electro−elastic active materials

ELECTRO−ELASTIC ACTIVE MATERIALS

PART II

1. DIELECTRICITY AND PIEZOELECTRICITY2 CRYSTAL STRUCTURE OF PIEZOELECTRICS2. CRYSTAL STRUCTURE OF PIEZOELECTRICS3. DIRECT AND CONVERSE PIEZOELECTRIC EFFECT4. PIEZOELECTRIC COUPLING COEFFICIENTS4 PI O CTRIC COUP ING CO ICI NTS5. CHARACTERISTICS OF COMMERCIAL PIEZOS6. COUPLED CONSTITUTIVE EQUATIONS7. ELECTROSTRICTIVE MATERIALS8. ANALOGIES IN FIELD PROBLEMS 9 VARIATIONAL PRINCIPLES9. VARIATIONAL PRINCIPLES10. FINITE ELEMENTS 11. APPLICATIONS

P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II

1. DIELECTRICITY AND PIEZOELECTRICITY (1)

IN THE DIELECTRIC (INSULATING) MATERIALS THE CONSTITUENT ATOMSARE IONIZED AND ARE EITHER POSITIVELY (CATIONS) OR NEGATIVELY(ANIONS) CHARGED(ANIONS) CHARGED.

THE ELECTRIC CHARGES ARE NOT FREE TO MOVE AND AN ELECTRONICCLOUD SURROUNDS EACH ATOM LOCALLYCLOUD SURROUNDS EACH ATOM LOCALLY.

UNDER THE EFFECT OF AN ELECTRIC FIELD CATIONS ARE ATTRACTED BYTHE CATODE AND ANIONS BY THE ANODE THE ELECTRONIC CLOUD ALSOTHE CATODE AND ANIONS BY THE ANODE. THE ELECTRONIC CLOUD ALSODEFORMS, CAUSING ELECTRIC DIPOLES.

ELECTRIC POLARIZATION (Uchino)

ELECTRONIC POLARIZATION + +−

E=0 E + −

IONIC POLARIZATION−

+ +− + +−


DIPOLE REORIENTATION + − + −

+ −

+ −

+ −

+ −

+ −

2. DIELECTRICITY AND PIEZOELECTRICITY (2)

(1) THE ABILITY OF CERTAIN CRYSTALLINE MATERIALS TO DEVELOP ANELECTRIC CHARGE THAT IS PROPORTIONAL TO A MECHANICAL STRESS. J.AND P. CURIE˘DIRECT PIEZOELECTRIC STRESS−1880

VERY SOON THE CONVERSE EFFECT WAS ALSO DISCOVERED TO BEINHERENT IN THESE MATERIALS A GEOMETRIC STRAIN IS DEVELOPEDINHERENT IN THESE MATERIALS: A GEOMETRIC STRAIN IS DEVELOPEDUPON THE APPLICATION OF A VOLTAGE.

THE VARIETY OF CRYSTALS THAT EXHIBIT A PIEZOELECTRIC BEHAVIORTHE VARIETY OF CRYSTALS THAT EXHIBIT A PIEZOELECTRIC BEHAVIORDOES NOT HAVE A CENTER OF SIMMETRY WITHIN THE CRYSTAL. THEABSENCE OF SIMMETRY GIVES RISE TO SPONTANEOUS POLARIZATION.

PIEZOELECTRICITY IS LIMITED TO 20 OUT OF 32 CRYSTAL CLASSES FORALL CRYSTALLINE MATERIALS.

(2) MOST PIEZOELECTRIC MATERIALS ARE ALSO FERROELECTRIC THEY(2) MOST PIEZOELECTRIC MATERIALS ARE ALSO FERROELECTRIC: THEYTRANSFORM TO A HIGH SYMMETRY NON PIEZOELECTRIC PHASE AT HIGHTEMPERATURES. THE TRANSFORMATION TEMPERATURE IS KNOWN AS


CURIE TEMPERATURE.

2. CRYSTAL STRUCTURE OF PIEZOELECTRICS

BARIUM TITANATE (BaTiO3) IS A TYPICAL FERROELECTRIC MATERIAL

0.036Å

+ Ba2+

_+ 0.061Å

Å

O2-

Ti4+

IONIC SHIFTS

0.12Å

THE IONIC SHIFTS PRODUCE A DIPOLE MOMENT IN THE CRYSTAL THE

Ti++

THE IONIC SHIFTS PRODUCE A DIPOLE MOMENT IN THE CRYSTAL THEINTENSITY OF WHICH IS PROPORTIONAL TO THE ELECTRIC CHARGE ANDITS POSITION SHIFT.

THE ORIGIN OF THIS SPONTANEOUS POLARIZATION AND THE POSSIBILITYOF MAINTAINING THE EQUILIBRIUM OF THE CRYSTAL LATTICE CAN BEASCRIBED TO THE EFFECTS OF THE LOCAL FIELD WHICH SURROUNDS


ASCRI TO TH FF CTS OF TH OCA FI WHICH SURROUN SEVERY CATION (+) AND ANION (−).

2. CRYSTAL STRUCTURE OF PIEZOELECTRICS (2)

FERROELECTRIC MATERIALS, AS SEEN BEFORE, EXHIBIT A SPONTANEOUSPOLARIZATION, DUE TO THE LACK OF SIMMETRY IN THEIR CRYSTALSTRUCTURE, WHICH IS LOST ABOVE THE CURIE TEMPERATURE.

IT IS IMPORTANT TO CONSIDER THAT THE SPONTANEOUS POLARIZATIONCAN BE REVERSED OR REORIENTED BY THE APPLICATION OF ANCAN BE REVERSED OR REORIENTED BY THE APPLICATION OF ANELECTRIC FIELD OF A CERTAIN INTENSITY (COERCITIVE FIELD).

FERROELECTRIC MATERIALS ARE PRESENT AS CRYSTALS(POLYCRYSTALS) AND CERAMICS.A CERAMIC IS AN AGGREGATE OF FERROELECTRIC SINGLE CRYSTALGRAINS (CRISTALLITES).( )

UNPOLED + DIPOLE

VECTOR−

VECTOR


CERAMICS NEED TO BEPOLARIZED

MACROSCOPIC DIPOLE VECTOR

EP

3. DIRECT AND CONVERSE PIEZOELECTRIC EFFECTTHE ORIGIN OF THE DIRECT EFFECT

LET US CONSIDER THE EFFECT ON A SINGLE CRYSTAL OF LEADTITANATE (PbTiO3) + + σ3

x3

Pb+ ΔP3=d33σ3

x1

Pb

O−

Ti−

Ti IS DISPL.

Ti

− −

ΔP =d σ

− −σ1+σ5ΔP3=d31σ1 −

+

5

ΔP1=d15σ5


++−

3. DIRECT AND CONVERSE PIEZOELECTRIC EFFECT THE ORIGIN OF THE CONVERSE EFFECT

LET US CONSIDER THE EFFECT ON A POLARIZED PZT (LEAD ZIRCONATETITANATE) CERAMIC. å1

x3−

e+

+−

+

++++++++++++++−

x3

E

å3

E<EP

E POLARIZATION FIELD DEFORMED CONFIGURATION

+−

+x1

ep−

−

− − − − − − − − − − − − − −

+x1

E E<EP

EP: POLARIZATION FIELDx3: POLARIZATION DIRECTIONE IS PARALLEL AND HAS THE SAME SIGN OF EP

DEFORMED CONFIGURATIONε1= d31E3

ε3= d33E3P 3 33 3x3

−

e+

+−

+

x3

++++++++++++++

+

γ13

d E

+−

+x1

ep−

+−

x1E

− − − − − − − − − − − − − −

+ −


E IS NORMAL TO EPγ13= d15E1

4. PIEZOELECTRIC COUPLING COEFFICIENTS (1)

THE DIRECT PIEZOELECTRIC EFFECT IS THE DEVELOPMENT OF ANELECTRIC CHARGE UPON THE APPLICATION OF A MECHANICAL STRESS.

P=dó

P: ELECTRIC POLARIZATION (CHARGE PER UNIT AREA)ó dó: MECHANICAL STRESS d: COUPLING COEFFICIENT

INSTEAD OF P, THE VARIABLE D (ELECTRIC DISPLACEMENT) IS OFTENUSED BEINGUSED, BEING

D=ξ0E+P=ξrξ0E

(ELECTRIC CONSTITUTIVE EQUATION FOR DIELECTRIC MATERIALS;î0=8.854*10−12 F/m VACUUM PERMITTIVITY; îr RELATIVE PERMITTIVITY)

FOR THE CONVERSE EFFECT THE STRAIN S PRODUCED BY AN APPLIEDC IC I IS GI N BYELECTRIC FIELD IS GIVEN BY

S=dES: GEOMETRIC STRAIN E: ELECTRIC FIELD


d: COUPLING COEFFICIENT (SAME OF DIRECT EFFECT)


ANOTHER IMPORTANT CONSTANT DESCRIBES THE ELECTRIC FIELDPRODUCED BY A STRESS. CONSIDER THE CONSTITUTIVE EQUATION:

D=ξ0E+P

IN THE ABSENCE OF AN EXTERNALLY APPLIED ELECTRIC FIELD

D=P=dσ

IF WE NOW EVALUATE THE ELECTRIC FIELD GENERATED BY THEóPOLARIZATION DUE TO THE PRESENCE OF ó

σξd

ξξD

Er0

GEN ==d

ξξξ r0

THE CONSTANT =g IS DEFINED AS VOLTAGE COEFFICIENT.

A HIGH d CONSTANT IS DESIRABLE FOR ACTUATOR MATERIALS INTENDED

ξd

A HIGH d CONSTANT IS DESIRABLE FOR ACTUATOR MATERIALS INTENDEDTO DEVELOP MOTION.

A HIGH g CONSTANT IS DESIRABLE FOR SENSOR MATERIALS WHERE HIGH


G g CO ST T S S O S SO T S GVOLTAGES ARE GENERATED FROM WEAK MECHANICAL STRESSES.


ANOTHER INTERESTING COEFFICIENT TO BE CONSIDERED IS THEELECTROMECHANICAL COUPLING FACTOR K, WHICH MEASURES THEFRACTION OF ELECTRICAL ENERGY CONVERTED TO MECHANICAL ENERGY.OF COURSE K<1.

W W :ENERGY CONVERTED TO MECHANICAL ENERGY

21

12

WWW

K+

=W1:ENERGY CONVERTED TO MECHANICAL ENERGYW1+W2:TOTAL ELECTRICAL ENERGY GIVEN TO THE MATERIALW2:ELECTRICAL ENERGY NOT CONVERTED

1 CHARGE WITH E1. CHARGE WITH E2. LET THE MATERIAL FREE TO EXPAND3. BLOCK MECHANICALLY ALONG X 4 DISCONNECT E

−x3

EP 4. DISCONNECT E5. RELEASE THE BLOCK AND MEASURE THE

WORK DONE BY A MECHANICAL LOAD DISPLACEMENT ALONG X (W )

+x1

EP

DISPLACEMENT ALONG X3 (W1)

T33

E33

2332

33 ξSd

K = SE: COMPLIANCE AT CONSTANT ELECTRIC FIELDîT: PERMITTIVITY AT CONSTANT STRESS


3333ξS

5. CHARACTERISTICS OF COMMERCIAL PIEZOS

QUARTZ LOW TEMPERATURE FORM OF SINGLE CRYSTAL SiO2

PVDF POLYVINYLIDENE FLUORIDE, A PIEZOELECTRIC POLYMERIC MATERIAL

PZT Pb(Zr,Ti)O3 PIEZOELECTRIC CERAMIC. THE 52/48 DESIGNATION ISCHOSEN FOR THE COMPOSITION AT THE PHASE BOUNDARY ONPbZrO /PbTiO DIAGRAMPbZrO3/PbTiO3 DIAGRAM

PZTL PZT DOPED WITH LANTHANUM

MATERIAL T (ÚC) d ( 1012C/N) ( 1014C/N)MATERIAL TCURIE (ÚC) d33 (x1012C/N) g33 (x1014C/N) εr

QUARTZ 573 −2.3 −57.5 4

PVDF 52/48 41 30 200.0 15

PZT 386 223 39.5 1500

PZTL 65 682 20.0 3400

HIGH g LOW d FOR PVDF SENSOR CAPABILITY


HIGH g LOW d FOR PVDF SENSOR CAPABILITYHIGH d FOR PZT WIDELY USED AS ACTUATOR

5. CHARACTERISTICS OF COMMERCIAL PIEZOS (2)

G1195DIELECTRIC CONSTANTS

(AT CONSTANT STRESS)

DENSITY CURIE TEMPERATURE

PIEZOELECTRIC STRAIN

COEFFICIENTSSTRESS) COEFFICIENTS

î33T/î0 î 11

T/î0 ñ TC d33 d31

1700 1700 7650 kg/m3 360 ÚC 360pm/V −180pm/V1700 1700 7650 kg/m3 360 ÚC 360pm/V −180pm/V

YOUNG’S MODULI TENSILE COMPRESSIVE COERCITIVE FIELD(AT CONSTANT

ELECTRIC FIELD)STRENGHT STRENGHT

C33E C11

E FT FC ECC33 C11 FT FC EC

49 GPa 63 GPa 77 MPa >500 MPa 1200 V/mm


6. GOVERNING EQUATIONSCONSTITUTIVE EQUATIONS

PIEZOELECTRICITY − FULL CONSTITUTIVE RELATIONS (LINEAR MODEL)PIEZOCERAMICS CAN BE REPRESENTED BY COUPLED ELECTRICAL−MECHANICAL EQUATIONS IN THE FORM:MECHANICAL EQUATIONS IN THE FORM:

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡TE

sddξ

SD

T

D= ξE + dT

S= dTE + sTCOUPLING TERMS

D: ELECTRIC

⎦⎣⎦⎣⎦⎣ TsdS S= d E + sT

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

2

1

151

151

2

1

EE

00d0000ξ00d000000ξ

DD

DISPLACEMENTE: ELECTRIC FIELDS: STRAIN

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

1

3

2

13121131

3331313

151

1

3

2

TE

000SSSd00000dddξ00

ξ

SD

T: STRESSî: PERMITTIVITYs: COMPLIANCE⎥

⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥

⎢⎢⎢⎢=

⎥⎥⎥⎥

⎢⎢⎢⎢

3

2

33131333

13111231

3

2

TTT

00S0000d0000SSSd00000SSSd00

SSS

s: COMPLIANCEd: PIEZOELECTRIC COUPLING⎥

⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ 6

5

4

66

5515

5515

6

5

4

TTT

S000000000S000000d00S0000d0

SSS


SYMMETRY OF THE MATRIX CONSERVATIVE FIELD

⎦⎣⎦⎣⎦⎣ 6666

6. GOVERNING EQUATIONS (2)

PIEZOELECTRICITY−GOVERNING EQUATIONS (LINEAR CASE) FOR APIEZOELECTRIC CONTINUUM OF VOLUME V AND BOUNDARY S IN A 3DSPACESPACE

τij,i + fjB = 0

1/2 ( )

MECHANICAL EQUILIBRIUM

STRAIN DISPLACEMENT RELATIONεij = 1/2 (ui,j + uj,i)

Di,i = 0

STRAIN DISPLACEMENT RELATION

MAXWELL’S EQUATION FOR THE

in V

Ei = - Φ,i

niτij = fjSQUASI STATIC ELECTRIC FIELD

NATURAL MECHANICAL CONDITIONS ON Sf

niDi = σS

ui = ūi

NATURAL ELECTRICAL CONDITIONS ON Só

ESSENTIAL MECHANICAL CONDITIONS ON SU

on S

Φ = Φ

τij = Cijhkεhk- ekij Ek

ESSENTIAL ELECTRICAL CONDITIONS ON SÖ

CONSTITUTIVE (ˆMECHANICAL˜)


j j j

Di = eihkεhk + ξij Ej RELATIONS (ˆELECTRICAL˜)

7. ELECTROSTRICTIVE MATERIALS

ELECTROSTRICTIVE MATERIALS − CONSTITUTIVE EQUATIONS

D = ξ l El + m ijTij E COUPLING TERMS AREDr ξrl El + mrnijTij En

Sij = mijkl Ek El + sijpq Tpq

COUPLING TERMS AREQUADRATIC FUNCTIONS OF Ei

D: ELECTRIC DISPLACEMENT T: STRESSD: ELECTRIC DISPLACEMENTE: ELECTRIC FIELD

T: STRESSS: STRAIN

î: PERMITTIVITY s: COMPLIANCE m: ELECTROSTR. COUPLING

−CERAMIC SIMILAR TOPIEZOELECTRICNEGLIGIBLE HYSTERESIS AND0,75

1

Ëx103

−NEGLIGIBLE HYSTERESIS ANDCREEP−STRAIN IS A QUADRATICFUNCTION OF FIELD

0,5

0, 5

FUNCTION OF FIELD−HIGH MODULUS OF ELASTICITY−PERFORMANCE SENSITIVE TO

0,25


TEMPERATURE0

-15 -5 5 15 E (KV/cm)

8. ANALOGIES IN FIELD PROBLEMS (1)

VARIABLES − STATE VARIABLES i=1,2,3DISPLACEMENTS ui (xi,t) u(x,y,z,t), v(…), w(…)STRAINS ( t)

ui STRAINS εij (xi,t) εx, εy, εz, γyz, γxz, γxySTRESSES σij (xi,t) σx, σy, σz, τyz, τxz, τxy−EXTERNAL ACTIONS on V, Su, Sf

Fi

i

x3VOLUME FORCES Xi (xi,t) X (x,y,z,t), Y(…), Z(…)SURFACE FORCES fi (xi,t) fx (x,y,z,t), fy(…), fz(…)APPLIED DISPLAC. ūi (xi,t) u (x,y,z,t), v(…), w(…)o

0Xσσσ

xzxyx =+∂

+∂

+∂

x1 x2

i ( i, ) ( ,y, , ), ( ), ( )

EQUATIONS ˘ EQUILIBRIUMσij j + Xi = 0 on V

o

0Xzyx

=+∂

+∂

+∂

u∂ vu ∂∂

σij,j + Xi 0 on Vniσij = fi on Sf

COMPATIBILITYε = ½(u + u )

xuε

x ∂∂

= xyγ

xy ∂+

∂=

u∂ uΣ∂

εij = ½(ui,j + uj,i)ui = ūi on Su


REPEATED INDEX = SUMMATIONj

i

j,i xu

u∂∂

=i

i

ij,i xu

Σu∂∂

=

8. ANALOGIES IN FIELD PROBLEMS (2)

SOLID MECHANICSVARIABLESDISPLACEMENTS ui

EQUATIONS

EQUILIBRIUM ó + f = 0DISPLACEMENTS uiSTRAINS åij

STRESSES óij

EXT VOLUME FORCES X

EQUILIBRIUM óij,i + fj = 0nióij = fj on Sf

COMPATIBILITY åij = ½(ui j + uj i)EXT. VOLUME FORCES XiEXT. SURF. FORCES fiAPPLIED DISPLACEMENT ûi

ij ( i,j j,i)ui = ûi on Su

∫∫∫ + dSu~fdVu~XdVε~σVIRTUAL WORK EQUATION L L∫∫∫ +=FS

iiiV

iijV

ij dSufdVuXdVεσVIRTUAL WORK EQUATION: Li = Le

HEAT CONDUCTION (ELECTROSTATICS)VARIABLES EQUATIONS

dTTEMPERATURE TELECTRIC POT ÖTEMP. GRAD. T,i

HEAT FLUX −qi,i = cMAXWELL Di’i = 0

dtdT

ˆDISPLAC.˜

ˆSTRAINS˜TEMP. GRAD. T,iELECTRIC FIELD EiHEAT FLUX qi

ELECTRIC DISPL D

gradT = T,i

MAXWELL gradΦ = E,i

ˆSTRAINS˜

ˆSTRESSES˜


ELECTRIC DISPL. DiVIRTUAL ˆWORK˜ EQUATION: ∫ ∫=

V S

*ii

σ

dSΦ~

σdVE~

D ó*: EXT.SURFACE CHARGE PER UNIT AREA

9. VARIATIONAL PRINCIPLES

VIRTUAL WORK (DISPLACEMENT VERSION) Lext = Lint

∫ ∫ ∫+= dSu~fdVu~XdVε~σ

óij: ACTUAL STRESSESXi, fi: ACTUAL EXTERNAL FORCES

∫ ∫ ∫+V V S

iiiiijijf

dSufdVuXdVεσ

VIRTUAL DISPLACEMENT (SMALL, ZERO ON Su)VIRTUAL STRAINS (COMPATIBLE WITH )

THE WEAK FORM OF EQUILIBRIUM EQUATION

iu~

ijε~ i

u~

0dVu~)Xσ( =+∫ i j=1 2 3THE WEAK FORM OF EQUILIBRIUM EQUATION:

DERIVATIVE OF A PRODUCT

0dVu)Xσ(i

Vij,ij

=+∫ i, j=1,2,3

j,iijj,iijij,iju~σ)u~σ(u~σ −=

DIVERGENCE THEOREM S=Su+Sf∫ ∫=V S

jiijj,iijdSnu~σdV)u~σ(

ijijfnσ = on Sf

FOR THE SIMMETRY OF ó

ijij f

0dVu~XdSu~fdVu~σV

iiS

iiV

j,iijF

=++− ∫∫∫

ε~σ)u~u~(1σu~σ +


FOR THE SIMMETRY OF óijiji,jj,iijj,iijεσ)uu(

2σuσ =+=

9. VARIATIONAL PRINCIPLES (2)

VIRTUAL WORK EQUATION: ∫∫∫ +=V

iiS

iiV

j,iijdVu~XdSu~fdVε~σ

F

STRESS SURFACE FORCE VOLUME FORCE

FIELD PROBLEMSVARIABLES

VSV F

V.STRAIN V.DISPLAC

STATE VAR. HEAT COND. MOISTURE ABS. ELECTROSTATICS

ˆDISPLACEMENT˜ ui

TEMPERATURE

T(xi,t)MOISTURE CONTENT

U(xi,t)ELECTRIC POTENTIAL

Φ(xi,t)

ˆSTRAIN˜ TEMP GRADIENT MOIST CONT GRAD ELECTRIC FIELDˆSTRAIN˜ TEMP. GRADIENT T,j(xi,t)

MOIST. CONT. GRAD.

U,j(xi,t)ELECTRIC FIELD

Ei(xi,t)

ˆSTRESS˜ HEAT FLUX qi(xi,t) MOISTURE FLOW qi(xi,t) ELECTRIC DISPL. D( t)Di(xi,t)

EXTERNAL ACTIONSˆVOLUME GENERATED HEAT PER

BGENERATED FLUID PER

BVOLUME DENS. ELECTR.

B 0FORCES˜ UNIT VOLUME qB UNIT VOLUME qB CHARGE σB=0

ˆSURFACE FORCES˜

ASSIGNED FLOW ON THE BOUNDARIES qS

ASSIGNED FLOW ON THE BOUNDARIES qS

ASSIGNED SURFACE DENS. CHARGE óS=óS


ˆAPPLIED DISPLAC.˜

ASSIGNED TEMP. ON THE BOUNDARIES T

ASSIGNED MOIST. CONT. ON THE BOUNDARIES Û

ASSIGNED ELECTRIC POTENTIAL Öi= Öi


FIELD PROBLEMS EQUATIONS HEAT COND ELECTROSTATICS

T∂EQUILIBRIUM 0Xσij,ij=+ c

tTcq

i,i−

∂∂

ρ=− 0Di,i=

xρX̂X &&−= nqqn = SDn σ=(DYNAMICS)

COMPATIBILITY εij = ½(ui,j+uj,i) grad T = T,j Ei = - Φ,i

iiixρXX

iiqqn

iiDn σ

VIRTUAL TEMPERATURES

ui = ūi T = Ti Φi = Φi

ˆSTRESS˜ ˆVOL. FORCE˜ ˆSURFACE FORCE˜

VIRTUAL TEMPERATURES EQUATION ∫ ∫∫ +=

V SS

VBii

dST~qdVT

~qdV,T

~q

Sq

ˆVIRTUAL STRAIN˜ ˆVIRTUAL DISPL.˜

NOTE:

[q] ≠ [qB]

VIRTUAL ELECTRIC ∫∫ −= dSΦ~

σdVED ób = 0


POTENTIAL EQUATION∫∫SσS

SV

iidSΦσdVED

ˆSTRESS˜ ˆSURFACE FORCE˜

b


VIRTUAL WORK: ∫∫∫ +=FS

iiV

iiV

ijij dSu~FdVu~XdVε~σ

VIRTUAL ELECTRIC POTENTIALS: ∫ ∫−=V S

Sii dSΦ~

σdVE~

DSσ

CONSTITUTIVE EQUATIONS

ELASTIC SOLID: hkijhkij εCσ = or hkijhkij σFε =

DIELECTRIC MATERIAL:

hkijhkij or hkijhkij

iiji EξD = ),Tkq( jiji −= k=THERMAL CONDUCTIVITIES

PIEZOELECTRIC SOLID: kkijhkijhkij EdσFε +=

EξσdD +=

THE ELECTRICAL AND MECHANICAL EQUATIONS OF THE ˆVIRTUAL WORK˜

jijklikli EξσdD +=


THE ELECTRICAL AND MECHANICAL EQUATIONS OF THE VIRTUAL WORKAPPROACH ARE COUPLED BY MEANS OF THE CONSTITUTIVE EQUATIONS.


FOR EVERY TIME t AND FOR ANY POSSIBLE CHOICE OF VIRTUALDISPLACEMENT δui SATISFYING THE ESSENTIAL BOUNDARY CONDITIONSTHE FOLLOWING RELATION HOLDS:THE FOLLOWING RELATION HOLDS:

WHERE t DENOTES THE GENERIC TIME, IS THEVIRTUAL STRAIN CORRESPONDING TO δ AND δ ARE THE

)uu(2/1 i,jj,iijt δ+δ=εδ∫ ∫ δ+δ∫ =εδτ tV tS

tSStti

BttV

tijtij

t

f iiiSdufVdufVd PRINCIPLE OF VIRTUAL WORKS

VIRTUAL STRAIN CORRESPONDING TO δui , AND δuis ARE THE

VIRTUAL DISPLACEMENTS ON TSf.

IN AN ANALOGOUS WAY THE FOLLOWING RELATIONS CAN BEIN AN ANALOGOUS WAY THE FOLLOWING RELATIONS CAN BEWRITTEN (PRINCIPLE OF VIRTUAL ELECTRIC POTENTIAL)

∫ δφσ∫ =δσtS

tss

ttV

tii

t SdVdEDWHERE δEi IS THE VIRTUAL ELECTRIC FIELD CORRESPONDING TO δφ ANDδφs IS THE VIRTUAL ELECTRIC POTENTIAL ON TSσ.

NO RESTRICTIONS ON COSTITUTIVE RELATIONS HAVE BEEN INTRODUCEDNO RESTRICTIONS ON COSTITUTIVE RELATIONS HAVE BEEN INTRODUCEDUP TO THIS POINT INTO THE VARIATIONAL PRINCIPLES. IN ABSENCE OFDYNAMICS EFFECTS THE TIME t HAS TO BE CONSIDERED AS A VARIABLETHAT DENOTES SUBSEQUENT APPLICATION OF LOADS OF DIFFERENT


THAT DENOTES SUBSEQUENT APPLICATION OF LOADS OF DIFFERENTINTENSITY AND THE CORRESPONDING DEFORMED CONFIGURATION.


THE PROPOSED SOLUTION PROCEDURE IS BASED ON THE FOLLOWINGSTEPS:

1) THE INCREMENTAL FORMULATION OF THE VARIATIONAL PRINCIPLESFROM THE RELATION THAT HOLDS AT THE TIME t+Δt:

t+Δt Re - t+Δt Ri =0WHERE Re ARE THE EXTERNAL FORCES (THAT ARE SUPPOSED TO BEKNOWN FOR EACH t) AND Ri ARE THE INTERNAL FORCES; A LINEARIZATIONIS ASSUMED t+Δt Ri t Ri ≅ tKΔu Δu = t+Δt u t uIS ASSUMED t+Δt Ri - t Ri ≅ tKΔui Δui = t+Δt ui - t ui

AND THEN FOR SUBSEQUENT APPROXIMATION THE FOLLOWING EQUATIONIS SOLVED: tKΔui = t+Δt Ri - t Ri2) THE USE OF THE TYPICAL FINITE ELEMENTS DESCRIPTIVE FUNCTIONFOR DISPLACEMENTS AND ELECTRIC POTENTIALS:

u(m) = H (m) uu φ(m) = Hφ(m) φφu Hu uu φ( ) Hφ φφ

3) THE SET UP OF A NUMERICAL PROCEDURE THAT ITERATES BETWEENTHE FINITE ELEMENT EQUATIONS THAT DESCRIBE THE CONVERSEPIEZOELECTRIC EFFECT AND THE ONES THAT DESCRIBE THE DIRECT


PIEZOELECTRIC EFFECT AND THE ONES THAT DESCRIBE THE DIRECTEFFECT.

10. FINITE ELEMENTSLINEAR OR LINEARIZED CASE

BSuuu uu

FFkuk +=φ+ φ

SFkuk φ+(A)

u Fkukφ

=φ+ φφφ

u AND φ ARE, RESPECTIVELY, THE NODAL DISPLACEMENTS AND THENODAL ELECTRIC POTENTIALS ANDNODAL ELECTRIC POTENTIALS AND

mmTmT dV)B()B(kk ∑ ∫

mmum Vm

Tmuuu dV)B(C)B(k ∑ ∫= mBm

m VmTmB dVf)H(Fu ∑ ∫=

mSmTSmS dSf)H(F ∑ ∫=

SUMMATIONS ARE EXTENDED TO ALL THE ELEMENTS OF THE

mm Vm

Tuuu dV)B(e)B(kk φφφ ∑ ∫−==

mmm Vm

Tm dV)B(e)B(k φφφφ ∑ ∫=

fm Sm dSf)H(Fu ∑ ∫=mSm

m SmTSmS dS)H(F σφ ∑ ∫ σ=

SUMMATIONS ARE EXTENDED TO ALL THE ELEMENTS OF THEDISCRETIZATION. THE SHAPE FUNCTIONS ARE INCLUDED IN THE MATRIXH ; B MATRICES ARE OBTAINED FROM H BY DERIVATION AND, IFPOSTMULTIPLIED FOR u, EXPRESS THE STRAINS.

IN THE NONLINEAR CASE THE EQUATIONS (A) CAN BE FORMALLYREGARDED AS CONCERNING NOT THE OVERALL VALUES OF THE


REGARDED AS CONCERNING NOT THE OVERALL VALUES OF THEVARIABLES BUT THEIR INCREMENT AT THE i−th STEP.

10. FINITE ELEMENTSCONVERSE EFFECT EQUATION

IF WE ASSUME THAT THE ELECTRIC POTENTIAL IS ASSIGNED FOREACH NODE THE SOLUTION CAN BE DIRECTLY WRITTEN AS FOLLOWS:EACH NODE, THE SOLUTION CAN BE DIRECTLY WRITTEN AS FOLLOWS:

u = kuu-1(Fu

B + FuS - kuφφ) (B)

THIS RELATION EXPRESSES IN ABSENCE OF EXTERNAL MECHANICALTHIS RELATION EXPRESSES, IN ABSENCE OF EXTERNAL MECHANICALLOADS, THE DISPLACEMENT FIELD GENERATED BY AN ASSIGNEDELECTRIC POTENTIAL.

IN THIS CASE THE EFFECT THAT THE STRESSES GENERATED BY THEDISPLACEMENT FIELD PRODUCE ON THE ELECTRIC POTENTIAL CAN BENEGLECTEDNEGLECTED.

THE FIRST MATRICIAL EQUATION (A) CAN BE RESOLVED UNCOUPLEDFROM THE SECONDFROM THE SECOND.


10. FINITE ELEMENTS DIRECT EFFECT EQUATIONS

ANALOGOUS CONSIDERATIONS CAN BE DONE FOR THE SECOND OFTHE (A) EQUATIONS IF THE NODAL DISPLACEMENTS u AREASSIGNED:

φ = kφφ−1(F φ

S − kφuu) (C)

IN THIS WAY THE EFFECT OF THE ELECTRIC FIELD ONDISPLACEMENT HAS BEEN NEGLECTEDDISPLACEMENT HAS BEEN NEGLECTED.


10. FINITE ELEMENTS SIMULTANEOUS SOLUTION OF THE SYSTEM

IN MANY ENGINEERING APPLICATIONS EQUATION (B) AND (C) CANB US I C Y O S U Y I C AN CON SBE USED DIRECTLY TO STUDY DIRECT AND CONVERSEPIEZOELECTRIC EFFECTS. IN GENERAL THE FULL COUPLINGBETWEEN THE EQUATIONS HAS TO BE ACCOUNTED FOR.

A POSSIBLE WAY TO GET THE COUPLED SOLUTION IS, OF COURSE,THE SIMULTANEOUS SOLUTION OF EQUATIONS (A).

THIS CAN GENERATE HIGH COMPUTING COSTS, ESPECIALLY IN ANONLINEAR CASE AND GENERALLY REQUIRES THE DEVELOPMENTOF ˆAD HOC˜ FINITE ELEMENTS FOR PIEZOELECTRIC ANALYSISOF AD HOC FINITE ELEMENTS FOR PIEZOELECTRIC ANALYSIS.


10. FINITE ELEMENTSAN ITERATIVE SOLUTION TECHNIQUE

1. SOLVE (C) ASSUMING u = 0 AND GETTING φ(1);2. SUBSTITUTE φ = φ(1) INTO (B) GETTING u(2);φ φ3. SOLVE (C) AGAIN, ASSUMING u = u(2), OBTAINING φ(3);4. COMPARE φ(3) WITH φ(1):

⏐ φ(3) − φ(1) ⏐/⏐ φ(1)⏐ ≤ β1;⏐ φ φ ⏐/⏐ φ ⏐ ≤ β1;5. SOLVE AGAIN (B) WITH φ = φ(3), GETTING u(5);6. COMPARE u(5) WITH u(2):

⏐ u(5) − u(2) ⏐/⏐ u(2) ⏐ ≤ β ;⏐ u(5) − u(2) ⏐/⏐ u(2) ⏐ ≤ β2;7. IF 4 AND 6 ARE NOT FULFILLED GO TO 3.

ONE OF THE PRINCIPAL ADVANTAGES OF THIS TECHNIQUE IS THAT IT ISEASY TO MODIFY AND USE COMMERCIAL FINITE ELEMENT PACKAGESALREADY SET UP FOR SOLID MECHANICS OR HEAT TRANSFER ALSO INTHE NONLINEAR CASE.THE NONLINEAR CASE.ANOTHER ADVANTAGE IS THAT A SIGNIFICANT REDUCTION OF THE SIZESOF THE PROBLEM CAN BE OBTAINED AS WELL AS MOST PROBABLY AREDUCTION IN COMPUTING TIMES ESPECIALLY FOR THE NONLINEAR


REDUCTION IN COMPUTING TIMES ESPECIALLY FOR THE NONLINEARCASE.

11. APPLICATIONS

A) LINEAR RESPONSE OF A RECTANGULAR 2D PIEZOELECTRICCONTINUUM TO APPLIED STRESSES AND ELECTRIC POTENTIALS(COMPARISON WITH A CLOSED FORM SOLUTION)

B) DEFLECTION OF A CANTILEVER BEAM BY MEANS OFB) DEFLECTION OF A CANTILEVER BEAM BY MEANS OFPIEZOELECTRIC ACTUATORS (NONLINEAR BEHAVIOR OF THEMATERIAL)

C) INFLUENCE OF PERIODIC GEOMETRIES OF ELECTRODES IN THEELECTRIC FIELD AND IN THE DEFORMATION OF PIEZOELECTRICLAYERS OF RECTANGULAR SECTIONLAYERS OF RECTANGULAR SECTION

D) INTERACTION BETWEEN A PIEZO FIBER AND AN EPOXY MATRIXIN A PIEZOELECTRIC FIBER COMPOSITE


electro−elastic active materials

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