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Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01,...

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Modelling Ferroelectric and Ferroelastic Hysteresis: Thermodynamical Consistency by Hysteresis Potentials Barbara Kaltenbacher, Alpen-Adria-Universit¨ at Klagenfurt joint work with Thomas Hegewald, University of Erlangen Manfred Kaltenbacher, Technical University of Vienna Pavel Krejˇ ı, Czech Academy of Scieces Berlin, April 2014 1
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Page 1: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Modelling Ferroelectric and Ferroelastic Hysteresis:Thermodynamical Consistency

by Hysteresis Potentials

Barbara Kaltenbacher, Alpen-Adria-Universitat Klagenfurt

joint work withThomas Hegewald, University of Erlangen

Manfred Kaltenbacher, Technical University of Vienna

Pavel Krejcı, Czech Academy of Scieces

Berlin, April 2014

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Page 2: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Overview

motivation

Preisach operators

a phenomenological model using Preisach operators

thermodynamic consistency and hysteresis potentials

a thermodynamically consistent material law forferroelectricity and ferroelasticity

well-posedness

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Page 3: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Piezoelectric Transducers

Direct effect: apply mechanical force → measure electric voltage

Indirect effect: impress electric voltage → observe mechanical displacement

Application Areas:

Ultrasound (medical imaging & therapy)

Force- and acceleration Sensors

Actor injection valves (common–rail Dieselengines)

SAW (surface-acoustic-wave) sensors

. . .

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Page 4: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Hysteresis in Piezoelectricity

e.g. ferroelectric hysteresis:dielectric displacement and mechanical strainat high electric field intensities (E ∼ 2MV /m):

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Page 5: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Piezoelectricity and Ferroelectricity on Unit Cell Level

Unit cell of BaTiO3 above (left) and below (right) Curie temperature Tc,

the latter exhibiting sponaneos polarization and strain

courtesy to M.Kamlah

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Page 6: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Grain and Domain Structure

Grains with same unit cell orientation

domains with same polarization direction

courtesy to M.Kamlah

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Page 7: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Poling Process

Orientation of the total polarization of the grains at initial state (left),due to a strong external electric field (middle)

and after switching it off, leading to a remanent polarization and strain

(right)

courtesy to M.Kamlah

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Page 8: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Grains

left: Nb doped PZT bulk grain structure (courtesy to CeramTech)right: Sputter deposited ZnO thin films with columnar grains

[Damjanovic 1998]

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Page 9: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Domains

Domain structure of single PZT grain with 90 degree and 180degree domain walls (courtesy to G.Schneider, TU Hamburg)

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Page 10: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Ferroelectricity

polarization hysteresis butterfly hysteresis

courtesy to M.Kamlah

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Page 11: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Ferroelasticity

stress-strain relation mechanical depolarization

courtesy to M.Kamlah

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Page 12: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Piezoelectricity – Ferroelectricity – Ferroelasticity

Piezoelectricity. . . linear coupling between electric and mechanical fields(reversible)

Ferroelectricity. . . external electric field influences polarization(irreversible, hysteretic)

Ferroelasticity. . . external mechanical field influences polarization(irreversible, hysteretic)

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Page 13: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Models of Ferroelectricity/Ferroelasticity1 Thermodynamically consistent models

macroscopic view, 2nd law of thermodynamicsBassiouny&Ghaleb’89, Kamlah&Bohle’01, Landis’04,Schroder&Romanowski’05, Su&Landis’07,Linnemann&Klinkel&Wagner’09, Mielke&Timofte’06,Alber&Kraynyukova’12

2 Micromechanical modelsconsider material on level of single grainsHuber&Fleck’01, Frohlich’01, Delibas&Arockiarajan&Seemann’05,Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

3 Phase field modelstransition between phases (domain wall motion)Wang&Kamlah&Zhang’10,Xu&Schrade&Muller&Gross&Granzow&Rodel’10,

4 Multiscale modelsSchroder&Keip’10, Zah&Liefer&Rosato&Miehe’10

5 Phenomenological models using hysteresis operatorsfrom input-output description for control purposesHughes&Wen’95, Kuhnen’01, Cimaa&Laboure&Muralt’02,Smith&Seelecke&Ounaies&Smith’03, Pasco&Berry04,Kuhnen&Krejcı’07, Ball&Smith&Kim&Seelecke’07,Hegewald&BK&MK&Lerch’08,’09

13

Page 14: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Models of Ferroelectricity/Ferroelasticity

1 Thermodynamically consistent modelsBassiouny&Ghaleb’89, Kamlah&Bohle’01, Landis’04,Schroder&Romanowski’05, Su&Landis’07,Linnemann&Klinkel&Wagner’09, Mielke&Timofte’06,Alber&Kraynyukova’12

2 Micromechanical modelsHuber&Fleck’01, Frohlich’01, Delibas&Arockiarajan&Seemann’05,Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

3 Phase field modelsWang&Kamlah&Zhang’10,Xu&Schrade&Muller&Gross&Granzow&Rodel’10,

4 Multiscale modelsSchroder&Keip’10, Zah&Liefer&Rosato&Miehe’10

5 Phenomenological models using hysteresis operatorsHughes&Wen’95, Kuhnen’01, Cimaa&Laboure&Muralt’02,Smith&Seelecke&Ounaies&Smith’03, Pasco&Berry04,Kuhnen&Krejcı’07, Ball&Smith&Kim&Seelecke’07,Hegewald&BK&MK&Lerch’08,’09

14

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Preisach Operators

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Hysteresis and Hysteresis Operators

input:

→t

output:

→t

↑ output

→ input

magnetics

piezoelectricity

plasticity

. . .

∗ memory

∗ Volterraproperty

∗ rateindependence

Krasnoselksii-Pokrovskii (1983), Mayergoyz (1991), Visintin (1994),

Krejcı (1996), Brokate-Sprekels (1996)

16

Page 17: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Simple Example I: The Relay

Rβ,α[v](t) = w(t)

=

+1 if v(t) > α or (w(ti ) = +1 ∧ v(t) > β)−1 if v(t) < β or (w(ti ) = −1 ∧ v(t) < α)

t ∈ [ti , ti+1]

t0, t1, t2, . . . sequence of local extrema of v,i.e., v monotone on [ti , ti+1].

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A Simple Example II: The Mechanical Play

Fr[v](t) = w(t) = maxv(t)−r,minv(t)+r,w(ti ) t ∈ [ti , ti+1]

Relation to Relay operator: Fr[v](t) = 12

∫∞−∞Rs−r,s+r[v](t) ds

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A Simple Example III: The Elastic-Plastic Element

v∼ strain

w∼ stress

r∼ yield stress

Sr[v](t) = w(t) = minr,max−r, v(t) t ∈ [ti , ti+1]

Relation to mechanical Play: Sr[v](t) = v(t)−Fr[v](t)

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Page 20: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A General Hysteresis Model: the Preisach Operator

weighted superposition of elementary relayswith Preisach weight function ω defined onPreisach plane S = S+ ∪ S−:

S−

S+

Pω[v](t) =

∫∫α,β∈S

ω(β, α)Rβ,α[v](t) d(α, β)

=

∫∫α,β∈S+(t)

ω(β, α) d(α, β)−∫∫

α,β∈S−(t)ω(β, α) d(α, β)

± high dimensionality

+ can model minor loops

+ can model saturation

+ highly efficient evaluation minor loop

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Page 21: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A General Hysteresis Model: the Preisach Operator

weighted superposition of elementary relayswith Preisach weight function ω defined onPreisach plane S = S+ ∪ S−:

S−

S+

Pω[v](t) =

∫∫α,β∈S

ω(β, α)Rβ,α[v](t) d(α, β)

=

∫∫α,β∈S+(t)

ω(β, α) d(α, β)−∫∫

α,β∈S−(t)ω(β, α) d(α, β)

± high dimensionality

+ can model minor loops

+ can model saturation

+ highly efficient evaluation

minor loop

20

Page 22: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A General Hysteresis Model: the Preisach Operator

weighted superposition of elementary relayswith Preisach weight function ω defined onPreisach plane S = S+ ∪ S−:

S−

S+

Pω[v](t) =

∫∫α,β∈S

ω(β, α)Rβ,α[v](t) d(α, β)

=

∫∫α,β∈S+(t)

ω(β, α) d(α, β)−∫∫

α,β∈S−(t)ω(β, α) d(α, β)

± high dimensionality

+ can model minor loops

+ can model saturation

+ highly efficient evaluation minor loop

20

Page 23: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Regularity and Monotonicity Properties of PreisachOperators

Lipschitz continuity on C [0,T ] if

∫ ∞0

sups∈R|ω(s−r , s +r)| dr <∞

Coercivity: P[v ]t(t)vt(t) ≥ µ|vt(t)|2 if 0 < µ ≤ ∂2E

Convexity: ∂∂t (vtP[v ]t) (t) ≤ 2vtt(t)P[v ]t(t) if sign(v∗−v∗)∂

22E(v∗, v

∗) ≤ 0

E(v∗, v∗) =

2∫∫

v∗≤β≤α≤v∗ ω(β, α) d(α, β) if v∗ ≤ v∗

E(−v∗,−v∗) = E(v∗, v∗) if v∗ > v∗

However. . .

P is not differentiable in a classical sense1

P is not a monotone operator

1[Brokate&Krejcı’14] generalized differentiability21

Page 24: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Phenomenological Model for Hysteresis inPiezoelectricity

S = S r + S i

~D = ~Dr + ~D i

S r = sEσ + dT~P~E

~Dr = d~Pσ + εσ~E

~D i = ~P = Pω[E ]e~P

S i = fS(~P)( 32 e~PeT~P

− 12 I )

d~P = fd(P) = PPsat

d

S r . . . reversible strain~Dr . . . reversible polarization

S i . . . irreversible strain~D i . . . irreversible polarization

σ. . . mech. stress~E . . . electr. field~P. . . polarization

sE . . . elast. coeff.

εσ . . . dielectr. coeff.

d . . . coupling coeff.

hysteresis identification: [Hegewald&B.K&M.K.&Lerch, J.Int.Mat.Sys.Struct.’08]

finite element formulation: [M.K&B.K.&Hegewald&Lerch J.Int.Mat.Sys.Struct.’09]

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Page 25: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Phenomenological Model for Hysteresis inPiezoelectricity

S = S r + S i

~D = ~Dr + ~D i

S r = sEσ + dT~P~E

~Dr = d~Pσ + εσ~E

~D i = ~P = Pω[E ]e~P

S i = fS(~P)( 32 e~PeT~P

− 12 I )

d~P = fd(P) = PPsat

d

S r . . . reversible strain~Dr . . . reversible polarization

S i . . . irreversible strain~D i . . . irreversible polarization

σ. . . mech. stress~E . . . electr. field~P. . . polarization

sE . . . elast. coeff.

εσ . . . dielectr. coeff.

d . . . coupling coeff.

good qualitative and quantitative reproduction of measurementshighly efficient in FEM simulations

. . . however, thermodynamic consistencycannot be shown for this model

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Page 26: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Phenomenological Model for Hysteresis inPiezoelectricity

S = S r + S i

~D = ~Dr + ~D i

S r = sEσ + dT~P~E

~Dr = d~Pσ + εσ~E

~D i = ~P = Pω[E ]e~P

S i = fS(~P)( 32 e~PeT~P

− 12 I )

d~P = fd(P) = PPsat

d

S r . . . reversible strain~Dr . . . reversible polarization

S i . . . irreversible strain~D i . . . irreversible polarization

σ. . . mech. stress~E . . . electr. field~P. . . polarization

sE . . . elast. coeff.

εσ . . . dielectr. coeff.

d . . . coupling coeff.

good qualitative and quantitative reproduction of measurementshighly efficient in FEM simulations

. . . however, thermodynamic consistencycannot be shown for this model

23

Page 27: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Bending Actuator

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Page 28: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Bending Actuator

Identified Preisach weight function:

25

Page 29: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Bending Actuator

comparison measurement – simulation with fitted Preisach operators:

26

Page 30: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Bending Actuator

comparison measurement – simulation for alternative input signal:

27

Page 31: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Revolving Drive

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Page 32: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulation Results: Revolving Drive

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Page 33: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Thermodynamic Consistency and

Hysteresis Potentials

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Page 34: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Thermodynamic Consistencyσ . . . mechanical stressε . . . mechanical strain~E . . . electic field~D. . . dielectic displacementW . . . work done by electric and mechanical forcesF . . . free energyD. . . energy dissipation

W (t1, t2) =

∫ t2

t1

ε(t) : σ(t) + ~D(t) · ~E (t) dt

2nd law of thermodynamics:

D(t1, t2) = W (t1, t2)− (F (t2)− F (t1)) ≥ 0

differential form:

ε : σ + ~D(t) · ~E − F ≥ 0

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Hysteresis Potentials

clockwise:

P[q](t)q(t)− ˙U [q](t) ≥ 0 a.e. ∀q ∈W 1,1(0,T )

counterclockwise:

q(t) ˙P[q](t)− ˙U [q](t) ≥ 0 a.e. ∀q ∈W 1,1(0,T )

q. . . internal stateP. . . hysteresis operatorU . . . hysteresis potential

32

Page 36: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Hysteresis Potentials for Preisach Operators

Theorem (e.g., [Brokate&Sprekels, 1996], [Krejcı, 1996])

P[v](t) = Pω[v](t) =

∫∫α,β∈S

ω(β, α)Rβ,α[v](t) d(α, β)

U [v](t) = 2

∫∫α,β∈S

(α + β)ω(β, α)Rβ,α[v](t) d(α, β)

with ω ≥ 0.Then P is a counterclockwise piecewise convex hysteresis operatorand U is a counterclockwise hysteresis potential for P.

. . . clockwise case via inverse of P.

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Page 37: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Hysteresis Potentials for Preisach Operators

Theorem (e.g., [Brokate&Sprekels, 1996], [Krejcı, 1996])

P[v](t) = Pω[v](t) =

∫∫α,β∈S

ω(β, α)Rβ,α[v](t) d(α, β)

U [v](t) = 2

∫∫α,β∈S

(α + β)ω(β, α)Rβ,α[v](t) d(α, β)

with ω ≥ 0.Then P is a counterclockwise piecewise convex hysteresis operatorand U is a counterclockwise hysteresis potential for P.

. . . clockwise case via inverse of P.

33

Page 38: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Thermodynamically Consistent Material Law

for Ferroelectricity and Ferroelasticity

34

Page 39: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Thermodynamically Consistent Material Lawsee [Davino&Krejcı&Visone’13] for magnetostriction

Ansatz:

σ = cε− eE + aP[q] + bU [q]

D = eε+ κE + cP[q] + dU [q]

F = c2ε

2 + κ2 E 2 + ξP[q] + ηU [q]

q = q(ε,E ) . . . internal variablea = a(ε,E ), b = b(ε,E ), c = c(ε,E ), d = d(ε,E ), ξ = ξ(ε,E ),η = η(ε,E ) . . . coefficient functionsU counterclockwise hysteresis potential for P

thermodynamically consistent choice (X , Y . . . Gibbs potentials):

a =∂X

∂εb =

∂Y

∂εc =− ∂X

∂Ed =− ∂Y

∂Eξ =X + cE η =Y + dE

q =− X

Y0 <Y

35

Page 40: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

A Thermodynamically Consistent Material Law

see [Davino&Krejcı&Visone’13] for magnetostriction

Ansatz:

σ = cε− eE +aP[q] + bU [q]

D = eε+ κE +cP[q] + dU [q]

F = c2ε

2 + κ2 E 2+ξP[q] + ηU [q]

q = q(ε,E ) . . . internal variablea = a(ε,E ), b = b(ε,E ), c = c(ε,E ), d = d(ε,E ), ξ = ξ(ε,E ),η = η(ε,E ) . . . coefficient functionsU counterclockwise hysteresis potential for P

linear part: piezoelectric couplingnonlinear hysteretic part: ferroelectricity and ferroelasticity

36

Page 41: Modelling Ferroelectric and Ferroelastic Hysteresis ... · Huber&Fleck’01, Fr ohlich’01, Delibas&Arockiarajan&Seemann’05, Belov&Kreher’06, Huber’06, McMeeking&Landis&Jimeneza’07

Simulations for some Simple Test Case

Ansatz:

σ = cε− eE + aP[q] + bU [q]

D = eε+ κE + cP[q] + dU [q]

F = c2ε

2 + κ2 E 2 + ξP[q] + ηU [q]

q . . . internal variableU counterclockwise hysteresis potential for P

thermodynamically consistent choice (X = −EY (ε), Y = Y (ε)):

a =− EY ′(ε) b =Y ′(ε) c =Y (ε) d =0

ξ =0 η =Y (ε)

q =E 0 <Yσ = cε− eE − Y ′(ε)(EP[E ]− U [E ])

D = eε+ κE + Y (ε)P[E ]

F = c2ε

2 + κ2 E 2 + Y (ε)U [E ]

37

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Simulations for some Simple Test Case

Ansatz:

σ = cε− eE + aP[q] + bU [q]

D = eε+ κE + cP[q] + dU [q]

F = c2ε

2 + κ2 E 2 + ξP[q] + ηU [q]

q . . . internal variableU counterclockwise hysteresis potential for P

thermodynamically consistent choice (X = −EY (ε), Y = Y (ε)):

a =− EY ′(ε) b =Y ′(ε) c =Y (ε) d =0

ξ =0 η =Y (ε)

q =E 0 <Yσ = cε− eE − Y ′(ε)(EP[E ]− U [E ])

D = eε+ κE + Y (ε)P[E ]

F = c2ε

2 + κ2 E 2 + Y (ε)U [E ]

37

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A Simple Test (I)

Preisach weight function:

38

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A Simple Test (I): Ferroelectricity

39

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A Simple Test (I): Ferroelectricity

40

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A Simple Test (I): Ferroelectricity

41

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A Simple Test (I): Ferroelectricity

42

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A Simple Test (I): Ferroelectricity and Ferroelasticity

43

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A Simple Test (II)

Preisach weight function:

44

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A Simple Test (II): Ferroelectricity

45

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A Simple Test (II): Ferroelectricity

46

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A Simple Test (II): Ferroelectricity

47

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A Simple Test (II): Ferroelectricity

48

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A Simple Test (II): Ferroelectricity and Ferroelasticity

49

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The Full PDE Model

ρd −∇Ts σ =0 Newton’s law

−∇D =0 Gauß’s law

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd d . . . mechanical displacement

E =∇φ − φ. . . electric potential ,

boundary conditions:

n · σ =fm on ∂Ω

φ =φe on Γ

n · D =fe on ∂Ω \ Γ .

50

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The Full PDE Model

ρd −∇Ts σ =0 Newton’s law

−∇D =0 Gauß’s law

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd d . . . mechanical displacement

E =∇φ − φ. . . electric potential ,

boundary conditions:

n · σ =fm on ∂Ω

φ =φe on Γ

n · D =fe on ∂Ω \ Γ .

50

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The Full PDE Model

ρd −∇Ts σ =0 Newton’s law

−∇D =0 Gauß’s law

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd d . . . mechanical displacement

E =∇φ − φ. . . electric potential ,

boundary conditions:

n · σ =fm on ∂Ω

φ =φe on Γ

n · D =fe on ∂Ω \ Γ .

50

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The Full PDE Model

ρd −∇Ts σ =0 Newton’s law

−∇D =0 Gauß’s law

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd d . . . mechanical displacement

E =∇φ − φ. . . electric potential ,

boundary conditions:

n · σ =fm on ∂Ω

φ =φe on Γ

n · D =fe on ∂Ω \ Γ .

50

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Well-posedness

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Well-posedness in the Context of Hysteresis Operators

challenges:

only very weak regularity properties

no monotonicity

no compactness

52

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Well-posedness via Thermodynamic Consistency?

ρd −∇Ts σ = 0 (Newton) (1)

−∇D = 0 ( ddt Gauss) (2)

Test (1) with d and (2) with φ:∫Ω

1

2

d

dt

(ρd2)

+ σ : ε+ D · E

dx =

∫∂Ω

fmd ds +

∫Γ

n · Dφ0 ds +

∫∂Ω\Γ

feφ ds

=: 〈f , (d , φ)〉V ∗,VFrom thermodynamic consistency

ε : σ + D(t) · E − F ≥ 0

and our ansatz for the free energy

F =c

2ε2 +

κ

2E 2 + ξP[q] + ηU [q]

we get:

d

dt

1

2|√ρd2|2H +

1

2|√cε|2H +

1

2|√κE |2H︸ ︷︷ ︸

=:E(t)

+ξP[q] + ηU [q]

≤ 〈f , (d , φ)〉V ∗,V

i.e., setting ξ = 0 and η ≥ 0 and U ≥ 0 decrease of energy E(t)

53

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Well-posedness via Thermodynamic Consistency?

ρd −∇Ts σ = 0 (Newton) (1)

−∇D = 0 ( ddt Gauss) (2)

Test (1) with d and (2) with φ:∫Ω

1

2

d

dt

(ρd2)

+ σ : ε+ D · E

dx =

∫∂Ω

fmd ds +

∫Γ

n · Dφ0 ds +

∫∂Ω\Γ

feφ ds

=: 〈f , (d , φ)〉V ∗,VFrom thermodynamic consistency

ε : σ + D(t) · E − F ≥ 0

and our ansatz for the free energy

F =c

2ε2 +

κ

2E 2 + ξP[q] + ηU [q]

we get:

d

dt

1

2|√ρd2|2H +

1

2|√cε|2H +

1

2|√κE |2H︸ ︷︷ ︸

=:E(t)

+ξP[q] + ηU [q]

≤ 〈f , (d , φ)〉V ∗,V

i.e., setting ξ = 0 and η ≥ 0 and U ≥ 0 decrease of energy E(t)

53

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∫Ω

1

2

d

dt

(ρd2)

+ σ : ε+ D · E

dx =

∫∂Ω

fmd ds +

∫Γ

n · Dφ0 ds +

∫∂Ω\Γ

feφ ds

=: 〈f , (d , φ)〉V ∗,VFrom thermodynamic consistency

ε : σ + D(t) · E − F ≥ 0

and our ansatz for the free energy

F =c

2ε2 +

κ

2E 2 + ξP[q] + ηU [q]

we get:

d

dt

1

2|√ρd2|2H +

1

2|√cε|2H +

1

2|√κE |2H︸ ︷︷ ︸

=:E(t)

+ξP[q] + ηU [q]

≤ 〈f , (d , φ)〉V ∗,V

i.e., setting ξ = 0 and η ≥ 0 and U ≥ 0 estimate of energy E(t) (decreasing for f = 0)

54

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d

dt

1

2|√ρd2|2H +

1

2|√cε|2H +

1

2|√κE |2H︸ ︷︷ ︸

=:E(t)

+ξP[q] + ηU [q]

≤ 〈f , (d , φ)〉V ∗,V

i.e., setting ξ = 0 and η ≥ 0 and U ≥ 0 estimate of energy E(t) (decreasing for f = 0)

However, time discretization and passage to the limit not possible:Hysteresis operators need C [0,T ] convergence of input q = q(ε,E ),but energy estimate only yields C [0,T ] boundedness of input q. . . lack of compactness

55

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d

dt

1

2|√ρd2|2H +

1

2|√cε|2H +

1

2|√κE |2H︸ ︷︷ ︸

=:E(t)

+ξP[q] + ηU [q]

≤ 〈f , (d , φ)〉V ∗,V

i.e., setting ξ = 0 and η ≥ 0 and U ≥ 0 estimate of energy E(t) (decreasing for f = 0)

However, time discretization and passage to the limit not possible:Hysteresis operators need C [0,T ] convergence of input q = q(ε,E ),but energy estimate only yields C [0,T ] boundedness of input q. . . lack of compactness

55

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Well-posedness via Mechanical Damping

ρd −∇Ts (k ε+ σ) =0

−∇D =0

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd

E =∇φ ,

weak form:

〈Λu,w〉V ∗,V + (AJu + ΠJu + Φ[Ju], Jw)H = 〈f ,w〉V ∗,V ∀w ∈ V

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Well-posedness via Mechanical Damping

ρd −∇Ts (k ε+ σ) =0

−∇D =0

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd

E =∇φ ,

weak form:

〈Λu,w〉V ∗,V + (AJu + ΠJu + Φ[Ju], Jw)H = 〈f ,w〉V ∗,V ∀w ∈ V

strong damping nonlinearity

57

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with

u = (d , φ) , w = (v , ψ) ,

V = H1(Ω)d+1 , H = L2(Ω)d(d+1)

2+d ,

Λ : V ∗ → V ∗ , Λ(d , φ) = (ρd , 0) ,

A : H → H , A(ε,E ) = (kε, 0)

J : V → H , J(d , φ) = (∇sd ,∇φ) ,

Π : H → H , Π(ε,E ) = (cε− eE , eT ε+ κE ) ,

Φ : C (0,T ; H)→ C (0,T ; H) ,

Φ[(ε,E )] = ((aP[q] + bU [q])M, (cP[q] + dU [q])p)

〈f (t),w〉V ∗,V =

∫∂Ω

(fm(t)v + fe(t)ψ) ds .

58

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An Abstract Well-posedness Result

〈Λu,w〉V ∗,V + (AJu + ΠJu + Φ[Ju], Jw)H = 〈f ,w〉V ∗,V ∀w ∈ V

assumptions:

(A1)

V → H → V ∗

Λ ∈ L(V ∗,V ∗) , 〈Λv , v〉V ∗,V = 12

ddtλ(v) for some continuous function

λ : D(λ) ⊆ V ∗ → R+0 , λ(0) = 0

A ∈ L(H,H) , (Ah, h)H ≥ α|h|2H for some α > 0

Π ∈ L(H,H) , (Πh, h)H = ddt

12π(h) + θ(h, h) for some continuous fctns

π ∈ H → R+0 , θ ∈ H → R , θ(h, h) ≤ Lθ

√π(h)‖h‖H , π(0) = 0

|Φ[h](t)− Φ[h](t)|H ≤ LΦ

(∫ t

0|h(τ)− ˙h(τ)|2H dτ + |h(0)− h(0)|2H

) 12

J ∈ L(V ,H)

59

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An Abstract Well-posedness Result

〈Λu,w〉V ∗,V + (AJu + ΠJu + Φ[Ju], Jw)H = 〈f ,w〉V ∗,V ∀w ∈ V(3)

Theorem

Under conditions (A1), for any T > 0, initial conditionsJu(0) ∈ H, u(0) ∈ D(λ), and right hand side f ∈ L2(0,T ; V ∗)such that the linearization of (3) at 0 has a solution ulin ∈W =v ∈ L2(0,T ; V ∗) : λ(v), π(Jv) ∈ C [0,T ] , Jv ∈ L2(0,T ; H),also the nonlinear equation (3) has a solution u ∈W , and thesolution is unique in a sufficiently small neighborhood of ulin.

Idea of proof: Banach’s Fixed Point Theorem

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Application to the Piezoelectric Model

ρd −∇Ts (k ε+ σ) =0

−∇D =0

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd

E =∇φ ,

(4)

Assume Lipschitz continuity pf Φ

|Φ[h](t)− Φ[h](t)|H ≤ LΦ

(∫ t

0|h(τ)− ˙h(τ)|2H dτ + |h(0)− h(0)|2H

) 12

(5)in

H = L2(Ω)d(d+1)

2+d

typically satisfied for Preisach hysteresis operators andpotentials cf. [Brokate&Sprekels’96,Krejcı’96].

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Application to the Piezoelectric Model

Corollary

Let ρ, k , l > 0, c , κ be positive definite tensors, and let

Φ : C (0,T ; L2(Ω)d(d+1)

2+d)→ C (0,T ; L2(Ω)

d(d+1)2

+d)

Φ[(ε,E )] = ((aP[q] + bU [q])M, (cP[q] + dU [q])p)

satisfy the Lipschitz condition (5).Then for any T > 0, initial conditions(d0, φ0) = (d(0), φ(0)) ∈ H1(Ω)d+1, d1 = d(0) ∈ L2(Ω)d, andboundary conditions (fm, fe) ∈ L2(0,T ; H−1/2(∂Ω)d+1), (4) has aweak solution (d , φ) ∈ H1([0,T ]; H1(Ω)d+1) withd ∈ C ([0,T ]; L2(Ω)d), and the solution is unique in a sufficientlysmall neighborhood of (dlin, φlin) solving the linearization of (4) at0 with initial conditions d0, φ0, d1.

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Well-posedness via Mechanical Viscosity

ρd −∇Ts (k ε+ σ) =0

−∇D =0

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd

E =∇φ

(6)

Corollary

Existence, uniqueness and stability of (d , φ) ∈ C ([0,T ]; H1(Ω)d+1)with d ∈ C ([0,T ]; L2(Ω)d) ∩ L2(0,T ; H1(Ω)d)

Idea of proof:elimination of φ via 2nd, 4th, and 6th equation in (6).

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Well-posedness via Mechanical Viscosity

ρd −∇Ts (k ε+ σ) =0

−∇D =0

σ =cε− eE + (aP[q] + bU [q])M

D =eT ε+ κE + (cP[q] + dU [q])p

ε =∇sd

E =∇φ

Corollary

Existence, uniqueness and stability of (d , φ) ∈ C ([0,T ]; H1(Ω)d+1)with d ∈ C ([0,T ]; L2(Ω)d) ∩ L2(0,T ; H1(Ω)d)

Idea of proof:elimination of φ via 2nd, 4th, and 6th equation.

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10th International Piezo WorkshopSept. 21-24, 2014

Vienna University of Technology

theory

numerical simulation (direct and inverse analysis)

advanced material characterization

fatigue, damage, cracks and reliability

manufacturing processes

performance of piezoelectric transducers

http://www.mec.tuwien.ac.at/piezo2014/piezoworkshop_2014/

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Thank you for your attention!

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