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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−
+ +− + +−
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
++−
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
− − − − − − − − − − − − − −
+ −
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
d: COUPLING COEFFICIENT (SAME OF DIRECT EFFECT)
4. PIEZOELECTRIC COUPLING COEFFICIENTS (2)
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
G g CO ST T S S O S SO T S GVOLTAGES ARE GENERATED FROM WEAK MECHANICAL STRESSES.
4. PIEZOELECTRIC COUPLING COEFFICIENTS (3)
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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˜)
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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˜
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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 +
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
ˆAPPLIED DISPLAC.˜
ASSIGNED TEMP. ON THE BOUNDARIES T
ASSIGNED MOIST. CONT. ON THE BOUNDARIES Û
ASSIGNED ELECTRIC POTENTIAL Öi= Öi
9. VARIATIONAL PRINCIPLES (3)
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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
POTENTIAL EQUATION∫∫SσS
SV
iidSΦσdVED
ˆSTRESS˜ ˆSURFACE FORCE˜
b
9. VARIATIONAL PRINCIPLES (4)
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 +=
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
THE ELECTRICAL AND MECHANICAL EQUATIONS OF THE VIRTUAL WORKAPPROACH ARE COUPLED BY MEANS OF THE CONSTITUTIVE EQUATIONS.
9. VARIATIONAL PRINCIPLES (5)
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
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THAT DENOTES SUBSEQUENT APPLICATION OF LOADS OF DIFFERENTINTENSITY AND THE CORRESPONDING DEFORMED CONFIGURATION.
9. VARIATIONAL PRINCIPLES (6)
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
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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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II
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.
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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.
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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.
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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
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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
P.GAUDENZI − ACTIVE MATERIALS AND INTELLIGENT STRUCTURES − PART II