Piezoelectricity - Stem2

1 PiezoelectricityJim Emery4/3 TheLinearPiezoelectricEquations .. Piezoelectric Ceramics Constants and One Dimensional Equa-tions .. Three Systems of Notation For The Piezoelectric Equations:Physics(IRE),ABAQUS,andANSYS .. Deriving the tensor Stress Form of the Piezoelectric EquationsFromtheStrainForm .. OrthotropicMaterial .. The Equality of Direct and Converse Piezo Coefficients .. Piezoelectric Current and Impedance From A Finite ElementCalculation .. The Conversion Example: Parameter Conversion Using Compliance Components Example: Parameter Conversion Using Four Independent Elas-ticityComponents .. ,OrthotropicElasticConstants .. Program Listing .. Location of Piezoelectric Information In ANSYS Manuals .. ANSYS Comparison For A Composite Piezoelectric Trans-ducer: Example IntroductionWe shall show how to derive the stress form of the piezoelectric equationsfrom the strain form, and we shall show how to find the clamped permittivitymatrix from the unclamped matrix.

2 These conversions are required for finiteelement programs because in such programs equations are solved for theelastic displacements, and so the strains rather than the stresses are theindependent variables. We shall do the basic derivation three times in thefollowing sections. We shall first treat the one dimensional equation. Inlater sections we shall treat the matrix case and the full tensor case. Thesederivations are conceptually the The Linear Piezoelectric EquationsA piezoelectric ceramic is a ferroelectric material. It exhibits hysteresis andhas nonlinear behavior. When this material is poled (placed in a strongelectric field), it attains a permanently polarized state. For a small variationin the electric field, it behaves approximately linearly near that linear piezoelectric equations are written in two different forms. Inthe more common form, which is called the strain form, the strain tensor ijand the electric polarization vectorPiare each written as linear functions ofthe electric field vectorEiand the stress tensor ij.

3 These equations are ij=Ekdkij+c ijkl kl,andPi= 0 ijEj+dijk the components of the piezoelectric tensor (strain coefficients).Thec ijklare the components of the inverse elastic tensor . The constant 0is the permittivity of free space. The ijare the components of the electricsusceptibility tensor . In the direct piezoelectric effect, when a piezoelectricmaterial is put under a stress, the material becomes electrically polarizedand surface charges appear. The direct piezoelectric effect isPi=dijk jk,which is obtained from the second equation, when the external electric fieldis the converse piezoelectric effect, when a piezoelectric material is putin an external electric field (voltages applied to electrodes), the materialexperiences a strain. The converse piezoelectric effect is ij=Ekdkij,which is obtained from the first equation, when there are no external forcesand the stress is fact that the coefficients for both the direct effect and the converseeffect are the same is a consequence of conservation of energy.

4 This fact canbe established by a thermodynamic argument (See Nye). Because of varioussymmetries in the tensor indices, the equations have matrix forms. Let usdefine an index functionf:(i, j) >kthat takes a pair of indices andproduces a single index. The functionftakes values as follows:f(1,1) =1,f(2,2) = 2,f(3,3) = 3,f(2,3) = 4,f(1,3) = 5,f(1,2) = 6 The function issymmetric,f(i, j)=f(j, i), and sof(3,2) = 4,f(3,1) = 5,f(2,1) = 6. Thisis the Physics and The Institute of Radio Engineers assignment function. Wedefine this function with a table:f123116526243543 This index assignment is the one most often used in elasticity theory toconvert the second rank stress and strain tensors to six component this is the usual convention, the following assignment defined byfunctiongis used sometimes (ABAQUS).g123114524263563A third index function is also used. The function, which we callhis usedby the finite element program !

5 These differences in index assignment will change the matricessuch asc, d, matrix components of the piezoelectric tensor aredi,f(i,j)and areeither equal to the tensor component itself, wheni=j,ortotwicethecomponent. This will be explained is a second form of the piezoelectric equations, which is called thestress stress tensor ijand the electric displacement vectorDi,areeachwritten as linear functions of the electric field vectorEiand the strain tensor ij. These equations are ij=cijkl kl Ekekij,andDi=eijk jk+ the components of the elastic tensor . Theeijkare the compo-nents of the piezoelectric tensor (stress coefficients). The ijare the compo-nents of the permittivity equations are suitable for finite element programs because in theseprograms the displacements and hence the strains are the independent vari-ables, whereas the boundary conditions are given as force loadings, that is asstresses.

6 It will be shown below that the piezoelectric stress coefficients andthe strain coefficients are related as follows:eijk= varies greatly in the field of Piezoelectricity . In a followingsection we will summarize the notation used in physics, and in two finiteelement programs ABAQUS and Piezoelectric Ceramics Constants and OneDimensional EquationsThe principal references for this section are: Jaffe, Cook and Jaffe,Piezo-electric Ceramics, and the Morgan Matroc documentGuide To ModernPiezoelectric original research on piezoelectric materials was conducted with par-allel plate capacitors, and the problem was treated as one dimensional. Sothe strain form of the equations considered were =Ed+c 1 P= 0 E+d ,where the quantities are in the 3 direction. From the definition of the electricdisplacementD=P+ 0E= 0( +1)E+d = E+d .So we have an alternate formD= E+d .Here is the permittivity at constant stress (unclamped).

7 The second set of piezoelectric equations (stress form) may be obtainedfrom the first set (strain form). Solving the first equation for the stress, wehave =c cdE=c this substitution for in the alternate form of the second equationof the first set, we get the second equation of the second E+d(c eE)= E+e ,=( de)E+dc = E+e ,where the piezoelectric coefficiente= is the permittivity atconstant stress (unclamped).6 The coefficients may be defined as partial derivatives. For example,d=( E) ,where the subscript means that is a function ofEand ,where is heldconstant in computing the partial derivative. Similarly we haved=( P )EThe constantgis defined byg=( E ) =( D) .We haveE=1 (D d )sog= ( E ) =d .The constanteis defined bye= ( E) =( D )EThe poled ceramic is an orthotropic material with a plane of symmetrywhose normal is in the poled direction. Further it is symmetric with respectto any rotation about the poling direction.

8 Performing a reflection throughthe symmetry plane, some of the strain components and stress componentsare maintained in sign and some others are changed in sign. Writing thevarious components of stress as functions of strain, in the two coordinatesystems, we find that the proper signs can be maintained only if some of theelastic coefficients are zero (see pp62-64 SokolnikoffMathematical Theoryof Elasticity). Carrying out rotation transformations about the poling axis(3 direction) we find further conditions on the elasticity coefficients. We haveonly the following independent elastic coefficients (Jaffe et al, p20)c11,c33,c44,c12, havec22=c11,c55=c44,c66=2(c11 c12),c23= which are not in the upper 3-dimensional submatrix or on themain diagonal are independent piezoelectric coefficients ared33,d31, haved32=d31,d24= other coefficients are independent dielectric coefficients areK11,K33We haveK22= other coefficients are permittivity is ij= electromechanical coupling constant is defined in terms of the ratioof stored electrical to stored mechanical energy.

9 LetQmechbe the electricalenergy converted to mechanical energy and letQinputbe the electrical energyinput. Then the electromechanical coupling factor isk2= is claimed (Jaffe et al p7) that the relation between the unclamped dielec-tric constantK and the clamped dielectric constantK isK =K (1 k2).Because 0<k<1, the clamped dielectric constant is smaller than theclamped constantK <K . The Stress Form of The Piezoelectric Ma-trixGiven the strain form of the piezoelectric equations, we can convert to thestress form and give expressions for the three independentetensor parame-ters. In matrix form we have =dTE+c 1 P= 0 E+d From the first equation multiplying by the elasticity matrixcwe havec =cdTE+ ,so solving for the stress matrix we have =c cdTE=c eE,where the matrixeis defined as the product ofcand the transpose ofde= haveeT=(cdT)T=cTd=cdThe last equality follows becausecis symmetric.

10 From the second equationD=P+ 0E= 0 E+d + 0E= 0( +I)E+d = E+d ,where is the permittivity at constant stress (unclamped permittivity).Substituting the expression for the stress from aboveD= E+d(c eE)=( de)E+dc 9= E+eT ,where = deis the permittivity at constant strain (clamped permittivity). In the previoussection we saw that the elastic matrixchas 5 independent parameters anddhas three independent parameters. We havec= c11c12c13000c12c11c13000c13c13c33000000c 44000000c440000002(c11 c12) ,andd= 0000d150000d1500d31d31d33000 .Then the piezoelectric stress matrix ise=cdT= c11c12c13000c12c11c13000c13c13c33000000c 44000000c440000002(c11 c12) 00d3100d3100d330d150d1500000 = 00c11d31+c12d31+c13d3300c12d31+c11d31+c1 3d3300c13d31+c13d31+c33d330c44d150c44d15 00000 10= 00e1300e1300e330e420e4200000 So the three independent piezoeparameters aree13,e33,e42.(Warning!Theematrix is sometimes defined as the transpose of thisematrix.)