equilibrium concentration
cintα = aα +
∆E + κγ
k(aα − aβ)
cintβ = aβ +
∆E + κγ
k(aα − aβ)
∆E = Ecoh−∆fel =(ε∼eα − ε∼
eβ
): σ∼β−(feα−feβ)
Pure elastic behaviour
∆Eelas = −(ε?)2
A
Elasto− plastic behaviour
∆Eplas = −2σ0αε?
[ln
(ε?
Aσ0α
)+ 1
]+A(σ0
α)2
where A =1− ναEα
Plasticity and Phase Transformation Kinetics 50/56
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79 / 97Growth of an elastic particle in an elastioplastic matrix
0 1 2 3 4 5 6 7 8 9
temps (s) 1e5
0
2
4
6
8
10
12
14
Δr/
r 0
chemical (analytic)
chemical (numeric)
elastic (analytic)
elastic (numeric)
plastic (analytic)
plastic (numeric)
Growth kinetics of a misfitting spherical precipitate in an infinite matrix
Plasticity and Phase Transformation Kinetics 51/56
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Plan
1 Formulation of a Phase Field/Diffusion/Mechanical ModelPrinciple of Phase Field MethodMechanical Phase-Field CouplingFinite Element Implementation
2 Phase-Field and "Homogenization"Multiphase Approach
Voigt/Taylor ModelReuss/Sachs Model
Elastoplastic Phase-Field CouplingCoherent Equilibrium
3 Plasticity and Phase Transformation KineticsOxidation of zirconiumGrowth of a Misfitting Spherical Precipitate
4 Conclusions and Future Work
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81 / 97 Main results
I A general framework has been proposed to combine standard phasefield approaches with a different complex linear or non-linear materialbehaviour for each phase.
The proposed framework offers a number of advantages:
Balance and constitutive equations are clearly separated in theformulation, which allows the application of any arbitrary form for thefree energy functional.
The formulation is shown to be well-suited for a finite elementformulation of the initial boundary value problems on finite sizespecimens with arbitrary geometries and for very general non-periodicor periodic boundary conditions.
The approach makes possible to mix different types of constitutiveequations for each phase.
The formulation allows the use of any arbitrary mixture rules taken fromwell-known homogenization theory, in the interface region.
Conclusions and Future Work 53/56
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82 / 97 Main results
II Programming and implementation of finite element constitutive equationswith different homogenization schemes in finite element code ZeBuLoNwhere specific classes have been defined following the philosophy ofobject oriented programming.
III Modelling and simulation of some elementary initial boundary valueproblems in both pure diffusion and coupled diffusion-elastoplasticity onfinite size specimens.
The different results demonstrate that the choice of such an interpolationscheme can have serious consequences on the predicted coherent phasediagram:
Reuss scheme is unacceptable for coupling mechanics and phasetransformation
Voigt/Taylor and Khachaturyan Models are equivalent
Conclusions and Future Work 54/56
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83 / 97 Future Work
Short-term prospects
Effect of coherent elastic strain on shape instabilities duringgrowth.
Use of other general homogenization schemes, like theHashin–Shtrikman procedure or the self–consistent method.
Anisotropic effects through the interface energies, the elasticcoefficients, or the material parameters.
Phase field and crystal plasticity
Conclusions and Future Work 55/56
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84 / 97 Future WorkLong-term prospects
Inheritance of plastic deformation during migration of phaseboundaries.
Mesh sensitivity and adaptive mesh
[Abrivard, 2009]
Conclusions and Future Work 56/56
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85 / 97Ammar, K., Appolaire, B., Cailletaud, G., Feyel, F., and Forest, S. (2009).Finite element formulation of a phase field model based on the concept ofgeneralized stresses.Computational Materials Sciences, 45:800–805.
Appolaire, B. and Gautier, E. (2003).Modelling of phase transformations in titanium alloys with a phase field model.Lecture Notes in Computational Science & Engineering, 32:196–201.
Earmme, Y., Johnson, W., and Lee, J. (1981).Plastic relaxation of the transformation strain energy of a misfitting sphericalprecipitate: Linear and power–law strain hardening.Metallurgical Transactions A, 12A:1521–1530.
Furtado, A., Castro, J., and Silva, A. (2006).Simulation of the solidification of pure nickel via the phase-field method.Materials Research, 9(4):349–356.
Gaubert, A., Finel, A., Le Bouar, Y., and Boussinot, G. (2008).Viscoplastic phase field modellling of rafting in ni base superalloys.In Continuum Models and Discrete Systems CMDS11, pages 161–166. MinesParis Les Presses.
Germain, P. (1973).The method of virtual power in continuum mechanics. Part 2: Microstructure.SIAM J. Appl. Math., 25:556–575.
Gurtin, M. (1996).
Conclusions and Future Work 56/56
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86 / 97Generalized Ginzburg–Landau and Cahn–Hilliard equations based on amicroforce balance.Physica D, 92:178–192.
Johnson, W. C. and Alexander, J. I. D. (1986).Interfacial conditions for thermomechanical equilibrium in two-phase crystals.Journal of Applied Physics, 9:2735–2746.
Khachaturyan, A. (1983).Theory of Structural Transformations in Solids.John Wiley & Sons, New York.
Kim, S., Kim, W., and Suzuki, T. (1998).Interfacial compositions of solid and liquid in a phase–field model with finiteinterface thickness for isothermal solidification in binary alloys.Physical Review E, 58(3):3316–3323.
Lee, J., Earmme, Y., Aaronson, H., and Russell, K. (1980).Plastic relaxation of the transformation strain energy of a misfitting sphericalparticle: Ideal plastic behavior.Metallurgical transactions A, 11A:1837–1847.
Schrade, D., Mueller, R., Xu, B., and Gross, D. (2007).Domain evolution in ferroelectric materials: A continuum phase field model andfinite element implementation.Computer Methods in Applied Mechanics and Engineering, 196:4365–4374.
Steinbach, I. and Apel, M. (2006).Multi phase field model for solid state transformation with elastic strain.
Conclusions and Future Work 56/56
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87 / 97Physica D, 217:153–160.
Ubachs, R., Schreurs, P., and Geers, M. (2005).Phase field dependent viscoplastic behaviour of solder alloys.International Journal of Solids and Structures, 42:2533–2558.
Conclusions and Future Work 57/56
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88 / 97Evolution equation for order parameter:
∇.ξ + π = −βφ+ α∆φ− ∂f0∂φ−∂f∂φ
= 0
= −βφ+ α∆φ− ∂f0∂φ− 1
2ε∼e :
∂C≈∂φ
: ε∼e − ∂ε∼
e
∂φ: C≈
: ε∼e = 0
At equilibrium (φ = 0) and in the case of homogeneous elasticity (∂C≈∂φ
= 0),
the phase field equation can be written as follows:
αd2φeq
dx2− ∂f0∂φeq
− ∂f
∂φeq= 0
αd2φeq
dx2− ∂f0∂φeq
− ∂ε∼e
∂φeq: C≈
: ε∼e = 0
Khachaturyan voigt/Taylor Sachs/Reuss
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= 0
σ∼ = σ∼α = σ∼βC≈ α
: ε∼eα = C
≈ β: ε∼
eβ
=⇒ ε∼eα = ε∼
eβ =⇒ ∂ε∼
e
∂φ= 0
Conclusions and Future Work 57/56
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89 / 97Evolution equation for order parameter:
∇.ξ + π = −βφ+ α∆φ− ∂f0∂φ−∂f∂φ
= 0
= −βφ+ α∆φ− ∂f0∂φ
:0
−1
2ε∼e :
∂C≈∂φ
: ε∼e − ∂ε∼
e
∂φ: C≈
: ε∼e = 0
At equilibrium (φ = 0) and in the case of homogeneous elasticity (∂C≈∂φ
= 0),
the phase field equation can be written as follows:
αd2φeq
dx2− ∂f0∂φeq
− ∂f
∂φeq= 0
αd2φeq
dx2− ∂f0∂φeq
− ∂ε∼e
∂φeq: C≈
: ε∼e = 0
Khachaturyan voigt/Taylor Sachs/Reuss
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= 0
σ∼ = σ∼α = σ∼βC≈ α
: ε∼eα = C
≈ β: ε∼
eβ
=⇒ ε∼eα = ε∼
eβ =⇒ ∂ε∼
e
∂φ= 0
Conclusions and Future Work 57/56
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90 / 97Evolution equation for order parameter:
∇.ξ + π = −βφ+ α∆φ− ∂f0∂φ−∂f∂φ
= 0
= −βφ+ α∆φ− ∂f0∂φ
:0
−1
2ε∼e :
∂C≈∂φ
: ε∼e
:0−∂ε∼
e
∂φ: C≈
: ε∼e = 0
At equilibrium (φ = 0) and in the case of homogeneous elasticity (∂C≈∂φ
= 0),
the phase field equation can be written as follows:
αd2φeq
dx2− ∂f0∂φeq
>
0
− ∂f
∂φeq= 0
αd2φeq
dx2− ∂f0∂φeq
:
0− ∂ε∼
e
∂φeq: C≈
: ε∼e = 0
Khachaturyan voigt/Taylor Sachs/Reuss
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= ε∼
?α − ε∼
?β = cste
∂ε∼e
∂φ= 0
σ∼ = σ∼α = σ∼βC≈ α
: ε∼eα = C
≈ β: ε∼
eβ
=⇒ ε∼eα = ε∼
eβ =⇒ ∂ε∼
e
∂φ= 0
Conclusions and Future Work 57/56
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91 / 97 Element residual vectorElement residual vector:
Re(u, c, φ) =
ReuRecReφ
Reu, Rec(c, φ) and Reφ(φ) are respectively the element residuals for the variationalformulation of classical mechanics, diffusion and phase field, defined as follow:
(Reu)i =
∫V e
(−[B∼e]ijσj + [Ne]ij fj
)dv + [Ne]ij tjds
(Rec)i =
∫V e
(Nei N
ej c
ej − [Be]ij Jj
)dv +
∫∂V e
Nei j ds
(Reφ)i =
∫V e
(Nei π(φ) − [Be]ij ξj
)dv +
∫∂V e
Nei ζ ds
The global residual vector can be obtained by assembling the element residuals for allfinite elements using the matrix assembly [Ae]:
R(φ) =N∑e=1
[Ae] . Re(φ) = 0
Conclusions and Future Work 58/56
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92 / 97 Variational formulations
Variational formulations:
=(u ?) =∫V−σ∼ : gradu ? dV+
∫Vf .u ? dV+
∫∂Vt .u ? dS = 0
=(c?) =∫Vcc? dV−
∫VJ .∇c? dV+
∫∂Vjc? dS = 0
=(φ?) =∫Vπφ? dV−
∫Vξ .∇φ? dV+
∫∂Vζφ? dS = 0
Conclusions and Future Work 59/56
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93 / 97 Stiffness matrixStiffness matrix
[Ket ] =
[Ke
t11] [0] [Ket13]
[0] [Ket22] [Ke
t23]
[Ket13] [Ke
t32] [Ket33]
et [δ] =
ucφ
(Ket11)ij =
∂(Re1)i∂uej
= −∫Ve
[∂σ∼∂u e
]ik
.[B∼e]kjdV
(Ket13)ij =
∂(Re1)i∂φej
= −∫Ve
[∂σ∼∂φe
]ik
.[B∼e]kjdV
(Ket31)ij =
∂(Re3)i∂uej
=∫Ve
[Ne]ik .[∂π
∂ue
]ik
dV
(Ket22)ij =
∂(Re2)i∂cej
=1
∆t
∫Ve
Nei .N
ej dV −
∫Ve
[Be]ik .[∂J
∂ce
]kj
dV
Conclusions and Future Work 60/56
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94 / 97 Stiffness matrixStiffness matrix
[Ket ] =
[Ke
t11] [0] [Ket13]
[0] [Ket22] [Ke
t23]
[Ket13] [Ke
t32] [Ket33]
et [δ] =
ucφ
(Ket23)ij =
∂(Re2)i∂φej
= −∫Ve
[Be]ik .[∂J
∂φe
]kj
dV
(Ket32)ij =
∂(Re3)i∂cej
=∫Ve
Nei .(
∂π
∂ce)j dV
(Ket33)ij =
∂(Re3)i∂φej
=∫Ve
Nei .(
∂π
∂φe)j dV −
∫Ve
[Be]ik .[∂ξ
∂φe
]kj
dV
Conclusions and Future Work 61/56
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95 / 97 Principle of Virtual PowerTwo virtual fields: order parameter φ? and displacement u?
Virtual power of internal generalized forces :
P(i)(u ?, φ?,V) = −∫V
(σ∼ : ∇u ? + ξ .∇φ? − πφ?) dV
Virtual power of external forces :
1. Virtual power of long range volume forces :
P(e)(u ?, φ?,V) =∫V
(f .u ? + γφ? + γ .∇φ?) dV
volumic density of force f .
2. Virtual power of generalized contact forces:
P(c)(u ?, φ?,V) =∫∂V
(t .u ? + ζφ?) dS
where t is a surface density of cohesion forces and ζ is asurface density of microtraction.
Conclusions and Future Work 62/56
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96 / 97 Principle of Virtual PowerPrinciple of virtual power P.P.V. :
∀u ?, ∀φ?, ∀D ⊂ V
P(i)(u ?, φ?,D) + P(c)(u ?, φ?,D) + P(e)(u ?, φ?,D) + P(a)(u ?, φ?,D) = 0
=(u ?, φ?) =
∫V
(∇.σ∼ + f ).u ? dV +
∫∂V
(−σ∼.n + t ).u ? dS
+
∫V
(∇.ξ + π)φ? dV +
∫∂V
(−ξ .n + ζ)φ? dS = 0
Equations of the phase-field/diffusion/mechanical model:
Local static equilibrium:∇.σ∼ + f = 0
σ∼.n = t
Balance of generalized stresses:∇.ξ + π = 0
ξ .n = ζ
Balance of mass: ∇.J + c = 0J .n = j
where J is the diffusion flux
Conclusions and Future Work 63/56
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97 / 97 Elastoplastic/Phase-Field/Diffusion modelState laws [Ammar et al., 2009] Complementary laws
σ∼ =∂f
∂ε∼ε∼∼p =
∂Ω∂σ∼
Ak =∂f
∂αkαk = − ∂Ω
∂Ak
ξ (φ) =∂f
∂∇φπdis = π +
∂f
∂φ
µ =∂f
∂c=∂f0∂c
J = −k∇µ = −k∇∂f0∂c
Ak are the thermodynamic forces associated with the set of internalvariables αk.
µ is the diffusion potential.
Ω is the dissipation potential.
ε∼p is the plastic strain.
πdis is the chemical force associated with the dissipative processes.
Conclusions and Future Work 64/56