1.3cm Numerical Modelling of Phase Transformation...

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

2 / 97 Why ?

Phase transformation-mechanics coupling:

Different morphological evolutions

Base Ni Co-Pt [Le Bouar] Cu-Al-Ni

Different kinetics

Different behaviours (TRIP)

2/56

3 / 97 Why ?

Phase transformation-mechanics coupling:

Different morphological evolutions

Ni-Al[Diologent,2004] Cu-Cd [Sullonen]

Different kinetics

Different behaviours (TRIP)

2/56

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]

3/56

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].

4/56

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

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

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

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.


Plan







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



F (φ,∇φ, c) =∫V


[fch(φ, c) +

α

2|∇φ|2

]dv


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



F (φ,∇φ, c) =∫V


[fch(φ, c) +

α

2|∇φ|2

]dv


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



F (φ,∇φ, c) =∫V


[fch(φ, c) +

α

2|∇φ|2

]dv


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


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


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)


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




∇.

(∂f

∂ε∼e

)= 0

Balance of mass:

c−∇.

[L(φ)

(∇∂f

∂c

)]= 0


−βφ− ∂f

∂φ+ ∇.

∂f

∂∇φ= 0


σ∼ =∂f

∂ε∼e

J = −L(φ)

(∇∂f

∂c

)π = −βφ− ∂f

∂φ

ξ =∂f

∂∇φ= α∇φ




∇. σ∼ = 0

Balance of mass:

c−∇.

[L(φ)

(∇∂f

∂c

)]= 0


−βφ− ∂f

∂φ+ ∇.

∂f

∂∇φ= 0


σ∼ =∂f

∂ε∼e

J = −L(φ)

(∇∂f

∂c

)π = −βφ− ∂f

∂φ

ξ =∂f

∂∇φ= α∇φ




∇. σ∼ = 0

Balance of mass:c−∇. J = 0


−βφ− ∂f

∂φ+ ∇.

∂f

∂∇φ= 0


σ∼ =∂f

∂ε∼e

J = −L(φ)

(∇∂f

∂c

)

π = −βφ− ∂f

∂φ

ξ =∂f

∂∇φ= α∇φ




∇. σ∼ = 0

Balance of mass:c−∇. J = 0

Balance of generalized stresses (Gurtin’s balance of microforces):π + ∇. ξ = 0


σ∼ =∂f

∂ε∼e

J = −L(φ)

(∇∂f

∂c

)π = −βφ− ∂f

∂φ

ξ =∂f

∂∇φ= α∇φ


Plan







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



-u?, c?, φ?

-

- 6

??

?


c−∇.J = 0

π + ∇. ξ = 0






[K] ==

[Ke

t11] [

Ket12] [

Ket13]

[Ke

t21] [

Ket22] [

Ket23]

[Ke

t13] [

Ket32] [

Ket33]





-u?, c?, φ?

-

-

6

??

?


c−∇.J = 0

π + ∇. ξ = 0






[K] ==

[Ke

t11] [

Ket12] [

Ket13]

[Ke

t21] [

Ket22] [

Ket23]

[Ke

t13] [

Ket32] [

Ket33]





-u?, c?, φ?

-

- 6

??

?


c−∇.J = 0

π + ∇. ξ = 0






[K] ==

[Ke

t11] [

Ket12] [

Ket13]

[Ke

t21] [

Ket22] [

Ket23]

[Ke

t13] [

Ket32] [

Ket33]




33 / 97


Plan







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

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


Plan







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


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)


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


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



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



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


Plan







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


Plan







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)



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



0.3 0.4 0.5 0.6 0.7c0

0.0

0.5

1.0

1.5

2.0A

Williams point

β α


boundary

incoherent α field

boundary



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



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



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



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



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)



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


boundaryincoherent α field

boundary



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



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



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



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


Plan







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

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

66 / 97


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).


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



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%


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



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%


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.


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

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

Plan







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


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



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




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



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



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


Plan







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

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


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


84 / 97 Future WorkLong-term prospects

Inheritance of plastic deformation during migration of phaseboundaries.

Mesh sensitivity and adaptive mesh

[Abrivard, 2009]


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).


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.


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.


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



∇.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


= 0),


αd2φeq

dx2− ∂f0∂φeq

− ∂f

∂φeq= 0

αd2φeq

dx2− ∂f0∂φeq

− ∂ε∼e

∂φeq: C≈

: ε∼e = 0


∂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



∇.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


= 0),


αd2φeq

dx2− ∂f0∂φeq

>

0

− ∂f

∂φeq= 0

αd2φeq

dx2− ∂f0∂φeq

:

0− ∂ε∼

e

∂φeq: C≈

: ε∼e = 0


∂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


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


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


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


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


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.


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


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.


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