Appendix A Piezoelectric Contitutive quation A.1 hree-dimenional Form of the Linear Piezoelectric Contitutive quation In general, poled piezoceramic (uch a PZ-5A and PZ-5H) are tranverely iotropic material. o be in agreement with the I Standard on Piezoelectricity [1], the plane of iotropy i defined here a the 12-plane (or the xy-plane). he piezoelectric material therefore exhibit ymmetry about the 3-axi (or the z-axi), which i the poling axi of the material. he field variable are the tre component ( ij ), train component (S ij ), electric field component ( k ), and the electric diplacement component (D k ). he tandard form of the piezoelectric contitutive equation can be given in four different form by taking either two of the four field variable a the independent variable. Conider the tenorial repreentation of the train electric diplacement form [1] where the independent variable are the tre component and the electric field component (and the remaining term are a defined in Section 1.4): S ij ijkl kl + d kij k D i d ikl kl + ε ik k (A.1) (A.2) which i the preferred form of the piezoelectric contitutive equation for bounded media (to eliminate ome of the tre component depending on the geometry and ome of the electric field component depending on the placement of the electrode). quation (A.1) and (A.2) can be given in matrix form a [ ] [ ][ ] S d t D d ε (A.3) where the upercript and denote that the repective contant are evaluated at contant electric field and contant tre, repectively, and the upercript t tand for the tranpoe. Piezoelectric nergy Harveting, Firt dition. Alper rturk and Daniel J. Inman. 20 John Wiley & Son, Ltd. Publihed 20 by John Wiley & Son, Ltd. ISBN: 978-0-470-68254-8
344 Appendix A he expanded form of quation (A.3) i S 3 S 4 S 5 D 1 D 2 12 13 0 0 0 0 0 d 31 12 13 0 0 0 0 0 d 31 13 13 0 0 0 0 0 d 2 0 0 0 55 3 0 0 0 d 15 0 4 0 0 0 0 55 0 d 15 0 0 5 0 0 0 0 0 66 0 0 0 0 0 0 0 d 15 0 ε 0 0 1 0 0 0 d 15 0 0 0 ε 0 2 d 31 d 31 d 0 0 0 0 0 ε 3 (A.4) where the contracted notation (i.e., Voigt notation: 1, 22 2, 3, 23 4, 13 5, 12 6) i ued o that the vector of train and tre component are S 3 S 4 S 5 S 2 S 23 23 22, 2 3 4 5 22 23 3 2 (A.5) herefore the hear train component in the contracted notation are the engineering hear train. It hould be noted from the elatic, piezoelectric, and dielectric contant in quation (A.4) that the ymmetrie of tranverely iotropic material behavior ( 22, d 31 d 32, etc.) are directly applied. A.2 Reduced quation for a hin Beam If the piezoelatic behavior of the thin tructure i to be modeled a a thin beam baed on the uler Bernoulli beam theory or Rayleigh beam theory, the tre component other than the one-dimenional bending tre are negligible o that 2 3 4 5 0 (A.6) Along with thi implification, if an electrode pair cover the face perpendicular to the 3-direction, quation (A.4) become ] [ ] [ ] d 31 1 d 31 ε 3 [ S1 (A.7)
Appendix A 345 which can be written a [ ][ ] 0 1 d 31 1 [ 1 d31 0 ε ][ S1 3 ] (A.8) herefore the tre electric diplacement form of the reduced contitutive equation for a thin beam i [ 1 ] [ ē 31 ē 31 ε S ][ S1 3 ] where the reduced matrix of the elatic, piezoelectric, and dielectric contant i (A.9) [ ] C ē 31 ē 31 ε S [ 0 d 31 1 ] 1 [ ] 1 d31 0 ε (A.10) Here and hereafter, an overbar denote that the repective contant i reduced from the threedimenional form to the plane-tre condition. In quation (A.10), 1, ē 31 d 31, ε S ε d2 31 (A.) where the upercript S denote that the repective contant i evaluated at contant train. A.3 Reduced quation for a Moderately hick Beam If the piezoelaticity of the tructure i to be modeled a a moderately thick beam baed on the imohenko beam theory, the tre component other than (the tre component in the axial direction) and 5 (the tranvere hear tre) are negligible o that 2 3 4 0 (A.12) i applied in quation (A.4). hen, 0 d 31 S 5 0 55 0 5 d 31 0 ε 3 which can be written a 0 0 1 0 d 31 0 55 0 5 0 1 0 S 5 d 31 0 1 0 0 ε 3 (A.13) (A.14)
346 Appendix A herefore the tre electric diplacement form of the reduced contitutive equation i 0 ē 31 5 0 55 0 S 5 (A.15) ē 31 0 ε S 3 Here, the reduced matrix of the elatic, piezoelectric, and permittivity contant i 0 ē 31 C 0 55 0 0 0 0 ē 31 0 ε S 55 0 d 31 0 1 1 0 d 31 0 1 0 0 0 ε 1 (A.16) where 1, 55 1 55, ē 31 d 31, ε S ε d2 31 (A.17) Note that the tranvere hear tre in quation (A.15) i corrected due to imohenko [2,3] 5 κ 55 S 5 (A.18) where κ i the hear correction factor [2 12]. A.4 Reduced quation for a hin Plate If the thin tructure i to be modeled a a thin plate (i.e., Kirchhoff plate) due to two-dimenional train fluctuation, the normal tre in the thickne direction of the piezoceramic and the repective tranvere hear tre component are negligible: quation (A.4) become which can be rearranged to give 12 0 0 12 0 0 0 0 66 0 d 31 d 31 0 1 3 4 5 0 12 0 d 31 12 0 d 31 0 0 66 0 d 31 d 31 0 ε 2 2 3 1 0 0 d 31 0 1 0 d 31 0 0 1 0 0 0 0 ε 3 (A.19) (A.20) (A.21)
Appendix A 347 he tre electric diplacement form of the reduced contitutive equation become 2 12 0 ē 31 12 0 ē 31 0 0 66 0 ē 31 ē 31 0 ε S 3 (A.22) where 12 0 ē 31 12 0 0 12 C 0 ē 31 0 0 66 0 12 0 0 0 0 66 0 ē 31 ē 31 0 ε S d 31 d 31 0 1 1 0 0 d 31 0 1 0 d 31 0 0 1 0 0 0 0 ε 1 (A.23) Here, the reduced elatic, piezoelectric, and permittivity contant are ( )( + 12 12) (A.24) 12 12 ( )( + 12 12) (A.25) 66 1 66 (A.26) ē 31 d 31 + 12 (A.27) ε S ε 2d2 31 + 12 (A.28) where the firt term (quation (A.24)) i the elatic contant that i related to the bending tiffne of a piezoelectric plate (accounting for the Poion effect) in the abence of torion. Reference 1. Standard Committee of the I Ultraonic, Ferroelectric, and Frequency Control Society (1987) I Standard on Piezoelectricity, I, New York. 2. imohenko, S.P. (1921) On the correction for hear of the differential equation for tranvere vibration of primatic bar. Philoophical Magazine, 41, 744 746. 3. imohenko, S.P. (1922) On the tranvere vibration of bar of uniform cro-ection. Philoophical Magazine, 43, 125 131. 4. Mindlin, R.D. (1951) hickne-hear and flexural vibration of crytal plate. Journal of Applied Phyic, 22, 316 323.
348 Appendix A 5. Mindlin, R.D. (1952) Forced thickne-hear and flexural vibration of piezoelectric crytal plate. Journal of Applied Phyic, 23, 83 88. 6. Cowper, G.R. (1966) he hear coefficient in imohenko beam theory. ASM Journal of Applied Mechanic,, 5 340. 7. Kaneko,. (1975) On imohenko correction for hear in vibrating beam. Journal of Phyic D: Applied Phyic, 8, 1927 1936. 8. Stephen, N.G. (1978) On the variation of imohenko hear coefficient with frequency. ASM Journal of Applied Mechanic, 45, 695 697. 9. Stephen, N.G. (1980) imohenko hear coefficient from a beam ubjected to gravity loading. ASM Journal of Applied Mechanic, 47, 121 127. 10. Stephen, N.G. and Hutchinon, J.R. (2001) Dicuion: hear coefficient for imohenko beam theory. ASM Journal of Applied Mechanic, 68, 959 961.. Hutchinon, J.R. (2001) Shear coefficient for imohenko beam theory. ASM Journal of Applied Mechanic, 68, 87 92. 12. Puchegger, S., Bauer, S., Loidl, D., Kromp, K., and Peterlik, H. (2003) xperimental validation of the hear correction factor. Journal of Sound and Vibration, 261, 177 184.