icmm tgzielinski piezoelectricity.slides

8/10/2019 ICMM TGZielinski Piezoelectricity.slides

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Introduction Equations of piezoelectricity Forms of constitutive law Comments and remarks

Fundamentals of PiezoelectricityIntroductory Course on Multiphysics Modelling

TOMASZ G. ZIELINSKI

Institute of Fundamental Technological ResearchWarsaw Poland

http://www.ippt.pan.pl/~tzielins/
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.ippt.pan.pl/~tzielins/http://www.ippt.pan.pl/~tzielins/mailto:[email protected]


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Outline

1

IntroductionThe piezoelectric effectSimple molecular model for explaining the piezoelectric effect


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Outline

1


2 Equations of piezoelectricityPiezoelectricity viewed as electromechanical coupling

Field equations of linear piezoelectricityBoundary conditionsFinal set of partial differential equations


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Outline

1




3 Forms of constitutive law

Forms of constitutive relationsTransformations for converting piezoelectric constitutive dataMatrix notation of piezoelectric constitutive relations

I t d ti E ti f i l t i it F f tit ti l C t d k


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Outline

1






4 Comments and remarks



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Outline

1









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Introduction: the piezoelectric effect

Observed phenomenon

Piezoelectricityis the ability of some materials to generate anelectric chargein response to applied mechanical stress. If thematerial is not short-circuited, the applied charge induces a voltageacross the material.



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Observed phenomenon


The piezoelectric effect is reversible, that is, all piezoelectric

materials exhibit:the direct piezoelectric effect the production of electricity when

stress is applied,the converse piezoelectric effect the production of stress and/or

strain when an electric field is applied.



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Observed phenomenon


the direct piezoelectric effect the production of electricity when

stress is applied,the converse piezoelectric effect the production of stress and/or

strain when an electric field is applied.

Some historical facts and etymology

The (direct) piezoelectric phenomenon was discovered in 1880 by thebrothers Pierre and Jacques Curie during experiments on quartz.

The existence of the reverse process was predicted by Lippmann

in 1881 and then immediately confirmed by the Curies.

The wordpiezoelectricitymeans electricity by pressure and is derived

from the Greekpiezein, which means to squeeze or press.



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q p y

Introduction: a simple molecular model

+

+

+

neutral molecule

Beforesubjecting the material to some

external stress:the centres of the negative andpositive charges of each moleculecoincide,

the external effects of the charges

are reciprocally cancelled,as a result, an electrically neutralmolecule appears.



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q p y


+

+

+

+

+

small dipole

Afterexerting some pressure on the

material:the internal structure is deformed,

that causes the separation of thepositive and negative centres of themolecules,

as a result, little dipoles aregenerated.



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+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Eventually:

the facing poles inside the materialare mutually cancelled,

a distribution of a linked chargeappears in the materials surfacesand the material is polarized,

the polarization generates an electricfield and can be used to transformthe mechanical energy of thematerials deformation into electricalenergy.



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Outline

1 IntroductionThe piezoelectric effectSimple molecular model for explaining the piezoelectric effect








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Equations of piezoelectricityPiezoelectricity viewed aselectromechanical coupling

Scalar, vector, and tensor quantities

ui[m]the mechanical displacements

V= J

C

the electric field potential

Sij

mm

the strain tensor

Ei

Vm =

NC

the electric field vector

Tij N

m2

the stress tensorDi

Cm2

the electric displacements

fi N

m3the mechanical body forces

q

Cm3

the electric body charge

kg

m3

the mass density

cijkl

Nm2

the elastic material constants

ij

Fm =

CV m

the dielectric constants

ELASTIC

DIELECTRIC

+



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Equations of piezoelectricityPiezoelectricity viewed aselectromechanical coupling



V= J

C


Sij

mm

the strain tensor

Ei

Vm =

NC


Tij N

m2

the stress tensorDi

Cm2


fi N


q

Cm3


kg

m3

the mass density

cijkl

Nm2


ekij C

m2

the piezoelectric constantsij

Fm =

CV m


ELASTIC

DIELECTRIC

+Piezoelectric Effects



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Equations of piezoelectricityField equations of linear piezoelectricity



V= J

C


Sij

mm

the strain tensor

Ei

Vm =

NC


Tij N

m2

the stress tensorDi

Cm2


fi N


q

Cm3


kg

m3

the mass density

cijkl

Nm2


ekij C

m2


Fm =

CV m


Equations of motion(Elastodynamics)

Tij|j+fi = ui

Gauss law(Electrostatics)

Di|iq= 0



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Equations of piezoelectricityField equations of linear piezoelectricity


ui

[m]the mechanical displacements

V= J

C


Sij

mm

the strain tensor

Ei

Vm =

NC


Tij N

m2

the stress tensorDi

Cm2


fi N


q

Cm3


kg

m3

the mass density

cijkl

Nm2


ekij C

m2


Fm =

CV m


Equations of motion(Elastodynamics)

Tij|j+fi = ui

Gauss law(Electrostatics)

Di|iq= 0

Kinematic relations(Mechanics)

Sij = 12 (ui|j+uj|i)

Maxwells law(Electricity)

Ei=

|i


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E ti f i l t i it


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Equations of piezoelectricityBoundary conditions


ui [m]the mechanical displacements

V= J

C


Sij

mm

the strain tensor

Ei

Vm =

NC


Tij

Nm2

the stress tensor

Di

Cm2


fi Nm

3 the mechanical body forcesq

Cm3


kg

m3

the mass density

cijkl

Nm2


ekij C

m2


Fm =

CV m


Boundary conditions are uncoupled

(essential) (natural)

mechanical : ui= ui or Tijnj= Fi

electric : = or Dini=Q

ui, the specified mechanical displacements [m]and electric potentialV

Fi, Q the specified surface forces

Nm2

and surface charge

Cm2

ni the outward unit normal vector components


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O li


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Outline

1 IntroductionThe piezoelectric effectSimple molecular model for explaining the piezoelectric effect







F f tit ti l ti


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Forms of constitutive relations

Stress

Nm2

Strain

mm

Charge

Cm2 T,D

e

cE=0, S=0 (S,E) S,D d

sE=0, T=0 (T,E) (voltage)

Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)


F f tit ti l ti


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Stress

Nm2

Strain

mm

Charge

Cm2 T,D

e

cE=0, S=0 (S,E) S,D d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)

Stress-Charge form:

T= cE=0: SeTE ,

D=e :S+S=0 E .


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Stress

Nm2

Strain

mm

Charge

Cm2 T,D

e

cE=0, S=0 (S,E) S,D d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)

Stress-Charge form:

T= cE=0: SeTE ,

D=e :S+S=0 E .

Stress-Voltage form:

T= cD=0: SqTD ,

E=q :S+1S=0 D .

Strain-Charge form:

S= sE=0: T+dTE ,

D=d :T+T=0 E .




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Stress

Nm2

Strain

mm

Charge

Cm2 T,D

e

cE=0, S=0 (S,E) S,D d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)

Stress-Charge form:

T= cE=0: SeTE ,

D=e :S+S=0 E .

Stress-Voltage form:

T= cD=0: SqTD ,

E=q :S+1S=0 D .

Strain-Charge form:

S= sE=0: T+dTE ,

D=d :T+T=0 E .

Strain-Voltage form:

S= sD=0: T+gTD ,

E=g :T+1T=0 D .


Forms of constitutive equations


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Stress

Nm2

Strain

mm

Charge C

m2 T,D

e

cE=0, S=0 (S,E) S,D d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)

Here, the followingtensors of constitutive coefficientsappear:

fourth-ordertensors ofelasticmaterial constants: c

Nm2

(stiffness coefficients) and s= c1 m2

N

(compliance coefficients)obtained in the absence of electric field (E=0) or charge (D=0);




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Stress

Nm2

Strain

mm

Charge C

m2 T,D

e

cE=0, S=0 (S,E) S,D d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)


fourth-ordertensors ofelasticmaterial constants: c

Nm2

(stiffness coefficients) and s= c1 m2

N

(compliance coefficients)obtained in the absence of electric field (E=0) or charge (D=0);

second-ordertensors ofdielectricmaterial constants:

Fm

(electric permittivity) and 1

mF

obtained in the absence of

mechanical strain (S=0) or stress (T=0);




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Stress

Nm2

Strain

mm

Charge C

m2 T,D

e

cE=0, S=0 (S,E) S,D

d


Voltage

Vm

T,E

qcD=0,

1S=0

(S,D) S,E g

sD=0,

1T=0

(T,D)(charge)

(strain) (stress)


third-order tensors ofpiezoelectriccoupling coefficients:

e Cm

2 the piezoelectric coefficients forStress-Chargeform,q

m2

C

the piezoelectric coefficients forStress-Voltageform,

d

CN

the piezoelectric coefficients forStrain-Chargeform,

g

NC

the piezoelectric coefficients forStrain-Voltageform.


Transformations of piezoelectric constitutive data


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1 Strain-Charge Stress-Charge:

cE=0 = s1E=0 , e=d : s1E=0 , S=0 = T=0d s

1E=0 d

T .




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cE=0 = s1E=0 , e=d : s1E=0 , S=0 = T=0d s

1E=0 d

T .

2 Strain-Charge Strain-Voltage:

sD=0 = sE=0dT

1T=0 d , g=

1T=0 d .




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cE=0 = s1E=0 , e=d : s1E=0 , S=0 = T=0d s

1E=0 d

T .


sD=0 = sE=0dT

1T=0 d , g=

1T=0 d .

3 Strain-Charge Stress-Voltage: . . .




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cE=0 = s1E=0 , e=d : s1E=0 , S=0 = T=0d s1E=0 dT .


sD=0 = sE=0dT

1T=0 d , g=

1T=0 d .


4 Stress-Charge Stress-Voltage:

cD=0 = cE=0eT

1S=0 e , q=

1S=0 e .




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sD=0 = sE=0dT

1T=0 d , g=

1T=0 d .



cD=0 = cE=0eT

1S=0 e , q=

1S=0 e .

5 Stress-Charge Strain-Voltage: . . .




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sD=0 = sE=0dT

1T=0 d , g=

1T=0 d .



cD=0 = cE=0eT

1S=0 e , q=

1S=0 e .

5 Stress-Charge Strain-Voltage: . . .6 Strain-Voltage Stress-Voltage:

cD=0 = s1D=0 , q=g : s

1D=0 ,

1S=0 =

1T=0+g s

1D=0 g

T .


Matrix notation of constitutive relations


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at otat o o co st tut e e at o s

Rule of change of subscripts

11 1 , 22 2 , 33 3 , 23 4 , 13 5 , 12 6 .




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11 1 , 22 2 , 33 3 , 23 4 , 13 5 , 12 6 .

Tij [T](61) , Sij [S](61) , Ei [Ei](31) , Di [Di](31) ,

cijkl [c](66) , sijkl [s](66) , ij [ij](33) , 1ij [

1ij ](33) ,

ekij [ek](36) , dkij [dk](36) , qkij [qk](36) , gkij [gk](36) .

(Here: i,j,k, l= 1,2,3, and,= 1,...6.)




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11 1 , 22 2 , 33 3 , 23 4 , 13 5 , 12 6 .

Tij [T](61) , Sij [S](61) , Ei [Ei](31) , Di [Di](31) ,

cijkl [c](66) , sijkl [s](66) , ij [ij](33) , 1ij [

1ij ](33) ,

ekij [ek](36) , dkij [dk](36) , qkij [qk](36) , gkij [gk](36) .

(Here: i,j,k, l= 1,2,3, and,= 1,...6.)

Strain-Charge form:

S(61) = s(66)T(61)+dT

(63)E(31) ,

D(31) =d(36)T(61)+(33)E(31) .

Stress-Charge form:

T(61) = c(66)S(61)eT

(63)E(31) ,

D(31)

= e(36)

S(61)

+(33)

E(31)

.




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Fororthotropicpiezoelectric materials there are9+5+3= 17material constants, and the matrices of material constants read:

c(66) =

c11 c12 c13 0 0 0

c22 c23 0 0 0

c33 0 0 0

c44 0 0

sym. c55 0

c66

,

e(36) =

0 0 0 0 e15 0

0 0 0 e24 0 0

e31 e32 e33 0 0 0

, (33) =

11 0 0

0 22 0

0 0 33

.




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Fororthotropicpiezoelectric materials there are9+5+3= 17material constants, and the matrices of material constants read:

c(66) =

c11 c12 c13 0 0 0

c22 c23 0 0 0

c33 0 0 0

c44 0 0

sym. c55 0

c66

,

e(36) =

0 0 0 0 e15 0

0 0 0 e24 0 0

e31 e32 e33 0 0 0

, (33) =

11 0 0

0 22 0

0 0 33

.

Many piezoelectric materials (e.g., PZT ceramics) can be treated astransversally isotropic. Then, there are only 10material constants,since4+2+1= 7of the orthotropic constants depend on the others:

c22 = c11 , c23 = c13 , c55 = c44 , c66 =c11c12

2,

e24 =e15 , e32 =e31 , 22 = 11 .


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Comments and final remarks


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Electromechanical coupling

Induced electric potential

Piezoelectric finite elements

Thermal analogy approach an approximation based on theresemblance between thermoelastic and converse piezoelectricconstitutive equations: the stress vs. strain and voltage equation

Tij = cijklSklemijEm = cijklSkldmklEm

with dmkl =emij c

1

ijkl

resembles the Hookes constitutive relation with initial strain S0kl

or initial temperature0

Tij = cijklSklS0kl= cijklSklkl

0

.Thus, that latter thermoelastic law can be used to approximatethe piezoelectric problem. In this case the thermal expansioncoefficients are determined as

kl =1

0S0kl where S

0kl =dmklEm =dmkl|m .

icmm tgzielinski piezoelectricity.slides

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